Chapter 1. Introduction 1. The Einstein equation 2. Semi-conformal maps and conformal foliations Notons que le modèle cosmologique FRW non déformé du départ a un champ électromagnétique F 0 trivial. Par suite, ce théorème souligne la pertinence de l'approche de déformation biconforme de ce travail : l'émergence des champs espacetemps non triviaux sur des modèles espace-temps obtenus en déformant les solutions d'Einstein classiques. 'quadrants' of Kruskal spacetime, discovered by M. Kruskal in 1960 [23]. The third type of spacetime that we use in this work is Friedmann-Robertson-Walker (FRW) spacetime M (k, I, f ). First studied by A. Friedmann in the 1920s [14, 15] and further developed by H. Robertson and by A.G. Walker in the 1930s [33-2

In the literature, the term "cosmological" is evidently associated with spacetime manifolds that model the entire universe, e.g. Friedmann-Robertson-Walker (FRW) spacetimes (see below). Indeed, for other types of spacetimes, Λ is customarily set to zero. However, in this work, we entertain the possibility of Λ being nonzero even for non-FRW spacetimes.

 In general, R n

6. Complex multiplication from R 4 \ {0} onto R 2 7. Projection from Schwarzschild spacetime onto S 2

Bibliography

Résumé en français

Cette thèse est consacrée à la résolution de l'équation d'Einstein en dimension 4 dans les contextes riemannien et lorentzien (plus spécifiquement, espace-temps). Nous obtenons des solutions g 1 par une méthode novatrice de déformation biconforme d'une métrique g 0 qui est solution a priori. Une telle déformation a lieu sur une variété métrique (M, g 0 ) munie d'un feuilletage conforme par des surfaces ; ce feuilletage est déterminé à son tour par une submersion semi-conforme de (M, g 0 ) dans une surface riemannienne.

Cette approche d'appliquer une déformation biconforme nous permet de réduire le problème de la construction de solutions d'Einstein en un système d'équations aux dérivées partielles (EDP) en deux paramètres : le paramètre de déformation σ orthogonal au feuilletage et le paramètre de déformation ρ tangent au feuilletage. L'autre caractéristique importante de cette approche est l'expression de la courbure de Ricci en termes des quantités géométriques associées au feuilletage.

Au coeur de la relativité générale est l'équation d'Einstein G := Ric -1 2 Sg = 8πT. La partie gauche est la courbure d'Einstein G de l'espace-temps ; g est la métrique, Ric la courbure de Ricci et S la courbure scalaire. Dans la partie droite, T est le tenseur énergie-impulsion total de l'espace-temps sous considération. Dans ce travail, le tenseur énergie-implusion total T est la somme de trois termes qui correspondent respectivement aux tenseurs énergie-implusion du fluide parfait, du champ électromagnétique et du vide : Nous utilisons deux outils principaux dans ce travail : applications semi-conformes (et ses feuilletages associés) et déformation biconforme. Une application lisse ϕ : 11 (M 4 , g 0 ) → (N 2 , h) d'une variété riemannienne ou lorentzienne dans une surface riemannienne s'appelle submersion semi-conforme avec dilatation λ 0 : M → (0, +∞) si

g 0 = λ 0 -2 ϕ * h
sur les espaces horizontaux H x = (ker dϕ x ) ⊥ , x ∈ M (les espaces verticaux sont V x = ker dϕ x ). Dans ce cas, les fibres de ϕ déterminent un feuilletage conforme sur M , c.à.d. le transport de Lie de l'espace normal le long des feuilles est conforme. Nous écrivons la métrique g 0 en termes de ses composantes horizontale et verticale :

g 0 = g H 0 + g V 0 .
Une déformation biconforme de g 0 est une métrique sur M de la forme

g = σ -2 g H 0 + ρ -2 g V 0 ,
où σ, ρ : M → (0, +∞) sont lisses ; les fonctions σ, ρ s'appellent les paramètres de déformation. Cette nouvelle métrique g a la même signature que g 0 , et l'application ϕ : (M, g) → (N, h) est une submersion semi-conforme avec dilatation λ = σλ 0 . Nous réalisons notre travail en trois étapes :

(1) Chapitre 3 : Nous calculons la courbure de Ricci d'une variété riemannienne ou lorentzienne (M, g) de dimension 4 munie d'un feuilletage conforme de dimension 2 provenant d'une submersion semi-conforme dans une surface riemannienne. Nous exprimons la courbure de Ricci en termes des quantités géométriques associées au feuilletage. (2) Chapitre 4 : Nous calculons le changement Ric 1 -Ric 0 dans la courbure de Ricci lorque nous passons d'une métrique initiale g 0 en une métrique déformée biconformément g 1 = σ -2 g H 0 + ρ -2 g V 0 , avec g 0 (et donc g 1 ) vérifiant aux conditions du premier point.

(3) Chapitre 5 : En commençant avec une solution d'Einstein a priori (M, g 0 ) qui vérifie aux conditions du premier point, nous construisons des solutions d'Einstein (M, g 1 ) en déformant biconformément g 0 . En particulier, nous appliquons la différence Ric 1 -Ric 0 calculée au deuxième point afin d'exprimer l'équation d'Einstein comme un système d'EDP dans les paramètres de déformation σ, ρ.

0.1. La coubure de Ricci. Soit (M 4 , g) une variété riemannienne ou lorentzienne (plus spécifiquement, espace-temps) orientée et soit (N 2 , h) une surface riemannienne orientée. Soit ϕ : (M, g) → (N, h) une submersion semi-conforme avec dilatation λ : M → (0, +∞). Soient H et V les projections orthogonales dans les espaces horizontaux et verticaux, respectivement.

Nous exprimons la courbure de Ricci de la métrique g en termes des quantités géométriques associées au feuilletage suivantes :

1. la dilatation λ ; 2. les premières formes fondamentales des distributions horizontales et verticales ; 3. la deuxième forme fondamentale A de la distribution horizontale ainsi que son adjoint A * ; 4. la deuxième forme fondamentale B de la distribution verticale et son adjoint B * ; 5. la courbure sectionnelle intrinsèque K ϕ des fibres de ϕ ; 6. la courbure moyenne ν de la distribution horizontale ; 7. la courbure moyenne µ des fibres de ϕ ; 8. la 1-forme d'intégrabilité ζ de la distribution horizontale.

Nous calculons la courbure Ric suivant ses trois composants :

Ric H = Ric(H•, H•), Ric V = Ric(V•, V•), Ric mixed = Ric(H•, V•) + Ric(V•, H•).
Nous énonçons le résultat principal du chapitre 3 : Théorème 0.1 (Théorème 13.3). La courbure de Ricci de la métrique g est donnée par

Ric = (λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ )) g H + εK ϕ g V + L µ g(H•, H•) + L ν g(V•, V•) -2ν 2 + 2ζ 2 -2 sym ν (B * ) + 2 sym ζ(B * • J•) + 4 sym(∇µ (V•, H•)) + 2 sym(∇ν (H•, V•)) -2 sym(∇ζ(JH•, V•)) + div 1 B 1 -2 sym div 2 B 1 -2C + C * .
0.2. Déformation biconforme. Soit (M 4 , g 0 ) une variété riemannienne ou lorentzienne (plus spécifiquement, espace-temps) orientée et soit (N 2 , h) une surface riemannienne orientée. Soit ϕ : (M, g 0 ) → (N, h) une submersion semi-conforme avec dilatation λ 0 : M → (0, +∞). Nous considérons la métrique déformée biconformément g = σ -2 g H 0 + ρ -2 g V 0 sur M avec paramètres de déformation σ, ρ : M → (0, +∞). Notons que g a la même signature que g 0 . De plus, ϕ : (M, g) → (N, h) est une submersion semiconforme avec dilatation λ = σλ 0 . Le calcul de la coubure de Ricci de l'étape précédante marche aussi bien pour (M, g 0 ) que pour (M, g), ce qui nous donne Ric 0 = Ric 0 (λ 0 , g H 0 , g V 0 , A 0 , A * 0 , B 0 , B * 0 , K ϕ 0 , ν 0 , µ 0 , ζ 0 ), Ric = Ric(λ, g H , g V , A, A * , B, B * , K ϕ , ν, µ, ζ).

Nous exprimons les quantités géométriques associées à g en termes de celles associées à g 0 et en termes des paramètres de déformation σ, ρ. Ensuite, donnée λ 0 , . . . , ζ 0 , nous obtenons Ric = Ric(σ, ρ). En particulier, nous exprimons les différences suivantes en termes des paramètres de déformation σ, ρ : Ric H -Ric H 0 , Ric V -Ric V 0 , Ric mixed -Ric mixed 0 , σ -2 S -S 0 , ρ -2 S -S 0 , où S et S 0 sont les courbures scalaires de g et g 0 , respectivement. Par exemple, nous montrons en section 15 que la première différence est donnée par le théorème suivant.

Théorème 0.2 (Théorème 15.12). Le changement Ric H -Ric H 0 dans la partie horizontale de la courbure de Ricci après une déformation biconforme g = σ -2 g H 0 + ρ -2 g V 0 est donné par Ric H -Ric H 0 = -2 g H 0 (grad 0 ln σ, grad 0 ln ρ) g H 0 -2σ -2 ρ 2 g V 0 (grad 0 ln σ, grad 0 ln σ) g H 0 -2(σ 2 ρ -2 -1)g 0 (ζ 0 0 , ζ 0 0 ) g H 0 -2σ -2 ρ 2 ν 0 (ln σ) g H 0 + (σ -2 ρ 2 -1) div 0 ν 0 0 g H 0 + div 0 Hd ln σ g H 0 + σ -2 ρ 2 div 0 Vd ln σ g H 0 + L H grad 0 ln ρ g 0 (H•, H•) + 4µ 0 0 Hd ln σ -4µ 0 0 Hd ln ρ + 4 Hd ln σ Hd ln ρ -2(Hd ln ρ) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] .

Nous précisons que dans l'accomplissement de cette étape en chapitre 4, nous sommes amené à calculer la différence dans les connexions Levi-Civita (théorème 14.13). 0.3. Solutions à l'équation d'Einstein. Soient les métriques g 0 , g comme dans l'étape précédante. Supposons que g 0 est une solution à l'équation d'Einstein G 0 = 8πT 0 ⇔ Ric 0 = 8πT 0 + 1 2 S 0 g 0 . Nous déterminons les conditions sur les paramètres de déformation σ, ρ sous lesquelles g est une solution à l'équation d'Einstein G = 8πT ⇔ Ric = 8πT + [START_REF] Baird | Biconformal equivalence between 3-dimensional Ricci solitons[END_REF] 2 Sg pour des paramètres physiques u, ρ, p, F, μ, Λ. Cette equation est équivalente au système

Ric H -Ric H 0 = 8π(T H -T H 0 ) + 1 2 (σ -2 S -S 0 )g H 0 , Ric V -Ric V 0 = 8π(T V -T V 0 ) + 1 2 (ρ -2 S -S 0 )g V 0 , Ric mixed -Ric mixed 0 = 8π(T mixed -T mixed 0 ),
où, de l'étape précédante, les parties gauches, ainsi que les différences σ -2 S -S 0 et ρ -2 S -S 0 , sont exprimées en termes de σ, ρ. En évaluant sur un repère orthonormé (par rapport à g 0 ), ce système devient un système d'EDP dans les paramètres de déformation σ, ρ. 0.3.1. Exemples provenant de l'espace euclidien de dimension 4. Nous mettons en oeuvre le plan ci-dessus pour M = R 4 , N = R 2 et ϕ(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 ) dans la section 18. Le système d'EDP qui en résulte est donné par le théorème 18.1. Par exemple, nous appliquons ce théorème général dans le cas σ = σ(x 1 ), ρ = ρ(x 1 ) :

Théorème 0.3 (Théorème 18.8). La métrique déformée biconformément maximale g = σ(x 1 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 1 ) -2 (dx 3 2 + dx 4 2 ), est une métrique d'Einstein si et seulement si l'un des cas suivants se produit :

• de courbure de Ricci nulle : IA. g est isométrique à la métrique euclidienne de dimension 4 ; IB. g est isométrique au produit de (a) la métrique tordue y 2 dx 2 +dy 2 sur le demi-plan supérieur et (b) la métrique euclidienne de dimension 2 ; III. g est homothétique à ds 2 = x -1/2 (dx 2 + dy 2 ) + x(dz 2 + dw 2 ), x > 0 ; • de courbure de Ricci non nulle : II. g est homothétique à la métrique hyperbolique de dimension 4 ;

IV. g est la métrique (18.21), où ρ est donné par (18.20), b ∈ R est donné par (18.11), c ∈ R est donné par (18.13), d ∈ R \ {0} et c 2 ∈ R.

Dans le point IB., g s'effondre asymptotiquement vers la métrique euclidienne de dimension 2. Dans le point IV., g s'effondre asymptotiquement soit vers la métrique euclidienne de dimension 2, soit vers une métrique qui est homothétique à la métrique hyperbolique de dimension 4. 0.3.2. Exemples provenant de l'espace-temps de Minkowski. Nous mettons en oeuvre le plan ci-dessus pour M = R 4 1 , N = R 2 et ϕ(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 ) dans la section 19. Le système d'EDP qui en résulte est donné par le théorème 19.2. Par exemple, nous appliquons ce théorème général dans le cas σ = σ(x 1 , x 3 ), ρ = ρ(x 4 ) :

Théorème 0.4 (Théorème 19.19). La métrique déformée biconformément maximale g = σ(x 1 , x 3 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 4 ) -2 (dx 3 2 -dx 4 2 ), avec domaine x 3 de σ(x 1 , •) un intervalle ouvert pour chaque x 1 , vérifie l'équation d'Einstein avec champ de flot u = ρ ∂ 4 et champ électromagnétique trivial si et seulement si l'un des cas suivants se produit :

• g est isométrique à 01. ds 2 = (dx 2 + dy 2 ) + (dz 2 -dt 2 ) ; 02. ds 2 = (y 2 dx 2 + dy 2 ) + (dz 2 -dt 2 ), y > 0 ; 03. ds 2 = (dx 2 + dy 2 ) + (t 2 dz 2 -dt 2 ), t > 0 ; 04. ds 2 = (y 2 dx 2 + dy 2 ) + (t 2 dz 2 -dt 2 ), y, t > 0 ; 11. ds 2 = D z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), where D ∈ (0, +∞) \ {1} ; 12. ds 2 = D z (y 2 dx 2 +dy 2 )+(dz 2 -dt 2 ), y > 0, where D ∈ (0, +∞)\{1} ; IB. ds 2 = z 2 sech 2 x (dx 2 + dy 2 ) + (dz 2 -dt 2 ), z > 0 ;

• g est homothétique à 05. ds 2 = sech 2 x (dx 2 + dy 2 ) + t -2 (dz 2 -dt 2 ), t > 0 ; 06. ds 2 = sech 2 x (dx 2 + dy 2 ) + csch 2 t (dz 2 -dt 2 ), t > 0 ; 07. ds 2 = sech 2 x (dx 2 + dy 2 ) + csc 2 t (dz 2 -dt 2 ), 0 < t < π ; 08. ds 2 = x -2 (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), x > 0 ; 09. ds 2 = csch 2 x (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), x > 0 ; 10. ds 2 = csc 2 x (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), 0 < x < π ; IIB. ds 2 = sech 2 x cos 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), -π 2 < z < π 2 ; IIIA. ds 2 = x -2 cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), x > 0 ; IIIC. ds 2 = csch 2 x cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), x > 0 ; IIID. ds 2 = csc 2 x cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), 0 < x < π ; IVB. ds 2 = sech 2 x sinh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), z > 0. Dans les points 01. à 10. et le point IB., le fluide parfait est vide-trivial. Dans les autres points, le fluide parfait est non vide-trivial mais la densité d'énergie et la pression sont des fonctions constantes. 0.3.3. Exemples de l'espace-temps Friedmann-Robertson-Walker (FRW) avec espace plat. Nous mettons en oeuvre le plan ci-dessus pour M = M (0, I, f ), N = R 2 et ϕ(x 1 , x 2 , x 3 , t) = (x 1 , x 2 ) dans la section 20. Le système d'EDP qui en résulte est une généralisation de celui de la section 19. Nous anticipons alors que les résultats seront des généralisations ce ceux obtenus auparavant.

Pourtant, il apparaît un nouveau phénomène -la condition de Hubble pour l'espace-temps (pseudo-)cosmologique déformé biconformément (M, g). Nous donnons une définition pragmatique de cette condition (dans le cas σ = σ(t), ρ = ρ(t)) dans la sous-section 20.2. Nous utilisons cette condition pour généraliser un résultat classique sur les singularités aux extrémité de la cosmologie FRW (théorème 20.10). Nous montrons également le résultat suivant : Théorème 0.5 (Théorème 20.7, théorème 20.8). Supposons que la métrique déformée biconformément

g = σ(t) -2 f (t) 2 (dx 1 2 + dx 2 2 ) + ρ(t) -2 (f (t) 2 dx 3 2 -dt 2 )
vérifie l'équation d'Einstein avec champ de flot u = ρ ∂ t , et telle que la condition de Hubble pour g est vérifiée. Si σ ou ρ est une fonction constante et l'autre ne l'est pas, alors le champ électromagnétique F est non trivial. Si en plus F a partie mixte nulle, alors F a un tenseur énergie-impulsion T em non trivial.

CHAPTER 1

Introduction

This thesis is devoted to the construction of 4-dimensional solutions to the Einstein equation in the Riemannian and Lorentzian (specifically, spacetime) contexts. We obtain such solutions g 1 by way of biconformal deformation of metrics g 0 that are solutions a priori. Such a deformation is done in the presence of a conformal foliation by surfaces on a 4-dimensional Riemannian or Lorentzian manifold (M, g 0 ); this foliation in turn arises as the fibers of a semi-conformal submersion from (M, g 0 ) to a Riemannian surface.

The novel approach of using biconformal deformation allows us to reduce the problem of constructing Einstein solutions to a system of partial differential equations (PDEs) in two parameters: the deformation parameter σ orthogonal to the foliation and the deformation parameter ρ tangent to it. The other significant feature of this approach is the expression of the Ricci curvature in terms of geometric quantitites associated to the foliation. G := Ric -1 2 Sg = 8πT. The left-hand side is the Einstein curvature G of the spacetime manifold; g is the metric, Ric is the Ricci curvature, and S is the scalar curvature. On the righthand side, T is the total energy-momentum tensor of the spacetime being modeled. Geometry determines G while physics determines T , and the Einstein equation asserts that these two are related in a very specific way.

In spite of the simplicity of (1.1), the Einstein equation translates to a nonlinear system of PDEs (called the Einstein equations) when written in terms of local coordinates; this system is difficult to solve in general. A general existence and uniqueness result was obtained by Y. Choquet-Bruhat and R. Geroch in 1969 by solving the Cauchy problem in general relativity: Any suitable initial data set has a unique maximal development that is a solution to the Einstein equation (see [9]).

In this work, the total energy-momentum tensor T is a sum of three terms:

(1.2) T = T matter + T em + T vacuum , the addends being the energy-momentum tensors of matter, electromagnetic field, and vacuum, respectively. The last addend in (1.2) is T vacuum (Λ) = -1 8π Λg = -ρ vacuum g, [START_REF] Baird | Biconformal equivalence between 3-dimensional Ricci solitons[END_REF] Compare with [17] (3.15) or [25] (17.11), and of course with Einstein himself [12] (2a). Note that some authors omit the factor 8π (e.g. [8] Subsection III.5.2 and [36] Definition 4.1.1).

17

where Λ ∈ R is the cosmological constant 2 and ρvacuum =1 8π Λ is the vacuum energy density of the spacetime ( [25] (17.12b)). The first addend in (1.2) is, in this work, the energy-momentum tensor T fluid of a perfect fluid (u, ρ, p):

T matter = T fluid (u, ρ, p) = (ρ + p)u 2 + pg, where the flow vector field u is unit and future-pointing timelike, the energy density ρ : M → R is smooth, and the pressure p : M → R is smooth ( [25] (5.21)). The middle addend in (1.2) is given in terms of coordinates by

T em αβ (F, μ) = -μ -1 (F αγ F γ β + 1 4 F γδ F γδ g αβ ),
where μ ∈ R \ {0} is the magnetic constant (also called the vacuum permeability constant) and F is the electromagnetic field, a closed 2-form 3 , of the spacetime (c.f. [17] (3.7) or [25] (5.22)). Equation (1.1) with T = T vacuum (Λ) is called the vacuum Einstein equation:

G = -Λg,
or equivalently

(1.3) Ric = Λg.
The simplest vacuum solution (with Λ = 0) is Minkowski spacetime R 4 1 , originally developed for the Maxwell equations by H. Minkowski in 1908 [24]. This spacetime is R 4 with line element [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] Minkowski spacetime is the setting of special relativity.

ds 2 = dx 2 + dy 2 + dz 2 -dt 2 of index 1.
A more complicated vacuum, Ricci-flat (but not flat) solution is Schwarzschild spacetime. Discovered by K. Schwarzschild in 1916 [37], this spacetime models the region around a static, spherical mass. Exterior Schwarzschild spacetime is

S 2 × (2M, +∞) × R with line element ds 2 = r 2 dΩ 2 + f (r) -1 dr 2 -f (r) dt 2 .
Here, (S 2 , dΩ 2 ) is the 2-sphere of radius 1, M ∈ (0, +∞) is the Schwarzschild mass, r ∈ (2M, +∞), t ∈ R, and

f (r) = 1 -2Mr -1 .
Meanwhile, interior Schwarzschild spacetime is

S 2 × (0, 2M) × R
with the same line element formula. The exterior and interior were joined as two 35, 40], FRW spacetime models the entire history of the universe. This spacetime is the warped product P f ×I with line element

ds 2 = f (t) 2 dΣ 2 -dt 2 .
Here, (P, dΣ 2 ) is a 3-dimensional Riemannian manifold of constant curvature k,

(I, -dt 2 ) is an interval of R 1

Semi-conformal maps and conformal foliations

We use two main tools in this work: semi-conformal maps (and the associated conformal foliations) and biconformal deformation. The first of these is the subject of this section.

In the Riemannian context, the notion of a semi-conformal map was first coined by B. Fuglede in 1978 in his characterization of harmonic morphisms [16]. This characterization was also independently obtained by T. Ishihara and by A. Bernard, E. Campbell, and A.M. Davie in 1979 [7, 21]. A harmonic morphism is a map that pulls back germs of harmonic functions to germs of harmonic functions. In the context of manifolds, a C 2 map between (semi-)Riemannian manifolds is a harmonic morphism if and only if it is harmonic and semi-conformal.

This second necessary condition of semi-conformality is defined as: A smooth map ϕ : (M m , g) → (N n , h) between semi-Riemannian manifolds is semi-conformal

at x ∈ M if there exists λ 2 (x) ∈ R such that g(dϕ * x (V ), dϕ * x (W )) = λ 2 (x)h(V, W ) for all V, W ∈ T ϕ(x) N , where dϕ * x : T ϕ(x) N → T x M is the adjoint of dϕ x . The map ϕ is semi-conformal if it is semi-conformal at every x ∈ M .
Note the the semi-conformality of ϕ is invariant under conformal changes of g or h. Moreover, there are various equivalent formulations of semi-conformality (see [4] Lemma 14.5.2). For example, if {e a } is a frame at x ∈ M and {f α } is a frame at ϕ(x), then ϕ is semi-conformal at x if and only if m a,b=1

g ab ϕ α a ϕ β b = λ 2 (x)h αβ
for all α, β ∈ {1, . . . , n}, where the quantities ϕ α a are given by the equation dϕ(e a ) = n α=1 ϕ α a f α . In this work, we are interested in the case m = 4, n = 2, g Lorentzian or Riemannian, and h Riemannian. In this case, each λ 2 (x) ∈ [0, +∞) and we indeed have λ(x) ∈ [0, +∞) such that λ 2 (x) = λ(x) 2 . The map λ : M → [0, +∞) is called the dilation of ϕ. [START_REF] Baird | Harmonic morphisms, conformal foliations and shear-free ray congruences[END_REF] When n = 2 and h is Riemannian, there are local isothermal coordinates on N that make it locally conformally equivalent to a an open subset of C. In particular, [START_REF] Baird | Harmonic morphisms, conformal foliations and shear-free ray congruences[END_REF] In [4], the square conformality λ 2 (x) at x ∈ M is denoted as Λ(x). However, we avoid doing this here and reserve the uppercase letter Λ for the cosmological constant (and related quantities) instead.

if U is an open subset of R 4 or R

+ ∂ϕ ∂x 3 2 + ε ∂ϕ ∂x 4 2 = 0, where ε = +1, if g is Riemannian, -1, if g is Lorentzian.
Semi-conformal submersions and conformal foliations are intimately linked. In the Riemannian context, a foliation F is conformal if Lie transport of the normal space along the leaves is conformal. In particular, if T F denotes tangent space to the leaves and N F denotes normal space, then F is conformal if and only if there

exists a : T F → R such that (L V g)(X, Y ) = a(V )g(X, Y ), V ∈ T F, X, Y ∈ N F. If ϕ : (M m , g) → (N n , h
) is a semi-conformal submersion between Riemannian manifolds, then its fibers form a conformal foliation on M . Conversely, if F is a conformal foliation on M and ψ :

U → R n × R m-n is a local foliated chart on M ,
then there is a conformal metric on the leaf space N of F| U such that the canonical projection ϕ : U → N is a semi-conformal submersion. The map a in the conformal foliation definition given above and the dilation λ of ϕ are related by

a = d(ln λ -2 )| V ,
where V = T F = ker dϕ. (See [4] Section 2.5.) Since the conformality of a foliation depends only on the metric of the horizontal space, the same is true for the case (M m , g) Lorentzian and (N 2 , h) Riemannian.

A relation between semi-conformal maps and spacetime fields is given by shearfree ray (SFR) congruences, which we now define. Let be a smooth foliation by null lines on an open subset U of Minkowski spacetime R 4 1 . Let u be a unit vector in R 3 , v = u + ∂ t (t being the time coordinate) be the tangent vector field of , and V = span{v}. Then the orthogonal complement V ⊥ of V is 3-dimensional and contains V ; let H be any complement of V in V ⊥ . The foliation is called a SFR congruence if Lie transport along V of vectors in H is conformal [6, 20, 32]. SFR congruences are fundamental in the twistorial description of massless spacetime fields.

The following result was obtained by P. Baird and J.C. Wood ([5] Theorem 2.3): Let ϕ : U → (N 2 , h) be a real-analytic harmonic morphism without critical points from an open subset of R 4 1 to a Riemannian surface. Then there is a SFR congruence on U such that each connected component of a fiber of ϕ is the union of parallel null lines of .

As a converse, we note the following: Let v = u + ∂ t be the tangent vector field of a SFR congruence in R 4 1 as above. Then the restriction of u to each spacelike slice t = t 0 gives a vector field that is tangent to a conformal foliation. Conversely, any conformal foliation by curves on an open subset of a spacelike slice t = t 0 determines a SFR congruence on a suitable open subset of R 4 1 . (See [1] Proposition 2.1.) In the Riemannian context, there is a strong connection between harmonic morphisms ϕ : (M 4 , g) → (N 2 , h) and anti-self-dual 4-dimensional Einstein manifolds [START_REF] Benn | A unified description of null and non-null shear-free congruences[END_REF] Then, by the characterization above, such a map is a harmonic morphism if and only if

∂ϕ ∂x 1 2 + ∂ϕ ∂x 2 2 + ∂ϕ ∂x 3 2 + ε ∂ϕ ∂x 4 2 = 0 and ∂ 2 ϕ ∂x 1 2 + ∂ 2 ϕ ∂x 2 2 + ∂ 2 ϕ ∂x 3 2 + ε ∂ 2 ϕ ∂x 4 2 = 0.
established by M. Ville and by J.C. Wood [39, 41]. In the case of 1-dimensional fibers, R. Pantilie showed that harmonic morphisms ϕ : (M 4 , g) → (N 3 , h) on 4-dimensional Einstein manifolds belong to certain types [31].

Biconformal deformation

We now address the other main tool used in this work: biconformal deformation. Let (M 4 , g 0 ) be either a Riemannian or Lorentzian manifold and let (N 2 , h) be a Riemannian surface. Let ϕ : (M, g 0 ) → (N, h) be a semi-conformal submersion with dilation λ 0 : M → (0, +∞), i.e.

g 0 = λ 0 -2 ϕ * h
on the horizontal spaces H x = (ker dϕ x ) ⊥ , x ∈ M (the vertical spaces are V x = ker dϕ x ). We write the metric g 0 in terms of its horizontal and vertical parts:

g 0 = g H 0 + g V 0 .
While g H 0 is always positive definite, the index of g V 0 is equal to the index of g 0 . A biconformal deformation of g 0 is a metric on M of the form

g = σ -2 g H 0 + ρ -2 g V 0
for some smooth deformation parameters 7 σ, ρ : M → (0, +∞); that is, g is obtained by performing individual deformations to the horizontal and vertical parts of g 0 . This new metric g has the same index as g 0 , and the map ϕ : (M, g) → (N, h) is a semi-conformal submersion with dilation λ = σλ 0 .

In the context of harmonic morphisms, the notion of biconformal deformations was introduced by X. Mo by establishing conditions on the deformation parameters σ, ρ that ensure that ϕ is still a harmonic morphism with respect to the biconformally deformed metric g [26-28]. Such deformations were used by L. Danielo to construct 4-dimensional Einstein metrics in the restrictive case of harmonic morphisms between Riemannian manifolds with 1-dimensional fibers [10, 11].

The combination of harmonicity and semi-conformality of ϕ : (M, g 0 ) → (N, h) (see previous section) form an overdetermined system and so restrict the geometric structures that can occur. In this work we dispense of the harmonicity condition, providing a degree of freedom for the deformation parameters σ, ρ restricted only by the Einstein equation. Though the resulting PDEs are complex in general, we are able to simplify these in a variety of interesting cases to construct significant examples of Einstein solutions.

We now provide a quick sketch of how we make use of our main tools of semiconformality and biconformal deformation. We begin with an Einstein solution, say Minkowski spacetime R 4 1 with metric

g 0 = dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 ,
which is a solution to the Einstein equation

G 0 = 8πT 0
with ρ0 = p0 = 0, F 0 = 0, and Λ 0 = 0. Endow R 4

1 with the 2-dimensional conformal foliation F obtained from the (semi-conformal submersion) projection

ϕ : (x 1 , x 2 , x 3 , x 4 ) ∈ R 4 1 → (x 1 , x 2 ) ∈ R 2
(the leaves of F being the fibers of ϕ). We biconformally deform g 0 by performing individual deformations to parts of g 0 tangent and orthogonal to F:

g = σ -2 (dx 1 2 + dx 2 2 ) + ρ -2 (dx 3 2 -dx 4 2
) [START_REF] Bernard | Brownian motion and generalized analytic and inner functions[END_REF] We point out that, in spite of our notation, the deformation parameter ρ and the energy densities ρ, ρvacuum have a priori nothing to do with each other.

for smooth positive-valued deformation parameters σ, ρ defined on an open subset of R 4

1 . The Einstein equation (1.1) for g is then equivalent to a system of PDEs written in terms of the geometric parameters σ, ρ and the physical parameters u, ρ, p, F, μ, Λ.

Plans and results

We carry out our work in three stages:

(1) Chapter 3: Compute the Ricci curvature of a 4-dimensional Riemannian or

Lorentzian manifold (M, g) endowed with a 2-dimensional conformal foliation coming from a semi-conformal submersion to a Riemannian surface. Express this Ricci curvature in terms of geometric quantities associated to the foliation. (2) Chapter 4: Compute the change Ric 1 -Ric 0 in the Ricci curvature as we go from an initial metric g 0 to a biconformally deformed metric g 1 = σ -2 g H 0 + ρ -2 g V 0 , with g 0 (and hence g 1 , too) satisfying the conditions in the first item.

(3) Chapter 5: Starting from an a priori Einstein solution (M, g 0 ) that satisfies the conditions in the first item, construct Einstein solutions (M, g 1 ) by biconformally deforming g 0 . In particular, use the difference Ric 1 -Ric 0 computed in the second item to express the Einstein equation as a system of PDEs in the deformation parameters σ, ρ.

These stages are discussed one by one below.

4.1. Ricci curvature. Let (M 4 , g) be an oriented Riemannian or Lorentzian (specifically, spacetime) manifold and let (N 2 , h) be an oriented Riemannian surface. Let ϕ : (M, g) → (N, h) be a semi-conformal submersion with dilation λ : M → (0, +∞), i.e. g = λ -2 ϕ * h on the horizontal spaces H x , x ∈ M . Let H and V denote projection onto the horizontal and vertical spaces, respectively.

We express the Ricci curvature Ric of the metric g in terms of the following geometric quantities associated to the semi-conformal submersion ϕ (or equivalently to the associated conformal foliation):

1. the dilation λ; 2. the first fundamental forms of the horizontal and vertical distributions; 3. the second fundamental form A of the horizontal distribution and its adjoint A * ; 4. the second fundamental form B of the vertical distribution and its adjoint B * ; 5. the intrinsic sectional curvature K ϕ of the fibers of ϕ; 6. the mean curvature ν of the horizontal distribution; 7. the mean curvature µ of the fibers of ϕ; 8. the integrability 1-form ζ of the horizontal distribution.

We break down in three portions this computation of Ric according to its three parts (addends):

Ric H = Ric(H•, H•), Ric V = Ric(V•, V•), Ric mixed = Ric(H•, V•) + Ric(V•, H•).
To give the reader an idea, we state here the main result of Chapter 3.

Ric = (λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ )) g H + εK ϕ g V + L µ g(H•, H•) + L ν g(V•, V•) -2ν 2 + 2ζ 2 -2 sym ν (B * ) + 2 sym ζ(B * • J•) + 4 sym(∇µ (V•, H•)) + 2 sym(∇ν (H•, V•)) -2 sym(∇ζ(JH•, V•)) + div 1 B 1 -2 sym div 2 B 1 -2C + C * .
4.2. Biconformal deformation. Let (M 4 , g 0 ) be an oriented Riemannian or Lorentzian (specifically, spacetime) manifold and let (N 2 , h) be an oriented Riemannian surface. Let ϕ : (M, g 0 ) → (N, h) be a semi-conformal submersion with dilation λ 0 : M → (0, +∞). Consider the biconformally deformed metric

g = σ -2 g H 0 + ρ -2 g V 0
on M with smooth deformation parameters σ, ρ : M → (0, +∞). Note that g has the same index as g 0 :

g is Riemannian if g 0 is; g is Lorentzian if g 0 is. Moreover, ϕ : (M, g) → (N, h) is a semi-conformal submersion with dilation λ = σλ 0 .
The Ricci curvature computation of the previous stage works for the metric g 0 as well as for the metric g, yielding

Ric 0 = Ric 0 (λ 0 , g H 0 , g V 0 , A 0 , A * 0 , B 0 , B * 0 , K ϕ 0 , ν 0 , µ 0 , ζ 0 ), Ric = Ric(λ, g H , g V , A, A * , B, B * , K ϕ , ν, µ, ζ).
We express the geometric quantities λ, . . . , ζ associated to g in terms of the geometric quantities λ 0 , . . . , ζ 0 associated to g 0 and the deformation parameters σ, ρ. For example, we derive the following two results. Lemma 4.2 (Lemma 14.9).

B = σ 2 ρ -2 (B 0 + g V 0 (•, •) H grad 0 ln ρ), B * = B * 0 + V • (Hd ln ρ)(•).
Lemma 4.3 (Lemma 15.5). K ϕ = ρ 2 (K ϕ 0 + ε div 0 Vd ln ρ + 2ε ν 0 (ln ρ)). Upon viewing λ 0 , . . . , ζ 0 as given objects, we get Ric = Ric(σ, ρ). Indeed, we express the following differences in terms of the deformation parameters σ, ρ:

Ric H -Ric H 0 , Ric V -Ric V 0 , Ric mixed -Ric mixed 0 , σ -2 S -S 0 , ρ -2 S -S 0 ,
where S and S 0 are the scalar curvatures of g and g 0 , respectively. For example, we show in Section 15 that the first difference is as given in the following theorem.

Theorem 4.4 (Theorem 15.12). The change Ric H -Ric H 0 in the horizontal part of Ricci curvature under a biconformal deformation

g = σ -2 g H 0 + ρ -2 g V 0 is given by Ric H -Ric H 0 = -2 g H 0 (grad 0 ln σ, grad 0 ln ρ) g H 0 -2σ -2 ρ 2 g V 0 (grad 0 ln σ, grad 0 ln σ) g H 0 -2(σ 2 ρ -2 -1)g 0 (ζ 0 0 , ζ 0 0 ) g H 0 -2σ -2 ρ 2 ν 0 (ln σ) g H 0 + (σ -2 ρ 2 -1) div 0 ν 0 0 g H 0 + div 0 Hd ln σ g H 0 + σ -2 ρ 2 div 0 Vd ln σ g H 0 + L H grad 0 ln ρ g 0 (H•, H•) + 4µ 0 0
Hd ln σ -4µ 0 0 Hd ln ρ + 4 Hd ln σ Hd ln ρ -2(Hd ln ρ) 2 . [START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF] The quantity K N is the sectional curvature of N , the operator J will be defined in Section 5, and the quantities B 1 , C, C * will be defined in Section 10.

We also mention that in accomplishing this stage in Chapter 4, we are naturally led to the difference between Levi-Civita connections: Theorem 4.5 (Theorem 14.13). The change ∇ -∇ 0 in Levi-Civita connection under a biconformal deformation g = σ -2 g H 0 + ρ -2 g V 0 is a (1, 2) tensor field given by Suppose that g 0 is an Einstein metric with constant of proportionality Λ 0 :

∇ -∇ 0 = (σ -2 ρ 2 -1)g H 0 (•, •) ν 0 + (σ 2 ρ -2 -1)B 0 • • + σ -2 ρ 2 g H 0 (•, •) V grad 0 ln σ + σ 2 ρ -2 g V 0 (•, •) H grad 0 ln ρ -(σ 2 ρ -2 -1)JH• ζ 0 (•) -(σ 2 ρ -2 -1)ζ 0 (•) JH• -H• (d ln σ)(V•) -V • (d ln ρ)(H•) -d ln σ(•) H• -d ln ρ(•) V• + d ln σ(JH•) JH• +ε d ln ρ(JV•) JV • .
Ric 0 = Λ 0 g 0 .
We determine conditions on the deformation parameters σ, ρ under which g is an Einstein metric:

Ric = Λg
for some constant of proportionality Λ. This equation is equivalent to the system

Ric H -Ric H 0 = (σ -2 Λ -Λ 0 )g H 0 , Ric V -Ric V 0 = (ρ -2 Λ -Λ 0 )g V 0 , Ric mixed -Ric mixed 0 = 0,
where, from the previous stage, the left-hand sides are expressed in terms of σ, ρ. Evaluating on orthonormal frame field (with respect to g 0 ) elements, this system becomes a system of PDEs in the deformation parameters σ, ρ.

4.3.2. Examples from Euclidean 4-space. We implement the above plan for M = R 4 , N = R 2 , and ϕ(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 ) in Section 18. The resulting system of PDEs is as in the following theorem.

Theorem 4.6 (Theorem 18.1). The biconformally deformed metric

g = σ -2 (dx 1 2 + dx 2 2 ) + ρ -2 (dx 3 2 + dx 4 2 )
is an Einstein metric if and only if the following system of nine PDEs holds:

0 = σ 2 Λ ii -ρ 2 Λ pp , i ∈ {1, 2}, p ∈ {3, 4}, 0 = Λ ab , a, b ∈ {1, 2, 3, 4}, a < b.
In this case, Ric = Λg, where

Λ = σ 2 Λ ii = ρ 2 Λ pp
for any i ∈ {1, 2} and p ∈ {3, 4}.

In the theorem, the quantities Λ ab are given by:

Λ ii = j (ln σ) jj + σ -2 ρ 2 q (ln σ) qq -2σ -2 ρ 2 q (ln σ) q 2 + 2(ln ρ) ii -2(ln ρ) i 2 + 2(ln σ) i (ln ρ) i -2(ln σ) i (ln ρ) i , i ∈ {1, 2}, Λ 12 = (ln ρ) 12 + (ln σ) 1 (ln ρ) 2 + (ln σ) 2 (ln ρ) 1 -(ln ρ) 1 (ln ρ) 2 , Λ pp = q (ln ρ) qq + σ 2 ρ -2 j (ln ρ) jj -2σ 2 ρ -2 j (ln ρ) j 2 + 2(ln σ) pp -2(ln σ) p 2 + 2(ln σ) p (ln ρ) p -2(ln σ) p (ln ρ) p , p ∈ {3, 4}, Λ 34 = (ln σ) 34 + (ln σ) 3 (ln ρ) 4 + (ln σ) 4 (ln ρ) 3 -(ln σ) 3 (ln σ) 4 , Λ ip = (ln σ) ip + (ln ρ) ip + 2(ln ρ) i (ln σ) p , i ∈ {1, 2}, p ∈ {3, 4}.
For example, we apply this general theorem to deduce the following result.

Theorem 4.7 (Theorem 18.8). The maximal, biconformally deformed metric

g = σ(x 1 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 1 ) -2 (dx 3 2 + dx 4 2 ),
is an Einstein metric if and only if one of the following holds:

• Ricci-flat: IA. g is isometric to the Euclidean 4-metric; IB. g is isometric to the product metric of (a) the warped metric y 2 dx 2 + dy 2 on the upper half-plane and (b) the Euclidean 2-metric; III. g is homothetic to ds 2 = x -1/2 (dx 2 + dy 2 ) + x(dz 2 + dw 2 ), x > 0;

• non-Ricci-flat: II. g is homothetic to the hyperbolic 4-metric;

IV. g is the metric (18.21), where ρ is given by (18.20), b ∈ R is given by (18.11), c ∈ R is given by (18.13), d ∈ R \ {0}, and c 2 ∈ R.

In item IB., g asymptotically collapses to the Euclidean 2-metric. In item IV., g asymptotically collapses either to the Euclidean 2-metric, or to a metric that is homothetic to the hyperbolic 4-metric.

4.3.3. Lorentzian context. Let the metrics g 0 , g be as in the previous stage and suppose that both are Lorentzian.

Suppose that g 0 is a solution to the Einstein equation

G 0 = 8πT 0 ⇔ Ric 0 = 8πT 0 + 1 2 S 0 g 0 .
We determine conditions on the deformation parameters σ, ρ under which g is a solution to the Einstein equation (4.1) G = 8πT ⇔ Ric = 8πT + 1 2 Sg for some physical parameters u, ρ, p, F, μ, Λ. Note that (4.1) is equivalent to the system

Ric H -Ric H 0 = 8π(T H -T H 0 ) + 1 2 (σ -2 S -S 0 )g H 0 , Ric V -Ric V 0 = 8π(T V -T V 0 ) + 1 2 (ρ -2 S -S 0 )g V 0 , Ric mixed -Ric mixed 0 = 8π(T mixed -T mixed 0 ),
where, from the previous stage, the left-hand sides, as well as the differences σ -2 S -S 0 and ρ -2 S -S 0 , are expressed in terms of σ, ρ. Evaluating on orthonormal frame field (with respect to g 0 ) elements, this system becomes a system of PDEs in the deformation parameters σ, ρ.

4.3.4.

Examples from Minkowski spacetime. We implement the above plan for ).

M = R 4 1 , N = R 2 ,
In the theorem, the quantities 8πT ab are given in terms of the deformation parameters σ, ρ, similar to the quantities Λ ab above (see Section 19).

This general theorem, when applied to a product of a Riemannian and a Lorentzian surface, gives the following result. Theorem 4.9 (Theorem 19.6). The biconformally deformed metric

g = σ(x 1 , x 2 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 3 , x 4 ) -2 (dx 3 2 -dx 4 2 )
satisfies the Einstein equation (4.1) with flow vector field u = ρ ∂ 4 and electromagnetic field F having zero mixed part if and only if either

sup ρ 2 q ε q (ln ρ) qq ≤ inf σ 2 j (ln σ) jj or sup σ 2 j (ln σ) jj ≤ inf ρ 2 q ε q (ln ρ) qq .
Example 4.10 (Example 19.7). The metric

g = e -2x 2 (dx 2 + dy 2 ) + e -2t 2 (dz 2 -dt 2 )
on R 4 satisfies the Einstein equation with μ ∈ (0, +∞) and F, u, ρ, p, Λ given/related by

F = ± 1 2 μπ -1 e -x 2 dx ∧ dy ± 1 2 μπ -1 e -t 2 dz ∧ dt, u = e t 2 ∂ t , ρ + p = 0, 8π p -Λ = -e 2x 2 + e 2t 2 .
Another application of the general theorem is in the case σ = σ(x 4 ), ρ a constant function:

F = 0, u = ∂ t , 8π ρ + 8π p = -3 2 t -2 , 8π p -Λ = -7 4 t -2 .
Yet another application of the general theorem is in the case σ = σ(x 1 , x 3 ), ρ = ρ(x 4 ): Theorem 4.12 (Theorem 19.19). The maximal, biconformally deformed metric

g = σ(x 1 , x 3 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 4 ) -2 (dx 3 2 -dx 4
ds 2 = csc 2 x (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), 0 < x < π; IIB. ds 2 = sech 2 x cos 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), -π 2 < z < π 2 ; IIIA. ds 2 = x -2 cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), x > 0; IIIC. ds 2 = csch 2 x cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), x > 0; IIID. ds 2 = csc 2 x cosh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), 0 < x < π; IVB. ds 2 = sech 2 x sinh 2 z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), z > 0.
In items 01. to 10. and item IB., the perfect fluid is vacuum-trivial [START_REF] Choquet-Bruhat | Global aspects of the Cauchy problem in general relativity[END_REF] . In the rest of the items, the perfect fluid is non-vacuum-trivial but the energy density and pressure are constant functions.

Other cases and examples are found in Section 19. 4.3.5. Examples from FRW spacetime with flat space. We also implement the plan for M = M (0, I, f ), N = R 2 , and ϕ(x 1 , x 2 , x 3 , t) = (x 1 , x 2 ) in Section 20. The general theorem, owing to the presence of the expansion f , is a generalization of the one in the previous subsubsection. We thus expect the results to be generalizations of those obtained beforehand.

There is one entirely new phenomenon, however -the Hubble condition for the biconformally deformed (pseudo-)cosmological spacetime10 (M, g). We give a sensible definition of this condition (in the case σ = σ(t), ρ = ρ(t)) in Subsection 20.2. We use this condition to generalize a classical result on endpoint singularities from FRW cosmology (Theorem 20.10). We also prove the following: Theorem 4.13 (Theorems 20.7 and 20.8). Suppose that the biconformally deformed metric

g = σ(t) -2 f (t) 2 (dx 1 2 + dx 2 2 ) + ρ(t) -2 (f (t) 2 dx 3 2 -dt 2 )
satisfies the Einstein equation (4.1) with flow vector field u = ρ ∂ t , and suppose that the Hubble condition for g holds. If σ or ρ is a constant function and the other is not, then the electromagnetic field F is nontrivial. If in addition F has zero mixed part, then F has nontrivial energy-momentum tensor contribution T em .

Note that the undeformed FRW cosmological model that we started with has trivial electromagnetic field F 0 . This theorem (along with the previous two examples) thus highlights the relevance of the biconformal deformation approach of this work: the emergence of nontrivial spacetime fields on spacetime models that are obtained by deforming classical Einstein solutions.

CHAPTER 2

Preliminaries

Geometric quantities associated to a semi-conformal submersion

In this chapter and the next, (M, g) is an oriented 4-dimensional Riemannian or Lorentzian (specifically, spacetime [START_REF] Danielo | Construction de métriques d'Einstein à partir de transformations biconformes[END_REF] ) manifold, (N, h) is an oriented Riemannian surface, and ϕ : (M, g) → (N, h) is a smooth semi-conformal submersion with smooth dilation λ : M → (0, +∞), i.e.

(5.1)

g(X, Y ) = λ(x) -2 h(dϕ x (X), dϕ x (Y )), x ∈ M, X, Y ∈ (ker dϕ x ) ⊥ ,
where dϕ x : T x M → T ϕ(x) N is the differential of ϕ at x ∈ M . Letting the vertical space V x and the horizontal space H x at x ∈ M be

V x = ker dϕ x , H x = V ⊥ x ,
respectively, we can write (5.1) as

g = λ -2 ϕ * h on H x × H x , x ∈ M,
where ϕ * h is the pull-back of h. [START_REF] Einstein | Die feldgleichungen der gravitation[END_REF] Note that the equation above also holds on the products H x × T x M . The horizontal distribution and vertical distribution of ϕ are the assignments x → H x and x → V x , respectively. Let H and V denote projection onto horizontal and vertical spaces, respectively. If (M, g) is a spacetime manifold, then we assume that there is a global timelike vector field T (i.e. a time-orientation) on (M, g) that is vertical, i.e. T = VT, or equivalently dϕ(T) = 0.

We associate to the semi-conformal submersion ϕ a number of geometric quantities. These quantities will be used in the computation of the Ricci curvature of g in Chapter 3. In Chapter 4, we will compute how these quantities change under a biconformal deformation of the metric, and such changes will be key in constructing solutions to the Einstein equation in Chapter 5.

While some of these geometric quantitites will be introduced in Section 10 (C, C * ), the principal ones are:

1. the dilation λ; 2. the first fundamental forms of the horizontal and vertical distributions, given by

g H = g(H•, H•), g V = g(V•, V•);
11 A spacetime manifold is by definition a 4-dimensional oriented, time-oriented Lorentzian manifold. [START_REF] Einstein | Die feldgleichungen der gravitation[END_REF] In view of the equation above, a semi-conformal map is called a horizontally weakly conformal map in [4]. where {e 1 , e 2 } is a (local) orthonormal frame field of the horizontal distribution. We make the following observations: 2' g V restriced to the vertical distribution is precisely the induced metric g ϕ on the fibers of ϕ. 3' A is symmetric if and only if the horizontal distribution is integrable. 4' a. B restricted to the vertical distribution is precisely the second fundamental form of the fibers of ϕ. b. B is symmetric since the vertical distribution is integrable. 6' By Corollary 7.3 below, (5.5) ν = V(grad ln λ); in view of this, set 14 ν ⊥ = H(grad ln λ). 8' a. By Corollary 7.4 below, [START_REF] Friedmann | Über die möglichkeit einer welt mit konstanter negativer krümmung des raumes[END_REF] (5.6)

ζ = g(∇ e1 e 2 , V•) = 1 2 g([e 1 , e 2 ], V•) = 1 2 V[e 1 , e 2
] . b. ζ = 0 if and only if the horizontal distribution is integrable. By the presence of the Lie bracket in (5.6), we see that the assignment (5.4) is independent of the signed orthonormal frame field {e 1 , e 2 }. Orthonormal frame fields are indispensable in this work, not just in making definitions but also as a computational tool. For example, we evaluate the difference G 1 -G 0 (between Einstein curvatures of (M, g 1 ) and (M, g 0 )) at orthonormal frame field elements in Chapter 5 to write the Einstein equation as a system of partial differential equations.

Let f = {f 1 , f 2 } be a (local) positive orthonormal frame field on (N, h) and e = {e 1 , e 2 , e 3 , e 4 } be a (local) positive orthonormal frame field on (M, g) such that

(1) e 1 , e 2 are horizontal vector fields with dϕ(e i ) = λf i for i ∈ {1, 2}, (2) e 3 , e 4 are vertical vector fields, and

(3) e 4 is timelike and future-pointing if g is Lorentzian. [START_REF] Einstein | On the relation between the expansion and the mean density of the universe[END_REF] The differences A-A * and B -B * are called O'Neill's fundamental tensors (see [30] Section 2). [START_REF] Friedmann | Über die krümmung des raumes[END_REF] Note that ν ⊥ does not mean the orthogonal complement of ν with respect to g. We are simply using the symbol ⊥ to remind ourselves that we are projecting onto the horizontal spaces. Let θ = {θ 1 , θ 2 , θ 3 , θ 4 } be the dual frame field of e, i.e. θ a = ε a e a for all a ∈ {1, 2, 3, 4}, where

ε 1 = ε 2 = ε 3 = 1, ε 4 = ε = +1, if g is Riemannian, -1, if g is Lorentzian.
The connection 1-forms associated to f and e are given by for all a ∈ {1, 2, 3, 4}.

ρ ij = h(∇ • f i , f j ), i, j ∈ {1, 2}, ω ab = g(∇ • e a ,
In terms of the frame fields e and θ, some of the geometric quantities listed above are given by 2"

g H = θ 1 2 + θ 2 2 , g V = θ 3 2 + ε θ 4 2 , 6" ν = 1 2 V(∇ e1 e 1 + ∇ e2 e 2 )
= e 3 (ln λ) e 3 + ε e 4 (ln λ) e 4 , ν ⊥ = e 1 (ln λ) e 1 + e 2 (ln λ) e 2 ,

7"

µ = 1 2 H(∇ e3 e 3 + ε∇ e4 e 4 ) =: µ 1 e 1 + µ 2 e 2 .

Notation and conventions

In this work, covariant differentiation is denoted by ∇ while Lie differentiation is denoted by L. Moreover, the musical isomorphisms are as follows: For a vector field X, X is the 1-form that is metrically equivalent to X, i.e. X = g(X, •). For a 1-form α, α is the vector field that is metrically equivalent to α, i.e. (α ) = α.

The following list enumerates some conventions that we follow in this work.

(1) We use the following sign and trace conventions for the Riemannian, Ricci, and sectional curvatures (of (M, g)):16

R(X, Y )Z = [∇ X , ∇ Y ]Z -∇ [X,Y ] Z, Ric(X, Y ) = trace g(R(X, •)•, Y ) = 4 a=1 ε a g(R(X, e a )e a , Y ), K(span{X, Y }) = K(X, Y ) = g(R(X, Y )Y, X) ({X, Y } a frame).
(2) The exterior product of two 1-forms α and β is given by

α ∧ β = α ⊗ β -β ⊗ α
while their symmetric product is given by

α β = sym α ⊗ β = 1 2 (α ⊗ β + β ⊗ α).
(3) We do not use the Einstein summation convention for repeated indices. Whenever we want to sum over an index α (repeated or otherwise), we will always explicitly write α . (4) We use the following convention on indices:

a, b, c ∈ {1, 2, 3, 4}, i, j, k ∈ {1, 2}, p, q, r ∈ {3, 4}. Set i = 2, if i = 1, 1, if i = 2, p = 4, if p = 3, 3, if p = 4;
similarly for j, k, q, r. For example, if

f 3 = f 4 = 0, then (6.1) ∇ ϕ ea dϕ(e b ) = ∇ ϕ ea λ (f b • ϕ) + λ∇ ϕ ea (f b • ϕ) = e a (ln λ) dϕ(e b ) + λ 2 ∇ fa f b
, where ∇ ϕ is the induced connection on the pull-back bundle ϕ -1 (T N ) → M . Now, we take the exterior product of the horizontal elements of θ and also that of the vertical elements; let

ξ H = θ 1 ∧ θ 2 , ξ V = θ 3 ∧ θ 4 .
Note that these assignments are independent of the signed orthonormal frame fields of the distributions. Moreover, ξ V is precisely the volume element of the fibers of ϕ (up to sign).)

Lemma 6.1. For 1-forms α, β, ξ H (α , β ) = α ∧ β (e 1 , e 2 ), ξ V (α , β ) = ε α ∧ β (e 3 , e 4 ).
Proof. Using the definitions, we have

ξ V (α , β ) = θ 3 (α )θ 4 (β ) -θ 3 (β )θ 4 (α ) = εα(e 3 )β(e 4 ) -εβ(e 3 )α(e 4 ) = ε α ∧ β (e 3 , e 4 ).
The proof of the first equation has the same steps, except that the indices 3, 4 are replaced by 1, 2, respectively, and ε is not involved.

Finally, we endow the horizontal and vertical spaces with almost (para)complex structures. Definition 6.2. Let J be the (1, 1) tensor field given by

Je 1 = e 2 , Je 2 = -e 1 , Je 3 = e 4 , Je 4 = -ε e 3 .
(Note that these assignments are independent of the signed orthonormal frame fields of the distributions.)

For a vector field X = a X a e a , JX = -X 2 e 1 + X 1 e 2 -εX 4 e 3 + X 3 e 4 = i (-1) i X i e i + p (-1) p ε p X p e p .
The tensor field J is an almost complex structure (specifically, a rotation by +π/2 about the origin) on individual horizontal spaces. The same is true for the vertical spaces if g is Riemannian. If g is Lorentzian, then J is an almost paracomplex structure (specifically, a reflection about the line through the origin with direction e 3 + e 4 ) on individual vertical spaces. The tensor field J will appear in our formula for the Ricci curvature in Theorem 13.3.

The following lemma is used throughout this chapter and the next.

Lemma 6.3.

(1) For vector fields X, Y ,

g(X, JY ) = -ξ H (X, Y ) -ε ξ V (X, Y ).
(2) For vector fields X, Y ,

g(JX, JY ) = g H (X, Y ) + ε g V (X, Y ).
(3) For a 1-form α,

αJ := α(J•) = α(e 2 ) θ 1 -α(e 1 ) θ 2 + α(e 4 ) θ 3 -ε α(e 3 ) θ 4 = i (-1) i α(e i ) θ i + p (-1) p ε p α(e p ) θ p = -(Jα ) .
Proof.

(1) Write X = a X a e a and Y = a Y a e a . Then

g(X, JY ) = g(X, -Y 2 e 1 + Y 1 e 2 -εY 4 e 3 + Y 3 e 4 ) = -X 1 Y 2 + X 2 Y 1 -εX 3 Y 4 + εX 4 Y 3 ,
from which the result follows by the definitions of ξ H and ξ V . (2) This follows from (1).

(3) This is a direct consequence of the definition of J.

In view of the last item of the previous lemma, we adopt the following conventions: If we want to output a vector field, we will write JX or Jα . If we want a 1-form, we will write X J or αJ.

Christoffel symbols

Before we compute the Ricci curvature of the metric g in the Chapter 3, we first compute its Levi-Civita connection. We do this by listing the Christoffel symbols of g with respect to the frame field e, by which we mean the scalar fields

ε b g(∇ ec e a , e b ) = ε b ω ab (e c ).
We start by using the symmetry of the second fundamental form ∇dϕ of ϕ, which is given by ∇dϕ(X, Y ) = ∇ ϕ X dϕ(Y ) -dϕ(∇ X Y ) for vector fields X, Y on M (see [4] Section 3.2). The equation ∇dϕ(e 3 , e 4 ) = ∇dϕ(e 4 , e 3 ) is just equivalent to the integrability of the vertical distribution, and so is not useful. The equations ∇dϕ(e i , e j ) = ∇dϕ(e j , e i ), ∇dϕ(e i , e p ) = ∇dϕ(e p , e i ), however, give us Lemmas 7.1 and 7.2 below, respectively. Before stating these lemmas, we calculate

λ 2 [f i , f i ] = λ 2 ρ i i (f i )f i -λ 2 ρ ii (f i )f i = ϕ * ρ i i (e i ) dϕ(e i ) -ϕ * ρ ii (e i ) dϕ(e i ) = - k ϕ * ρ ii (e k ) dϕ(e k ). (7.1) Lemma 7.1. g(∇ ei e i , e j ) = e j (ln λ) -δ ij e i (ln λ) + ϕ * ρ ij (e i ).
Proof. Using (6.1), ∇dϕ(e i , e j ) = ∇ ϕ ei dϕ(e j ) -dϕ(∇ ei e j ) = e i (ln λ) dϕ(e j ) + λ 2 ∇ fi f j -dϕ(∇ ei e j ) and similarly for ∇dϕ(e j , e i ). By the symmetry of ∇dϕ,

dϕ([e i , e j ]) = e i (ln λ) dϕ(e j ) -e j (ln λ) dϕ(e i ) + λ 2 [f i , f j ].
Then, using (7.1),

g(∇ ei e j , e i ) = g([e i , e j ], e i ) = δ ij e i (ln λ) -e j (ln λ) -ϕ * ρ ij (e i ),
from which the result immediately follows.

Lemma 7.2 (Cf. [1] Lemma 2.1). g([e p , e i ], e j ) = δ ij e p (ln λ).

Proof. We have ∇dϕ(e i , e p ) = -dϕ(∇ ei e p ) and, using (6.1), ∇dϕ(e p , e i ) = e p (ln λ) dϕ(e i ) -dϕ(∇ ep e i ).

By the symmetry of ∇dϕ,

dϕ([e p , e i ]) = e p (ln λ) dϕ(e i ),
from which the result immediately follows using semi-conformality.

if i = j, (-1) j ζ(e p ), if i = j, (2) g(∇ ei e i , e i ) = e i (ln λ) + (-1) i ϕ * ρ 12 (e i ), (3) g(∇ ep e i , e i ) = (-1) i ζ(e p ), (4)
g(∇ ep e i , e q ) = -B ep e q (e i ) = -B * ep e i (e q ), (5) g(∇ ea e p , e p ) = (-1) p ω 34 (e a ).

Proof.

(1) The first branch is by either Lemma 7.2 or Corollary 7.3, and the second is by the definition of ζ and Corollary 7.3.

(2) This is from Lemma 7.1.

(3) This is from Lemma 7.2 and item (1). ( 4) This is from the definitions of B and B * .

(5) This is from the definition of ω 34 .

To end this section, in view of (5.2) and (5.3), we record these other "permutations": Corollary 7.7.

H∇ H• e i = (ν ⊥ J(•) + ϕ * ρ 12 (•)) Je i , H∇ V• e i = -ζ(•) Je i , V∇ • e p = ε ω 34 (•) Je p .
Proof. This is a consequence of Corollary 7.6. For example, for a vector field X = a X a e a ,

H∇ HX e i = j X j H∇ ej e i = j X j g(∇ ej e i , e i ) e i = j X j (-1) i +j (e j (ln λ) + (-1) j ϕ * ρ 12 (e j )) e i = (ν ⊥ J(X) + ϕ * ρ 12 (X)) Je i .

Divergences

Aside from the Christoffel symbols of Section 7, we will also need some divergence terms in the computation of Ricci curvature in Chapter 3. In particular, we calculate the divergences of grad ln λ, ν, ν ⊥ , µ, ζ, B ep e q , and B * eq e i in this section. Recall that for any 1-form α, But the sum inside α in the last term is given by the following lemma.

Lemma 8.1. The following hold:

(

1) i ∇ ei e i = J(ϕ * ρ 12 ) + grad ln λ + ν. (2) p ε p ∇ ep e p = ε JVω 34 + 2µ. (3)
a ε a ∇ ea e a = J(ϕ * ρ 12 ) + ε JVω 34 + grad ln λ + ν + 2µ. (This result tells us how to write the sums i ∇ ei e i -J(ϕ * ρ 12 ) , p ε p ∇ ep e pε JVω 34 , and a ε a ∇ ea e a -J(ϕ * ρ 12 ) -ε JVω 34 invariantly.) Going back to (8.1), and using Lemmas 8.1 and 6.3, we obtain Corollary 8.2. For any 1-form α,

Proof. Using Corollary 7.6, i ∇ ei e i = H i ∇ ei e i + V i ∇ ei e i = (e 1 (ln λ) + ϕ * ρ 21 (e 2 )) e 1 + (e 2 (ln λ) + ϕ * ρ 12 (e 1 )) e 2 + 2ν = -ϕ * ρ 12
div α = a ε a e a (α(e a ))+ξ H (α , (ϕ * ρ 12 ) )+ξ V (α , ω 34 )-α(grad ln λ)-α(ν)-2α(µ).
(This result tells us how to write the sum a ε a e a (α(e a )) + ξ H (α , (ϕ

* ρ 12 ) ) + ξ V (α , ω 34 ) invariantly.) In particular, if α is horizontal, then div α = i e i (α(e i )) + ξ H (α , (ϕ * ρ 12 ) ) -α(ν ⊥ ) -2α(µ); if α is vertical, then div α = p ε p e p (α(e p )) + ξ V (α , ω 34 ) -2α(ν).
We can now record the divergences mentioned at the beginning of this section.

Corollary 8.3. The following hold:

(1) ∆ ln λ = a ε a e a (e a (ln λ))

+ ξ H (ν ⊥ , (ϕ * ρ 12 ) ) + ξ V (ν, Vω 34 ) -ν ⊥ 2 - 2g(ν, ν) -2µ(ln λ), (2) div ν = p ε p e p (e p (ln λ)) + ξ V (ν, Vω 34 ) -2g(ν, ν), (3) div ν ⊥ = i e i (e i (ln λ)) + ξ H (ν ⊥ , (ϕ * ρ 12 ) ) -ν ⊥ 2 -2µ(ln λ), (4) div µ = i e i (µ i ) + ξ H (µ, (ϕ * ρ 12 ) ) -µ(ln λ) -2 µ 2 , (5) div ζ = p ε p e p (ζ(e p )) + ξ V (ζ , ω 34 ) -2ζ(ν), (6) div B ep e q = i e i (B ep e q (e i ))+ξ H (B ep e q , (ϕ * ρ 12 ) )-ν ⊥ (B ep e q )-2µ (B ep e q ), (7)
div B * eq e i = p ε p e p (B * eq e i (e p )) + ξ V (B * eq e i , ω 34 ) -2ν (B * eq e i ).

Sectional curvature of a surface

In this section, we express the (intrinsic) sectional curvatures K ϕ of the fibers of ϕ and K N of the surface N in terms of the frame field e. The former will be used in the proof of Lemma 15.5 and the latter in Section 11.

We start with a general semi-Riemannian surface (S, g S ) with index 0 or 1. If {a 1 , a 2 } is a (local) orthonormal frame field on S, giving the connection 1-form

α 12 = g S (∇ • a 1 , a 2 ) = -[a 1 , a 2 ] ,
then the sectional curvature K S of S is given by

K S = g S (R(a 1 , a 2 )a 2 , a 1 ) = g S (∇ a1 ∇ a2 a 2 , a 1 ) -g S (∇ a2 ∇ a1 a 2 , a 1 ) -g S (∇ [a1,a2] a 2 , a 1 ) = -a 1 (α 12 (a 2 )) + a 2 (α 12 (a 1 )) + α 12 [a 1 , a 2 ] = -2 dα 12 (a 1 , a 2 ),
and its Ricci curvature Ric S is given by

Ric S = ε S K S g S , where ε S = g S (a 2 , a 2 ).
For example, if S is a fiber of ϕ and a i = e i+2 (so that α 12 = Vω 34 ):

Lemma 9.1. The intrinsic sectional curvature K ϕ of the fibers of ϕ is given by

K ϕ = - p ε p e p (ω 34 Je p ) -g V (ω 34 , ω 34 ).
And if S = N and a i = f i (so that α 12 = ρ 12 ):

Lemma 9.2. The sectional curvature K N of N is given by

λ 2 K N = -e 1 (ϕ * ρ 12 (e 2 ))+e 2 (ϕ * ρ 12 (e 1 ))+ξ H (ν ⊥ , (ϕ * ρ 12 ) )-(ϕ * ρ 12 ) 2 = -2 d(ϕ * ρ 12 )(e 1 , e 2 ).
Proof. We know that

K N = -f 1 (ρ 12 (f 2 )) + f 2 (ρ 12 (f 1 )) -ρ 12 2 .
We compose these addends with ϕ:

f i (ρ 12 (f i )) = λ -1 dϕ(e i )(ρ 12 (f i )) = λ -1 e i (λ -1 ϕ * ρ 12 (e i )) = λ -2 e i (ϕ * ρ 12 (e i )) -λ -2 e i (ln λ)ϕ * ρ 12 (e i ), -f 1 (ρ 12 (f 2 )) + f 2 (ρ 12 (f 1 )) = -λ -2 e 1 (ϕ * ρ 12 (e 2 )) + λ -2 e 2 (ϕ * ρ 12 (e 1 )) + λ -2 d ln λ ∧ ϕ * ρ 12 (e 1 , e 2 ) = -λ -2 e 1 (ϕ * ρ 12 (e 2 )) + λ -2 e 2 (ϕ * ρ 12 (e 1 )) + λ -2 ξ H (ν ⊥ , (ϕ * ρ 12 ) ), -ρ 12 2 = - i (ρ 12 (f i )) 2 = -λ -2 i (ϕ * ρ 12 (e i )) 2 = -λ -2 (ϕ * ρ 12 ) 2 .
The first equation now follows upon multiplication by λ 2 . On the other hand, by using Corollary 7.5 and Lemma 6.3, we have

-2 d(ϕ * ρ 12 )(e 1 , e 2 ) = -e 1 (ϕ * ρ 12 (e 2 )) + e 2 (ϕ * ρ 12 (e 1 )) + ϕ * ρ 12 (Jν ⊥ -(ϕ * ρ 12 ) ) = -e 1 (ϕ * ρ 12 (e 2 )) + e 2 (ϕ * ρ 12 (e 1 )) + ξ H (ν ⊥ , (ϕ * ρ 12 ) ) -(ϕ * ρ 12 ) 2 .
In Lemma 2.1 of Appendix B, the sectional curvature K N is expressed in terms of the dilation λ, the mean curvature µ, and the conformality factor associated to isothermal coordinates on N . The first equation in the lemma shows that the horizontal distribution is umbilic by definition ([4] Definition 2.5.6): for a horizontal vector field X with X = 1,

A = g H (•, •) ν + ξ H (•, •) ζ , A * = H• ν (•) + JH• ζ(•),
A X X = ν, which is independent of X.
Now, we calculate the divergences of a tensor field that is metrically equivalent to the second fundamental form B. These divergences will be used in the computation of the vertical and mixed parts of Ricci curvature in Sections 12 and 13, respectively. Definition 10.2. Let B 1 be the (3, 0) tensor field metrically equivalent to the second fundamental form B that is obtained by lowering the contravariant slot of B and making it the first covariant slot: for vector fields X, Y, Z,

B 1 (X, Y, Z) = g(X, B Y Z).
Let div 1 B 1 and div 2 B 1 be the divergences of B 1 with respect to the first and second (or third) slots, respectively: for vector fields X, Y , 17

div 1 B 1 (X, Y ) = trace ∇ • B 1 (•, X, Y ), div 2 B 1 (X, Y ) = trace ∇ • B 1 (X, •, Y ) = trace ∇ • B 1 (X, Y, •). Lemma 10.3. div 1 B 1 = p,q i e i (g(e i , B ep e q )) θ p θ q + (ϕ * ρ 12 J)B -ν ⊥ (B) -2µ (B) -2 sym ν (B * ) + 2 sym ζ(B * • J•) -2ε sym ω 34 (B • J•) = p,q (div B ep e q ) θ p θ q -2 sym ν (B * ) + 2 sym ζ(B * • J•) -2ε sym ω 34 (B • J•).
Proof. Using Lemmas 8.1 and 6.3,

div 1 B 1 = a ε a ∇ ea B 1 (e a , •, •) = i e i (g(e i , B)) - a ε a g(H∇ ea e a , B) - i g(e i , B ∇e i • •) - i g(e i , B • ∇ ei •) = i e i (g(e i , B)) + (ϕ * ρ 12 J)B -ν ⊥ (B) -2µ (B) - i g(e i , B ∇e i • •) - i g(e i , B • ∇ ei •) = p,q i e i (g(e i , B ep e q )) θ p ⊗ θ q + (ϕ * ρ 12 J)B -ν ⊥ (B) -2µ (B) - a,q i g(e i , B ∇e i ea e q ) θ a ⊗ θ q - p,b i g(e i , B ep ∇ ei e b ) θ p ⊗ θ b . But - a,q i
g(e i , B ∇e i ea e q ) θ a ⊗ θ q =j,q i g(e i , B ∇e i ej e q ) θ j ⊗ θ q p,q i g(e i , B ∇e i ep e q ) θ p ⊗ θ q , where - [START_REF] Hawking | The large-scale structure of space-time, Cambridge Monographs on Mathematical Physics[END_REF] The divergence div 2 B 1 (Y, X) here is the same as the divergence div B 2 (X, Y ) in [2].

j,q i g(e i , B ∇e i ej e q ) θ j ⊗ θ q = - i g(e i , B Ae i •1 • 2 ) = - i g(e i , B •2 (g(e i , • 1 ) ν -g(e i , J• 1 ) ζ )) = -g(• 1 , B ν • 2 ) + g(J• 1 , B ζ • 2 ) = -ν (B * •2 • 1 ) + ζ(B * •2 J• 1 ) using Lemma 10.1 and - p,q i g(e i , B ∇e i ep e q ) θ p ⊗ θ q = - p,q i (-1) p ε p ω 34 (e i )g(e i , B e p e q ) θ p ⊗ θ q = -ε i ω 34 (e i )g(e i , B J•1 • 2 ) = -ε ω 34 (B J•1 • 2 ).
Similarly,

- p,b i g(e i , B ep ∇ ei e b ) θ p ⊗ θ b = -ν (B * •1 • 2 ) + ζ(B * •1 J• 2 ) -ε ω 34 (B •1 J• 2 ).
Before stating a similar formula for div 2 B 1 in Lemma 10.5, we introduce the tensor fields C and C * . These geometric quantities, along with those in Section 5, will be used in the computation of Ricci curvature in Chapter 3.

Definition 10.4. Let C and C * be the (0, 2) tensor fields given by

C = p ε p g(B ep •, B ep •) = g(trace B * B •, •), C * = p ε p g(B * ep •, B * ep •) = g(trace B B * •, •).
Note that

C = i g(B * • e i , B * • e i ) = p,i ε p (B * ep e i ) 2 , C * = p,q ε p ε q (B ep e q ) 2
also.

Lemma 10.5.

div 2 B 1 = i,q p
ε p e p (g(e p , B * eq e i ))

θ i ⊗ θ q -2ν (B * •2 • 1 ) + ζ(B * •2 J• 1 ) + ε ω 34 J(B * •2 • 1 ) -ε ω 34 (B * J•2 • 1 ) -C + C * = i,q (div B * eq e i ) θ i ⊗ θ q + ζ(B * •2 J• 1 ) -ε ω 34 (B * J•2 • 1 ) -C + C * .
Proof. Using Lemmas 8.1 and 6.3,

div 2 B 1 = a ε a ∇ ea B 1 (• 1 , e a , • 2 ) = p ε p e p (g(• 1 , B ep • 2 )) - a ε a g(• 1 , B V∇e a ea • 2 ) - p ε p g(∇ ep • 1 , B ep • 2 ) - p ε p g(• 1 , B ep ∇ ep • 2 ) = p ε p e p (g(e p , B * •2 • 1 )) + ε ω 34 J(B * •2 • 1 ) -2ν (B * •2 • 1 ) - p ε p g(∇ ep • 1 , B ep • 2 ) - p ε p g(• 1 , B ep ∇ ep • 2 ) = i,q p
ε p e p (g(e p , B * eq e i ))

θ i ⊗ θ q + ε ω 34 J(B * •2 • 1 ) -2ν (B * •2 • 1 ) - a,q p ε p g(∇ ep e a , B ep e q ) θ a ⊗ θ q - i,b p ε p g(e i , B ep ∇ ep e b ) θ i ⊗ θ b . But - a,q p ε p g(∇ ep e a , B ep e q ) θ a ⊗ θ q = - i,q p ε p g(∇ ep e i , B ep e q ) θ i ⊗ θ q - r,q p ε p g(∇ ep e r , B ep e q ) θ r ⊗ θ q = - i,q p ε p g(H∇ ep e i , B ep e q ) θ i ⊗ θ q -C,
where

- i,q p ε p g(H∇ ep e i , B ep e q ) θ i ⊗ θ q = i,q p ε p ζ(e p )g(Je i , B ep e q ) θ i ⊗ θ q = p ε p ζ(e p )g(J•, B ep •) = ζ(B * •2 J• 1 ) using Corollary 7.7. Moreover, - i,b p ε p g(e i , B ep ∇ ep e b ) θ i ⊗ θ b = - i,j p ε p g(e i , B ep ∇ ep e j ) θ i ⊗ θ j - i,q p ε p g(e i , B ep ∇ ep e q ) θ i ⊗ θ q = C * - i,q p ε p g(B * ep e i , ∇ ep e q ) θ i ⊗ θ q ,
where i,q p ε p g(B * ep e i , ∇ ep e q ) θ i ⊗ θ q = -ε i,q p (-1) q ε q ε p ω 34 (e p )g(B * ep e i , e q ) θ i ⊗ θ q = -ε

p ε p ω 34 (e p )g(B * J•2 • 1 , e p ) = -ε ω 34 (B * J•2 • 1 )
. The divergence div 2 B 1 has horizontal part C * and vertical part -C (see items (3) and (4) of the following corollary).

Corollary 10.6. The following hold:

(1)

div 1 B 1 (H•, •) = -ν (B * •2 • 1 ) + ζ(B * •2 J• 1 ), (2) div 1 B 1 (•, H•) = -ν (B * ) + ζ(B * • J•), (3) div 2 B 1 (V•, •) = -C, (4) div 2 B 1 (•, H•) = C * .
Next, we take the traces of div 1 B 1 , div 2 B 1 , C, and C * . In particular, trace div 1 B 1 will be used in the calculation of the scalar curvature of the metric g in Chapter 3.

Lemma 10.7. The following hold:

(1) trace div

1 B 1 = 2 div µ , ( 2
) trace div 2 B 1 = 0.
Proof.

(1) This follows from Corollary 8.

* = p,q ε p ε q B ep e q 2 = p,i ε p g(B * ep e i , B * ep e i ).
As written in the next corollary, if g is Riemannian, then the triviality of trace C implies the triviality of B. The same is true in the Lorentzian case if the "mixed vertical part" of B is also trivial. Finally, we record an identity (Corollary 10.11) that will be used in the computation of the vertical part of Ricci curvature in Section 12.

Lemma 10.10.

K(e 3 , e 4 ) -

K ϕ = B e3 e 4 2 -g(B e3 e 3 , B e4 e 4 ) = 1 2 ε trace C -2ε µ 2 . Proof.
The first equation is just the Gauss equation applied to the fibers of ϕ. The second equation is from a direct calculation:

trace C -4 µ 2 = p,q ε p ε q B ep e q 2 -g p ε p B ep e p ,
q ε q B eq e q = 2ε ( B e3 e 4 2 -g(B e3 e 3 , B e4 e 4 )).

Corollary 10.11. εK(e 3 , e 4 )

g V = εK ϕ g V -2µ (B) + C.
Proof. This follows from the first equation in the previous lemma and by evaluating the tensor field -2µ (B) + C on orthonormal frame field elements.

With these preliminaries, we can now proceed to the computation of Ricci curvature in Chapter 3 and of changes under biconformal deformation of the metric in Chapter 4.

CHAPTER 3

Ricci curvature

As in the previous chapter, here (M, g) is an oriented 4-dimensional Riemannian or Lorentzian (specifically, spacetime) manifold, (N, h) is an oriented Riemannian surface, and ϕ : (M, g) → (N, h) is a smooth semi-conformal submersion with smooth dilation λ : M → (0, +∞), i.e.

g = λ -2 ϕ * h on H x × H x , x ∈ M.
(If (M, g) is a spacetime manifold, then we assume that there is a global timelike vector field on (M, g) that is vertical.) The (local) frame fields f , e, θ and the connection 1-forms ρ ij , ω ab are as in the previous chapter, as well.

We compute the Ricci curvature Ric of the metric g in this chapter by using the auxiliary frame fields f , e, θ. We express this Ricci curvature in terms of the geometric quantities presented in Sections 5 and 10. Note that

Ric = Ric H + Ric V + Ric mixed ,
where

Ric H = Ric(H•, H•), Ric V = Ric(V•, V•), Ric mixed = Ric(H•, V•) + Ric(V•, H•),
called the horizontal part, vertical part, and mixed part of Ric, respectively. We compute these three parts of Ric in the three sections of this chapter, and record them in Lemmas 11.1, 12.2, and 13.2. We write down the full Ricci curvature Ric of g in Theorem 13.3, and also the scalar curvature S of g in Theorem 12.4. [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés, Fondations[END_REF] The following results are used throughout this chapter: Lemma 6.

Ric H = i,j a ε a g(R(e i , e a )e a , e j ) θ i θ j = K(e1,e2)
g(R(e 1 , e 2 )e 2 , e 1 ) g H + [IP P J] i,j p ε p g(R(e i , e p )e p , e j ) θ i θ j . [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés, Fondations[END_REF] An alternative computation of the Ricci curvature Ric, by first assuming that N = R 2 and then using isothermal coordinates to generalize to any Riemannian surface N , is outlined in Appendix B.

43

We first look at K(e 1 , e 2 ) g H . We have g(∇ e1 ∇ e2 e 2 , e 1 ) = e 1 (g(∇ e2 e 2 , e 1 )) -g(∇ e2 e 2 , ∇ e1 e 1 ) = e 1 (e 1 (ln λ)) -e 1 (ϕ * ρ 12 (e 2 )) -g(ν, ν), -g(∇ e2 ∇ e1 e 2 , e 1 ) = -e 2 (g(∇ e1 e 2 , e 1 )) + g(∇ e1 e 2 , ∇ e2 e 1 ) = e 2 (e 2 (ln λ)) + e 2 (ϕ * ρ 12 (e 1 ))

-g(ζ , ζ ), -g(∇ [e1,e2] e 2 , e 1 ) = -g(∇ Jν ⊥ -(ϕ * ρ12) e 2 , e 1 ) -g(∇ 2ζ e 2 , e 1 ) = (ν ⊥ J + ϕ * ρ 12 )(Jν ⊥ -(ϕ * ρ 12 ) ) -2g(ζ , ζ ) = -ν ⊥ 2 -(ϕ * ρ 12 ) 2 + 2 ξ H (ν ⊥ , (ϕ * ρ 12 ) ) -2g(ζ , ζ ).
Hence, using Lemma 9.2,

K(e 1 , e 2 ) = -e 1 (ϕ * ρ 12 (e 2 )) + e 2 (ϕ * ρ 12 (e 1 )) + ξ H (ν ⊥ , (ϕ * ρ 12 ) ) -(ϕ * ρ 12 ) 2 + i e i (e i (ln λ)) + ξ H (ν ⊥ , (ϕ * ρ 12 ) ) -ν ⊥ 2 -g(ν, ν) -3 g(ζ , ζ ) = λ 2 K N + div ν ⊥ + 2µ(ln λ) -g(ν, ν) -3 g(ζ , ζ ).
Next we look at [IP P J]. We have i,j p

ε p g(∇ ei ∇ ep e p , e j ) θ i θ j = i,j g(∇ ei 2µ, e j ) θ i θ j + i,j g(∇ ei εJVω 34 , e j ) θ i θ j = L µ g(H•, H•) -ε i,j g(Jω 34 , A ei e j ) θ i θ j = L µ g(H•, H•) -ε i,j g(Jω 34 , δ ij ν + (-1) j δ i j ζ ) θ i θ j = L µ g(H•, H•) + ξ V (ν, Vω 34 ) g H , - i,j p ε p g(∇ ep ∇ ei e p , e j ) θ i θ j = - i,j p ε p e p (g(∇ ei e p , e j )) θ i θ j + i,j p ε p g(∇ ei e p , ∇ ep e j ) θ i θ j = i,j p ε p e p (g(e p , A ei e j )) θ i θ j - i,j p ε p g(A * ei e p , H∇ ep e j ) θ i θ j - i,j p ε p g(V∇ ei e p , B * ep e j ) θ i θ j = i,j p ε p e p (g(e p , δ ij ν + (-1) j δ i j ζ )) θ i θ j - i,j p ε p g(e p (ln λ) e i + (-1) i ζ(e p ) e i -(-1) j ζ(e p ) e j ) θ i θ j - i,j p ε p g((-1) p ε p ω 34 (e i ) e p , B * ep e j ) θ i θ j = (div ν -ξ V (ν, Vω 34 ) + 2g(ν, ν) + g(ζ , ζ )) g H , and
- i,j p ε p g(∇ [ei,ep] e p , e j ) θ i θ j = - i,j p ε p g(∇ -ep(ln λ) ei e p , e j ) θ i θ j - i,j p ε p g(∇ B * ep ei+ε ω34(ei) Jep e p , e j ) θ i θ j = - i,j p ε p e p (ln λ) g(e p , A ei e j ) θ i θ j - i,j p ε p g(B * ep e i + ε ω 34 (e i ) Je p , B * ep e j ) θ i θ j = - i,j p ε p e p (ln λ) g(e p , δ ij ν + (-1) j δ i j ζ ) θ i θ j - i,j p ε p g(B * ep e i + (-1) p ε p ω 34 (e i ) e p , B * ep e j ) θ i θ j = -g(ν, ν) g H -C * .
Thus,

Ric H = (λ 2 K N + div ν ⊥ + 2µ(ln λ) -g(ν, ν) -3 g(ζ , ζ )) g H + L µ g(H•, H•) + ξ V (ν, Vω 34 ) g H + (div ν -ξ V (ν, Vω 34 ) + 2g(ν, ν) + g(ζ , ζ )) g H -g(ν, ν) g H -C * .
Simplification gives us the next lemma.

Lemma 11.1. The horizontal part

Ric H = Ric(H•, H•) of the Ricci curvature Ric of g is given by 19 Ric H = (λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ )) g H + L µ g(H•, H•) -C * . Since 1 2 trace L µ g(H•, H•) = i (∇ ei µ )(e i ) = i e i (µ i ) - i µ (∇ ei e i ) = (div µ -ξ H (µ, (ϕ * ρ 12 ) ) + µ(ln λ) + 2 µ 2 ) -µ (J(ϕ * ρ 12 ) + ν ⊥ ) = div µ + 2 µ 2 ,
then the "horizontal part" of the scalar curvature S of g is as follows.

Lemma 11.2. The trace of Ric H is given by

1 2 trace Ric H = λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ ) + div µ + 2 µ 2 -1 2 trace C.

Vertical part

The vertical part Ric V = Ric(V•, V•) of the Ricci curvature Ric of g is given by Ric V = p,q a ε a g(R(e p , e a )e a , e q ) θ p θ q = ε K(e3,e4) g(R(e 3 , e 4 )e 4 , e 3 ) g V + [P IIQ] p,q i g(R(e p , e i )e i , e q ) θ p θ q .

19 General Lie derivatives of g are discussed in Appendix A.

From Corollary 10.11, εK(e 3 , e 4 ) g V = εK ϕ g V -2µ (B) + C.

For [P IIQ], we start with p,q i g(∇ ep ∇ ei e i , e q ) θ p θ q = p,q g(∇ ep (J(ϕ * ρ 12 ) + ν ⊥ ), e q ) θ p θ q + p,q g(∇ ep 2ν, e q ) θ p θ q =p,q g(J(ϕ

* ρ 12 ) + ν ⊥ , B ep e q ) θ p θ q + L ν g(V•, V•) = (ϕ * ρ 12 J)B -ν ⊥ (B) + L ν g(V•, V•).
Remark 12.1. The motivation for using

L ν g(V•, V•) is the presence of L µ g(H•, H•)
in our formula for Ric H . If instead we want to use L grad ln λ g, we can decompose i ∇ ei e i as the sum of J(ϕ * ρ 12 ) -ν ⊥ (which is still horizontal) and 2 grad ln λ in the above calculation. This gives us

p,q i g(∇ ep ∇ ei e i , e q ) θ p θ q = (ϕ * ρ 12 J)B + ν ⊥ (B) + L grad ln λ g(V•, V•).
Equating the two expressions (or from a direct calculation), we get

L ν g(V•, V•) = 2ν ⊥ (B) + L grad ln λ g(V•, V•).
Using Lemma 10.3, p,q i g(∇ ei ∇ ep e i , e q ) θ p θ q =p,q i e i (g(∇ ep e i , e q )) θ p θ q + p,q i g(∇ ep e i , ∇ ei e q ) θ p θ q = p,q i e i (g(e i , B ep e q )) θ p θ qp,q i g(H∇ ep e i , A * ei e q ) θ p θ q p,q i g(B * ep e i , V∇ ei e q ) θ p θ q

= div 1 B 1 -(ϕ * ρ 12 J)B + ν ⊥ (B) + 2µ (B) + 2 sym ν (B * ) -2 sym ζ(B * • J•) + 2ε sym ω 34 (B • J•) - p,q i g(-ζ(e p ) Je i , e q (ln λ) e i + ζ(e q ) Je i ) θ p θ q - p,q i g(B * ep e i , ε ω 34 (e i ) Je q ) θ p θ q = div 1 B 1 -(ϕ * ρ 12 J)B + ν ⊥ (B) + 2µ (B) + 2ζ 2 + ε sym ω 34 (B • J•) + 2 sym ν (B * ) -2 sym ζ(B * • J•). Finally, p,q i g(∇ [ei,ep] e i , e q ) θ p θ q = p,q i g(∇ -ep(ln λ) ei e i , e q ) θ p θ q + p,q i g(∇ B * ep ei+ε ω34(ei) Jep e i , e q ) θ p θ q = - p,q
e p (ln λ) g(2ν, e q ) θ p θ qp,q i g(e i , B eq (B * ep e i + ε ω 34 (e i ) Je p ))

θ p θ q = -2ν 2 -C -ε sym ω 34 (B • J•).
Thus,

Ric V = εK ϕ g V -2µ (B) + C + (ϕ * ρ 12 J)B -ν ⊥ (B) + L ν g(V•, V•) + div 1 B 1 -(ϕ * ρ 12 J)B + ν ⊥ (B) + 2µ (B) + 2ζ 2 + ε sym ω 34 (B • J•) + 2 sym ν (B * ) -2 sym ζ(B * • J•) -2ν 2 -C -ε sym ω 34 (B • J•).
Simplification gives us the next lemma.

Lemma 12.2. The vertical part Ric V = Ric(V•, V•) of the Ricci curvature Ric of g is given by 20 Ric V = εK ϕ g V + L ν g(V•, V•) -2ν 2 + 2ζ 2 + 2 sym ν (B * ) -2 sym ζ(B * • J•) + div 1 B 1 = εK ϕ g V + L grad ln λ g(V•, V•) -2ν 2 + 2ζ 2 + 2ν ⊥ (B) + 2 sym ν (B * ) -2 sym ζ(B * • J•) + div 1 B 1 . Since 1 2 trace L ν g(V•, V•) = p ε p (∇ ep ν )(e p ) = p ε p e p (e p (ln λ)) - p ε p ν (∇ ep e p ) = (div ν -ξ V (ν, Vω 34 ) + 2g(ν, ν)) -ν (εJVω 34 )
= div ν + 2g(ν, ν), and 1 2 trace div 1 B 1 = div µ by Lemma 10.7, then the "vertical part" of the scalar curvature S of g is as follows.

Lemma 12.3. The trace of Ric V is given by

1 2 trace Ric V = εK ϕ + div ν + g(ν, ν) + g(ζ , ζ ) + div µ .
Lemmas 11.2 and 12.3 lead to the following theorem.

Theorem 12.4. The scalar curvature S of g is given by

1 2 S = λ 2 K N + εK ϕ + 2µ(ln λ) + 2 µ 2 + g(ν, ν) -g(ζ , ζ ) + 2 div µ + 2 div ν + div ν ⊥ -1 2 trace C.

Mixed part

The mixed part Ric

mixed = Ric(H•, V•) + Ric(V•, H•) of the Ricci curvature Ric of g is given by 1 2 Ric mixed = i,p a ε a g(R(e i , e a )e a , e p ) θ i θ p = [IJJP ]
i,p j g(R(e i , e j )e j , e p ) θ i θ p + [P QQI] i,p q ε q g(R(e p , e q )e q , e i ) θ i θ p . We first look at [IJJP ]. We start with i,p j g(∇ ei ∇ ej e j , e p )

θ i θ p = i,p g(∇ ei (J(ϕ * ρ 12 ) + ν ⊥ ), e p ) θ i θ p + i,p g(∇ ei 2ν, e p ) θ i θ p = i,p g(J(ϕ * ρ 12 ) + ν ⊥ , A * ei e p ) θ i θ p + 2 sym(∇ν (H•, V•)) = i,p g(J(ϕ * ρ 12 ) + ν ⊥ , ν (e p ) e i + ζ(e p ) Je i ) θ i θ p + 2 sym(∇ν (H•, V•)) = ν ⊥ ν + ν ⊥ J ζ -ϕ * ρ 12 J ν + ϕ * ρ 12 ζ + 2 sym(∇ν (H•, V•)).
Remark 13.1. The motivation for using ∇ν (H•, V•) is the presence of ∇µ (V•, H•) later on. If instead we want to use ∇d ln λ, we can again decompose j ∇ ej e j as the sum of J(ϕ * ρ 12 ) -ν ⊥ and 2 grad ln λ. This gives us

i,p j g(∇ ei ∇ ej e j , e p ) θ i θ p = -ν ⊥ ν -ν ⊥ J ζ -ϕ * ρ 12 J ν + ϕ * ρ 12 ζ + 2 sym(∇d ln λ(H•, V•)).
Equating the two expressions (or from a direct calculation), we get

sym(∇ν (H•, V•)) = -ν ⊥ ν -ν ⊥ J ζ + sym(∇d ln λ(H•, V•)).
j δ i j ζ(e p )) θ i θ p - i,p j g((ν ⊥ J(e i ) + ϕ * ρ 12 (e i )) Je j , e p (ln λ) e j + ζ(e p ) Je j ) θ i θ p + i,p j g(δ ij ν + (-1) j δ i j ζ , ε ω 34 (e j ) Je p ) θ i θ p = - i,p e i (e p (ln λ)) θ i θ p - i,p (Je i )(ζ(e p )) θ i θ p -2 i,p (ν ⊥ J(e i ) + ϕ * ρ 12 (e i ))ζ(e p ) θ i θ p + i,p ε ω 34 (e i )ν (Je p ) θ i θ p + i,p ε ω 34 (Je i )ζ(Je p ) θ i θ p = - i,p e i (e p (ln λ)) θ i θ p - i,p (Je i )(ζ(e p )) θ i θ p -2 ν ⊥ J ζ -2 ϕ * ρ 12 ζ + ε ω 34 H ν J + ε ω 34 JH ζJ. But - i,p e i (e p (ln λ)) θ i θ p = - i,p (∇ ei d ln λ)(e p ) θ i θ p - i,p d ln λ(∇ ei e p ) θ i θ p = -sym(∇d ln λ(H•, V•)) + i,p ν ⊥ (A * ei e p ) θ i θ p - i,p ν (V∇ ei e p ) θ i θ p = -sym(∇d ln λ(H•, V•)) + i,p ν ⊥ (ν (e p )e i + ζ(e p )Je i ) θ i θ p - i,p ν (ε ω 34 (e i )Je p ) θ i θ p = -sym(∇d ln λ(H•, V•)) + ν ⊥ ν + ν ⊥ J ζ -ε ω 34 H ν J = -sym(∇ν (H•, V•)) -ε ω 34 H ν J and - i,p (Je i )(ζ(e p )) θ i θ p = - i,p (∇ Jei ζ)(e p ) θ i θ p - i,p ζ(∇ Jei e p ) θ i θ p = -sym(∇ζ(JH•, V•)) - i,p ζ(V∇ Jei e p ) θ i θ p = -sym(∇ζ(JH•, V•)) - i,p ζ(εω 34 J(e i )Je p ) θ i θ p = -sym(∇ζ(JH•, V•)) -ε ω 34 JH ζJ, so - i,p j g(∇ ej ∇ ei e j , e p ) θ i θ p = -sym(∇ν (H•, V•)) -sym(∇ζ(JH•, V•)) -2 ν ⊥ J ζ -2 ϕ * ρ 12 ζ.
Continuing,

- i,p j g(∇ [ei,ej ] e j , e p ) θ i θ p = - i,p g(∇ [ei,e i ] e i , e p ) θ i θ p = - i,p g(∇ [e1,e2] Je i , e p ) θ i θ p = - i,p g(∇ Jν ⊥ -(ϕ * ρ12) Je i , e p ) θ i θ p - i,p g(∇ 2ζ Je i , e p ) θ i θ p = - i,p g(A Jν ⊥ -(ϕ * ρ12) Je i , e p ) θ i θ p + i,p g(B * 2ζ Je i , e p ) θ i θ p = - i,p (g(Jν ⊥ -(ϕ * ρ 12 ) , Je i )e p (ln λ) + g(Jν ⊥ -(ϕ * ρ 12 ) , e i )ζ(e p )) θ i θ p + 2 sym ζ(B * • J•) = -(Jν ⊥ -(ϕ * ρ 12 ) ) J ν -(Jν ⊥ -(ϕ * ρ 12 ) ) ζ + 2 sym ζ(B * • J•) = -ν ⊥ ν + ν ⊥ J ζ + ϕ * ρ 12 J ν + ϕ * ρ 12 ζ + 2 sym ζ(B * • J•). Next we look at [P QQI]. We first have i,p q ε q g(∇ ep ∇ eq e q , e i ) θ i θ p = i,p g(∇ ep 2µ, e i ) θ i θ p + i,p g(∇ ep εJVω 34 , e i ) θ i θ p = 2 sym(∇µ (V•, H•)) -ε sym ω 34 J(B * ).
Moreover, using Lemma 10.5, i,p q ε q g(∇ eq ∇ ep e q , e i ) θ i θ p =i,p q ε q e q (g(∇ ep e q , e i )) θ i θ p + i,p q ε q g(∇ ep e q , ∇ eq e i ) θ i θ p =i,p q ε q e q (g(e q , B * ep e i )) θ i θ p + i,p q ε q g(B ep e q , H∇ eq e i ) θ i θ p i,p q ε q g(V∇ ep e q , B * eq e i )

θ i θ p = -sym div 2 B 1 -2 sym ν (B * ) + sym ζ(B * • J•) + ε sym ω 34 J(B * ) -ε sym ω 34 (B * J• •) -C + C * + i,p q ε q g(B ep e q , -ζ(e q ) Je i ) θ i θ p - i,p q ε q g((-1) q ε q ω 34 (e p ) e q , B * eq e i ) θ i θ p = -sym div 2 B 1 -2 sym ν (B * ) + ε sym ω 34 J(B * ) -ε sym ω 34 (B * J• •) -C + C * .
Finally,

- i,p q ε q g(∇ [ep,eq] e q , e i ) θ i θ p = - i,p ε p g(∇ [ep,e p ] e p , e i ) θ i θ p = - i,p ε g(∇ [e3,e4] Je p , e i ) θ i θ p = - i,p ε g(∇ -Vω34 Je p , e i ) θ i θ p = ε sym ω 34 (B * J• •).
Thus,

1 2 Ric mixed = ν ⊥ ν + ν ⊥ J ζ -ϕ * ρ 12 J ν + ϕ * ρ 12 ζ + 2 sym(∇ν (H•, V•)) -sym(∇ν (H•, V•)) -sym(∇ζ(JH•, V•)) -2 ν ⊥ J ζ -2 ϕ * ρ 12 ζ -ν ⊥ ν + ν ⊥ J ζ + ϕ * ρ 12 J ν + ϕ * ρ 12 ζ + 2 sym ζ(B * • J•) + 2 sym(∇µ (V•, H•)) -ε sym ω 34 J(B * ) -sym div 2 B 1 -2 sym ν (B * ) + ε sym ω 34 J(B * ) -ε sym ω 34 (B * J• •) -C + C * + ε sym ω 34 (B * J• •).
Simplification gives us the next lemma.

Lemma 13.2. The mixed part

Ric mixed = Ric(H•, V•) + Ric(V•, H•) of the Ricci curvature Ric of g is given by 21 1 2 Ric mixed = 2 sym(∇µ (V•, H•)) + sym(∇ν (H•, V•)) -sym(∇ζ(JH•, V•)) -2 sym ν (B * ) + 2 sym ζ(B * • J•) -sym div 2 B 1 -C + C * = -ν ⊥ ν -ν ⊥ J ζ + sym(∇d ln λ(H•, V•)) + 2 sym(∇µ (V•, H•)) -sym(∇ζ(JH•, V•)) -2 sym ν (B * ) + 2 sym ζ(B * • J•) -sym div 2 B 1 -C + C * .
At last, we can now write down the full Ricci curvature Ric of g in terms of geometric quantities associated to the semi-conformal submersion ϕ.

Theorem 13.3. The Ricci curvature Ric of the metric g is given by Ric

= (λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ )) g H + εK ϕ g V + L µ g(H•, H•) + L ν g(V•, V•) -2ν 2 + 2ζ 2 -2 sym ν (B * ) + 2 sym ζ(B * • J•) + 4 sym(∇µ (V•, H•)) + 2 sym(∇ν (H•, V•)) -2 sym(∇ζ(JH•, V•)) + div 1 B 1 -2 sym div 2 B 1 -2C + C * .
We will compute how this Ricci curvature changes under biconformal deformation of the metric in the next chapter. This change will then be used to construct solutions to the Einstein equation in Chapter 5.

CHAPTER 4

Biconformal deformation

In this chapter and the next (except in Section 16), (M, g 0 ) is an oriented 4-dimensional Riemannian or Lorentzian (specifically, spacetime) manifold, (N, h) is an oriented Riemannian surface, and ϕ : (M, g 0 ) → (N, h) is a smooth semiconformal submersion with smooth dilation λ 0 : M → (0, +∞). (If (M, g 0 ) is a spacetime manifold, then we assume that there is a global future-pointing timelike vector field T 0 on (M, g 0 ) that is vertical.) The (local) frame fields e 0 = {e 0

1 , e 0 2 , e 0 3 , e 0 4 } and θ 0 = {θ 0 1 , θ 0 2 , θ 0 3 , θ 0 4 } are such that the conditions in Section 5 are satisfied.

Let g be a biconformal deformation of the metric g 0 , i.e. let

g = σ -2 g H 0 + ρ -2 g V 0
for some smooth deformation parameters σ, ρ : M → (0, +∞). [START_REF] Kobayashi | Foundations of differential geometry[END_REF] Then the (0, 2) tensor field g is a metric [START_REF]Maximal extension of Schwarzschild metric[END_REF] on M having the same index 24 as g 0 : g is Riemannian if g 0 is; g is Lorentzian if g 0 is. (If g 0 is a spacetime metric, then we declare the global timelike vector field T = ρT 0 on (M, g) to be future-pointing; this makes g a spacetime metric with future "in the same direction" as that of g 0 .) With respect to this new metric, the smooth map ϕ : (M, g) → (N, h) is a semi-conformal submersion with dilation λ = σλ 0 .

The notions of horizontality and verticality are the same for the two metrics g 0 and g. Let e i = σ e 0 i , e p = ρ e 0 p and

θ i = σ -1 θ 0 i , θ p = ρ -1 θ 0
p . Then e = {e 1 , e 2 , e 3 , e 4 } and θ = {θ 1 , θ 2 , θ 3 , θ 4 } are (local) frame fields that satisfy the conditions in Section 5. Defining the (1, 1) tensor fields J 0 and J as in Section 6, we see that J 0 = J. [START_REF] Kobayashi | Foundations of differential geometry[END_REF] If σ = ρ, then the metric g is a conformal deformation of the metric g 0 . [START_REF]Maximal extension of Schwarzschild metric[END_REF] The symmetry of g 0 implies that of g. For nondegeneracy, let x ∈ M and let m 0 = h v be a matrix representation of g 0 at x, where h, v are nonsingular 2 × 2 real matrices. A matrix representation of g at x is

m = σ -2 h ρ -2 v ,
whose determinant is det m = σ -4 ρ -4 det m 0 = 0.

24 See the definition of e = {e 1 , e 2 , e 3 , e 4 } (given e 0 = {e 0 1 , e 0 2 , e 0 3 , e 0 4 }) below.

53

All of the above tell us that the discussion in Chapters 2 and 3 holds for (M, g 0 ) and for (M, g). We write a "0" sub(super)script for quantities associated to g 0 (e.g. B * 0 , div 0 ν 0 ⊥ 0 ) and leave those associated to g unsub(super)scripted (e.g. B * , div ν ⊥ ). In Sections 14 and 15, we express relevant unsub(super)scripted quantities in terms of the sub(super)scripted ones and the deformation parameters σ, ρ. The main result of Section 14 (Theorem 14.13) is a formula for the change ∇ -∇ 0 in Levi-Civita connection under biconformal deformation of the metric, in terms of the sub(super)scripted geometric quantities and σ, ρ. The main results of Section 15 (Theorems 15.12, 15.14, 15.16, 15.19, 15.21) are formulas for the changes

Ric H -Ric H 0 , Ric V -Ric V 0 , Ric mixed -Ric mixed 0 (13.1)
in the parts of Ricci curvature under biconformal deformation and for the differences

σ -2 S -S 0 , ρ -2 S -S 0 (13.2)
under such deformation. These five differences will be used in Chapter 5 to construct solutions to the Einstein equation (see (17.1) and (17.4)).

Change in connection

The goal in this chapter is to compute the change Ric -Ric 0 in Ricci curvature and the differences σ -2 S -S 0 , ρ -2 S -S 0 under biconformal deformation of the metric. To do this, we express the unsub(super)scripted addends of Ric in Theorem 13.3 in terms of sub(super)scripted quantities and the deformation parameters σ, ρ. At some point, we will have done enough calculations to be able to write down the change ∇ -∇ 0 in Levi-Civita connection under biconformal deformation (Theorem 14.13); this section is the series of calculations up to the that point.

We start with simple calculations (trace and the musical isomorphisms) that yield expressions for ν ⊥ , ν, ζ as corollaries. Then, we continue with A, A * , B, B * , µ, ω 34 before stating Theorem 14.13. The next lemma is used throughout this section. Lemma 14.2. For a vector field X and a 1-form α,

X = σ -2 (HX) 0 + ρ -2 (VX) 0 , α = σ 2 (Hα) 0 + ρ 2 (Vα) 0 .
Proof. We have

X = g(HX, •) + g(VX, •) = σ -2 g 0 (HX, •) + ρ -2 g 0 (VX, •) = σ -2 (HX) 0 + ρ -2 (VX) 0 and α = i α(e i ) e i + p α(e p ) ε p e p = σ 2 i α(e 0 i ) e 0 i + ρ 2 p α(e 0 p ) ε p e 0 p = σ 2 (Hα) 0 + ρ 2 (Vα) 0 .
Corollary 14.3. For a smooth map f : M → R,

grad f = σ 2 H grad 0 f + ρ 2 V grad 0 f.
Corollary 14.4.

ν ⊥ = σ 2 (ν ⊥ 0 + H grad 0 ln σ), ν = ρ 2 (ν 0 + V grad 0 ln σ).
Corollary 14.5. g(ν, ν) = ρ 2 (g 0 (ν 0 , ν 0 )+2ν 0 (ln σ)+g V 0 (grad 0 ln σ, grad 0 ln σ)).

Corollary 14.6.

ζ = σ 2 ρ -2 ζ 0 .
Proof. Using (5.6), we have

ζ = 1 2 V[e 1 , e 2 ] = 1 2 σ 2 V[e 0 1 , e 0 2 ] = 1 2 σ 2 ρ -2 V[e 0 1 , e 0 2 ] 0 = σ 2 ρ -2 ζ 0 . Corollary 14.7. g(ζ , ζ ) = σ 4 ρ -2 g 0 (ζ 0 0 , ζ 0 0 ).
Now, the second fundamental forms and their adjoints. Lemma 14.8.

A = A 0 + (σ -2 ρ 2 -1)g H 0 (•, •) ν 0 + σ -2 ρ 2 g H 0 (•, •) V grad 0 ln σ, A * = A * 0 + (σ 2 ρ -2 -1)JH• ζ 0 (•) + H• (Vd ln σ)(•).
Proof. Using Lemma 10.1, we have

A = g H (•, •) ν -g H (•, J•) ζ = σ -2 ρ 2 g H 0 (•, •) (ν 0 + V grad 0 ln σ) -g H 0 (•, J•) ζ 0 0 = A 0 + (σ -2 ρ 2 -1)g H 0 (•, •) ν 0 + σ -2 ρ 2 g H 0 (•, •) V grad 0 ln σ and A * = H• ν (•) + JH• ζ(•) = H• (ν 0 0 + Vd ln σ)(•) + JH• σ 2 ρ -2 ζ 0 (•) = A * 0 + (σ 2 ρ -2 -1)JH• ζ 0 (•) + H• (Vd ln σ)(•).
Recall the Koszul formula characterizing the Levi-Civita connection (see, for example, [22] Section 4.2):

2g(∇ X Y, Z) = X(g(Y, Z)) + Y (g(Z, X)) -Z(g(X, Y )) -g(X, [Y, Z]) + g(Y, [Z, X]) + g(Z, [X, Y ])
for vector fields X, Y, Z. Lemma 14.9.

B = σ 2 ρ -2 (B 0 + g V 0 (•, •) H grad 0 ln ρ), B * = B * 0 + V • (Hd ln ρ)(•).
Proof. We start by applying the Koszul formula: 2g(B ep e q , e i ) = g(e p , [e i , e q ]) + g(e q , [e i , e p ]) = g(ρe 0 p , σρ[e 0 i , e 0 q ]) + g(ρe 0 q , σρ[e 0 i , e 0 p ]) + g(ρe 0 p , σ(e 0 i ρ)e 0 q ) + g(ρe 0 q , σ(e 0 i ρ)e 0 p ) = σg 0 (e 0 p , [e 0 i , e 0 q ]) + σg 0 (e 0 q , [e 0 i , e 0 p ]) + 2σ(e 0 i ln ρ)g 0 (e 0 p , e 0 q ) = 2σg 0 (B 0 e 0 p e 0 q , e 0 i ) + 2σd ln ρ(e 0 i ) g 0 (e 0 p , e 0 q ) = 2σ 2 ρ -2 g(B 0 ep e q , e i ) + 2ρ -2 g V 0 (e p , e q ) Hd ln ρ(e i ).

(14.1)

Then by tensoriality,

B ep e q = σ 2 ρ -2 B 0 ep e q + ρ -2 g V 0 (e p , e q ) Hd ln ρ. Upon raising, we get B ep e q = σ 2 ρ -2 B 0 ep e q + σ 2 ρ -2 g V 0 (e p , e q ) H grad 0 ln ρ, and the first equation follows by tensoriality.

From (14.1), we have g(B * ep e i , e q ) = ρ -2 g 0 (B 0 ep e q , e i ) + ρ -2 g 0 (e p , e q ) Hd ln ρ(e i ) = ρ -2 g 0 (B * 0 ep e i , e q ) + ρ -2 g 0 (e p , e q ) Hd ln ρ(e i ) = g(B * 0 ep e i , e q ) + g(e p , e q ) Hd ln ρ(e i ). Then by tensoriality,

B *
ep e i = B * 0 ep e i + e p Hd ln ρ(e i ). The second equation follows by raising and then by tensoriality.

Corollary 14.10. µ = σ 2 (µ 0 + H grad 0 ln ρ).

Corollary 14.11.

µ 2 = σ 2 ( µ 0 2 0 + 2µ 0 (ln ρ) + H grad 0 ln ρ 2 0
). The last piece that we need for the computation of the change ∇ -∇ 0 is the difference ω 34 -ω 0 34 .

Lemma 14.12.

Hω 34 = Hω 0 34 , Vω 34 = Vω 0 34 + (Vd ln ρ)J.

Proof. We apply the Koszul formula: In view of Corollary 7.6, all Christoffel symbols with respect to the frame field e, and thus the Levi-Civita connection ∇, of g are now given in terms of quantities associated to g 0 and the deformation parameters σ, ρ.

Theorem 14.13. The change ∇ -∇ 0 in Levi-Civita connection under a biconformal deformation g = σ -2 g H 0 + ρ -2 g V 0 is a (1, 2) tensor field given by

∇ -∇ 0 = (σ -2 ρ 2 -1)g H 0 (•, •) ν 0 + (σ 2 ρ -2 -1)B 0 • • + σ -2 ρ 2 g H 0 (•, •) V grad 0 ln σ + σ 2 ρ -2 g V 0 (•, •) H grad 0 ln ρ -(σ 2 ρ -2 -1)JH• ζ 0 (•) -(σ 2 ρ -2 -1)ζ 0 (•) JH• -H• (d ln σ)(V•) -V • (d ln ρ)(H•) -d ln σ(•) H• -d ln ρ(•) V• + d ln σ(JH•) JH• +ε d ln ρ(JV•) JV • .
Proof. Let X and Y be vector fields and write Y = a Y a e a = a Y a 0 e 0 a . Note that Y i 0 = σY i and Y p 0 = ρY p . Using Corollary 7.7,

∇ X Y = +A X Y -A * X Y + B X Y -B * X Y + i X(Y i ) e i + (ν ⊥ J(X) + ϕ * ρ 12 (X) -ζ(X)) JHY H∇ X HY + p X(Y p ) e p + ε ω 34 (X) JVY V∇ X VY = ∇ 0 X Y addends +A 0 X Y +(σ -2 ρ 2 -1)g H 0 (X, Y ) ν 0 + σ -2 ρ 2 g H 0 (X, Y ) V grad 0 ln σ -A * 0 X Y -(σ 2 ρ -2 -1)JHX ζ 0 (Y ) -HX(d ln σ)(VY ) +B 0 X Y +(σ 2 ρ -2 -1)B 0 X Y + σ 2 ρ -2 g V 0 (X, Y ) H grad 0 ln ρ -B * 0 X Y -VX(d ln ρ)(HY ) + i X(Y i 0 ) e 0 i + (ν 0 ⊥ 0 J(X) + ϕ * ρ 12 (X) -ζ 0 (X)) JHY -d ln σ(X) HY + d ln σ(JHX) JHY -(σ 2 ρ -2 -1)ζ 0 (X) JHY + p X(Y p 0 ) e 0 p + ε ω 0 34 (X) JVY -d ln ρ(X) VY + ε d ln ρ(JVX) JVY.

Change in Ricci curvature

We continue expressing unsub(super)scripted quantities in terms of sub(super)scripted ones and the deformation parameters in this section. The objective is to obtain formulas for the differences (13.1) and (13.2). To this end, we calculate how the addends of the right-hand sides of Theorems 13.3 and 12.4 change under biconformal deformation of the metric.

We start with divergences of 1-forms.

Lemma 15.1. For a 1-form α on M , div Hα = σ 2 (div 0 Hα -2 Hα(grad 0 ln ρ)), div Vα = ρ 2 (div 0 Vα -2 Vα(grad 0 ln σ)).

Proof. Using Corollary 8.2, we have

div Hα = i e i (α(e i )) -αJ(ϕ * ρ 12 ) -α(ν ⊥ ) -2α(µ) = σ 2 i e 0 i (α(e 0 i )) + σ 2 i e 0 i (ln σ)α(e 0 i ) -σ 2 αJ(ϕ * ρ 12 ) 0 -σ 2 α(ν ⊥ 0 ) -σ 2 α(H grad 0 ln σ) -2σ 2 α(µ 0 ) -2σ 2 α(H grad 0 ln ρ) = σ 2 div
-2ρ 2 α(ν 0 ) -2ρ 2 α(V grad 0 ln σ) = ρ 2 div 0 Vα -2ρ 2 Vα(grad 0 ln σ). Corollary 15.2. div µ = σ 2 (div 0 µ 0 0 + div 0 Hd ln ρ -2µ 0 (ln ρ) -2 H grad 0 ln ρ 2 0 ), div ν ⊥ = σ 2 (div 0 ν 0 ⊥ 0 + div 0 Hd ln σ -2ν ⊥ 0 (ln ρ) -2g H 0 (grad 0 ln σ, grad 0 ln ρ)), div ν = ρ 2 (div 0 ν 0 0 + div 0 Vd ln σ -2ν 0 (ln σ) -2g V
0 (grad 0 ln σ, grad 0 ln σ)). Then, C, C * and the sectional curvature K ϕ . Lemma 15.3.

C = σ 2 ρ -2 (C 0 + 2 d ln ρ(B 0 ) + H grad 0 ln ρ 2 0 g V 0 ), C * = C * 0 + 4µ 0 0 Hd ln ρ + 2(Hd ln ρ) 2 .
Proof. We have

C = p ε p g(B ep • 1 , B ep • 2 ) = σ 4 ρ -4 p ε p g(B 0 ep • 1 +g V 0 (e p , • 1 ) H grad 0 ln ρ, B 0 ep • 2 +g V 0 (e p , • 2 ) H grad 0 ln ρ) = σ 2 ρ -2 p ε p g 0 (B 0 e 0 p • 1 +g V 0 (e 0 p , • 1 ) H grad 0 ln ρ, B 0 e 0 p • 2 +g V 0 (e 0 p , • 2 ) H grad 0 ln ρ) = σ 2 ρ -2 (C 0 + 2 d ln ρ(B 0 ) + H grad 0 ln ρ 2 0 g V 0 )
. and

C * = p ε p g(B * ep • 1 , B * ep • 2 ) = p ε p g(B * 0 ep • 1 +d ln ρ(H• 1 ) e p , B * 0 ep • 2 +d ln ρ(H• 2 ) e p ) = p ε p g 0 (B * 0 e 0 p • 1 +d ln ρ(H• 1 ) e 0 p , B * 0 e 0 p • 2 +d ln ρ(H• 2 ) e 0 p ) = C * 0 + 4µ 0 0 Hd ln ρ + 2(Hd ln ρ) 2 .
Corollary 15.4. trace C = σ 2 (trace 0 C 0 + 4µ 0 (ln ρ) + 2 H grad 0 ln ρ 2 0 ).

Lemma 15.5. Moreover,

K ϕ = ρ 2 (K ϕ 0 + ε div 0 Vd ln ρ + 2ε ν 0 (ln ρ)).
-g V (ω 34 , ω 34 ) = -ρ 2 g V 0 (ω 0 34 0 -JV grad 0 ln ρ, ω 0 34 0 -JV grad 0 ln ρ) = -ρ 2 g V 0 (ω 0 34 0 , ω 0 34 0 ) +2ρ 2 ω 0 34 (JV grad 0 ln ρ) -ερ 2 g V 0 (grad 0 ln ρ, grad 0 ln ρ) cancelled out later .
Thus, by Lemma 9.1,

K ϕ = - p ε p e p (ω 34 Je p ) -g V (ω 34 , ω 34 ) = -ρ 2 p ε p e 0 p (ω 0 34 Je 0 p ) + ερ 2 div 0 Vd ln ρ + 2ερ 2 ν 0 (ln ρ) -ρ 2 g V 0 (ω 0 34 0 , ω 0 34 0 ) = ρ 2 K ϕ 0 + ερ 2 div 0 Vd ln ρ + 2ερ 2 ν 0 (ln ρ).
Now, the Lie derivatives of g and some parts of covariant derivatives of 1-forms.

Lemma 15.6. For a smooth map f : M → R and a vector field X on M ,

L f X g = f (σ -2 L X g H 0 + ρ -2 L X g V 0 -2σ -2 X(ln σ) g H 0 -2ρ -2 X(ln ρ) g V 0 ) + 2 df (σ -2 HX 0 + ρ -2 VX 0 ).
Proof. We have

L f X g = f L X g + 2 df X = f (σ -2 L X g H 0 + ρ -2 L X g V 0 + X(σ -2 )g H 0 + X(ρ -2 )g V 0 ) + 2 df (σ -2 HX 0 + ρ -2 VX 0 ). Corollary 15.7. L µ g(H•, H•) = L µ0 g 0 (H•, H•) + L H grad 0 ln ρ g 0 (H•, H•) -2µ 0 (ln σ) g H 0 -2 g H 0 (grad 0 ln σ, grad 0 ln ρ) g H 0 + 4 µ 0 0 Hd ln σ + 4 Hd ln σ Hd ln ρ, L ν g(V•, V•) = L ν0 g 0 (V•, V•) + L V grad 0 ln σ g 0 (V•, V•) -2ν 0 (ln ρ) g V 0 -2 g V 0 (grad 0 ln σ, grad 0 ln ρ) g V 0 + 4ν 0 0 Vd ln ρ + 4 Vd ln σ Vd ln ρ.
Proof. By the previous lemma, for a smooth map f : M → R and a vector field X on M ,

L f X g(H•, H•) = f σ -2 (L X g 0 (H•, H•) -2X(ln σ) g H 0 ) + 2σ -2 Hdf HX 0 and L f X g(V•, V•) = f ρ -2 (L X g 0 (V•, V•) -2X(ln ρ) g V 0 ) + 2ρ -2
Vdf VX 0 . Using f = σ 2 and X = µ 0 + H grad 0 ln ρ (so that f X = µ) in the first equation gives us the formula for L µ g(H•, H•), while using f = ρ 2 and X = ν 0 + V grad 0 ln σ (so that f X = ν) in the second equation gives us the formula for L ν g(V•, V•).

Lemma 15.8. For a vector field Z on M ,

∇HZ (V•, H•) = σ -2 ∇ 0 HZ 0 (V•, H•) + σ -2 (σ 2 ρ -2 -1) ζ 0 ⊗ HZ 0 J -σ -2 Vd ln σ ⊗ HZ 0 , ∇VZ (H•, V•) = ρ -2 ∇ 0 VZ 0 (H•, V•) -ρ -2 Hd ln ρ ⊗ VZ 0 .
Proof. A great deal of simplification occurs in Theorem 14.13 once we consider the forms of the left-hand sides of the equations in the lemma: for vector fields X, Z on M ,

H∇ VX HZ = H∇ 0 VX HZ -(σ 2 ρ -2 -1)ζ 0 (X) JHZ -Vd ln σ(X) HZ, V∇ HX VZ = V∇ 0
HX VZ -Hd ln ρ(X) VZ. Then we introduce Y , Y a vector field on M , into the mix:

(∇ VX HZ )(HY ) = (∇ 0 VX HZ)(HY ) + (σ 2 ρ -2 -1) ζ 0 (X) HZ J(Y ) -Vd ln σ(X) HZ (Y ), (∇ HX VZ )(VY ) = (∇ 0 HX VZ)(VY ) -Hd ln ρ(X) VZ (Y )
. Finally, we change from to 0 in the right-hand sides:

(∇ VX HZ )(HY ) = σ -2 (∇ 0 VX HZ 0 )(HY ) + σ -2 (σ 2 ρ -2 -1) ζ 0 (X) HZ 0 J(Y ) -σ -2 Vd ln σ(X) HZ 0 (Y ), (∇ HX VZ )(VY ) = ρ -2 (∇ 0 HX VZ 0 )(VY ) -ρ -2 Hd ln ρ(X) VZ 0 (Y ).
Corollary 15.9.

∇µ (V•, H•) = ∇ 0 µ 0 0 (V•, H•) + ∇ 0 Hd ln ρ(V•, H•) + Vd ln σ ⊗ (µ 0 0 + Hd ln ρ) + (σ 2 ρ -2 -1) ζ 0 ⊗ (µ 0 0 + Hd ln ρ)J, ∇ν (H•, V•) = ∇ 0 ν 0 0 (H•, V•) + ∇ 0 Vd ln σ(H•, V•) + Hd ln ρ ⊗ (ν 0 0 + Vd ln σ), ∇ζ(JH•, V•) = σ 2 ρ -2 ∇ 0 ζ 0 (JH•, V•) + 2 σ 2 ρ -2 (Hd ln σ)J ⊗ ζ 0 -σ 2 ρ -2 (Hd ln ρ)J ⊗ ζ 0 .
Proof. By the previous lemma,

∇µ (V•, H•) = ∇ 0 µ 0 0 (V•, H•) + ∇ 0 Hd ln ρ(V•, H•) + σ -2 Vdσ 2 ⊗ (µ 0 0 + Hd ln ρ) -Vd ln σ ⊗ (µ 0 0 + Hd ln ρ) + (σ 2 ρ -2 -1) ζ 0 ⊗ (µ 0 0 + Hd ln ρ)J.
But σ -2 Vdσ 2 = 2Vd ln σ. The other two equations are proven similarly.

Lastly, the divergences div 1 B 1 and div 2 B 1 .

Lemma 15.10.

div 1 B 1 = σ 2 ρ -2 div 0 1 B 0 1 + σ 2 ρ -2 div 0 Hd ln ρ g V 0 -2σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 -2σ 2 ρ -2 d ln ρ(B 0 ) + 2(σ 2 ρ -2 -1) sym ν 0 0 (B * 0 ) -2 sym d ln σ(B * 0 ) -2ν 0 0 Hd ln ρ -2 Vd ln σ Hd ln ρ + 2σ 2 ρ -2 ζ 0 (Hd ln ρ)J.
Proof. Since B ep e q = B 0 e 0 p e 0 q 0 + ε p δ pq Hd ln ρ, then div B ep e q = σ 2 (div 0 B 0 e 0 p e 0 q 0 -2 d ln ρ(B 0 e 0 p e 0 q )+ε p δ pq div 0 Hd ln ρ-2 ε p δ pq H grad 0 ln ρ 2 0 ).

Hence,

p,q (div B ep e q ) θ p θ q = σ 2 ρ -2 p,q (div 0 B 0 e 0 p e 0 q 0 ) θ 0 p θ 0 q -2σ 2 ρ -2 d ln ρ(B 0 ) + σ 2 ρ -2 div 0 Hd ln ρ g V 0 -2σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 .
The other addends of div 1 B 1 (see Lemma 10.3) are

-2 sym ν (B * ) = -2 sym(ν 0 0 + Vd ln σ)(B * 0 + V • (Hd ln ρ)(•)) = -2 sym ν 0 0 (B * 0 ) -2 sym d ln σ(B * 0 ) -2ν 0 0 Hd ln ρ -2 Vd ln σ Hd ln ρ, 2 sym ζ(B * • J•) = 2σ 2 ρ -2 sym ζ 0 (B * 0 • J • +V • (Hd ln ρ)J(•)) = 2σ 2 ρ -2 sym ζ 0 (B * 0 • J•) + 2σ 2 ρ -2 ζ 0 (Hd ln ρ)J, -2ε sym ω 34 (B • J•) = -2εσ 2 ρ -2 sym ω 0 34 (B 0 • J • +g V 0 (•, J•) H grad 0 ln ρ) = -2εσ 2 ρ -2 sym ω 0 34 (B 0 • J•). Thus, by Lemma 10.3, div 1 B 1 = p,q (div B ep e q ) θ p ⊗ θ q -2 sym ν (B * ) + 2 sym ζ(B * • J•) -2ε sym ω 34 (B • J•) = σ 2 ρ -2 addends of div 0 1 B 0 1 p,q (div 0 B 0 e 0 p e 0 q 0 ) θ 0 p ⊗ θ 0 q -2σ 2 ρ -2 d ln ρ(B 0 ) + σ 2 ρ -2 div 0 Hd ln ρ g V 0 -2σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 + σ 2 ρ -2 • -2 sym ν 0 0 (B * 0 ) +2(σ 2 ρ -2 -1) sym ν 0 0 (B * 0 ) -2 sym d ln σ(B * 0 ) -2ν 0 0 Hd ln ρ -2 Vd ln σ Hd ln ρ + σ 2 ρ -2 • 2 sym ζ 0 (B * 0 • J•) +2σ 2 ρ -2 ζ 0 (Hd ln ρ)J + σ 2 ρ -2 • -2ε sym ω 0 34 (B 0 • J•) .
Lemma 15.11.

div 2 B 1 = div 0 2 B 0 1 -(d ln σ + d ln ρ)(B * 0 •2 • 1 ) + (σ 2 ρ -2 -1)ζ 0 (B * 0 •2 J• 1 ) -ε d ln ρ(JB * 0 J•2 • 1 ) + ∇ 0 Hd ln ρ(V• 2 , H• 1 ) -2 Hd ln ρ ⊗ ν 0 0 + (σ 2 ρ -2 -1)(Hd ln ρ)J ⊗ ζ 0 -Hd ln ρ ⊗ Vd ln σ -σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 -2σ 2 ρ -2 d ln ρ(B 0 ) -(σ 2 ρ -2 -1)C 0 + 4µ 0 0 Hd ln ρ + 2(Hd ln ρ) 2 .
Proof. We do some preliminary calculations:

• d(σρ -1 ) = σρ -1 (d ln σ -d ln ρ), • d(σρ -1
Hd ln ρ(e 0 i ))(e 0 q ) = [Hd ln ρ(e 0 i ) d(σρ -1 ) + σρ -1 d(Hd ln ρ(e 0 i ))](e 0 q ) = σρ -1 Hd ln ρ(e 0 i )[Vd ln σ(e 0 q ) -Vd ln ρ(e 0 q )] + σρ -1 [∇ 0 Hd ln ρ(e 0 q , e 0 i ) + Hd ln ρ(H∇ 0 e 0 q e 0 i )] = σρ -1 [Hd ln ρ(e 0 i )Vd ln σ(e 0 q ) -Hd ln ρ(e 0 i )Vd ln ρ(e 0 q )] + σρ -1 [∇ 0 Hd ln ρ(e 0 q , e 0 i ) -(Hd ln ρ)J(e 0 i )ζ 0 (e 0 q )],

• div 0 e 0 q 0 = a ε a (∇ 0 e 0 a e 0 q 0 )(e 0 a ) = a ε a g 0 (∇ 0 e 0 a e 0 q , e 0 a ) = i g 0 (∇ 0 e 0 i e 0
q , e 0 i ) + ε q g 0 (∇ 0 e 0 q e 0 q , e 0 q ) = -2 Vd ln λ 0 (e 0 q ) + ε Vω 0 34 J(e 0 q ),

• for a smooth map f : M → R and a 1-form α on M , div 0 (f α) = f div 0 α + df (α 0 ).

Now, since

B * eq e i = σρ -1 (B * 0 e 0 q e 0 i 0 + Hd ln ρ(e 0 i ) e 0 q 0 ), then div B * eq e i = div(σρ -1 B * 0

e 0 q e 0 i 0 ) + div(σρ -1 Hd ln ρ(e 0 i ) e 0 q 0 ) = ρ 2 [div 0 (σρ -1 B * 0 e 0 q e 0 i 0 ) -2σρ -1 d ln σ(B * 0 e 0 q e 0 i )] + ρ 2 [div 0 (σρ -1 Hd ln ρ(e 0 i ) e 0 q 0 ) -2σρ -1 Hd ln ρ(e 0 i )Vd ln σ(e 0 q )] = ρ 2 [σρ -1 div 0 B * 0 e 0 q e 0 i 0 + d(σρ -1 )(B * 0 e 0 q e 0 i ) -2σρ -1 d ln σ(B * 0 e 0 q e 0 i )] + ρ 2 [σρ -1
Hd ln ρ(e 0 i ) div 0 e 0 q 0 + d(σρ -1 Hd ln ρ(e 0 i ))(e 0 q ) -2σρ -1 Hd ln ρ(e 0 i )Vd ln σ(e 0 q )]. Hence, i,q

(div B * eq e i ) θ i ⊗ θ q = i,q (div 0 B * 0 e 0 q e 0 i 0 ) θ 0 i ⊗ θ 0 q -(d ln σ + d ln ρ)(B * 0 •2 • 1 ) + Hd ln ρ ⊗ (-2 Vd ln λ 0 + ε Vω 0 34 J) + Hd ln ρ ⊗ Vd ln σ -Hd ln ρ ⊗ Vd ln ρ + ∇ 0 Hd ln ρ(V• 2 , H• 1 ) -(Hd ln ρ)J ⊗ ζ 0 -2 Hd ln ρ ⊗ Vd ln σ.
The other addends of div 2 B 1 (see Lemma 10.5) are [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] . Thus, by Lemma 10.5,

ζ(B * •2 J• 1 ) = σ 2 ρ -2 ζ 0 (B * 0 •2 J • 1 +V • 2 (Hd ln ρ)J• 1 ) = σ 2 ρ -2 ζ 0 (B * 0 •2 J• 1 ) + σ 2 ρ -2 (Hd ln ρ)J ⊗ ζ 0 , -ε ω 34 (B * J•2 • 1 ) = -ε (ω 0 34 + (Vd ln ρ)J)(B * 0 J•2 • 1 +JV • 2 (Hd ln ρ)(• 1 )) = -ε ω 0 34 (B * 0 J•2 • 1 ) -ε d ln ρ(JB * 0 J•2 • 1 ) -ε Hd ln ρ ⊗ Vω 0 34 J + Hd ln ρ ⊗ Vd ln ρ, -C = -σ 2 ρ -2 C 0 -2σ 2 ρ -2 d ln ρ(B 0 ) -σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 , C * = C * 0 + 4µ 0 0 Hd ln ρ + 2(Hd ln ρ)
div 2 B 1 = i,q (div B * eq e i ) θ i ⊗ θ q + ζ(B * •2 J• 1 ) -ε ω 34 (B * J•2 • 1 ) -C + C * = addends of div 0 2 B 0 1 i,q (div 0 B * 0 e 0 q e 0 i 0 ) θ 0 i ⊗ θ 0 q -(d ln σ + d ln ρ)(B * 0 •2 • 1 ) - 2
Hd ln ρ ⊗ Vd ln ρ + ∇ 0 Hd ln ρ(V• 2 , H• 1 ) -(Hd ln ρ)J ⊗ ζ 0 combine +ζ 0 (B * 0 •2 J• 1 ) +(σ 2 ρ -2 -1)ζ 0 (B * 0 •2 J• 1 ) +σ 2 ρ -2 (Hd ln ρ)J ⊗ ζ 0 combine -ε ω 0 34 (B * 0 J•2 • 1 ) -ε d ln ρ(JB * 0 J•2
Hd ln ρ ⊗ Vd ln ρ -C 0 -(σ 2 ρ -2 -1)C 0 -2σ 2 ρ -2 d ln ρ(B 0 ) -σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 +C * 0 +4µ 0 0 Hd ln ρ + 2(Hd ln ρ) 2 .
We now have all that we need to compute the change Ric -Ric 0 in Ricci curvature under biconformal deformation of the metric. We perform this computation in three steps; the horizontal, vertical, and mixed parts of Ric -Ric 0 are recorded in Theorems 15.12, 15.14, and 15.16, respectively.

Theorem 15.12. The change Ric H -Ric H 0 in the horizontal part of Ricci curvature under a biconformal deformation g = σ -2 g H 0 + ρ -2 g V 0 is given by

Ric H -Ric H 0 = -2 g H 0 (grad 0 ln σ, grad 0 ln ρ) g H 0 -2σ -2 ρ 2 g V 0 (grad 0 ln σ, grad 0 ln σ) g H 0 -2(σ 2 ρ -2 -1)g 0 (ζ 0 0 , ζ 0 0 ) g H 0 -2σ -2 ρ 2 ν 0 (ln σ) g H 0 + (σ -2 ρ 2 -1) div 0 ν 0 0 g H 0 + div 0 Hd ln σ g H 0 + σ -2 ρ 2 div 0 Vd ln σ g H 0 + L H grad 0 ln ρ g 0 (H•, H•) + 4µ 0 0 Hd ln σ -4µ 0 0 Hd ln ρ + 4 Hd ln σ Hd ln ρ -2(Hd ln ρ) 2 .
Proof. Using Lemma 11.1 as a guide, the addends of Ric H -Ric H 0 are ( ( ( ( ( ( ( ( ( ( (

• λ 2 K N g H -λ 2 0 K N g H 0 = 0, • div ν ⊥ g H -div 0 ν 0 ⊥ 0 g H 0 = (div 0 Hd ln σ- 2ν ⊥ 0 (ln ρ)-
2g H 0 (grad 0 ln σ, grad 0 ln ρ)) g H 0 , • div ν g H -div 0 ν 0 0 g H 0 = (σ -2 ρ 2 -1) div 0 ν 0 0 g H 0 + σ -2 ρ 2 (div 0 Vd ln σ -2ν 0 (ln σ) -2g V 0 (grad 0 ln σ, grad 0 ln σ)) g H 0 ,
• Adding these gives the formula in the theorem.

div 1 B 1 -div 0 1 B 0 1 = (σ 2 ρ -2 -1) div 0 1 B 0 1 + σ 2 ρ -2 div 0 Hd ln ρ g V 0 -2σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 -2σ 2 ρ -2 d ln ρ(B 0 ) + 2(σ 2 ρ -2 -1) sym ν 0 0 (B * 0 ) - ( (
Note that by Lemma 10.3, the last line of the previous lemma is equal to

(σ 2 ρ -2 -1) p,q (div 0 B 0 e 0 p e 0 q 0 ) θ 0 p θ 0 q -2ε sym ω 0 34 (B 0 • J•) . Since L H grad 0 ln σ g 0 (V•, V•) = -2 d ln σ(B 0 )
by Theorem 1.6 of Appendix A, then we have the following corollary for the change Ric V -Ric V 0 when the deformation is conformal.

Corollary 15.15. Under a conformal deformation g = σ -2 g 0 , the change Ric V -Ric V 0 in the vertical part of Ricci curvature is given by Ric

V -Ric V 0 = -2 g 0 (grad 0 ln σ, grad 0 ln σ) g V 0 + ∆ 0 ln σ g V 0 + L grad 0 ln σ g 0 (V•, V•) + 2(Vd ln σ) 2 .
Theorem 15.16. The change Ric mixed -Ric mixed 0 in the mixed part of Ricci curvature under a biconformal deformation g = σ -2 g H 0 + ρ -2 g V 0 is given by

1 2 (Ric mixed -Ric mixed 0 ) = sym(∇ 0 Vd ln σ(H•, V•)) + sym(∇ 0 Hd ln ρ(V•, H•)) -(σ 2 ρ -2 -1) sym(∇ 0 ζ 0 (JH•, V•)) -sym d ln σ(B * 0 ) + sym d ln ρ(B * 0 ) + (σ 2 ρ -2 -1) sym ζ 0 (B * 0 • J•) + ε sym d ln ρ(JB * 0 J• •) + 2(σ 2 ρ -2 -1) µ 0 0 J ζ 0 + 2µ 0 0 Vd ln σ + Hd ln ρ ν 0 0 -2 σ 2 ρ -2 (Hd ln σ)J ζ 0 + (4σ 2 ρ -2 -1) (Hd ln ρ)J ζ 0 + 2 Hd ln ρ Vd ln σ.
Proof. Using Lemma 13.2 as a guide, the addends of [START_REF] Baird | Biconformal equivalence between 3-dimensional Ricci solitons[END_REF] 2 (Ric mixed -Ric mixed 0 ) are • ( ( ( ( ( ( ( ( ( ( ( 2g H 0 (grad 0 ln σ, grad 0 ln ρ), • -σ -2 1 2 trace C + 1 2 trace 0 C 0 = -2µ 0 (ln ρ) -H grad 0 ln ρ 2 0 . Adding these gives the formula in the theorem.

• 2 sym(∇µ (V•, H•)) -2 sym(∇ 0 µ 0 0 (V•, H•)) = 2 sym(∇ 0 Hd ln ρ(V•, H•)) + 2 Vd ln σ (µ 0 0 + Hd ln ρ) + 2(σ 2 ρ -2 -1) ζ 0 (µ 0 0 + Hd ln ρ)J, • sym(∇ν (H•, V•)) -sym(∇ 0 ν 0 0 (H•, V•)) = sym(∇ 0 Vd ln σ(H•, V•)) + Hd ln ρ (ν 0 0 + Vd ln σ), • -sym(∇ζ(JH•, V•)) + sym(∇ 0 ζ 0 (JH•, V•)) = -(σ 2 ρ -2 -1) sym(∇ 0 ζ 0 (JH•, V•)) -2 σ 2 ρ -2 (Hd ln σ)J ζ 0 + σ 2 ρ -2 (Hd ln ρ)J ζ 0 , • -2 sym ν (B * ) + 2 sym ν 0 0 (B * 0 ) = -2
2 sym ζ(B * • J•) -2 sym ζ 0 (B * 0 • J•) = 2(σ 2 ρ -2 -1) sym ζ 0 (B * 0 • J•) + 2σ 2 ρ -2 ζ 0 (Hd ln ρ)J, • -σ -2 g(ζ , ζ ) + g 0 (ζ 0 0 , ζ 0 0 ) = -(σ 2 ρ -2 -1)g 0 (ζ 0 0 , ζ 0 0 ), • σ -2 2 div µ -2 div 0 µ 0 0 = 2 div 0 Hd ln ρ - 4µ 0 (ln ρ) -4 H grad 0 ln ρ 2 0 , • σ -2 2 div ν -2 div 0 ν 0 0 = 2(σ -2 ρ 2 -1) div 0 ν 0 0 + 2σ -2 ρ 2 div 0 Vd ln σ -4σ -2 ρ 2 ν 0 (ln σ) -4σ -2 ρ 2 g V 0 (grad 0 ln σ, grad 0 ln σ), • σ -2 div ν ⊥ -div 0 ν 0 ⊥ 0 = div 0 Hd ln σ- 2ν ⊥ 0 (ln ρ)-
Corollary 15.20. Under a conformal deformation g = σ -2 g 0 , 1 2 (σ -2 S -S 0 ) = -3g 0 (grad 0 ln σ, grad 0 ln σ) + 3 div 0 d ln σ. Theorem 15.21. Under a biconformal deformation ( ( ( ( ( ( ( ( ( ( ( (

g = σ -2 g H 0 + ρ -2 g V 0 , 1 2 (ρ -2 S -S 0 ) = (σ 2 ρ -2 -1)λ 2 0 K N -3σ 2 ρ -2 H grad 0 ln ρ 2 0 -3 g V 0 (grad 0 ln σ, grad 0 ln σ) + 2(σ 2 ρ -2 -1) µ 0 2 0 -(σ 4 ρ -4 -1)g 0 (ζ 0 0 , ζ 0 0 ) + 2(σ 2 ρ -2 -1)µ 0 (ln λ 0 ) + 2σ 2 ρ -2 µ 0 (ln σ) -2σ 2 ρ -2 µ 0 (ln ρ) -2ν 0 (ln σ) + 2ν 0 (ln ρ) -1 2 (σ 2 ρ -2 -1) trace 0 C 0 + 2(σ 2 ρ -2 -1) div 0 µ 0 0 + (σ 2 ρ -2 -1) div 0 ν 0 ⊥ 0 + σ 2 ρ -2 div 0 Hd ln σ + 2 div 0 Vd ln σ + 2σ 2 ρ -2 div 0 Hd ln ρ + div 0 Vd ln ρ. Proof. The addends of 1 2 (ρ -2 S -S 0 ) are • ρ -2 λ 2 K N -λ 2 0 K N = (σ 2 ρ -2 -1)λ 2 0 K N , • ρ -2 εK ϕ -εK ϕ 0 = div 0 Vd ln ρ + 2ν 0 (ln ρ), • ρ -2 2µ(ln λ) -2µ 0 (ln λ 0 ) = 2σ
2σ 2 ρ -2 g H 0 (grad 0 ln σ, grad 0 ln ρ), • ρ -2 2 µ 2 -2 µ 0 2 0 = 2(σ 2 ρ -2 -1) µ 0 2 0 + ( ( ( ( ( ( ( 4σ 2 ρ -2 µ 0 (ln ρ)+2σ 2 ρ -2 H grad 0 ln ρ 2 0 , • ρ -2 g(ν, ν) -g 0 (ν 0 , ν 0 ) = 2ν 0 (ln σ) + g V 0 (grad 0 ln σ, grad 0 ln σ), • -ρ -2 g(ζ , ζ ) + g 0 (ζ 0 0 , ζ 0 0 ) = -(σ 4 ρ -4 -1)g 0 (ζ 0 0 , ζ 0 0 ), • ρ -2 2 div µ -2 div 0 µ 0 0 = 2(σ 2 ρ -2 -1) div 0 µ 0 0 + 2σ 2 ρ -2 div 0 Hd ln ρ - ( ( ( ( ( ( ( 4σ 2 ρ -2 µ 0 (ln ρ) -4σ 2 ρ -2 H grad 0 ln ρ 2 0 , • ρ -2 2 div ν -2 div 0 ν 0 0 = 2 div 0 Vd ln σ-4ν 0 (ln σ)-4g V 0 (grad 0 ln σ, grad 0 ln σ), • ρ -2 div ν ⊥ -div 0 ν 0 ⊥ 0 = (σ 2 ρ -2 -1) div 0 ν 0 ⊥ 0 + σ 2 ρ -2 div 0 Hd ln σ - ( ( ( ( ( ( ( 2σ 2 ρ -2 ν ⊥ 0 (ln ρ) - ( (
2σ 2 ρ -2 g H 0 (grad 0 ln σ, grad 0 ln ρ), • -ρ -2 1 2 trace C + 1 2 trace 0 C 0 = -1 2 (σ 2 ρ -2 -1) trace 0 C 0 -2σ 2 ρ -2 µ 0 (ln ρ) - σ 2 ρ -2 H grad 0 ln ρ 2 0 .
Adding these gives the formula in the theorem.

The differences (13.1) and (13.2) will be key in the next chapter in the construction of solutions to the Einstein equation (see (17.1) and (17.4)). For example, we will evaluate the differences (13.1) at orthonormal frame field elements (with respect to g 0 ) to write the Einstein equation as a system of partial differential equations in the deformation parameters σ, ρ (see, for example, Theorem 18.1).

CHAPTER 5

Solutions to the Einstein equation

As in the previous chapter, here (except in Section 16) (M, g 0 ), (M, g) are oriented 4-dimensional manifolds, either both Riemannian or both Lorentzian (specifically, spacetime), with the metric g being a biconformal deformation of the metric g 0 :

g = σ -2 g H 0 + ρ -2 g V 0
for some smooth deformation parameters σ, ρ : M → (0, +∞). (In the Lorentzian context, we assume that there are global future-pointing timelike, vertical vector fields T 0 , ρT 0 on (M, g 0 ), (M, g), respectively.) The manifold (N, h) is an oriented Riemannian surface, and ϕ : M → (N, h) is a smooth map that is a semi-conformal submersion both with respect to g 0 and with respect to g. With respect to g 0 , the dilation of ϕ is λ 0 : M → (0, +∞); with respect to g, the dilation of ϕ is λ = σλ 0 : M → (0, +∞). The (local) frame fields e 0 = {e 0 1 , e 0 2 , e 0 3 , e 0 4 } and θ 0 = {θ 0 1 , θ 0 2 , θ 0 3 , θ 0 4 } on (M, g 0 ) are such that the conditions in Section 5 are satisfied. We write a "0" sub(super)script for quantities associated to g 0 and leave those associated to g unsub(super)scripted.

We assume (except in Section 16) that the metric g 0 is a solution to the (sub(super)scripted) Einstein equation. In Section 17, we write the condition "the metric g is a solution to the (unsub(super)scripted) Einstein equation" in terms of the deformation parameters σ, ρ. This reformulation will be used in Sections 18, 19, and 20 to construct solutions to the Einstein equation, by way of biconformally deforming the Euclidean 4-space, the Minkowski spacetime, and the FRW spacetimes (with flat space), respectively. In each of these sections, we also give general results regarding such solutions. Before we do these, we first discuss the Einstein equation in Section 16.

The Einstein equation

In this section, we specify what we mean by the "Einstein equation" and enumerate some solutions to this equation in the Lorentzian context. Here, (M, g) denotes an oriented 4-dimensional Riemannian or Lorentzian (specifically, spacetime) manifold. We call a spacetime manifold (M, g) (or a spacetime metric g) an Einstein solution if it satisfies the Einstein equation ( 16.1) with T as in (16.2). A spacetime model is a quadruple (M, g), (u, ρ, p), (F, μ), Λ, where (M, g) is a solution to (16.1) with total energy-momentum tensor

T (u, ρ, p, F, μ, Λ) = T fluid (u, ρ, p) + T em (F, μ) + T vacuum (Λ) (see,
ds 2 = dx 2 + dy 2 + dz 2 -dt 2
of index 1. [START_REF] Robertson | Kinematics and world-structure[END_REF] Minkowski spacetime is the setting of special relativity. 16.1.2. Schwarzschild spacetime. A more complicated vacuum, Ricci-flat (but not flat) solution is Schwarzschild spacetime, which models the region around a static, spherical mass. Exterior Schwarzschild spacetime is

S 2 × (2M, +∞) × R with line element ds 2 = r 2 dΩ 2 + f (r) -1 dr 2 -f (r) dt 2 .
Here, (S 2 , dΩ 2 ) is the 2-sphere of radius 1, M ∈ (0, +∞) is the Schwarzschild mass, r ∈ (2M, +∞), t ∈ R, and

f (r) = 1 -2Mr -1 . Meanwhile, interior Schwarzschild spacetime is S 2 × (0, 2M) × R
with the same line element formula. (See Section 7 of Appendix D.) [START_REF] Penrose | Spinors and space-time[END_REF] In view of footnote 29, a solution to (16.1) with T em = 0 and a vacuum-trivial perfect fluid is a vacuum solution. [START_REF] Robertson | Kinematics and world-structure[END_REF] In general, R n 16.1.3. FRW spacetime. The third type of spacetime that we use in this work is Friedmann-Robertson-Walker (FRW) spacetime M (k, I, f ) (following the notation in [29] Chapter 12), which models the entire history of the universe. The spacetime is the warped product P f ×I with line element

ds 2 = f (t) 2 dΣ 2 -dt 2 .
Here, (P, dΣ 2 ) is a 3-dimensional Riemannian manifold of constant curvature k, Riemannian manifold (M, g) satisfies this equation for some Λ, then we call (M, g) a (Riemannian) Einstein manifold and g a (Riemannian) Einstein metric. That is, an Einstein manifold (metric, respectively) is a Riemannian manifold (metric, respectively) that formally satisfies (16.1) with T = T vacuum (Λ) (see (16.4)).

Biconformal deformation and the Einstein equation

For the rest of this chapter, (M, g 0 ), (M, g), (N, h), ϕ, λ 0 , e 0 , θ 0 are as in the previous chapter. In particular, the metric g is a biconformal deformation of the metric g 0 :

g = σ -2 g H 0 + ρ -2 g V 0
for some smooth deformation parameters σ, ρ : M → (0, +∞). As remarked in the previous chapter, g has the same index as g 0 : g is Riemannian if g 0 is; g is Lorentzian if g 0 is. And as in the previous chapter, we write a "0" sub(super)script for quantities associated to g 0 and leave those associated to g unsub(super)scripted. We assume for the rest of this chapter that the metric g 0 is a solution to the Einstein equation

G 0 = 8πT 0 (with T 0 = T vacuum 0 (Λ 0 ) if g 0 is Riemannian -see (16. 4
)). In this section, we write the condition "the metric g is a solution to the Einstein equation

G = 8πT
(with T = T vacuum (Λ) if g is Riemannian)" in terms of the deformation parameters σ, ρ. This reformulation will be used in Sections 18 to 20 to construct solutions to the Einstein equation and to give general results regarding such solutions. [START_REF] Robertson | Kinematics and world-structure II[END_REF] Here, it is important to note that Ric = Lg, with L : M → R smooth, implies that L is a constant function. This is because the equation Ric = Lg is equivalent to the equation G = -Lg (see (16.4)), and so 0 = div G = div(-Lg) = -dL. 17.1. Riemannian context. Suppose that g 0 is an Einstein metric with constant of proportionality Λ 0 : Ric 0 = Λ 0 g 0 . Then g is an Einstein metric with constant of proportionality Λ -Ric = Λg -if and only if the following system holds:

35 Ric H -Ric H 0 = (σ -2 Λ -Λ 0 )g H 0 , (17.1a) Ric V -Ric V 0 = (ρ -2 Λ -Λ 0 )g V 0 , (17.1b) Ric mixed -Ric mixed 0 = 0. (17.1c)
Note that the left-hand sides are expressed in terms of the deformation parameters σ, ρ via Theorems 15.12, 15.14, and 15.16. Thus, this is a system in σ, ρ and Λ.

Let M e be the open subset of M where the frame field e 0 is defined. [START_REF] Sachs | General relativity for mathematicians[END_REF] In view of (17.1), let the functions Λ ab : M e → R be given by (17.2) (Ric -Ric 0 )(e 0 a , e 0 b ) = Λ ab -Λ 0 g 0 (e 0 a , e 0 b ). Then the functions Λ ab are written in terms of σ, ρ. Evaluating the system (17.1) on elements of e 0 gives us the following theorem.

Theorem 17.1. The biconformally deformed metric g = σ -2 g H 0 + ρ -2 g V 0 is an Einstein metric on M e if and only if the following system of nine equations holds:

0 = σ 2 Λ ii -ρ 2 Λ pp , i ∈ {1, 2}, p ∈ {3, 4}, 0 = Λ ab , a, b ∈ {1, 2, 3, 4}, a < b.
In this case, Ric = Λg, where

Λ = σ 2 Λ ii = ρ 2 Λ pp
for any i ∈ {1, 2} and p ∈ {3, 4}. This theorem will be used in Section 18 (see Theorem 18.1). 17.2. Lorentzian context. Suppose that g 0 is a solution to the Einstein equation G 0 = 8πT 0 ⇔ Ric 0 = 8πT 0 + 1 2 S 0 g 0 with T 0 = T fluid 0 (u 0 , ρ0 , p0 ) + T em 0 (F 0 , μ0 ) + T vacuum 0 (Λ 0 ). Then g is a solution to the Einstein equation

(17.3) G = 8πT ⇔ Ric = 8πT + 1 2 Sg -with T = T fluid (u, ρ, p) + T em (F, μ) + T vacuum (Λ) -if
and only if the following system holds:

Ric H -Ric H 0 = 8π(T H -T H 0 ) + 1 2 (σ -2 S -S 0 )g H 0 , (17.4a) Ric V -Ric V 0 = 8π(T V -T V 0 ) + 1 2 (ρ -2 S -S 0 )g V 0 , (17.4b) Ric mixed -Ric mixed 0 = 8π(T mixed -T mixed 0
), (17.4c) Note that the left-hand sides, as well as the differences σ -2 S -S 0 and ρ -2 S -S 0 , are expressed in terms of the deformation parameters σ, ρ via Theorems 15.12, 15.14, 15.16, 15.19, and 15.21. Note also that the energy-momentum tensors T fluid , T em , T vacuum depend on the metric g, which is expressed in terms of σ, ρ.

Thus, this is a system in the geometric parameters σ, ρ and the physical parameters u, ρ, p, F, μ, Λ. Before evaluating the system (17.4) on elements of the frame field e 0 , we first do so with the total energy-momentum tensor T (under certain hypotheses). For brevity, let

T fv = T fluid (u, ρ, p) + T vacuum (Λ).
If the flow vector field u is e 4 = ρ e 0 4 (which is a reasonable requirement if u 0 = e 0 4 ), then 8πT fv (e 0 a , e 0 b ) = (8π ρ + 8π p)

    0 0 0 0 0 0 0 0 0 ρ -2     + (8π p -Λ)     σ -2 0 0 0 σ -2 0 0 ρ -2 0 -ρ -2     =     σ -2 (8π p -Λ) 0 0 0 σ -2 (8π p -Λ) 0 0 ρ -2 (8π p -Λ) 0 ρ -2 (8π ρ + Λ)     .
In general, if u = a u 0 a e 0 a , then

8πT fv (e 0 a , e 0 b ) = (8π ρ + 8π p)      σ -4 u 0 1 2 σ -4 u 0 1 u 0 2 σ -2 ρ -2 u 0 1 u 0 3 -σ -2 ρ -2 u 0 1 u 0 4 σ -4 u 0 2 2 σ -2 ρ -2 u 0 2 u 0 3 -σ -2 ρ -2 u 0 2 u 0 4 ρ -4 u 0 3 2 -ρ -4 u 0 3 u 0 4 ρ -4 u 0 4 2      + (8π p -Λ)     σ -2 0 0 0 σ -2 0 0 ρ -2 0 -ρ -2     .
As for T em (F, μ), it is given by (16.3), where (in terms of the frame field e 0 )

F 2 = 2σ 4 F (e 0 1 , e 0 2 ) 2 + 2σ 2 ρ 2 F (e 0 1 , e 0 3 ) 2 + 2σ 2 ρ 2 F (e 0 2 , e 0 3 ) 2 -2σ 2 ρ 2 F (e 0 1 , e 0 4 ) 2 -2σ 2 ρ 2 F (e 0 2 , e 0
i ) = -σ 2 F (e 0 1 , e 0 2 ) 2 - q ε q ρ 2 F (e 0 i , e 0 q ) 2 ,
trace * (e 0 1 , e 0 2 ) =q ε q ρ 2 F (e 0 1 , e 0 q )F (e 0 2 , e 0 q ), trace * (e 0 p , e 0 p ) = -

j σ 2 F (e 0 j , e 0 p ) 2 -ε p ρ 2 F (e 0 3 , e 0 4 ) 2 ,
trace * (e 0 3 , e 0 4 ) =j σ 2 F (e 0 j , e 0 3 )F (e 0 j , e 0 4 ), trace * (e 0 i , e 0 p ) = (-1) i σ 2 F (e 0 1 , e 0 2 )F (e 0 i , e 0 p ) -(-1) p ε p ρ 2 F (e 0 i , e 0 p )F (e 0 3 , e 0 4 ).

From now on, we set

F ab = F (e 0 a , e 0 b )
(not F (e a , e b )). We assume that the mixed part F mixed of the electromagnetic field F is zero; that is, (in terms of e 0 ) 37

F = F H F 12 θ 0 1 ∧ θ 0 2 + F V F 34 θ 0 3 ∧ θ 0 4 , (17.5) dF = 0. Then 38 8πT em (e 0 a , e 0 b ) = 8π μ-1     σ 2 F 12 2 0 0 0 σ 2 F 12 2 0 0 -ρ 2 F 34 2 0 ρ 2 F 34 2     -4π μ-1 (σ 4 F 12 2 -ρ 4 F 34 2 )     σ -2 0 0 0 σ -2 0 0 ρ -2 0 -ρ -2     .
Now, we consider the full sum [START_REF] Ville | Harmonic morphisms from Einstein 4-manifolds to Riemann surfaces[END_REF] T (u, ρ, p, F, μ, Λ) = T fluid (u, ρ, p) + T em (F, μ) + T vacuum (Λ).

Note that (on the open subset of M where the frame field e 0 is defined) 8π ρ + 8π p = ρ 4 (u 0 3 2 + u 0 4 2 ) -1 (8πT (e 0 3 , e 0 3 ) + 8πT (e 0 4 , e 0 4 )).

If u = e 4 or 8πT (e 0 3 , e 0 3 ) + 8πT (e 0 4 , e 0 4 ) = 0, then 8πT (e 0 a , e 0 b ) = (8π ρ + 8π p)

    0 0 0 0 0 0 0 0 0 ρ -2     + (8π p -Λ)     σ -2 0 0 0 σ -2 0 0 ρ -2 0 -ρ -2     + 8π μ-1     σ 2 F 12 2 0 0 0 σ 2 F 12 2 0 0 -ρ 2 F 34 2 0 ρ 2 F 34 2     -4π μ-1 (σ 4 F 12 2 -ρ 4 F 34 2 )     σ -2 0 0 0 σ -2 0 0 ρ -2 0 -ρ -2     ,
37 If (17.5) holds and the frame field e 0 is a coordinate vector field [START_REF] Ville | Harmonic morphisms from Einstein 4-manifolds to Riemann surfaces[END_REF] If u is vertical and F has zero mixed part, then T fluid (u, ρ, p) = 0 if and only if (u, ρ, p) is trivial,

{∂ 1 , ∂ 2 , ∂ 3 , ∂ 4 }, then the Maxwell equation dF = 0 is equivalent to ∂pF 12 = ∂ i F 34 = 0. 38 If F = F H + F V , then T em (±F H ± F V , μ) = T em (F, μ).
T em (F, μ) = 0 if and only if F is trivial, T vacuum (Λ) = 0 if and only if Λ is trivial.
which is equivalent to the system 8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = σ 2 1 2 (8πT (e 0 1 , e 0 1 ) + 8πT (e 0 2 , e 0 2 )) -ρ 2 8πT (e 0 3 , e 0 3 ), (17.6e) Back to the system (17.4). As in the Riemannian context, let M e be the open subset of M where the frame field e 0 is defined. [START_REF] Walker | On Milne's theory of world-structure[END_REF] In view of (17.4), let the functions T ab : M e → R be given by (Ric -Ric 0 )(e 0 i , e 0 j ) = 8πT ij -8πT 0 (e 0 i , e 0 j ) + 1 2 (σ -2 S -S 0 )g 0 (e 0 i , e 0 j ), (17.7a) (Ric -Ric 0 )(e 0 p , e 0 q ) = 8πT pq -8πT 0 (e 0 p , e 0 q ) + 1 2 (ρ -2 S -S 0 )g 0 (e 0 p , e 0 q ), (17.7b) (Ric -Ric 0 )(e 0 i , e 0 p ) = 8πT ip -8πT 0 (e 0 i , e 0 p ) (17.7c) with T ab = T ba . Then the functions T ab are written in terms of the deformation parameters σ, ρ. Evaluating the system (17.4) on elements of e 0 , we see that, by construction, the metric g satisfies the Einstein equation (17.3) on M e if and only if

T ab = T (e 0
a , e 0 b ). This, with the help of (17.6), gives us the following two theorems.

Theorem 17.2. Suppose that (17.8) 0 = 8πT 33 + 8πT 44 .

The biconformally deformed metric g = σ -2 g H 0 + ρ -2 g V 0 satisfies the Einstein equation (17.3) on M e with magnetic constant μ and electromagnetic field F = F 12 θ 0 1 ∧ θ 0 2 + F 34 θ 0 3 ∧ θ 0 4 having zero mixed part [START_REF] Wood | Harmonic morphisms and Hermitian structures on Einstein 4-manifolds[END_REF] if and only if the following system of eight equations holds: In this case, the flow vector field u can be any unit, future-pointing timelike vector field, and the energy density ρ, pressure p, and cosmological constant Λ are related by 42 8π ρ + 8π p = ρ 2 (8πT 33 + 8πT 44 ), (17.10a) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ). (17.10b) Theorem 17.3. The biconformally deformed metric g = σ -2 g H 0 + ρ -2 g V 0 satisfies the Einstein equation (17.3) on M e with flow vector field u = ρ e 0 4 , magnetic constant μ, and electromagnetic field F = F 12 θ 0 1 ∧ θ 0 2 + F 34 θ 0 3 ∧ θ 0 4 having zero [START_REF] Walker | On Milne's theory of world-structure[END_REF] In Sections 19 and 20, Me = M . [START_REF] Wood | Harmonic morphisms and Hermitian structures on Einstein 4-manifolds[END_REF] Here and in Theorem 17. and20.1). We end this section with some remarks on these theorems.

8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = σ 2 1 2 (8πT 11 + 8πT 22 ) -ρ 2 8πT 33 , (17.9a
8π p -Λ = ρ 2 8πT 33 + 4π μ-1 (σ 4 F 12 2 + ρ 4 F 34
The pair (17.10) indicates that the cosmological constant Λ ∈ R is a degree of freedom. The magnetic constant μ ∈ R\{0} is, in a sense, also a degree of freedom: for any α ∈ R\{0}, the triple (F 12 , F 34 , μ) can be replaced by (±αF 12 , ±αF 34 , α 2 μ) in (17.9a) (see footnotes 31 and 38).

In (17.9a) the right-hand side is nonnegative-valued if μ ∈ (0, +∞) and is nonpositive-valued if μ ∈ (-∞, 0). Hence, if the right-hand side attains a positive value and a negative value, then g cannot satisfy the Einstein equation on M e with u = ρ e 0 4 and F mixed = 0. If in addition (17.8) holds, then g cannot satisfy the Einstein equation on M e with any admissible u and F mixed = 0.

18. Solutions from Euclidean 4-space 18.1. Generalities. Let (M, g 0 ) be either Euclidean 4-space R 4 or Minkowski spacetime R [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] 1 and let (N, h) be R 2 . Let ϕ : (M, g 0 ) → (N, h) be the projection

ϕ : (x 1 , x 2 , x 3 , x 4 ) → (x 1 , x 2 ).
In rectangular coordinates, the metrics g 0 and h are given by

g 0 = dx 1 2 + dx 2 2 + dx 3 2 + ε dx 4 2 , h = dx 1 2 + dx 2 2 ,
respectively. From these we see that ϕ is a semi-conformal submersion with dilation

λ 0 = 1. Let e 0 a = ∂ a .
Note that

ν 0 = ν ⊥ 0 = µ 0 = ζ 0 0 = 0, B 0 = B * 0 = 0. Let ρ0 = p0 = 0, F 0 = 0, Λ 0 = 0,
so that (M, g 0 ) satisfies the Einstein equation in both the Riemannian and Lorentzian contexts. Theorems 15.12, 15.14, and 15.16 give us the differences

Ric H -Ric H 0 = -2 j (ln σ) j (ln ρ) j g H 0 -2σ -2 ρ 2 q ε q (ln σ) q 2 g H 0 + j (ln σ) jj g H 0 + σ -2 ρ 2 q ε q (ln σ) qq g H 0 + 2 j Hd(ln ρ) j dx j + 4 Hd ln σ Hd ln ρ -2(Hd ln ρ) 2 , Ric V -Ric V 0 = -2σ 2 ρ -2 j (ln ρ) j 2 g V 0 -2 q ε q (ln σ) q (ln ρ) q g V 0 + σ 2 ρ -2 j (ln ρ) jj g V 0 + q ε q (ln ρ) qq g V 0 + 2 q Vd(ln σ) q dx q + 4 Vd ln σ Vd ln ρ -2(Vd ln σ) 2 ,
and

Ric mixed -Ric mixed 0 = 2 sym(∇ 0 Vd ln σ(H•, V•)) + 2 sym(∇ 0 Hd ln ρ(V•, H•))
+ 4 Hd ln ρ Vd ln σ, respectively. Evaluating Ric -Ric 0 on the frame field elements e 0 a , we obtain (17.2)). Meanwhile, Theorems 15.19 and 15.21 give us the differences

Λ ii = (Ric -Ric 0 )(e 0 i , e 0 i ) = j (ln σ) jj + σ -2 ρ 2 q ε q (ln σ) qq -2σ -2 ρ 2 q ε q (ln σ) q 2 + 2(ln ρ) ii -2(ln ρ) i 2 + 2(ln σ) i (ln ρ) i -2(ln σ) i (ln ρ) i , Λ 12 = (Ric -Ric 0 )(e 0 1 , e 0 2 ) = 2(ln ρ) 12 + 2(ln σ) 1 (ln ρ) 2 + 2(ln σ) 2 (ln ρ) 1 -2(ln ρ) 1 (ln ρ) 2 , Λ pp = (Ric -Ric 0 )(e 0 p , e 0 p ) = ε p q ε q (ln ρ) qq + ε p σ 2 ρ -2 j (ln ρ) jj -2ε p σ 2 ρ -2 j (ln ρ) j 2 + 2(ln σ) pp -2(ln σ) p 2 + 2(ln σ) p (ln ρ) p -2ε(ln σ) p (ln ρ) p , Λ 34 = (Ric -Ric 0 )(e 0 3 , e 0 4 ) = 2(ln σ) 34 + 2(ln σ) 3 (ln ρ) 4 + 2(ln σ) 4 (ln ρ) 3 -2(ln σ) 3 (ln σ) 4 , Λ ip = (Ric -Ric 0 )(e 0 i , e 0 p ) = (ln σ) ip + (ln ρ) ip + 2(ln ρ) i (ln σ) p (see
1 2 (σ -2 S -S 0 ) = -3 j (ln ρ) j 2 -3σ -2 ρ 2 q ε q (ln σ) q 2 + j (ln σ) jj + 2σ -2 ρ 2 q ε q (ln σ) qq + 2 j (ln ρ) jj + σ -2 ρ 2 q ε q (ln ρ) qq and 1 2 (ρ -2 S -S 0 ) = -3σ 2 ρ -2 j (ln ρ) j 2 -3 q ε q (ln σ) q 2 + σ 2 ρ -2 j (ln σ) jj + 2 q ε q (ln σ) qq + 2σ 2 ρ -2 j (ln ρ) jj + q ε q (ln ρ) qq ,
respectively. The rest of this section deals with the Riemannian context (while the next section deals with the Lorentzian context). We restate Theorem 17.1 in this setting.

Theorem 18.1. The biconformally deformed metric

(18.1) g = σ -2 (dx 1 2 + dx 2 2 ) + ρ -2 (dx 3 2 + dx 4 2 )
is an Einstein metric if and only if the following system of nine partial differential equations holds:

0 = σ 2 Λ ii -ρ 2 Λ pp , i ∈ {1, 2}, p ∈ {3, 4}, 0 = Λ ab , a, b ∈ {1, 2, 3, 4}, a < b.
In this case, Ric = Λg, where

(18.2) Λ = σ 2 Λ ii = ρ 2 Λ pp
for any i ∈ {1, 2} and p ∈ {3, 4}.

We now discuss three cases that simplify the system in the previous theorem.

18.2. σ horizontal, ρ vertical. If σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ), then the metric g is the product metric of the two ("unlifted") surface metrics σ -2 g H 0 and ρ -2 g V 0 . The respective Ricci curvatures of these surface metrics are K H σ -2 g H 0 and K V ρ -2 g V 0 , where the sectional curvatures K H , K V are given by

K H = σ 2 j (ln σ) jj , K V = ρ 2 q (ln ρ) qq ,
respectively. Then the Ricci curvature of the product metric g is the sum of the lifts of these two surface Ricci curvatures:

Ric = K H g H + K V g V .
This agrees with our computation above for the difference Ric -Ric 0 , which gives the same expression for Ric when σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ). In view of this expression, we see that g is an Einstein metric with Ric = Λg if and only if

Λ = K H = K V .
This is precisely what Theorem 18.1 reduces to in the case σ = σ(x 1 , x 2 ), ρ = ρ(x 3 , x 4 ).

Theorem 18.2. Suppose that σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ). The biconformally deformed metric (18.1) is an Einstein metric if and only if

σ 2 j (ln σ) jj = ρ 2 q (ln ρ) qq .
In this case, Ric = Λg, where

(18.3) Λ = σ 2 j (ln σ) jj = ρ 2 q (ln ρ) qq .
Moreover, g is Ricci-flat if and only if ln σ is harmonic 43 if and only if ln ρ is harmonic.

Example 18.3. By the previous theorem, the product metric

g = σ(x 1 , x 2 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 3 , x 4 ) -2 (dx 3 2 + dx 4 2 ),
where

σ(x 1 , x 2 ) = e α1x1+α2x2 , ρ(x 3 , x 4 ) = e α3x3+α4x4 for some α 1 , α 2 , α 3 , α 4 ∈ R, on R 4 is Ricci-flat. Example 18.4. If σ = σ(x i ), then (18.3) becomes (18.4) Λ = -σ i 2 + σσ ii .
(If ρ = ρ(x p ), then the following discussion also applies for ρ.) I. Λ = 0. In this case, σ = e a0 e a1xi

for some a 0 , a 1 ∈ R.

Suppose that Λ = 0. In particular, σ is not a constant function. 43 Here and for the rest of the section, "harmonic" means "harmonic with respect to the metric g 0 ".

II. σ ii vanishes everywhere, Λ < 0. In this case,

σ = (-Λ) 1/2 |x i -c 2 |
for some c 2 ∈ R.

In particular, the product metric

g = x i -2 (dx 1 2 + dx 2 2 ) + x p -2 (dx 3 2 + dx 4 2 )
of two hyperbolic 2-metrics, on

{(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x i , x p > 0},
is an Einstein metric with Λ = -1.

Suppose that σ ii is not identically zero. Over an open interval where σ i and σ ii do not vanish 44 , σ has an inverse

x i = x i (σ); let v(σ) = σ i (x i (σ)).
We rewrite (18.4) using σ as the independent variable and v as the dependent variable:

-v 2 + σvv σ = Λ.

This differential equation has solution

v 2 = ηe 2c1 σ 2 -Λ,
where η ∈ {-1, +1} and c 1 ∈ R. Then, in terms of the independent variable x i and the dependent variable σ,

± e c1 η(e c1 σ) 2 -Λ = e c1 σ i .

III. η = +1, Λ > 0.

In this case, (18.5) can be rewritten as

±e c1 (e c1 σ) 2 -(Λ 1/2 ) 2 = e c1 σ i .
This gives the solution

σ = Λ 1/2 e -c1 cosh[e c1 (x i -c 2 )],
where c 2 ∈ R.

IV. η = +1, Λ < 0. In this case, (18.5) can be rewritten as

±e c1 (e c1 σ) 2 + ((-Λ) 1/2 ) 2 = e c1 σ i .
This gives the solution

σ = (-Λ) 1/2 e -c1 | sinh[e c1 (x i -c 2 )]|,
where c 2 ∈ R.

V. η = -1, Λ < 0.
In this case, (18.5) can be rewritten as

±e c1 ((-Λ) 1/2 ) 2 -(e c1 σ) 2 = e c1 σ i .
This gives the solution

σ = (-Λ) 1/2 e -c1 | sin[e c1 (x i -c 2 )]|,
where c 2 ∈ R. 44 We will see from items III., IV., and V. that σ ii does not vanish anywhere, and that σ i vanishes at at most one point, at which the forms of σ given in the items still hold.

In particular, for a fixed Λ ∈ (-∞, 0), the product metric

g = σ(x 1 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 3 ) -2 (dx 3 2 + dx 4 2 ),
where

σ(x 1 ) = (-Λ) 1/2 sinh x 1 , ρ(x 3 ) = (-Λ) 1/2 sin x 3 , on {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 1 > 0, 0 < x 3 < π}
is an Einstein metric with Ric = Λg.

18.3. σ horizontal, ρ horizontal.

Theorem 18.5. Suppose that σ = σ(x 1 , x 2 ) and ρ = ρ(x 1 , x 2 ). The biconformally deformed metric (18.1) is an Einstein metric if and only if the following system of three partial differential equations holds:

0 = j (ln σ) jj + (ln ρ) ii -(ln ρ) i i + 2(ln ρ) i 2 + 2(ln σ) i (ln ρ) i -2(ln σ) i (ln ρ) i , i ∈ {1, 2}, (18.6a
) 0 = (ln ρ) 12 + (ln σ) 1 (ln ρ) 2 + (ln σ) 2 (ln ρ) 1 -(ln ρ) 1 (ln ρ) 2 .
(18.6b)

In this case, Ric = Λg, where

Λ = σ 2   j (ln ρ) jj -2 j (ln ρ) j 2   .
Moreover,

(18.7) 0 = j (ln σ) jj + j (ln ρ) j 2 ,
and g is Ricci-flat if ln σ is harmonic if and only if ρ is a constant function.

Example 18.6. If σ = σ(x i ) and ρ = ρ(x i ), then the metric (18.1) is a (Ricciflat) Einstein metric if and only if σ = e a0 e a1xi for some a 0 , a 1 ∈ R and ρ is a constant function.

Example 18.7. If σ = σ(x i ) and ρ = ρ(x i ), then (18.6b) is automatically satisfied while (18.6a) is equivalent to the pair

0 = (ln σ) ii + (ln ρ) ii + 2(ln σ) i (ln ρ) i , (18.8a) 0 = (ln σ) ii -(ln ρ) ii + 2(ln ρ) i 2 -2(ln σ) i (ln ρ) i . (18.8b)
Using (18.7) in either of these equations gives us (18.9) 0 = (ln ρ) ii + (ln ρ) i (ln σ 2 ρ -1 ) i .

Moreover, we can rewrite (18.8b) as (18.10) 0 = (ln σρ -1 ) ii -2(ln σρ -1 ) i (ln ρ) i .

We see that (18.8) is equivalent to the pair (18.9)-(18.10).

I. If ρ is a constant function, then σ = e a0 e a1xi for some a 0 , a 1 ∈ R. This gives us the Ricci-flat product metric of (a) the warped metric y 2 dx 2 + dy 2 on the upper half-plane and (b) the Euclidean 2-metric.

Suppose that ρ is not a constant function. Over an interval where (ln ρ) i does not vanish (i.e. ρ i does not vanish) 45 , we integrate (18.9) to get

|(ln ρ) i | = e b σ -2 ρ,
where b ∈ R, or equivalently (18.11)

σ 2 = e b ρ 2 |ρ i | -1 .
In particular, (18.12)

(ln σρ -1 ) i = -1 2 (ln |ρ i |) i = -1 2 ρ i -1 ρ ii .
II. If σρ -1 is a constant function, say σρ -1 = e -b1/2 for some b 1 ∈ R, then by (18.11)

σ = e b+b1/2 |x i -b 2 |, ρ = e b+b1 |x i -b 2 |, Λ = -3e 2b+b1
for some b 2 ∈ R.

This gives us, when b = b 1 = b 2 = 0, the hyperbolic 4-metric

g = x i -2 (dx 1 2 + dx 2 2 + dx 3 2 + dx 4 2 ) on {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x i > 0} with Λ = -3.
Suppose that σρ -1 is not a constant function. Over a subinterval where (ln σρ -1 ) i does not vanish (i.e. ρ ii does not vanish by (18.12)) 46 , we integrate (18.10) to obtain (18.13) (ln σρ -1 ) i = ∓e c ρ 2

for some c ∈ R. By (18.12) and (18.13), (18.14)

ρ i = ± 2 3 e c (ρ 3 -d 3 )
for some d ∈ R; the + sign (-sign, respectively) is chosen here if the -sign (+ sign, respectively) is chosen in (18.13). III. d = 0. In this case, integrating (18.14) gives us

σ = 3 1/4 e (2b-c)/4 |x i -c 2 | 1/4 , ρ = √ 3 2 e -c/2 |x i -c 2 | -1/2 , Λ = 0 for some c 2 ∈ R.
Choosing c 2 = 0 and the pair b, c such that

3 1/4 e (2b-c)/4 = √ 3 2 e -c/2 = 1, we see that the metric g = x i -1/2 (dx 1 2 + dx 2 2 ) + x i (dx 3 2 + dx 4 2 ) on {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x i > 0}
is Ricci-flat. 45 We will see later that ρ i does not vanish anywhere. 46 We will see later that ρ ii does not vanish anywhere.

IV. d = 0. In this case, integrating (18.14) over a subinterval where ρ does not attain the value 47 d gives us (18.15)

η 2 3 e c (x i -c 2 ) = ln |ρ -d| 1/3d 2 (ρ 2 + dρ + d 2 ) 1/6d 2 - 1 √ 3d 2 tan -1 ρ + 1 2 d √ 3 2 d
where η ∈ {-1, +1} and c 2 ∈ R. Again, the + sign (-sign, respectively) is chosen here if the -sign (+ sign, respectively) is chosen in (18.13). That is, in view of (18.12) and (18.14), (18.16)

η = +1, if ρ i (ρ -d) > 0, -1, if ρ i (ρ -d) < 0.
Treating ρ as the independent variable momentarily, let H(ρ) be the righthand side of (18.15) multiplied by η. By (18.14), (18.17)

H ρ = η(ρ 3 -d 3 ) -1 .
Moreover, (18.18)

H(0) = -π 6 √ 3d 2 η and H → η(-∞) as ρ → d, (18.19a)
H → -π 2 √ 3d 2 η if d > 0, + π 2 √ 3d 2 η if d < 0
as ρ → +∞. (18.19b) Now, back to the x i -interval of integration. Over this interval, by (18.17) and (18.16), the sign of H ρ (ρ) is given by

H ρ (ρ)          > 0, if ρ -d > 0 and ρ i > 0, < 0, if ρ -d > 0 and ρ i < 0, > 0, if ρ -d < 0 and ρ i > 0, < 0, if ρ -d < 0 and ρ i < 0,
and so H has an inverse H -1 over the corresponding ρ-interval. Indeed, (18.20) ρ = H -1 ( 2 3 e c (x i -c 2 )) over the x i -interval. With the help of (18.18) and (18. 19), we see that when c 2 = 0 and c = ln [START_REF] Baird | Ray congruences that generate conformal foliations[END_REF] 2 the maximal x i -domain (over which ρ and σρ -1 are not constant functions) of the positive-valued solution (18.20) of the differential equation (18.14) with d = 0 is given by Table 18.1. Corresponding plots are shown in Figure 18.1.

By (18.11),

σ 2 = 3 2 e b-c ρ 2 |ρ 3 -d 3 | -1 , so that (18.21) g = ρ -2 [ 2 3 e c-b |ρ 3 -d 3 |(dx 1 2 + dx 2 2 ) + (dx 3 2 + dx 4 2 )].
Moreover, Λ = 2e b+c d 3 sgn(ρ -d).

For the families (2, 1), (

) of Table 18.1, ρ → d as x i → ∞, so the metric g asymptotically collapses to the Euclidean 2-metric. For the 47 We will see later that ρ does not attain the value d anywhere.

Table 18.1. Maximal x i -domains of the c 2 = 0 representatives of the six families of positive-valued solutions (18.20) of the differential equation (18.14) (with d = 0, c = ln [START_REF] Baird | Ray congruences that generate conformal foliations[END_REF] 2 ), classified according to sgn d, sgn ρ -d, and sgn ρ i .

ρ i > 0 ρ i < 0 d < 0 ρ -d > 0 -π 6 √ 3d 2 , π 2 √ 3d 2 -π 2 √ 3d 2 , π 6 √ 3d 2 d > 0 ρ -d > 0 -∞, -π 2 √ 3d 2 π 2 √ 3d 2 , +∞ d > 0 ρ -d < 0 π 6 √ 3d 2 , +∞ -∞, -π 6 √ 3d 2 families (1, 1), (1, 2), (3, 1), (3, 2)
, by (18.14), the Taylor expansion of ρ about

x i = x 0 i := 3 2 e -c H(0) + c 2 (where ρ(x 0 i ) = 0) 48 is ρ = -η 2 3 e c d 3 (x i -x 0 i ) + O((x i -x 0 i ) 4 ). Then g = ρ -2 [ 2 3 e c-b |d| 3 (dx 1 2 + dx 2 2 ) + (dx 3 2 + dx 4 2 )] + 2 3 e c-b sgn(ρ -d) ρ(dx 1 2 + dx 2 2 ),
where

ρ -2 -9 4 e -2c d -6 (x i -x 0 i ) -2 = O(x i -x 0 i ) 1 + O((x i -x 0 i ) 3 ) → 0 as x i → x 0 i and 2
3 e c-b sgn(ρ -d) ρ → 0 as x i → x 0 i . Thus, the metric g asymptotically collapses to the metric

ds 2 = 9 4 e -2c d -6 (x i -x 0 i ) -2 [ 2 3 e c-b |d| 3 (dx 1 2 + dx 2 2 ) + (dx 3 2 + dx 4 2 )],
which is homothetic to the hyperbolic 4-metric.

The previous example gives us the next theorem.

Theorem 18.8. The maximal, biconformally deformed metric

g = σ(x 1 ) -2 (dx 1 2 + dx 2 2 ) + ρ(x 1 ) -2 (dx 3 2 + dx 4 2 ),
is an Einstein metric if and only if one of the following holds:

• Ricci-flat: IA. g is isometric to the Euclidean 4-metric; IB. g is isometric to the product metric of (a) the warped metric y 2 dx 2 + dy 2 on the upper half-plane and (b) the Euclidean 2-metric; III. g is homothetic to

ds 2 = x -1/2 (dx 2 + dy 2 ) + x(dz 2 + dw 2 ), x > 0; • non-Ricci-flat:
II. g is homothetic to the hyperbolic 4-metric;

IV. g is the metric (18.21), where ρ is given by (18.20), b ∈ R is given by (18.11), c ∈ R is given by (18.13), d ∈ R \ {0}, and c 2 ∈ R.

In item IB., g asymptotically collapses to the Euclidean 2-metric. In item IV., g asymptotically collapses either to the Euclidean 2-metric, or to a metric that is homothetic to the hyperbolic 4-metric.

(a) Family (1, 1) of Table 18.1.

(b) Family (1, 2) of Table 18.1.

(c) Family (2, 1) of Table 18.1.

(d) Family (2, 2) of Table 18.1.

(e) Family (3, 1) of Table 18.1.

(f) Family (3, 2) of Table 18.1. 18.4. σ vertical, ρ horizontal. This is the simplest of the three cases.

Lemma 18.9. Suppose that σ = σ(x 3 , x 4 ) and ρ = ρ(x 1 , x 2 ).

(1) If the biconformally deformed metric (18.1) is an Einstein metric, then there exists Γ ∈ [0, +∞) such that

Γ = σ -2 q (ln σ) q 2 = ρ -2 j (ln ρ) j 2 .
(2) The biconformally deformed metric (18.1) is a (Ricci-flat) Einstein metric in some neighborhood of R 4 if and only if both σ and ρ are constant functions in some neighborhood of R 4 .

Proof.

(1) Equation ( 18.2) implies that

σ 2   j (ln ρ) jj - j (ln ρ) j 2 + σ -2 ρ 2 q (ln σ) qq -2σ -2 ρ 2 q (ln σ) q 2   = ρ 2   q (ln σ) qq - q (ln σ) q 2 + σ 2 ρ -2 j (ln ρ) jj -2σ 2 ρ -2 j (ln ρ) j 2   ,
which gives the desired result. (2) The key idea here is that 0 = (ln ρ) i (ln σ) p for all (i, p) in an open box if and only if either σ or ρ is a constant function in the box. Then, apply the previous item.

As a corollary, we have the following theorem.

Theorem 18.10. The only biwarped products of two Euclidean surfaces that are Einstein manifolds are neighborhoods of Euclidean 4-space.

8πT ii = -σ -2 ρ 2 q ε q (ln σ) qq + σ -2 ρ 2 q ε q (ln σ) q 2 (19.1a) -2(ln ρ) i i + (ln ρ) i 2 + 3(ln ρ) i 2 -σ -2 ρ 2 q ε q (ln ρ) qq + 2(ln σ) i (ln ρ) i -2(ln σ) i (ln ρ) i , i ∈ {1, 2}, 8πT 12 = 2(ln ρ) 12 + 2(ln σ) 1 (ln ρ) 2 + 2(ln σ) 2 (ln ρ) 1 -2(ln ρ) 1 (ln ρ) 2 , (19.1b)
8πT pp = -ε p σ 2 ρ -2 j (ln ρ) jj + ε p σ 2 ρ -2 j (ln ρ) j 2 (19.1c) + 2(ln σ) p p + (ln σ) p 2 -3(ln σ) p 2 -ε p σ 2 ρ -2 j (ln σ) jj + 2 q (ln σ) q (ln ρ) q , p ∈ {3, 4}, 8πT 34 = 2(ln σ) 34 + 2(ln σ) 3 (ln ρ) 4 + 2(ln σ) 4 (ln ρ) 3 -2(ln σ) 3 (ln σ) 4 , (19.1d)
8πT ip = (ln σ) ip + (ln ρ) ip + 2(ln ρ) i (ln σ) p , i ∈ {1, 2}, p ∈ {3, 4} (19.1e)
with T ab = T ba (see (17.7)). For future reference (see (19.3), (19.5a), (19.5b), and (19.6a)), (19.1a) is equivalent to the pair

1 2 (8πT 11 + 8πT 22 ) = -σ -2 ρ 2 q ε q (ln σ) qq + σ -2 ρ 2 q ε q (ln σ) q 2 - j (ln ρ) jj + 2 j (ln ρ) j 2 -σ -2 ρ 2 q ε q (ln ρ) qq , 1 2 (8πT 11 -8πT 22 ) = (ln ρ) 11 -(ln ρ) 22 -(ln ρ) 1 2 + (ln ρ) 2 2 + 2(ln σ) 1 (ln ρ) 1 -2(ln σ) 2 (ln ρ) 2 ,
while (19.1c) implies that

1 2 (8πT 33 + 8πT 44 ) = q (ln σ) qq - q (ln σ) q 2 + 2 q (ln σ) q (ln ρ) q .
Note that a 2-form on R 4 with zero mixed part is closed if and only if it is of the form

(19.2) F = F 12 (x 1 , x 2 ) dx 1 ∧ dx 2 + F 34 (x 3 , x 4 ) dx 3 ∧ dx 4
(see footnote 37). We restate Theorems 17.2 and 17.3 in this Lorentzian setting.

Theorem 19.1. Suppose that

(19.3) 0 = q (ln σ) qq - q (ln σ) q 2 + 2 q (ln σ) q (ln ρ) q .
The biconformally deformed metric

(19.4) g = σ -2 (dx 1 2 + dx 2 2 ) + ρ -2 (dx 3 2 -dx 4 2 )
satisfies the Einstein equation (17.3) with magnetic constant μ and electromagnetic field (19.2) having zero mixed part 49 if and only if the following system of eight partial differential equations holds: In this case, the flow vector field u can be any unit, future-pointing timelike vector field, and the energy density ρ, pressure p, and cosmological constant Λ are related by 50 8π ρ + 8π p = ρ 2 (8πT 33 + 8πT 44 ), (19.6a) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ). (19.6b) Theorem 19.2. The biconformally deformed metric (19.4) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ 4 , magnetic constant μ, and electromagnetic field (19.2) having zero mixed part if and only if the system (19.5) holds. In this case, the energy density ρ, pressure p, and cosmological constant Λ are related by the equations (19.6). 49 Here, F 12 can be replaced by -F 12 and F 34 by -F 34 . The same is true for the theorems, corollaries, and examples in the rest of this section. 50 Equation (19.3) implies that 8πT 33 + 8πT 44 = 0, but we write the general formula for 8π ρ + 8π p (see Theorem 19.2).

8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = σ 2 1 2 (8πT 11 + 8πT 22 ) -ρ 2 8πT 33 , (19.5a
8π p -Λ = ρ 2 8πT 33 + 4π μ-1 (σ 4 F 12 2 + ρ 4 F 34
Corollary 19.3. Suppose that the expression σ 2 1 2 (8πT 11 + 8πT 22 ) -ρ 2 8πT 33 satisfies the following conditions:

(1) It is always nonnegative or always nonpositive.

(2) The square root of its absolute value is smooth.

(

F 12 2 = σ -4 σ 2 j (ln σ) jj -α 8π μ-1 , (19.7a)
F 34 2 = ρ -4 ρ 2 q ε q (ln ρ) qq -α -8π μ-1 . (19.7b)
In this case, the flow vector field u can be any unit, future-pointing timelike vector field, and the energy density ρ, pressure p, and cosmological constant Λ are related by

ρ + p = 0, 8π p -Λ = -1 2 σ 2 j (ln σ) jj -1 2 ρ 2 q ε q (ln ρ) qq .
Moreover, sup ρ 2 q ε q (ln ρ) qq ≤ α ≤ inf σ 2 j (ln σ) jj , if μ ∈ (0, +∞), sup σ 2 j (ln σ) jj ≤ α ≤ inf ρ 2 q ε q (ln ρ) qq , if μ ∈ (-∞, 0), and F = 0 if and only if

σ 2 j (ln σ) jj = ρ 2 q ε q (ln ρ) qq = α.
Proof. Equations (19.3), (19.5b), and (19.5c) are automatically satisfied. Equation (19.5a), on the other hand, are satisfied if and only if there exists α ∈ R such that

σ 2 j (ln σ) jj -8π μ-1 σ 4 F 12 2 = α = ρ 2 q ε q (ln ρ) qq + 8π μ-1 ρ 4 F 34 2 .
Note that if σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ) such that 51 σ 2 j (ln σ) jj , ρ 2 q ε q (ln ρ) qq ∈ R,

51 Note that we have already seen from Example 18.4 how to solve differential equations of this type when σ = σ(x i ) and ρ = ρ(xp).

then g satisfies the Einstein equation with a horizontal electromagnetic field F = F H or a vertical electromagnetic field F = F V (trivial electromagnetic field F = 0 if the two constants are equal), and a vacuum-trivial perfect fluid.

Example 19.5. Let σ(x 1 ) = cosh x 1 , ρ(x 4 ) = sinh x 4 , x 4 > 0.

Then

σ 2 j (ln σ) jj = ρ 2 q ε q (ln ρ) qq = 1.
By the previous theorem, the metric

g = (sech 2 x 1 ) (dx 1 2 + dx 2 2 ) + (csch 2 x 4 ) (dx 3 2 -dx 4 2 ) on {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 4 > 0}
satisfies the Einstein equation with F = 0, any admissible u, and ρ = -p = -1 8π (Λ -1). Theorem 19.6. Suppose that σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ). The biconformally deformed metric (19.4) (ln σ) jj ≤ inf ρ 2 q ε q (ln ρ) qq .

Proof. The forward direction is a direct consequence of the previous theorem. For the backward direction, suppose that either (19.8) or (19.9) holds. If either inequality is strict, then we can choose α ∈ R and μ ∈ R \ {0} such that both the right-hand sides of (19.7a) and (19.7b) are always strictly positive, so that their square roots are smooth. If both (19.8) and (19.9) are equalities, then σ 2 j (ln σ) jj = ρ 2 q ε q (ln ρ) qq ∈ R, and the remark preceding the previous example applies.

Example 19.7. Let α 1 , α 2 ∈ [0, +∞) and α 4 ∈ R such that α 1 + α 2 + α 4 > 0. Let σ(x 1 , x 2 ) = e α1x1 2 +α2x2 2 , ρ(x 4 ) = e α4x4 2 .
Then In this case, the flow vector field u can be any unit, future-pointing timelike vector field, and the energy density ρ, pressure p, and cosmological constant Λ are related by

σ 2 j (ln σ) jj = 2(α 1 + α 2 )e 2(α1x1 2 +α2x2 2 ) , ρ 2 q ε q (ln ρ) qq = -2α 4 e 2α4x4 2 , with sup x4∈R ρ 2 q ε q (ln ρ) qq = -2α 4 < 2(α 1 + α 2 ) = inf
ρ + p = 0, 8π p -Λ = σ 2   -1 2 j (ln σ) jj - j (ln ρ) jj + 3 2 j (ln ρ) j 2   . Moreover, (19.11)
F 34 2 ∈   0, inf σ 2 j (ln σ) jj + j (ln ρ) j 2 8π μ-1 ρ 4   .
If σ = σ(x 1 , x 2 ) and ρ is a constant function, then the equations (19.10b) and (19.10c) are automatically satisfied.

(1) If ln σ is harmonic 52 , then the metric (19.4) Then the biconformally deformed metric (19.4) satisfies the Einstein equation (17.3) with electromagnetic field having zero mixed part and any admissible flow vector field. 52 Here and for the rest of the section, "harmonic" means "harmonic with respect to the metric g 0 ".

Proof. The case where j (ln σ) jj is always zero is just the first item of the preceding discussion. For the other cases, the equations (19.12) for some c 1 , c 2 ∈ R. Equation (19.13a) becomes

σ 4 F 12 2 + e 4c1 (x i -c 2 ) 4 F 34 2 = σ 2 8π μ-1 (x i -c 2 ) 2 .
Then μ ∈ (0, +∞) and, by (19.11),

F 34 2 ∈ 0, σ 2 8π μ-1 e 4c1 inf |x i -c 2 | 2 .
In particular, by choosing i = 1 and c 1 = c 2 = 0 in (19.14) and σ = 1, the warped metric

g = dx 1 2 + dx 2 2 + x 2 2 (dx 3 2 -dx 4 2 )
on (R × (0, +∞)) × R 2 1 satisfies the Einstein equation with any admissible u, μ ∈ (0, +∞), and F, ρ, p, Λ given/related by

F = ± 1 8π μ-1 x 2 dx 1 ∧ dx 2 , ρ + p = 0, 8π p -Λ = 1 2 x 2 -2 .
In other words, the metric

ds 2 = dx 2 + e 2y (dy 2 + dz 2 -dt 2 )
on R 4 satisfies the Einstein equation with any admissible u, μ ∈ (0, +∞), and F, ρ, p, Λ given/related by ) = e 4c σ -6 (x i -d) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] .

F = ± 1 8π μ-1 dx ∧ dy, ρ + p = 0, 8π p -Λ =
In particular, by choosing i = 1, σ = 1, and c = d = 0, the warped metric

g = dx 1 2 + dx 2 2 + x 1 2 (dx 3 2 -dx 4 2 ) on ((0, +∞) × R) × R 2
1 satisfies the Einstein equation with any admissible u and F, μ, ρ, p, Λ given/related by

F = x 1 -1 dx 1 ∧ dx 2 , μ = 8π, ρ + p = 0, 8π p -Λ = 1 2 x 1 -2 .
If σ = ρ (so the deformation is conformal) is not a constant function, then the system (19.15) is equivalent to the pair σ = ρ = e c |x i -d|, c, d ∈ R, (19.17)

F 12 = F 13 = 0.
That is, the conformally deformed metric

g = σ(x i ) -2 (dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 ),
where σ is not a constant function, satisfies the Einstein equation with F mixed = 0 if and only if F = 0 and σ has the form (19.17). In this case, ρ, p, Λ are related by ρ + p = 0, 8π p -Λ = 3e 2c .

In particular, by choosing i = 1 and c = d = 0 in (19.17), the hyperbolic Minkowski 4-metric

g = x 1 -2 (dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 ) on {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 |x 1 > 0}
satisfies the Einstein equation with F = 0, any admissible u, and ρ = -p = -1 8π (Λ + 3). In other words, the warped metric

ds 2 = dx 2 + (e -x ) 2 (dy 2 + dz 2 -dt 2 ) on R × R 3
1 satisfies the Einstein equation with said parameters.

The previous example gives us the next theorem.

Theorem 19.13. The maximal, conformally deformed metric

g = σ(x 1 ) -2 (dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 )
satisfies the Einstein equation (17.3) with electromagnetic field having zero mixed part if and only if g is isometric to the Minkowski 4-metric or g is homothetic to the hyperbolic Minkowski 4-metric (with the "hyperbolic coordinate" being one of the spatial coordinates). In this case, the electromagnetic field is trivial and the perfect fluid is vacuum-trivial.

19.4. σ vertical, ρ vertical. In the case σ = σ(x 3 , x 4 ), ρ = ρ(x 3 , x 4 ), we make the following observations:

(1) The system (19.5) becomes ε q (ln σ) q 2 + q ε q (ln ρ) qq , which is an analogue of (19.10a). (4) The component F 12 is a constant function.

8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = ρ 2 - q (ln σ) qq + 2(ln σ) 4 2 - q ε q (ln ρ) qq -2 q (ln σ) q (ln ρ) q , (19.18a
(5) If σ is a constant function, then we have a discussion similar to the one leading to Corollary 19.10. ( 6) Equation (19.18b) is obviously not an analogue of (19.10b), but (19.3), if it does hold, is. Equation (19.3) gives us an analogue of (19.13b) when σ = σ(x p ) and ρ = ρ(x p ), and of (19.15b) when σ = σ(x p ) and ρ = ρ(x p ). for some c 1 , c 2 ∈ R. In particular, by choosing c 1 = c 2 = 0 above and ρ = 1, the warped metric

g = (x 4 1/2 ) 2 (dx 1 2 + dx 2 2 ) + dx 3 2 -dx 4 2
on the product of R 2 and upper half-R 2 1 satisfies the Einstein equation with F = 0 and u, ρ, p, Λ given/related by

u = ∂ 4 , 8π ρ + 8π p = -3 2 x 4 -2 , 8π p -Λ = -7 4 x 4 -2 .
In the next example and theorem, we discuss how the biconformally deformed metric (19.4) Suppose that F = 0, that u = σ∂ 4 , and that ρ, p, Λ are related by the above equations. Let In particular, we can write

x 4 = x 4 (t), t ∈ t(J) =: I.
We then see that the metric g can be written as

g = f (t) 2 (dx 1 2 + dx 2 2 + dx 3 2 ) -dt 2 ,
where

f (t) = σ(x 4 (t)) -1 , t ∈ I,
and the flow vector field u as u = ∂ t . Hence, the triple

M (0, I, f ) = R 3 f ×I, (u, ρ, p), Λ
is a FRW cosmological model.

This example gives us the following theorem.

Theorem 19.16. The conformally deformed metric

g = σ(x 4 ) -2 (dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 )
gives rise to a FRW cosmological model with flat space, cosmological time t given by (19.19)- (19.20), and expansion f given by f (t) = σ(x 4 (t)) -1 .

19.5. σ vertical, ρ horizontal. This case reduces to either of the previous two cases.

Theorem 19.17. Suppose that σ = σ(x 3 , x 4 ) and ρ = ρ(x 1 , x 2 ) such that the biconformally deformed metric (19.4) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ 4 , magnetic constant μ, and electromagnetic field (19.2) having zero mixed part. Then either of the following hold:

(1) The function σ is constant and the system (19.10) holds. The metric g satisfies the Einstein equation with any admissible flow vector field, the same magnetic constant, and the same electromagnetic field. (2) The function ρ is constant and the system (19.18) holds.

Proof. By (19.5c), 0 = (ln σ) p (ln ρ) i for all i ∈ {1, 2} and p ∈ {3, 4}. This implies that either σ or ρ is a constant function.

19.6. σ spatial, ρ temporal. If σ = σ(x 1 , x 2 , x 3 ) and ρ = ρ(x 4 ), then the system (19.5) (1) α = 0.

(2) ln h is harmonic.

(3) f 3 vanishes everywhere (i.e. f is a constant function). (4) f 3 vanishes somewhere. 53 The choice of h depends on the choice of b: If a is another number in the domain of x 3 , then

σ = h(x 1 , x 2 ) exp a b f (w) dw • exp x 3 a f (w) dw .
Suppose that (19.23) holds. If α = 0, then by (19.22)

σ = h(x 1 , x 2 )e f (b)(x3-b)
with ln h harmonic (see items 11. and 12. of Theorem 19.19 for the case f (b) = 0).

Suppose that α ∈ R \ {0}. From (19.23), (19.24)

ρ 2 exp -2 x3 b f (w) dw = αf 3 -1 .
In particular, f 3 always has the same sign as α. Using (19.24) in (19.22), we have

σ = ρ|α| -1/2 h(x 1 , x 2 )|f 3 (x 3 )| 1/2 , so that g = ρ -2 |α|h(x 1 , x 2 ) -2 |f 3 (x 3 )| -1 (dx 1 2 + dx 2 2 ) + ρ -2 (dx 3 2 -dx 4 2 ).
Taking the ln and then differentiating both sides of (19.24), we get

(19.25) -2f = -f 3 -1 f 33 .
Integration yields

(19.26) c = f 3 -f 2
for some c ∈ R. for some d ∈ R.

II. 0 < c < f 3 . In this case, (19.27) can be rewritten as

±f 3 1/2 (f 3 1/2 ) 2 -(c 1/2 ) 2 = 1 2 f 3 -1/2 f 33 .
Upon integration, we find that

f 3 1/2 = c 1/2 sec[c 1/2 (x 3 -d 1 )]
for some d 1 ∈ R. Squaring and integrating both sides, we get

f = c 1/2 tan[c 1/2 (x 3 -d 1 )] + d 2 for some d 2 ∈ R. III. c < f 3 < 0.
In this case, (19.27) can be rewritten as

±(-f 3 ) 1/2 ((-c) 1/2 ) 2 -((-f 3 ) 1/2 ) 2 = -1 2 (-f 3 ) -1/2 f 33 .
Upon integration, we find that

(-f 3 ) 1/2 = (-c) 1/2 sech[(-c) 1/2 (x 3 -d 1 )]
for some d 1 ∈ R. Squaring and integrating both sides, we get

f = -(-c) 1/2 tanh[(-c) 1/2 (x 3 -d 1 )] + d 2 for some d 2 ∈ R. IV. c < 0 < f 3 .
In this case, (19.27) can be rewritten as

±f 3 1/2 (f 3 1/2 ) 2 + ((-c) 1/2 ) 2 = 1 2 f 3 -1/2 f 33 .
Upon integration, we find that

f 3 1/2 = ±(-c) 1/2 csch[(-c) 1/2 (x 3 -d 1 )]
for some d 1 ∈ R. Squaring and integrating both sides, we get Note that h cannot be a constant function.

f = -(-c) 1/2 coth[(-c)
A. h ii vanishes everywhere, α < 0. In this case,

h = (-α) 1/2 |x i -c 2 |
for some c 2 ∈ R.

Suppose that h ii is not identically zero. Over an open interval where h i and h ii do not vanish 54 , h has an inverse

x i = x i (h); let v(h) = h i (x i (h)).
We rewrite (19.28) using h as the independent variable and v as the dependent variable:

-v 2 + hvv h = α. This differential equation has solution

v 2 = ηe 2c1 h 2 -α,
where η ∈ {-1, +1} and c 1 ∈ R. Then, in terms of the independent variable x i and the dependent variable h, (19.29) ± e c1 η(e c1 h) 2 -α = e c1 h i .

B. η = +1, α > 0.

In this case, (19.29) can be rewritten as

±e c1 (e c1 h) 2 -(α 1/2 ) 2 = e c1 h i .
This gives the solution

h = α 1/2 e -c1 cosh[e c1 (x i -c 2 )],
where c 2 ∈ R.

C. η = +1, α < 0. In this case, (19.29) can be rewritten as

±e c1 (e c1 h) 2 + ((-α) 1/2 ) 2 = e c1 h i .
This gives the solution

h = (-α) 1/2 e -c1 | sinh[e c1 (x i -c 2 )]|,
where c 2 ∈ R. 54 We will see from items B., C., and D. that h ii does not vanish anywhere, and that h i vanishes at at most one point, at which the forms of h given in the items still hold.

In items 01. to 10. and item IB., the perfect fluid is vacuum-trivial. In the rest of the items, the perfect fluid is non-vacuum-trivial but the energy density and pressure are constant functions.

h -2 j (ln h) j 2 = σ -2 f 3 exp 2 x3 b f (w) dw = α
for some α ∈ [0, +∞). The following are equivalent in (19.32):

(1) α = 0.

(2) h is a constant function.

(3) f 3 vanishes everywhere (i.e. f is a constant function). (4) f 3 vanishes somewhere. Suppose that (19.32) Meanwhile, Theorems 15.19 and 15.21 give us the differences

1 2 (σ -2 S -S 0 ) = 3(σ -2 ρ 2 -1)((ln f ) tt + 2(ln f ) t 2 ) -3f -2 j (ln ρ) j 2 -3σ -2 ρ 2 (f -2 (ln σ) 3 2 -(ln σ) t 2 ) -2σ -2 ρ 2 (ln f ) t (ln σ) t + 2σ -2 ρ 2 (ln f ) t (ln ρ) t + f -2 j (ln σ) jj + 2σ -2 ρ 2 (f -2 (ln σ) 33 -(ln σ) tt -3(ln f ) t (ln σ) t ) + 2f -2 j (ln ρ) jj + σ -2 ρ 2 (f -2 (ln ρ) 33 -(ln ρ) tt -3(ln f ) t (ln ρ) t ) and 1 2 (ρ -2 S -S 0 ) = -3σ 2 ρ -2 f -2 j (ln ρ) j 2 -3(f -2 (ln σ) 3 2 -(ln σ) t 2 ) -2(ln f ) t (ln σ) t + 2(ln f ) t (ln ρ) t + σ 2 ρ -2 f -2 j (ln σ) jj + 2(f -2 (ln σ) 33 -(ln σ) tt -3(ln f ) t (ln σ) t ) + 2σ 2 ρ -2 f -2 j (ln ρ) jj + (f -2 (ln ρ) 33 -(ln ρ) tt -3(ln f ) t (ln ρ) t ),
respectively.

Let the functions T ab : M → R be given by

8πT ii = -σ -2 ρ 2 (f -2 (ln σ) 33 -(ln σ) tt -3(ln f ) t (ln σ) t ) (20.3a) + σ -2 ρ 2 (f -2 (ln σ) 3 2 -(ln σ) t 2 ) -2f -2 (ln ρ) i i + f -2 (ln ρ) i 2 + 3f -2 (ln ρ) i 2 -σ -2 ρ 2 (f -2 (ln ρ) 33 -(ln ρ) tt -3(ln f ) t (ln ρ) t ) + 2f -2 (ln σ) i (ln ρ) i -2f -2 (ln σ) i (ln ρ) i -2σ -2 ρ 2 (ln f ) t (ln ρ) t -σ -2 ρ 2 (2(ln f ) tt + 3(ln f ) t 2 ), i ∈ {1, 2}, 8πT 12 = 2f -2 (ln ρ) 12 + 2f -2 (ln σ) 1 (ln ρ) 2 + 2f -2 (ln σ) 2 (ln ρ) 1 -2f -2 (ln ρ) 1 (ln ρ) 2 , (20.3b
)

8πT 33 = -σ 2 ρ -2 f -2 j (ln ρ) jj + σ 2 ρ -2 f -2 j (ln ρ) j 2 (20.3c) + 2(ln σ) tt + f -2 (ln σ) 3 2 -3(ln σ) t 2 -σ 2 ρ -2 f -2 j (ln σ) jj + 2f -2 (ln σ) 3 (ln ρ) 3 + 2(ln σ) t (ln ρ) t + 6(ln f ) t (ln σ) t -2(ln f ) t (ln ρ) t -2(ln f ) tt -3(ln f ) t 2 , 8πT 44 = σ 2 ρ -2 f -2 j (ln ρ) jj -σ 2 ρ -2 f -2 j (ln ρ) j 2 (20.3d) + 2f -2 (ln σ) 33 -3f -2 (ln σ) 3 2 + (ln σ) t 2 + σ 2 ρ -2 f -2 j (ln σ) jj + 2f -2 (ln σ) 3 (ln ρ) 3 + 2(ln σ) t (ln ρ) t -4(ln f ) t (ln σ) t -2(ln f ) t (ln ρ) t + 3(ln f ) t 2 , 8πT 34 = 2f -1 (ln σ) 3t + 2f -1 (ln σ) 3 (ln ρ) t + 2f -1 (ln σ) t (ln ρ) 3 -2f -1 (ln σ) 3 (ln σ) t (20.3e) -2f -1 (ln f ) t (ln ρ) 3 , 8πT i3 = f -2 (ln σ) i3 + f -2 (ln ρ) i3 + 2f -2 (ln ρ) i (ln σ) 3 , i ∈ {1, 2}, (20.3f)
8πT i4 = f -1 (ln σ) it + f -1 (ln ρ) it + 2f -1 (ln ρ) i (ln σ) t -2f -1 (ln f ) t (ln ρ) i , i ∈ {1, 2}. ( 20
) = -σ -2 ρ 2 (f -2 (ln σ) 33 -(ln σ) tt -3(ln f ) t (ln σ) t ) + σ -2 ρ 2 (f -2 (ln σ) 3 2 -(ln σ) t 2 ) -f -2 j (ln ρ) jj + 2f -2 j (ln ρ) j 2 -σ -2 ρ 2 (f -2 (ln ρ) 33 -(ln ρ) tt -3(ln f ) t (ln ρ) t ) -2σ -2 ρ 2 (ln f ) t (ln ρ) t -σ -2 ρ 2 (2(ln f ) tt + 3(ln f ) t 2 ), 1 2 (8πT 11 -8πT 22 ) = f -2 (ln ρ) 11 -f -2 (ln ρ) 22 -f -2 (ln ρ) 1 2 + f -2 (ln ρ) 2 2 + 2f -2 (ln σ) 1 (ln ρ) 1 -2f -2 (ln σ) 2 (ln ρ) 2 ,
while (20.3c) and (20.3d) imply that

1 2 (8πT 33 + 8πT 44 ) = f -2 (ln σ) 33 + (ln σ) tt -f -2 (ln σ) 3 2 -(ln σ) t 2 + 2f -2 (ln σ) 3 (ln ρ) 3 + 2(ln σ) t (ln ρ) t + (ln f ) t (ln σ) t -2(ln f ) t (ln ρ) t -(ln f ) tt .
Note that a 2-form on M with zero mixed part is closed if and only if it is of the form (20.4)

F = F 12 (x 1 , x 2 , t) θ 0 1 ∧ θ 0 2 + F 34 (x 3 , t) θ 0 3 ∧ θ 0 4 . with ∂ ∂t F 12 = -2(ln f ) t F 12 .
In a (x 1 , x 2 , t)-region where F 12 does not vanish, this partial differential equation has solution

F 12 (x 1 , x 2 , t) = ±f (t) -2 e C(x1,x2
) for some smooth C. In particular, if F 12 = F 12 (t), then the Maxwell equation dF = 0 implies that

F 12 = kf -2
for some k ∈ R.

We restate Theorem 17.3 in this setting.

Theorem 20.1. The biconformally deformed metric

(20.5) g = σ -2 f 2 (dx 1 2 + dx 2 2 ) + ρ -2 (f 2 dx 3 2 -dt 2 )
satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t , magnetic constant μ, and electromagnetic field (20.4) having zero mixed part 56 if and only if the following system of eight partial differential equations holds:

8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = σ 2 1 2 (8πT 11 + 8πT 22 ) -ρ 2 8πT 33 , (20.6a
d dt (f ∆ 3 s) t=t0 = H 0 f (t 0 )∆ 3 s holds, where ∆ 3 s represents usual distance in R 3 , i.e. H 0 = (ln f ) t | t=t0 .
Now for (M, g). 57 Assume that σ = σ(t) and ρ = ρ(t), that σ and ρ have the same domain K, and that t 0 ∈ K ⊆ I. Let τ = τ (t) be an antiderivative of ρ -1 .

56 Here, F 12 can be replaced by -F 12 and F 34 by -F 34 . The same is true for the theorems, corollary, lemma, and examples in the rest of this section. 57 Here, M can be replaced by an open submanifold of M , depending on the domains of the deformation parameters σ, ρ.

Since ρ -1 > 0 on the t-interval K, then we can write t = t(τ ) on the τ -interval J := τ (K). Moreover, we can write the biconformally deformed metric g as

g = f (τ ) 2 (σ(τ ) -2 (dx 1 2 + dx 2 2 ) + ρ(τ ) -2 dx 3 2 ) -dτ 2 .
If (M, g) satisfies the Einstein equation, then we call the coordinate τ the (pseudo-)cosmological time of (M, g) (up to an additive constant). (If σ = ρ, then τ is indeed cosmological time.) To formulate a suitable Hubble condition for the biconformally deformed spacetime (M, g), we simply emulate (20.8). We say that (M, g) (or just g) satisfies the Hubble condition if there is a Hubble constant

H 1 ∈ R such that d dτ [f (σ -2 ∆ 2 s 2 + ρ -2 ∆ 1 s 2 ) 1/2 ] τ =τ0 = H 1 f (τ 0 )(σ(τ 0 ) -2 ∆ 2 s 2 + ρ(τ 0 ) -2 ∆ 1 s 2 ) 1/2 ,
where τ 0 = τ (t 0 ) and ∆ 2 s and ∆ 1 s respectively represent usual distance in R 2 and in R 1 (using the appropriate coordinates). This can be written as

H 1 -H 0 ρ(τ 0 ) = -(σ(τ 0 ) -2 ∆ 2 s 2 +ρ(τ 0 ) -2 ∆ 1 s 2 ) -1 (σ(τ 0 ) -2 (ln σ) τ | τ =τ0 ∆ 2 s 2 +ρ(τ 0 ) -2 (ln ρ) τ | τ =τ0 ∆ 1 s 2 ).
This in turn is equivalent to (20.9)

H 1 -H 0 ρ(τ 0 ) = -(ln σ) τ | τ =τ0 = -(ln ρ) τ | τ =τ0 . But (ln σ) τ | τ =τ0 = ρ(t 0 ) (ln σ) t | t=t0 , (ln ρ) τ | τ =τ0 = ρ(t 0 ) (ln ρ) t | t=t0 .
Hence, g satisfies the Hubble condition for some Hubble constant

H 1 ∈ R if and only if (ln σ) t | t=t0 = (ln ρ) t | t=t0 ; in this case, H 1 = ρ(t 0 ) (ln f σ -1 ) t t=t0 = ρ(t 0 ) (ln f ρ -1 ) t t=t0 and (ln σ) t | t=t0 = (ln ρ) t | t=t0 = -ρ(t 0 ) -1 (H 1 -H 0 ρ(t 0 )).
For example, any conformal deformation of the metric g 0 satisfies the Hubble condition.

+ 2ρ 2 ((ln f ) t (ln σ) t -2(ln f ) t (ln ρ) t -(ln f ) tt ), 8π p -Λ = ρ 2 ( 3 2 (ln σ) tt -2(ln σ) t 2 + 1 2 (ln ρ) tt + (ln σ) t (ln ρ) t ) (20.11b) + ρ 2 ( 9 2 (ln f ) t (ln σ) t -1 2 (ln f ) t (ln ρ) t -2(ln f ) tt -3(ln f ) t 2 ),
respectively.

Lemma 20.5. Suppose that σ = σ(t) and ρ = ρ(t) such that the biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t and electromagnetic field F = 0.

(1) If ρ is a constant function and σ is not, then

σ -2 = ±e c f -3 -d
for some c, d ∈ R.

(2) If σ is a constant function and ρ is not, then

ln ρ = ±e c f -3 -d for some c, d ∈ R.
(Here, f -3 denotes a particular antiderivative of f -3 .) In both items, the derivative of the nonconstant function does not vanish anywhere.

Proof.

(1) Equation (20.10) becomes 0 = (ln σ) tt -2(ln σ) t 2 + 3(ln f ) t (ln σ) t .

Over an interval where σ t does not vanish,

σ t = ∓ 1 2 e c f -3 σ 3
for some c ∈ R. We then see that σ t does not vanish anywhere on the domain of σ.

(2) Equation (20.10) becomes 0 = (ln ρ) tt + 3(ln f ) t (ln ρ) t .

Over an interval where ρ t does not vanish, ρ t = ±e c f -3 ρ for some c ∈ R. We then see that ρ t does not vanish anywhere on the domain of ρ.

Example 20.6. Consider the Einstein-de Sitter cosmological model M (0, (0, +∞), t 2/3 ), (∂ t , ρ0 , p0 ), Λ 0 , where 8π ρ0 = 4 3 t -2 , 8π p0 = 0, Λ 0 = 0.

Note that there is a big bang singularity at t = 0 since lim

t→0 + f = 0, lim t→0 + f t = +∞,
and (hence by (20.1b)) lim t→0 + ρ0 = +∞. (1) If ρ = 1 and F = 0, then in view of the first item of the previous lemma we set σ = t 1/2 . We verify that the metric

g = t 4/3 (t -1 (dx 1 2 + dx 2 2 ) + dx 3 2 ) -dt 2
on R 3 × (0, +∞) satisfies the Einstein equation with u = ∂ t , F = 0, and ρ, p, Λ related by

8π ρ + 8π p = 1 2 t -2 , 8π p -Λ = 1 4 t -2 ,
i.e. 8π ρ + Λ = 8π p -Λ = 1 4 t -2 . Note that ρ -p = -2ρ vacuum .

(If Λ = 0, then 8π ρ = 8π p = 1 4 t -2 and so deforming the horizontal part of the metric g 0 by the parameter σ has the effect of decreasing the energy density and increasing the pressure.) Note that there is still a 'big bang singularity', in the sense that

lim t→0 + t 2/3 = 0, lim t→0 + d dt [t 2/3 ] = +∞, lim t→0 + t (4/3-1)/2 = 0, lim t→0 + d dt [t (4/3-1)/2 ] = +∞, lim t→0 + ρ = +∞. ( 2
) If σ = 1 and F = 0, then in view of the second item of the previous lemma we set ρ ± = e ±1/t .

We verify that the metric [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ) -e ∓2/t dt 2 on R 3 × (0, +∞) satisfies the Einstein equation with u = ∂ t , F = 0, and ρ, p, Λ related by 8π ρ + 8π p = 4 3 e ±2/t t -3 (t ± 2), 8π p -Λ = ± [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] 3 e ±2/t t -3 , i.e. 8π ρ + Λ = [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] 3 e ±2/t t -3 (t ± 1), 8π p -Λ = ± 4 3 e ±2/t t -3 . From these we get 8π ρ -8π p + 2Λ = [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] 3 e ±2/t t -2 . If Λ ∈ (-∞, 0], then ρ is always greater than p. If Λ ∈ (0, +∞) and ρ = ρ + , then ρ is eventually taken over by p. If Λ ∈ (0, +∞) and ρ = ρ -, then either p is always greater than ρ (except possibly at a single point), or p starts out being greater than ρ, then is overtaken by ρ for some time, and then again becomes greater than ρ until the rest of time. Whatever the value of Λ is, if ρ = ρ + , then lim t→0 + ρ = +∞.

g = t 4/3 (dx 1 2 + dx 2 2 + e ∓2/t dx 3
Theorem 20.7. Suppose that σ = σ(t) and ρ = ρ(t), such that the biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t , magnetic constant μ, and electromagnetic field (20.4) having zero mixed part, and such that the Hubble condition (20.9) holds. If σ or ρ is a constant function and the other is not, then the electromagnetic field F = F (t) is nontrivial, with nontrivial energy-momentum tensor contribution T em = T em (F, μ).

Proof. If σ or ρ is a constant function and the other is not, then by the Hubble condition σ t (t 0 ) = ρ t (t 0 ) = 0. The previous lemma then shows that the electromagnetic field F cannot be trivial. And by footnote 39, the energy-momentum tensor contribution T em (F, μ) cannot be trivial, as well.

Theorem 20.8. Suppose that σ = σ(t) and ρ = ρ(t), such that the biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t , and such that the Hubble condition (20.9) holds. If σ or ρ is a constant function and the other is not, then the electromagnetic field is nontrivial.

Proof. If the mixed part F mixed of the electromagnetic field F is nontrivial, then we are done. On the other hand, if F mixed is trivial, then the previous theorem shows that F is nontrivial.

Example 20.9. If σ or ρ is a constant function and the other is not, the nontriviality of F does not imply the Hubble condition. For instance, in the Einstein-de Sitter example, if σ = f = t 2/3 and ρ = 1, then (20.10) holds with µ ∈ (0, +∞),

F 12 = 0, F 34 = ± 1 6 √ π μ-1 t -1
, yet the Hubble condition fails since σ t does not vanish anywhere.

The undeformed FRW cosmological model has trivial electromagnetic field F 0 by definition. By Corollary 20.3, a spacetime model that is a conformal deformation of the original model has trivial electromagnetic field F cd , as well (as long as we require F cd to have zero mixed part). However, by Theorem 20.8, a spacetime model that is a biconformal deformation of the original model, if σ or ρ is a constant function and the other is not, has nontrivial electromagnetic field F bd . Moreover, if F bd has zero mixed part, then it has nontrivial energy-momentum tensor contribution T em by Theorem 20.7. Theorems 20.7 and 20.8 thus highlight the relevance of the biconformal deformation approach of this work: the emergence of nontrivial spacetime fields on spacetime models that are obtained by deforming classical Einstein solutions. (See also e.g. Examples 19.7 and 19.14.) 20.4. Endpoint singularities. Finally, we talk about endpoint singularities in the context of a biconformally deformed FRW cosmological model. Let

t = inf K ∈ R ∪ {-∞}, t = sup K ∈ R ∪ {+∞}. If t ∈ R (t ∈
R, respectively), then we say that there is an initial singularity at t (final singularity at t , respectively).

The last result of this chapter is a generalization of Theorem 5.1 of Appendix C:

Theorem 20.10. Suppose that σ = σ(t) and ρ = ρ(t), such that the biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t and electromagnetic field (20.4) having zero mixed part, and such that the Hubble condition (20.9) holds. Suppose that the energy density ρ, pressure p, and vacuum energy density ρvacuum satisfy (ρ + 3p -2ρ vacuum ) + 3 4π ρ 2 (ln σ) t 2 > 0, and that the Hubble constant H 1 > 0. Suppose that either (a) σ is a constant function and (ln ρ) tt -5(ln f ) t (ln ρ) t ≤ 0, or (b) ρ is a constant function and 5(ln σ) tt + 11(ln f ) t (ln σ) t ≤ 0.

Then the following hold:

(1) t ∈ R with t 0 -H 1 + (ln ρ) t | t=t0 -1 < t < t 0 .

(2) Either f t > 0 on K, or f achieves a maximum after t 0 and t ∈ R.

Proof. From (20.12), 8πρ -2 (ρ + 3p -2ρ vacuum ) + 6(ln σ) t 2 = 5(ln σ) tt + (ln ρ) tt + 6(ln σ) t (ln ρ) t + 11(ln f ) t (ln σ) t -5(ln f ) t (ln ρ) t -6f -1 f tt .

Then the condition on ρ, p, ρvacuum , along with either (a) or (b), implies that f tt < 0 on K. Meanwhile, if σ or ρ is a constant function, then H 1 > 0 implies f t (t 0 ) > 0. Hence, t must be finite, and in particular t 0 -H 0 -1 < t < t 0 , as in Theorem 5.1 of Appendix C. But H 0 = H 1 + (ln ρ) t | t=t0 .

APPENDIX A

Metric Lie derivatives

Note: As in Chapters 2 and 3, here (M, g) is an oriented 4-dimensional Riemannian or Lorentzian (specifically, spacetime) manifold, (N, h) is an oriented Riemannian surface, and ϕ : (M, g) → (N, h) is a smooth semi-conformal submersion with smooth dilation λ : M → (0, +∞). (If (M, g) is a spacetime manifold, then we assume that there is a global timelike vector field on (M, g) that is vertical.) The (local) frame fields f , e, θ and the connection 1-forms ρ ij , ω ab (i ∈ {1, 2}, p ∈ {3, 4}) are as in Chapters 2 and 3, as well. The convention on indices stated in Section 6 is followed.

Metric Lie derivatives

In this appendix, we compute the Lie derivative of the metric g in the direction of an arbitrary vector field on M . Although our formula (see Theorem 1.6) is not written invariantly -there are terms involving ϕ * ρ 12 , ω 34 , θ a -, it is nevertheless useful in deriving formulas (written invariantly) for how the horizontal and vertical parts of Ricci curvature change under conformal deformation of the metric (Corollaries 15.13 and 15.15, respectively). The proof of Theorem 1.6 is a string of lemmas (Lemmas 1.1 to 1.4) and a corollary (Corollary 1.5). Lemma 1.3. For a vector field X = a X a e a on M , L X θ i = -X(ln λ) θ i + X i d ln λ + (-1) i ϕ * ρ 12 (X) θ i + (-1) i X i ϕ * ρ 12 + dX i , ε p L X θ p = 2ζ(e p )X JH + (B ep X) -(B * ep X) + (-1) p ω 34 (X) θ p + (-1) p X p ω 34 + ε p dX p .

Proof. We have ε a L X θ a = ε a X(θ a •) -ε a θ a [X, •] = 2ε a dθ a (X, •) + ε a dX a , and the result follows upon using Lemma 1.2. For the first three terms on the right side of the second equation, we additionally use Lemma 6.3.

Lemma 1.4. For a vector field X and a 1-form α on M , L X α 2 = 2 α L X α.

Proof. For vector fields Y and Z on M ,

L X α 2 (Y, Z) = X(α(Y )α(Z)) -α(L X Y )α(Z) -α(Y )α(L X Z) = α(Y )(X(α(Z)) -α(L X Z)) + α(Z)(X(α(Y )) -α(L X Y )) = α(Y )(L X α)(Z) + α(Z)(L X α)(Y ) = 2 α L X α (Y, Z).
Corollary 1.5. For a vector field X = a X a e a on M , That is, α 12 = (d ln τ )J (where here J is defined on S similarly as in Definition 6.2); if (S, g S ) = (N, h) and f i = a i , then (2.1) ϕ * ρ 12 = (ϕ * d ln τ )J.

Note that

i ε S i ∇ ∂i ∂ i = 0, i ε S i ∇ ai a i = grad g S ln τ = τ 2 i ε S i (ln τ ) i ∂ i .
These help us calculate the Laplacian of a smooth map f : S → R:

∆ g Sf = τ 2 i ε S i f ii .
Now, we calculate the sectional curvature K S of S (on S ). Since

∇ ∂1 ∇ ∂2 ∂ 2 = ∇ ∂1 (Γ 1 22 ∂ 1 + Γ 2 22 ∂ 2 ) = Γ 1 22,1 ∂ 1 + Γ 1 22 Γ 1 11 ∂ 1 + Γ 2 22 Γ 1 12 ∂ 1 + [∂ 2 terms], then g S (∇ ∂1 ∇ ∂2 ∂ 2 , ∂ 1 ) = τ -2 (ε S (ln τ ) 11 -ε S (ln τ ) 1 2 + (ln τ ) 2 2 ).
Similarly,

-g S (∇ ∂2 ∇ ∂1 ∂ 2 , ∂ 1 ) = -τ -2 (-(ln τ ) 22 + (ln τ ) 2 2 -ε S (ln τ ) 1 2 ).
Hence,

K S = τ 4 (g S (∇ ∂1 ∇ ∂2 ∂ 2 , ∂ 1 ) -g S (∇ ∂2 ∇ ∂1 ∂ 2 , ∂ 1 )) = ε S ∆ g S ln τ. (2.2)
For the rest of this appendix, (S, g S ) = (N, h) and f i = a i . Recall that the isothermal coordinate system ψ = (x 1 , x 2 ) is defined on the open subset N of N . This finishes the proof of the lemma.

Ricci curvature

Recall that ϕ : (M 4 , g) → (N 2 , h) is a smooth semi-conformal submersion with smooth dilation λ : M → (0, +∞). Case II: N a general Riemannian surface. Let the isothermal coordinate system ψ = (x 1 , x 2 ) : N → ψ(N ) (with ϕ -1 (N ) = ∅) be as in the previous section.

Consider the map φ := ψ • ϕ : ϕ -1 (N ) → R 2 . Note that, on ϕ -1 (N ), the horizontal spaces with respect to ϕ and φ are the same (similarly for the vertical spaces). One implication of this is that φ is a semi-conformal submersion with dilation λτ . Another is that, on ϕ -1 (N ), the tensor fields B, B * , K ϕ , ν, µ, ζ, B 1 , C, C * associated to φ are the same as those that are associated to ϕ. 59 Finally, using Lemma 2.1 in this equation yields (3.1a). 59 The mean curvature ν is invariant, since ν = V(grad ln(λτ )) by (5.5) and grad(ln τ • ϕ) is horizontal. (This can also be seen by the definition ν = 1 2 trace A and the fact that the second fundamental form A is invariant.) The tensor field ν ⊥ , however, is not invariant, since ν ⊥ = H(grad ln(λτ )) by definition and grad(ln τ • ϕ) is nonzero in general. Theorem 4.3. (c.f. [17] (5.11)-(5.12) or [25] (27.39a)- (27.39b), and [29] Theorem 12.11 for the case Λ = 0) The Einstein equation G = 8πT is equivalent to the pair

8π(ρ + ρvacuum ) = 3f -2 f t 2 + 3kf -2 , 8π(p -ρvacuum ) = -f -2 f t 2 -2f -1 f tt -kf -2 ,
or equivalently to the pair Another consequence is, if ρ, p, ρvacuum > 0, then (by (4.1b)) ρ and p decelerate the expansion of space while ρvacuum , and hence Λ, accelerates such expansion 60 .

The Hubble condition and endpoint singularities

The Hubble condition is the condition that the present rate of change of the distance between any two points in space is proportional to the present distance between those two points. 61 Fix a present time t 0 ∈ I. Then the Hubble condition in the context of a FRW spacetime M (k, I, f ) is the existence of a Hubble constant H 0 ∈ R such that d dt (f ∆s) t=t0 = H 0 f (t 0 )∆s (where ∆s represents distance in P ), i.e.

(5.1)

H 0 = f (t 0 ) -1 f t (t 0 ).
Hence, a FRW spacetime satisfies the Hubble condition so long as we set the Hubble constant H 0 according to (5.1) (which we do). 62 63 The inverse H 0 -1 of the Hubble constant is called the Hubble time. Let t = inf I ∈ R ∪ {-∞}, t = sup I ∈ R ∪ {+∞}. If t ∈ R (t ∈ R, respectively), then we say that there is an initial singularity at t (final singularity at t , respectively). Theorem 5.1. Suppose that ρ + 3p -2ρ vacuum > 0 and H 0 > 0 in a FRW cosmological model. Then the following hold:

(1) t ∈ R with t 0 -H 0 -1 < t < t 0 .

(2) Either f t > 0 on I, or f achieves a maximum after t 0 and t ∈ R. Example 5.3. The Einstein-de Sitter cosmological model is the triple M (0, (0, +∞), t 2/3 ), (∂ t , ρ0 , p0 ), Λ 0 , 60 Indeed, the cosmological constant Λ is one of the candidates for dark energy. 61 Compare with the Hubble law [19].

62 Alternatively, we define the Hubble parameter H : I → R as H = (ln f )t = f -1 ft, and set H 0 = H(t 0 ). 63 The fact that a FRW spacetime satisfies the Hubble condition (for an appropriately chosen Hubble constant) is seen as evidence that the universe can be modeled by such a spacetime. where 8π ρ0 = 4 3 t -2 , 8π p0 = 0, Λ 0 = 0 (see [13]). There is a big bang singularity at t = 0 since lim t→0 + f = 0, lim t→0 + f t = +∞.

APPENDIX D

Some other basic semi-conformal submersions (See the note at the beginning of Appendix A, but replace unsub(super)scripted quantities associated to the metric g with sub(super)scripted quantities associated to the metric g 0 .)

6. Complex multiplication from R 4 \ {0} onto R 2 Let (M, g 0 ) be Euclidean 4-space R 4 with the origin removed and let (N, h) be R 2 . Let ϕ : (M, g 0 ) → (N, h) be multiplication of complex numbers (not both zero): ϕ : x = (z 1 ; z 2 ) = (x 1 , x 2 ; x 3 , x 4 ) → (Re z 1 z 2 ; Im z 1 z 2 ) = (x 1 x 3 -x 2 x 4 ; x 1 x 4 +x 2 x 3 ).

In rectangular coordinates, the metrics g 0 and h are given by

g 0 = dx 1 2 + dx 2 2 + dx 3 2 + ε dx 4 2 , h = dx 1 2 + dx 2 2 ,
respectively. Note that the Jacobian

L(x)     x 3 -x 4 x 1 -x 2 x 4 x 3 x 2 x 1 x 1 x 2 -x 3 -x 4 -x 2 x 1 x 4 -x 3         ∂ 1 ∂ 2 ∂ 3 ∂ 4     , f i = ∂ i .
It is easily checked that dϕ(e 0 i ) = x f i . Hence, ϕ is a semi-conformal submersion with dilation λ 0 = x . Let Λ 0 = 0.

T

  = T fluid (u, ρ, p) + T em (F, μ) + T vacuum (Λ), où u = le champ de flot, ρ = la densité d'énergie, p = la pression, F = le champ électromagnétique, μ = la constante magnétique, Λ = la constante cosmologique. Dans le contexte riemannien, nous appelons l'équation Ric = Λg, où Λ ∈ R, l'équation d'Einstein avec constante de proportionnalité Λ. Une variété riemannienne (M, g) qui vérifie cette équation pour un certain Λ s'appelle une variété d'Einstein; g s'appelle une métrique d'Einstein. C'est à dire une variété d'Einstein est une variété riemannienne qui vérifie l'équation G = 8πT avec T = T vacuum (Λ).

1. The Einstein equation 1 . 1 . 1 ( 1

 1111 Lorentzian context. At the heart of general relativity is the Einstein equation

1 ,

 1 and f : I → (0, +∞) is the smooth expansion. For a FRW cosmological model, u = ∂ t and F = 0. (See Appendix C.) 1.2. Riemannian context. In the Riemannian context, we call the equation Ric = Λg, where Λ ∈ R, the Einstein equation with constant of proportionality Λ. That is, a Riemannian solution to the Einstein equation -a Riemannian Einstein manifold -is a Riemannian manifold that formally satisfies (1.1) with T = T vacuum (Λ) (see (1.3)).

Theorem 4 . 1 (

 41 Theorem 13.3). The Ricci curvature Ric of the metric g is given by[START_REF] Choquet-Bruhat | General relativity and the Einstein equations[END_REF]

4. 3 .

 3 Solutions to the Einstein equation. 4.3.1. Riemannian context. Let the metrics g 0 , g be as in the previous stage and suppose that both are Riemannian.

Example 4 . 11 ( 2 1

 4112 Example 19.14). The warped metric g = t(dx 2 + dy 2 ) + dz 2 -dt 2 on the product of R 2 and upper half-R satisfies the Einstein equation with F, u, ρ, p, Λ given/related by

29

 29

3 . 13 B

 313 the second fundamental form A of the horizontal distribution and its adjoint A * , given by A X Y = V∇ HX HY, (5.2a) A * X Y = -H∇ HX VY ; (5.2b) 4. the second fundamental form B of the vertical distribution and its adjoint B * , given by X Y = H∇ VX VY, (5.3a) B * X Y = -V∇ VX HY ; (5.3b) 5. the intrinsic sectional curvature K ϕ of the fibers of ϕ; 6. the mean curvature ν of the horizontal distribution, given by ν = 1 2 trace A; 7. the mean curvature µ of the fibers of ϕ, given by µ = 1 2 trace B; 8. the integrability 1-form ζ of the horizontal distribution, given (locally) by (5.4) ζ = (A e1 e 2 ) ,

= 4 b=1ε

 4 e b ), a, b ∈ {1, 2, 3, 4}, respectively. The connection matrices ρ = [ρ ij ] and ω = [ω ab ] are skew-symmetric and ∇f = ρ ⊗ f , ∇e a b ω ab ⊗ e b

Corollary 7 . 3 (Corollary 7 . 5 .

 7375 Cf. (5.5), cf.[1] Corollary 2.2). g(∇ ei e j +∇ ej e i , e p ) = 2δ ij e p (ln λ).Proof. By Lemma 7.2, 2δ ij e p (ln λ) = g([e p , e i ], e j ) + g([e p , e j ], e i ) = g(∇ ep e i , e j ) -g(∇ ei e p , e j ) + g(∇ ep e j , e i ) -g(∇ ej e p , e i ) = 0 + g(e p , ∇ ei e j ) + g(e p , ∇ ej e i ). Corollary 7.4 (Cf. (5.6)). g(∇ ei e i , e p ) = 1 2 g([e i , e i ], e p ). The Lie brackets of the frame field elements, are given by H[e 1 , e 2 ] = Jν ⊥ -(ϕ * ρ 12 ) , V[e 1 , e 2 ] = 2ζ , H[e i , e p ] = -e p (ln λ) e i , V[e i , e p ] = B * ep e i + ε ω 34 (e i ) Je p , [e 3 , e 4 ] = -Vω 34 . Proof. Using Lemma 7.1, we have H[e 1 , e 2 ] = -(e 2 (ln λ) + ϕ * ρ 12 (e 1 )) e 1 + (e 1 (ln λ) + ϕ * ρ 21 (e 2 )) e 2 , from which the first equation follows. Equation (5.6) gives the second equation while the third equation comes from Lemma 7.2. The fourth equation is just by definitions. Now, g([e 3 , e 4 ], e i ) = 0 = -Vω 34 (e i ), g([e 3 , e 4 ], e 3 ) = g(∇ e3 e 4 , e 3 ) = -Vω 34 (e 3 ), g([e 3 , e 4 ], e 4 ) = -g(∇ e4 e 3 , e 4 ) = -Vω 34 (e 4 ). Hence, [e 3 , e 4 ] = -Vω 34 , which gives us the last equation. Finally, we list the Christoffel symbols of the metric g with respect to the frame field e, which collectively give the Levi-Civita connection of g. Corollary 7.6. The Christoffel symbols ε b g(∇ ec e a , e b ) are given by (1) g(∇ ei e j , e p ) = e p (ln λ),

  ∇α(e a , e b ) = e a (α(e b )) -α(∇ ea e b ). Then 2 dα(e a , e b ) = ∇α(e a , e b ) -∇α(e b , e a ) = e a (α(e b )) -e b (α(e a )) -α[e a , e b ]. This tells us that the sums e 1 (α(e 2 )) -e 2 (α(e 1 )), e 3 (α(e 4 )) -e 4 (α(e 3 )) are independent of the signed frame fields of the distributions. Moreover, (8.1) div α = trace ∇α = a ε a e a (α(e a )) -α a ε a ∇ ea e a .

(e 2 )

 2 e 1 + ϕ * ρ 12 (e 1 ) e 2 + grad ln λ + ν and p ε p ∇ ep e p = V p ε p ∇ ep e p + H p ε p ∇ ep e p = -ε ω 34 (e 4 ) e 3 + ε ω 34 (e 3 ) e 4 + 2µ, from which the result follows.

10 .

 10 The second fundamental forms Corollary 7.6 has the following consequence: Lemma 10.1. The second fundamental form A and its adjoint A * are given by

  Using item (1) of Corollary 7.6, A ei e j = p ε p g(∇ ei e j , e p ) e p = p ε p e p (ln λ), if i = j, (-1) j ζ(e p ), if i = j e p and A * ei e p = -g(∇ ei e p , e i ) e i -g(∇ ei e p , e i ) e i = e p (ln λ) e i + (-1) i ζ(e p ) e i . The two equations now follow from tensoriality.

  3 and Lemma 10.3. (2) This follows from Lemma 10.5. Lemma 10.8. trace C = trace C

Corollary 10 . 9 .

 109 Suppose that trace C = trace C * = 0. If either (1) g is Riemannian, or (2) g is Lorentzian and B e3 e 4 = 0, then B = B * = 0 and the fibers of ϕ are totally geodesic.

20

  See Remark 12.1.

  ej ∇ ei e j , e p ) θ i θ p =i,p j e j (g(∇ ei e j , e p )) θ i θ p + i,p j g(∇ ei e j , ∇ ej e p ) θ i θ p =i,p j e j (g(A ei e j , e p )) θ i θ pi,p j g(H∇ ei e j , A * ej e p ) θ i θ p + i,p j g(A ei e j , V∇ ej e p ) θ i θ p =i,p j e j (δ ij e p (ln λ) + (-1)

Lemma 14 . 1 .

 141 For a (0, 2) tensor field D, trace D = σ 2 trace 0 D H +ρ 2 trace 0 D V . Of course, trace g = trace 0 g 0 = 4 -[common index of g and g 0 ].

2Hω 34 ( 4 ,

 344 e i ) = 2g(∇ ei e 3 , e 4 ) = g(e 3 , [e 4 , e i ]) + g(e 4 , [e i , e 3 ]) = σ g 0 (e 0 3 , [e 0 4 , e 0 i ]) + σ g 0 (e 0 i ) and 2Vω 34 (e p ) = 2g(∇ ep e 3 , e 4 ) = -g(e p , [e 3 , e 4 ]) + g(e 3 , [e 4 , e p ]) + g(e 4 , [e p , e 3 ]) = -ρ g 0 (e 0 p , p ) + 2(Vd ln ρ)Je p . The two equations now follow by tensoriality.

0 34 0

 34 Hα -2σ 2 Hα(grad 0 ln ρ) and div Vα = p ε p e p (α(e p )) -ε αJ(Vω 34 ) -2α(ν) = ρ 2 p ε p e 0 p (α(e 0 p )) + ρ 2 p ε p e 0 p (ln ρ)α(e 0 p ) -ερ 2 αJ(Vω 0 ) + ερ 2 αJ(JV(grad 0 ln ρ))

Proof. 34 0= -ρ 2 p 2 p= -ρ 2 p ε p e 0 p (ω 0 34 = -ρ 2 p ε p e 0 p (ω 0 34

 3422234234 First, using (8.1) and Lemma 8.1, div 0 Vd ln ρ = p ε p e 0 p (e 0 p (ln ρ)) -d ln ρ (εJVω 0 + 2ν 0 ) = p ε p e 0 p (e 0 p (ln ρ)) +ε ω 0 34 (JV grad 0 ln ρ) -2ν 0 (ln ρ). Then p ε p e p (ω 34 Je p ) =p ε p e p (ω 0 34 Je p ) + ε p ε p e p (e p (ln ρ)) ε p e 0 p (ω 0 34 Je 0 p ) + ερ 2 p ε p e 0 p (e 0 p (ln ρ)) previous calculation -ρ ε p e 0 p (ln ρ) ω 0 34 Je 0 p + ερ 2 pε p e 0 p (ln ρ) e 0 p (ln ρ) Je 0 p ) + ερ 2 div 0 Vd ln ρ -ρ 2 ω 0 34 (JV grad 0 ln ρ) combine + 2ερ 2 ν 0 (ln ρ) -ρ 2 ω 0 34 (JV grad 0 ln ρ) combine +ερ 2 g V 0 (grad 0 ln ρ, grad 0 ln ρ) Je 0 p ) + ερ 2 div 0 Vd ln ρ + 2ερ 2 ν 0 (ln ρ) -2ρ 2 ω 0 34 (JV grad 0 ln ρ) + ερ 2 g V 0 (grad 0 ln ρ, grad 0 ln ρ) cancelled out later .

2

  Vd ln σ Hd ln ρ + ( ( ( ( ( ( ( ( ( ( 2σ 2 ρ -2 ζ 0 (Hd ln ρ)J.

2

  sym d ln σ(B * 0 ) -( ( ( ( ( ( ( 2ν 0 0 Hd ln ρ -( ( ( ( ( ( ( ( ( Vd ln σ Hd ln ρ,

  [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ρ -2 (µ 0 + H grad 0 ln ρ)(ln λ 0 + ln σ) -2µ 0 (ln λ 0 ) = 2(σ 2 ρ -2 -1)µ 0 (ln λ 0 ) + 2σ 2 ρ -2 µ 0 (ln σ) + ( ( ( ( ( ( ( 2σ 2 ρ -2 ν ⊥ 0 (ln ρ)

16. 1 .

 1 Lorentzian context. At the heart of general relativity is the Einstein equation[START_REF] Misner | Gravitation[END_REF]

25

  Compare with[17] (3.15) or[25] (17.11), and of course with Einstein himself[12] (2a). Note that some authors omit the factor 8π (e.g.[8] Subsection III.5.2 and[36] Definition 4.1.1). 69 where trace * F (•, * ) ⊗ F ( * , •) = a g(e a , e a ) F (•, e a ) ⊗ F (e a , •), F 2 = a,b g(e a , e a )g(e b , e b )F (e a , e b ) 2 .

(

  I, -dt 2 ) is an interval of R 1 1 , and f : I → (0, +∞) is the smooth expansion. A FRW cosmological model is a spacetime model M (k, I, f ), (∂ t , ρ, p), (0, μ), Λ, which we simply denote as a triple M (k, I, f ), (∂ t , ρ, p), Λ. (See Appendix C.) 16.2. Riemannian context. In the Riemannian context, we call the equation Ric = Λg, where Λ ∈ R, the Einstein equation with constant of proportionality 34 Λ. If a

4 ) 2 -2ρ 4 F

 424 (e 0 3 , e 0 4 ) 2 and, denoting trace * F (•, * ) ⊗ F ( * , •) simply by trace * , trace * (e 0 i , e 0

) 0 = 8πT 11 -

 11 8πT 22 , (17.9b) 0 = 8πT ab , a ∈ {1, 2, 3, 4}, a < b. (17.9c)

A. a 1

 1 = 0. This gives us the Euclidean 4-metric. B. a 1 = 0.

Figure 18 . 1 .

 181 Figure 18.1. Plots of the c 2 = 0 representatives of the six families of positive-valued solutions (18.20) (with maximal x i -domains) of the differential equation (18.14) (with d = ±1, c = ln 3 2 ), as classified in Table 18.1. In each plot the grid is [-2, 2] × [0, 4]. In the appropriate plots lim xi→±π/6 √ 3 ρ = 0, lim xi→±π/2 √ 3 ρ = +∞,

19 .

 19 Solutions from Minkowski spacetime 19.1. Generalities. Going back to the general discussion in Subsection 18.1, let the functions T ab : M → R be given by

) 0 = 8πT 11 -

 11 8πT 22 , (19.5b) 0 = 8πT ab , a ∈ {1, 2, 3, 4}, a < b. (19.5c)

  ln σ) 34 + (ln σ) 3 (ln ρ) 4 + (ln σ) 4 (ln ρ) 3 -(ln σ) 3 (ln σ) 4 . (19.18c) (2) Equation (19.18c) is an analogue of (19.10c). In particular, the metric (19.4) with σ = σ(x 3 ) and ρ = ρ(x 4 ) satisfies the Einstein equation with F = 0 if and only if (at least) one of σ or ρ is a constant function and the ln of the other one is harmonic. (3) The right-hand side of (19.18a) is a bit more complicated than that of (19.10a). Equation (19.3) does not necessarily hold, but if it does (19.18a) becomes 8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = -ρ 2 q

Example 19 . 14 .

 1914 Suppose that σ = σ(x p ) and ρ is a constant function. Then F 12 = F 34 = 0 in (19.18a) if and only if 0 = (ln σ) 33 , if p = 3, 0 = -(ln σ) 44 + 2(ln σ) 4 2 , if p = 4; let us concentrate on the case p = 4. If σ is not a constant function, then σ = e c1 |x 4 -c 2 | -1/2

  can give rise to a FRW cosmological model M (0, I, f ), (u, ρ, p), Λ with flat space. (See Appendix C.) Example 19.15. Suppose that σ = σ(x 4 ), ρ = ρ(x 4 ), and, indeed, σ = ρ. Let J ⊆ R be the x 4 -domain. By Theorem 19.2, the conformally deformed metric g = σ(x 4 ) -2 (dx 1 2 + dx 2 2 + dx 3 2 -dx 4 2 ) satisfies the Einstein equation with flow vector field u = σ∂ 4 and electromagnetic field F having zero mixed part if and only if F = 0. (A FRW cosmological model has trivial electromagnetic field.) In this case, ρ, p, Λ are related by 8π ρ + 8π p = 2σ 2 ((ln σ) 44 + (ln σ) 4 2 ), 8π p -Λ = σ 2 (2(ln σ) 44 -(ln σ) 4 2 ).

(

  19.19) t = t(x 4 ), x 4 ∈ J,

( 2 j(

 2 19.22) σ = h(x 1 , x 2 ) exp x3 b f (w) dw for some positive-valued function h and real number b in the domain of x 3 . In this case, the right-hand side of (19.21a) turns into exp 2 x3 bf (w) dw h 2 j (ln h) jj -ρ 2 f 3 + ρ 2 (ln ρ) 44 .Suppose that ρ is a constant function and that(19.22) holds. Then in (19.21a)F 12 = F 34 = 0 if and only if (19.23) h ln h) jj = ρ 2 f 3 exp -2 x3 b f (w) dw = αfor some α ∈ R. The following are equivalent in (19.23):

  holds. If α = 0, then by (19.31) ρ = h 0 e f (b)(x3-b) with h 0 the common value of h (see item 03. of Theorem 19.21 for the case f (b) = 0). Suppose that α ∈ (0, +∞). From (19.32), (19.33) σ -2 exp 2 x3 b f (w) dw = αf 3 -1 .55 See equation(19.22) and footnote 53.

) 0 = 8πT 11 -

 11 8πT 22 , (20.6b) 0 = 8πT ab , a ∈ {1, 2, 3, 4}, a < b. (20.6c)In this case, the energy density ρ, pressure p, and cosmological constant Λ are related by 8π ρ + 8π p = ρ 2 (8πT 33 + 8πT 44 ), (20.7a)8π p -Λ = ρ 2 8πT 33 + 4π μ-1 (σ 4 F 12 2 + ρ 4 F34[START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ). (20.7b) 20.2. Hubble conditions. Before we go any further, we make sense of the Hubble condition for the biconformally deformed spacetime (M, g).

Fix a present time

  t 0 ∈ I. The Hubble constant H 0 ∈ R of the spacetime (M, g 0 ) is such that the Hubble condition(20.8)

Lemma 1. 1 . 4 = e 2 ( 4 = (e 2 (

 14242 ω 12 = ν ⊥ J -ζ + ϕ * ρ 12 , ω 1p = e p (ln λ) θ 1 -ζ(e p ) θ 2 -(B * ep e 1 ) , ω 2p = ζ(e p ) θ 1 + e p (ln λ) θ 2 -(B * ep e 2 ) .Proof. This is a direct consequence of the equationω ab = c g(∇ ec e a , e b ) θ c ,and of Corollary 7.6. For example,ω 12 = (e 2 (ln λ) + ϕ * ρ 12 (e 1 )) θ 1 + (-e 1 (ln λ) + ϕ * ρ 12 (e 2 )) θ 2 -ζ(e 3 ) θ 3 -ζ(e 4 ) θ ln λ) θ 1 -e 1 (ln λ) θ 2 -ζ + ϕ * ρ 12 .For the next lemma, we state one of the Cartan structure equations:dθ a = ε a c ω ac ∧ θ c (1.1) (see, for example,[38] Proposition 7.14 for the Riemannian case).Lemma 1.2.dθ i = -d ln λ ∧ θ i + (-1) i ϕ * ρ 12 ∧ θ i , ε p dθ p = -ζ(e p ) ξ H + j (B * ep e j ) ∧ θ j + (-1) p ω 34 ∧ θ p = -ζ(e p ) ξ Hq (B ep e q ) ∧ θ q + (-1) p ω 34 ∧ θ p .115Proof. Using the structure equation (1.1) and Lemma 1.1, we obtaindθ 1 = ω 12 ∧ θ 2 + ω 13 ∧ θ 3 + ω 14 ∧ θ ln λ) θ 1 -e 1 (ln λ) θ 2 -ζ + ϕ * ρ 12 ) ∧ θ 2 + (e 3 (ln λ) θ 1 -ζ(e 3 ) θ 2 -(B * e3 e 1 ) ) ∧ θ 3 + (e 4 (ln λ) θ 1 -ζ(e 4 ) θ 2 -(B * e4 e 1 ) ) ∧ θ 4 = -d ln λ ∧ θ 1 + ϕ * ρ 12 ∧ θ 2 , dθ 2 = ω 21 ∧ θ 1 + ω 23 ∧ θ 3 + ω 24 ∧ θ 4 = (-e 2 (ln λ) θ 1 + e 1 (ln λ) θ 2 + ζ -ϕ * ρ 12 ) ∧ θ 1 + (ζ(e 3 ) θ 1 + e 3 (ln λ) θ 2 -(B * e3 e 2 ) ) ∧ θ 3 + (ζ(e 4 ) θ 1 + e 4 (ln λ) θ 2 -(B * e4 e 2 ) ) ∧ θ 4 = -ϕ * ρ 12 ∧ θ 1 -d ln λ ∧ θ 2 , dθ 3 = ω 31 ∧ θ 1 + ω 32 ∧ θ 2 + ω 34 ∧ θ 4 = (-e 3 (ln λ) θ 1 + ζ(e 3 ) θ 2 + (B * e3 e 1 ) ) ∧ θ 1 + (-ζ(e 3 ) θ 1 -e 3 (ln λ) θ 2 + (B * e3 e 2 ) ) ∧ θ 2 + ω 34 ∧ θ 4 = -2ζ(e 3 ) θ 1 ∧ θ 2 + (B * e3 e 1 ) ∧ θ 1 + (B * e3 e 2 ) ∧ θ 2 + ω 34 ∧ θ 4 ,andε dθ 4 = ω 41 ∧ θ 1 + ω 42 ∧ θ 2 + ω 43 ∧ θ 3 = (-e 4 (ln λ) θ 1 + ζ(e 4 ) θ 2 + (B * e4e1 ) ) ∧ θ 1 + (-ζ(e 4 ) θ 1 -e 4 (ln λ) θ 2 + (B * e4 e 2 ) ) ∧ θ 2 -ω 34 ∧ θ 3 = -2ζ(e 4 ) θ 1 ∧ θ 2 + (B * e4 e 1 ) ∧ θ 1 + (B * e4 e 2 ) ∧ θ 2 -ω 34 ∧ θ 3 .

1 2 L 2 L

 22 X g H = -X(ln λ) g H + HX d ln λ -X JH ϕ * ρ 12 + i dX i θ i , 1 X g V = 2 X JH ζ + sym X (B * ) -X (B) -ε X JV ω 34 + p ε p dX p θ p .

Lemma 2 . 1 . 2 i(ln τ ) i 2 + λ 2 τ 2 i(

 21222 If ϕ -1 (N ) = ∅, then on that neighborhoodλ 2 K N = ∆ g (ln τ • ϕ) + 2µ(ln τ • ϕ). Proof. Since the 1-form d(ln τ • ϕ) is horizontal, then by Corollary 8.2 ∆ g (ln τ •ϕ) = i e i (d(ln τ •ϕ)(e i ))+ϕ * ρ 12 (J grad g (ln τ •ϕ))-(grad g ln λ)(ln τ •ϕ)-2µ(ln τ •ϕ).Using the fact that dϕ(e i ) = λf i andf i = τ ∂ i , we calculate i e i (d(ln τ • ϕ)(e i )) = λτ i e i (ln λ) (ln τ ) i + λ 2 τ ln τ ) ii ,where the last addend is just λ 2 K N by (2.2). Similarly, we calculate-(grad g ln λ)(ln τ • ϕ) = -λτ i e i (ln λ) (ln τ ) i . By (2.1), ϕ * ρ 12 (J grad g (ln τ • ϕ)) = -(ϕ * d ln τ )(grad g (ln τ • ϕ)). But ϕ * d ln τ = d(ln τ • ϕ) = λτ i (ln τ ) i θ i , so ϕ * ρ 12 (J grad g (ln τ • ϕ)) = -λ 2 τ 2 i (ln τ ) i 2 .

Theorem 3 . 1 . 1 2

 311 The Ricci curvature Ric of g is given byRic H = (λ 2 K N + ∆ ln λ + 2µ(ln λ) -2g(ζ , ζ )) g H + L µ g(H•, H•) -C * , (3.1a) Ric V = εK ϕ g V + L ν g(V•, V•) (3.1b) -2ν 2 + 2ζ 2 + 2 sym ν (B * ) -2 sym ζ(B * • J•) + div 1 B 1 , Ric mixed = 2 sym(∇µ (V•, H•)) + sym(∇ν (H•, V•)) -sym(∇ζ(JH•, V•)) (3.1c) -2 sym ν (B * ) + 2 sym ζ(B * • J•) -sym div 2 B 1 -C + C * .Proof. Case I: N = R 2 . Performing the computation in Chapter 3 with ρ 12 = 0 and K N = 0, we obtain equations (3.1).

  Applying the previous case, we already get (3.1b) and (3.1c), while for the horizontal part we get Ric H = (∆ ln(λτ ) + 2µ(ln(λτ )) -2g(ζ , ζ )) g H + L µ g(H•, H•) -C * .

Table 4 . 1 .H 3 , R 3 , or S 3 B

 41333 Ingredients of a FRW cosmological model Ingredients Conditions I. SET-UP A. Space P . Time interval I open interval in R II. GEOMETRY A. Expansion f smooth positive-valued function on I III. PHYSICS A. Energy density ρ smooth real-valued function on I B. Pressure p smooth real-valued function on I C. Cosmological constant Λ real number, Λ = 8π ρvacuum (C. Vacuum energy density ρvacuum ) (real number, ρvacuum = 1 8π Λ) IV. EINSTEIN EQUATION G = 8πTLet F = 0 be the electromagnetic field, Λ ∈ R be the cosmological constant, and ρvacuum = 1 8π Λ be the vacuum energy density. Then the energy-momentum tensors of electromagnetic field and vacuum areT em = 0, T vacuum = -18π Λg = -ρ vacuum g, respectively. The total energy-momentum tensor isT = T matter + T em + T vacuum = (ρ + p)u 2 + (p -ρvacuum )g.Finally, we relate the geometry and the physics to complete our cosmological model. The triple M (k, I, f ), (u, ρ, p), Λ is called a FRW cosmological model if the Einstein equation G = 8πT holds. In this case, the coordinate t is called cosmological time and the perfect fluid (u, ρ, p) is called a cosmological fluid . The ingredients of a FRW cosmological model are summarized in Table 4.1.

8π 3 (

 3 ρ + ρvacuum ) = f -2 f t 2 + kf -2 , (4.1a) -4π 3 (ρ + 3p -2ρ vacuum ) = f -1 f tt . (4.1b)In particular, the energy density ρ and pressure p are functions of cosmological time only.Note that this system can also be written as8π(ρ + ρvacuum ) = 3(ln f ) t 2 + 3kf -2 , 8π(p -ρvacuum ) = -2(ln f ) tt -3(ln f ) t 2 -kf -2 ,or equivalently as8π 3 (ρ + ρvacuum ) = (ln f ) t 2 + kf -2 , 4π(ρ + p) = -(ln f ) tt + kf -2 .A consequence of Theorem 4.3 is that the FRW cosmological model with P = R 3 (k = 0), I = R, f = 1, and Λ = 0 is just the Minkowski vacuum model.

Definition 5 . 2 .

 52 An initial singularity at t is called a big bang singularity if lim t→t + f = 0 and lim t→t + f t = +∞, and is called a physical singularity if lim t→t + ρ = +∞. Similarly, a final singularity at t is called a big crunch singularity if lim t→t - t = -∞, and is called a physical singularity if lim t→t - ρ = +∞. By (4.1a), a big bang (or big crunch) singularity of a FRW cosmological model is a physical singularity.

x 3 -x 4 x 1 -x 2 x 4 x 3 x 2 x 1 has rank 2

 312 everywhere on M . Let 64

64

  We use three norm notations here: |z| = absolute value of the complex number z,x = |z 1 | 2 + |z 2 | 2 =absolute value of the element x = (z 1 ; z 2 ) of M , X 0 = norm of the vector X on M with respect to the metric g 0 .

  127

  [START_REF] Baird | Harmonic morphisms between Riemannian manifolds[END_REF] 1 , then ϕ : U → C is semi-conformal if and only

	if 6	∂ϕ ∂x 1	2	+	∂x 2 ∂ϕ	2

  and ϕ(x 1 , x 2 , x 3 , x 4 ) = (x 1 , x 2 ) in Section 19. The resulting system of PDEs is as in the following theorem. and electromagnetic field F = F 12 dx 1 ∧ dx 2 + F 34 dx 3 ∧ dx 4 having zero mixed part if and only if the following system of eight PDEs holds:

	Theorem 4.8 (Theorem 19.2). The biconformally deformed metric
		g = σ -2 (dx 1	2 + dx 2	2 ) + ρ -2 (dx 3	2 -dx 4	2 )
	satisfies the Einstein equation (4.1) with flow vector field u = ρ ∂ 4 , magnetic con-
	stant μ, 8π μ-1 (σ 4 F 12	2 + ρ 4 F 34	2 ) = σ 2 1 2 (8πT 11 + 8πT 22 ) -ρ 2 8πT 33 ,
			0 = 8πT 11 -8πT 22 ,
			0 = 8πT ab ,		a ∈ {1, 2, 3, 4}, a < b.
	In this case, the energy density ρ, pressure p, and cosmological constant Λ are
	related by
		8π ρ + 8π p = ρ 2 (8πT 33 + 8πT 44 ),
		8π p -Λ = ρ 2 8πT 33 + 4π μ-1 (σ 4 F 12	2 + ρ 4 F 34	2

  [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ), with the x 3 -domain of σ(x 1 , •) an open interval for each x 1 , satisfies the Einstein equation (4.1) with flow vector field u = ρ ∂ 4 and trivial electromagnetic field if and only if one of the following holds:• g is isometric to 01.ds 2 = (dx 2 + dy 2 ) + (dz 2 -dt 2 ); 02. ds 2 = (y 2 dx 2 + dy 2 ) + (dz 2 -dt 2 ), y > 0; 03. ds 2 = (dx 2 + dy 2 ) + (t 2 dz 2 -dt 2 ), t > 0; 04. ds 2 = (y 2 dx 2 + dy 2 ) + (t 2 dz 2 -dt 2 ),y, t > 0; 11. ds 2 = D z (dx 2 + dy 2 ) + (dz 2 -dt 2 ), where D ∈ (0, +∞) \ {1}; 12. ds 2 = D z (y 2 dx 2 +dy 2 )+(dz 2 -dt 2 ), y > 0, where D ∈ (0, +∞)\{1}; IB. ds 2 = z 2 sech 2 x (dx 2 + dy 2 ) + (dz 2 -dt 2 ), z > 0; • g is homothetic to 05. ds 2 = sech 2 x (dx 2 + dy 2 ) + t -2 (dz 2 -dt 2 ), t > 0; 06. ds 2 = sech 2 x (dx 2 + dy 2 ) + csch 2 t (dz 2 -dt 2 ), t > 0; 07. ds 2 = sech 2 x (dx 2 + dy 2 ) + csc 2 t (dz 2 -dt 2 ), 0 < t < π; 08. ds 2 = x -2 (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), x > 0; 09. ds 2 = csch 2 x (dx 2 + dy 2 ) + sech 2 t (dz 2 -dt 2 ), x > 0; 10.

  [START_REF] Friedmann | Über die möglichkeit einer welt mit konstanter negativer krümmung des raumes[END_REF] See Corollary 7.5 below for the horizontal part of [e 1 , e 2 ].

  part if and only if the system (17.9) holds. In this case, the energy density ρ, pressure p, and cosmological constant Λ are related by the equations (17.10).

	Theorems 17.2 and 17.3 will be used in Sections 19 and 20 (see Theorems 19.1,
	19.2,

3, F 12 can be replaced by -F 12 and F 34 by -F 34 (see (17.9a), and also footnote 38). 42 Here, 8πT 33 + 8πT 44 = 0 by assumption, but we write the general formula for 8π ρ + 8π p (see Theorem 17.3).

mixed

  ) Either dividing it by σ 4 yields a function of x 1 , x 2 only, or dividing it by ρ 4 yields a function of x 3 , x 4 only. If the system (19.5b)-(19.5c) holds, then the metric (19.4) satisfies the Einstein equation (17.3) with u = ρ ∂ 4 and eitherF = F 12 (x 1 , x 2 ) dx 1 ∧ dx 2 or F = F 34 (x 3 , x 4 ) dx 3 ∧ dx 4 . (If (19.3) also holds, then u can be any admissible flow vector field.) In this case, ρ, p, Λ are related by(19.6).Theorem 19.4. Suppose that σ = σ(x 1 , x 2 ) and ρ = ρ(x 3 , x 4 ). The biconformally deformed metric(19.4) satisfies the Einstein equation (17.3) with magnetic constant μ and electromagnetic field (19.2) having zero mixed part if and only if there exists α ∈ R such that

	The rest of this section deals with six cases that simplify the system (19.5)
	(or (19.5) plus (19.3)).
	19.2. σ horizontal, ρ vertical.

  satisfies the Einstein equation (17.3) with electromagnetic field having zero mixed part if and only if either

	(19.8)	sup ρ 2	ε q (ln ρ) qq ≤ inf σ 2	(ln σ) jj
		q	j
	or
	(19.9)	sup σ 2
		j

  12 + (ln σ) 1 (ln ρ) 2 + (ln σ) 2 (ln ρ) 1 -(ln ρ) 1 (ln ρ) 2 .

	constant μ and electromagnetic field (19.2) having zero mixed part if and only if
	the following system of three partial differential equations holds:
	(19.10a)
									
	8π μ-1 (σ 4 F 12	2 + ρ 4 F 34	2 ) = σ 2		(ln σ) jj +	(ln ρ) j	2	 ,
				j	j
	(19.10b)
			0 = (ln ρ) 11 -(ln ρ) 22 -(ln ρ) 1	2 + (ln ρ) 2	2 + 2(ln σ) 1 (ln ρ) 1 -2(ln σ) 2 (ln ρ) 2 ,
	0 = (ln ρ) (19.10c)
						x1,x2∈R	σ 2	j	(ln σ) jj .

  satisfies the Einstein equation with F mixed = 0 if and only if F is trivial. (2) If j (ln σ) jj is either always nonnegative or always nonpositive, and is nonzero somewhere, then the metric (19.4) satisfies the Einstein equation with F mixed = 0 if and only if μ ∈ R\{0} has the same sign as j (ln σ) jj Corollary 19.10. Suppose that σ = σ(x 1 , x 2 ) and ρ is a constant function such that (1) j (ln σ) jj is always nonnegative or always nonpositive, and (2) | j (ln σ) jj | 1/2 is smooth.

		(at points where it is nonzero),
	(19.12a)	F 34	2 ∈ 0, ρ -4 inf	σ 2	j (ln σ) jj 8π μ-1	,
	(19.12b)	F 12	2 = σ -4 σ 2	j (ln σ) jj 8π μ-1	-ρ 4 F 34	2 .
		In this case, F is nontrivial.
	This discussion leads to the next corollary.

  allow us to choose appropriate components F 12 and F 34 . Equation (19.13c) implies that either σ or ρ is a constant function. (We see that F 12 = F 34 = 0 if and only if (ln σ) ii = 0 and ρ is a constant function.) If ρ is not a constant function, then by (19.13b) (19.14) ρ = e c1 |x i -c 2 | -1

	Example 19.11. If σ = σ(x i ) and ρ = ρ(x i ), then the system (19.10) is
	equivalent to the system
	(19.13a)	8π μ-1 (σ 4 F 12	2 + ρ 4 F 34	2 ) = σ 2 ((ln σ) ii + (ln ρ) i	2 ),
	(19.13b)			0 = (ln ρ) i i -(ln ρ) i	2 ,
	(19.13c)			0 = (ln σ) i (ln ρ) i .

  [START_REF] Baird | Biconformal equivalence between 3-dimensional Ricci solitons[END_REF] 2 e -2y . = e c σ -2 ρ 2 for some c ∈ R. Equation (19.15a) then becomes8π μ-1 (e 2c |ρ i | -2 F 12 2 + F 34 2 ) = 1 2 e c ρ -2 |ρ i | -1 (2(ln ρ) ii -(ln |ρ i |) ii + 2(ln ρ) i 2 ).

	Example 19.12. If σ = σ(x i ) and ρ = ρ(x i ), then the system (19.10) becomes 8π μ-1 (σ 4 F 12 2 + ρ 4 F 34 2 ) = σ 2 ((ln σ) ii + (ln ρ) i 2 ), (19.15a) 0 = (ln ρ) ii -(ln ρ) i 2 + 2(ln σ) i (ln ρ) i . (19.15b) If ρ is not a constant function, then (19.15b) implies that (19.16) dx 1 2 + dx 2 2 by a factor of e c/2 |ρ i | -1/2 , then adding dx 3 2 -dx 4 2 (i.e. taking the product metric with R 2 1 ), and finally conformally deforming the sum by a factor of ρ. If σ is a constant function, then (19.16) has solution ρ = e -c σ 2 |x i -d| -1 , where d ∈ R, and (19.15a) becomes |ρ i | Notice in this case that the metric (19.4) is obtained by first conformally deforming 8π μ-1 (e 4c σ -4 (x i -d) 4 F 12 2 + F 34 2

  jj -ρ 2 (ln σ) 33 + ρ 2 (ln ρ) 44 ,If for each (x 1 , x 2 ) the x 3 -domain of σ(x 1 , x 2 , •) is an open interval,then (19.21b) holds if and only if either σ = σ(x 1 , x 2 ) (see Theorems 19.4 and 19.6; see items 01. to 10. of Theorem 19.19) or ρ is a constant function. Equation (19.21c) holds if and only if (ln σ) 3 is a function of x 3 alone, say

	becomes
	8π μ-1 (σ 4 F 12 (ln σ) (19.21a) 2 + ρ 4 F 34 2 ) = σ 2
		j
	(19.21b)	0 = (ln σ) 3 (ln ρ) 4 ,
	(19.21c)	0 = (ln σ) i3 ,	i ∈ {1, 2}.
		(ln σ) 3 = f (x 3 ),
	which means that 53

  Since the right-hand side of (19.26) is just the right-hand side of (19.3), then c = 0 if and only if (19.3) holds. By (19.25) and (19.26),

	(19.27)	± f 3 f 3 -c = 1 2 f 33 .
	I. c = 0 (i.e. (19.3) holds).
	By (19.26),
		f = -(x 3 -d) -1

  1/2 (x 3 -d 1 )] + d 2 for some d 2 ∈ R. Example 19.18. If h = h(x i ) in the above discussion (with α ∈ R \ {0}), then

	from (19.23) the function h satisfies the differential equation
	(19.28)	-h i	2 + hh

ii = α.

  19.7. σ temporal, ρ spatial. If σ = σ(x 4 ) and ρ = ρ(x 1 , x 2 , x

							3 ), then the
	system (19.5) becomes
	(19.30a)
	8π μ-1 (σ 4 F 12	2 + ρ 4 F 34	2 ) = -ρ 2 (ln σ) 44 + 2ρ 2 (ln σ) 4	2 -ρ 2 (ln ρ) 33 + σ 2	(ln ρ) j	2 ,
							j
	(19.30b)
				0 = (ln ρ) 11 -(ln ρ) 22 -(ln ρ) 1	2 + (ln ρ) 2	2 ,
	(19.30c)
				0 = (ln ρ) 12 -(ln ρ) 1 (ln ρ) 2 ,
	(19.30d)
				0 = (ln σ) 4 (ln ρ) a ,			a ∈ {1, 2, 3},
	(19.30e)
				0 = (ln ρ) i3 ,			i ∈ {1, 2}.
	which means that 55
				x3
	(19.31)			ρ = h(x 1 , x 2 ) exp	f (w) dw
				b
	for some positive-valued function h and real number b in the domain of x 3 . In this
	case, the right-hand side of (19.30a) turns into
		x3
	h 2 exp 2	f (w) dw (-(ln σ) 44 + 2(ln σ) 4	2 -f 3 ) + σ 2	(ln h) j	2 ,
		b					j
	and ρ can be replaced by h in (19.30b) and in (19.30c).
	Suppose that σ is a constant function (so (19.3) is satisfied) and that (19.31)
	holds. Then in (19.30a) F 12 = F 34 = 0 if and only if
	(19.32)

Equation (19.30d) holds if and only if either ρ is a constant function (see Example 19.14; see items 01. and 02. of Theorem 19.21) or σ is a constant function. Equation (19.30e) holds if and only if (ln ρ) 3 is a function of x 3 alone, say (ln ρ) 3 = f (x 3 ),

  Motivated by the Hubble condition, we suppose that σ = σ(t) and ρ = ρ(t). (-(ln σ) tt +2(ln σ) t 2 +(ln ρ) tt -2(ln σ) t (ln ρ) t -3(ln f ) t (ln σ) t +3(ln f ) t (ln ρ) t ),

		Then (20.6b) and (20.6c) are automatically
	satisfied, while (20.6a) becomes
	(20.10)
	8π μ-1 (σ 4 F 12 2 ) = ρ 2 and equations (20.7) become 2 +ρ 4 F 34
	(20.11a)
	8π ρ + 8π p = 2ρ 2 ((ln σ) tt -(ln σ) t	2 + 2(ln σ) t (ln ρ) t )

20.3

. σ temporal, ρ temporal.

is n-dimensional Minkowski space (with index 1).

Vacuum-triviality will be defined in Section 16.

Here, M can be replaced by an open submanifold of M , depending on the domains of the deformation parameters σ, ρ.

Note that some authors use a different sign convention for the Riemannian curvature, i.e. the negative of what is used in this work. (See, for example, [29] Lemma 3.35.) This sign convention, however, still leads to the Ricci curvature as written here.

See Remark 13.1.

See footnote 34.

In Section 18, Me = M .

Here, we extend the solution ρ of the differential equation (18.14) at x i = x 0 i and beyond.

Acknowledgements

I am indebted to my thesis supervisor Prof. P. Baird for his invaluable mentorship and for his comments and suggestions on this manuscript.

I am thankful to Drs. R. Gicquaud, J.A. Valiente Kroon, J. Nicolas, and M. Ville for being the examinateurs of this thesis, and to the first two for being the rapporteurs.

I thank Profs. B. Franke and E. Humbert for being part of my Comité de suivi individuel.

I am grateful to the Institute of Mathematics, University of the Philippines Diliman for supporting my PhD education, in particular to Profs. C.P. Arceo, J. Basilla, J.M. Balmaceda, C.R. Canlubo, J.M. Escaner, J.E. Lope, F. Nemenzo, N. Reyes, M. Roque, and A.C. Velasco. I am also grateful to Profs. R.M. Gamot, V. Kobayashi, R. Lemence, J. Nable, L.L. Nalangan, M.A. Mata, and M. Tan.

I thank my mathematical sisters Drs. E. Ghandour and Z.G. Hanna for their support.

I thank the Philippine Commission on Higher Education and Campus France for financing my PhD studies in France.

• 2µ(ln λ) g H -2µ 0 (ln λ 0 ) g H 0 = 2(µ 0 + H grad 0 ln ρ)(ln λ 0 + ln σ) g H 0 -2µ 0 (ln λ 0 ) g H 0 = ( 2µ 0 (ln σ) + 2ν ⊥ 0 (ln ρ) + ( ( ( ( ( ( ( ( ( ( (

2µ 0 (ln σ) g H 0 -2g H 0 (grad 0 ln σ, grad 0 ln ρ) g H 0 + 4µ 0 0 Hd ln σ + 4 Hd ln σ Hd ln ρ,

• -C * + C * 0 = -4µ 0 0 Hd ln ρ -2(Hd ln ρ) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] . Adding these gives the formula in the theorem.

Since

L V grad 0 ln σ g 0 (H•, H•) = -2ν 0 (ln σ) g H 0 by Theorem 1.6 of Appendix A, then we have the following corollary for the change Ric H -Ric H 0 when the deformation is conformal.

Corollary 15.13. Under a conformal deformation g = σ -2 g 0 , the change Ric H -Ric H 0 in the horizontal part of Ricci curvature is given by Ric H -Ric H 0 = -2 g 0 (grad 0 ln σ, grad 0 ln σ) g H 0 + ∆ 0 ln σ g H 0 + L grad 0 ln σ g 0 (H•, H•) + 2(Hd ln σ) 2 .

Theorem 15.14. The change Ric V -Ric V 0 in the vertical part of Ricci curvature under a biconformal deformation g = σ -2 g H 0 + ρ -2 g V 0 is given by Ric V -Ric V 0 = -2σ 2 ρ -2 H grad 0 ln ρ 2 0 g V 0 -2 g V 0 (grad 0 ln σ, grad 0 ln ρ) g V 0 + σ 2 ρ -2 div 0 Hd ln ρ g V 0 + div 0 Vd ln ρ g V 0 + L V grad 0 ln σ g 0 (V•, V•) -2σ 2 ρ -2 d ln ρ(B 0 ) + 2(σ 4 ρ -4 -1)ζ 0 2 -4ν 0 0 Vd ln σ + 4ν 0 0 Vd ln ρ + 4 Vd ln σ Vd ln ρ -2(Vd ln σ) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] + (σ 2 ρ -2 -1) div 0 1 B 0 1 + 2(σ 2 ρ -2 -1) sym ν 0 0 (B * 0 ) -2(σ 2 ρ -2 -1) sym ζ 0 (B * 0 • J•).

Proof. Using Lemma 12.2 as a guide, the addends of Ric V -Ric V 0 are • -C + C 0 = ( ( ( ( ( ( ( (

( ( ( ( ( ( ( 4µ 0 0 Hd ln ρ + 2(Hd ln ρ) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] . Adding these gives the formula in the theorem.

Since the mixed part of [START_REF] Baird | Biconformal equivalence between 3-dimensional Ricci solitons[END_REF] 2 L grad 0 ln σ g 0 is Corollary 15.17. Under a conformal deformation g = σ -2 g 0 , the change Ric mixed -Ric mixed 0 in the mixed part of Ricci curvature is given by

Hd ln σ Vd ln σ. Summarizing the previous three corollaries, we get the change Ric -Ric 0 in Ricci curvature under a conformal deformation of the metric (c.f. [18] Section 6.1 for the Riemannian case).

Corollary 15.18. Under a conformal deformation g = σ -2 g 0 , the change Ric -Ric 0 in Ricci curvature is given by Ric -Ric 0 = -2 g 0 (grad 0 ln σ, grad 0 ln σ) g 0 + ∆ 0 ln σ g 0 + L grad 0 ln σ g 0 + 2(d ln σ) 2 .

Aside from the change Ric -Ric 0 , we will also need formulas for the differences σ -2 S -S 0 and ρ -2 S -S 0 in the next chapter (see (17.4)).

Theorem 15.19. Under a biconformal deformation

where on the left-hand side G is the Einstein curvature 26 of the spacetime manifold (M, g), and on the right-hand side T is the total energy-momentum tensor of the spacetime being modeled. Geometry determines G while physics determines T , and the Einstein equation asserts that these two are related in a very specific way. In this work, the total energy-momentum tensor T is a sum of three terms:

the addends being the energy-momentum tensors of matter, electromagnetic field, and vacuum, respectively. The last addend in (16.2) is

where Λ ∈ R is the cosmological constant [START_REF] Mo | Horizontally conformal maps and harmonic morphisms[END_REF] and ρvacuum = 1 8π Λ is the vacuum energy density of the spacetime ( [25] (17.12b)). The first addend in (16.2) is, in this work, the energy-momentum tensor T fluid of a perfect fluid (u, ρ, p):

where the flow vector field u is unit and future-pointing timelike, the energy density 28 ρ : M → R is smooth, and the pressure p : M → R is smooth ( [25] (5.21)). [START_REF] O'neill | Semi-Riemannian geometry with applications to relativity[END_REF] The middle addend in (16.2) is given in terms of coordinates by

where μ ∈ R \ {0} is the magnetic constant (also called the vacuum permeability constant) and F is the electromagnetic field , a closed 2-form [START_REF] O'neill | The fundamental equations of a submersion[END_REF] , of the spacetime (c.f. [17] (3.7) or [25] (5.22)). [START_REF] Pantilie | Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds[END_REF] In terms of a (local) orthonormal frame field {e 1 , e 2 , e 3 , e 4 }, [START_REF] Mo | On the geometry of horizontally homothetic maps and harmonic morphisms[END_REF] Note that Ric and G have the same formal expression in terms of the other:

2 C(G)g, where C denotes metric contraction of a (0, 2) tensor field. [START_REF] Mo | Horizontally conformal maps and harmonic morphisms[END_REF] In the literature, the term "cosmological" is evidently associated with spacetime manifolds that model the entire universe, e.g. Friedmann-Robertson-Walker (FRW) spacetimes (see below). Indeed, for other types of spacetimes, Λ is customarily set to zero. However, in this work, we entertain the possibility of Λ being nonzero even for non-FRW spacetimes. [START_REF] Mo | Harmonic morphisms via deformation of metrics for horizontally conformal maps[END_REF] We point out that, in spite of our notation, the energy densities ρ, ρvacuum and the deformation parameter ρ have a priori nothing to do with each other. [START_REF] O'neill | Semi-Riemannian geometry with applications to relativity[END_REF] Note that for any Λ ∈ R, admissible flow vector fields u, v, and constant function ρ : M → R,

In view of this, we call a perfect fluid (u, ρ, p) trivial if ρ = p = 0 (see also footnote 39), and vacuum-trivial if ρ is a constant function and p = -ρ. A trivial perfect fluid is vacuum-trivial. [START_REF] O'neill | The fundamental equations of a submersion[END_REF] In this work, we are only concerned with the first Maxwell equation dF = 0, and not with the second equation d( F ) = J where J is the current 1-form. [START_REF] Pantilie | Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds[END_REF] Note that

By the previous theorem, the metric

on R 4 satisfies the Einstein equation with F mixed = 0 and any admissible u. Indeed, we can use α = α 1 + α 2 -α 4 and μ ∈ (0, +∞) in Theorem 19.4. For example, if α 1 = 1, α 2 = 0, α 4 = 1, then g satisfies the Einstein equation with magnetic constant μ, and F, ρ, p, Λ given/related by

Then

Because of the given inequality, either (19.8) or (19.9) holds. Hence, by the previous theorem, the metric

on a connected open subset of R 4 where σ and ρ are positive-valued, satisfies the Einstein equation with F mixed = 0 and any admissible u. Using α = 0 in Theorem 19.4 and choosing μ ∈ R \ {0} according to the theorem, we see that g satisfies the Einstein equation with magnetic constant μ, and F, ρ, p, Λ given/related by

19.3. σ horizontal, ρ horizontal.

Theorem 19.9. Suppose that σ = σ(x 1 , x 2 ) and ρ = ρ(x 1 , x 2 ). The biconformally deformed metric (19.4) satisfies the Einstein equation (17.3) with magnetic

In this case, (19.29) can be rewritten as

This gives the solution

where c 2 ∈ R.

Here are some specific examples with ρ = 1 and i = 1:

The metric

satisfies the Einstein equation with F = 0, any admissible flow vector field, and ρ

The metric

The metric

satisfies the Einstein equation with F = 0 and u, ρ, p, Λ given/related by

satisfies the Einstein equation with F = 0 and u, ρ, p, Λ given/related by

satisfies the Einstein equation with F = 0 and u, ρ, p, Λ given/related by

satisfies the Einstein equation with F = 0 and u, ρ, p, Λ given/related by

The discussion in this subsection leads to the following theorem.

Theorem 19.19. The maximal, biconformally deformed metric

with the x 3 -domain of σ(x

In particular, f 3 is always positive. Using (19.33) in (19.31), we have

Taking the ln and then differentiating both sides of (19.33), we get

Integration yields

for some c ∈ (0, +∞). I. By (19.34) and (19.35),

Upon integration, we find that

for some d 1 ∈ R. Squaring and integrating both sides, we get

Example 19.20. If h = h(x i ) in the above discussion (with α ∈ (0, +∞)), then by (19.32)

for some c 1 ∈ R; this also satisfies (19.30b) and (19.30c).

IA. In particular, by choosing

we see that the metric

satisfies the Einstein equation with F = 0, any admissible flow vector field, and ρ = -p = -ρ vacuum .

The discussion in this subsection leads to the following theorem.

Theorem 19.21. The maximal, biconformally deformed metric IA. ds 2 = (dx 2 + dy 2 ) + x 2 sech 2 z (dz 2 -dt 2 ), x > 0. In items 01., 03., and IA., the perfect fluid is vacuum-trivial. In item 02., the perfect fluid approaches vacuum-triviality as t → +∞, and the energy density approaches +∞ and the pressure approaches -∞ as t → 0 + . 20. Solutions from FRW spacetimes with flat space 20.1. Generalities. (See Appendix C.) Let (M, g 0 ) be FRW spacetime M (0, I, f ) = R 3 f ×I with expansion f and let (N, h) be R 2 . Let ϕ : (M, g 0 ) → (N, h) be the projection

In rectangular coordinates, the metrics g 0 and h are given by

respectively. From these we see that ϕ is a semi-conformal submersion with dilation

Let Before we continue, we state that the Christoffel symbols of g 0 with respect to the coordinate system {x 1 , x 2 , x 3 , t} are given by

These symbols help us in determining that, for example,

Theorems 15.12, 15.14, and 15.16 give us the differences

Hd(ln ρ) j dx j + 4 Hd ln σ Hd ln ρ -2(Hd ln ρ) [START_REF] Baird | Four-dimensional Einstein metrics from biconformal deformations[END_REF] ,

-2(ln f ) t Hd ln ρ θ 0 4 + 4 Hd ln ρ Vd ln σ, respectively. Evaluating Ric -Ric 0 on the frame field elements e 0 a , we obtain

For future reference (see proof of Theorem 20.10), we note that the system (20.11) is equivalent to

, which in turn imply that 8πρ -2 (ρ + 3p -2ρ vacuum ) = 5(ln σ) tt -6(ln σ) t 2 + (ln ρ) tt + 6(ln σ) t (ln ρ) t (20.12)

Theorem 20.2. Suppose that σ = σ(t) and ρ = ρ(t). The biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t , magnetic constant μ, and electromagnetic field (20.4) having zero mixed part if and only if

satisfying (20.10). In this case, the energy density ρ, pressure p, and cosmological constant Λ are related by the equations (20.11).

Proof. Equation (20.10) implies that F 12 = F 12 (t) and F 34 = F 34 (t), and in particular, by the discussion preceeding Theorem 20.1, F 12 = kf -2 for some k ∈ R.

Corollary 20.3. The conformal deformation

of a FRW spacetime metric satisfies the Einstein equation (17.3) with flow vector field u = σ∂ t and electromagnetic field F having zero mixed part if and only if F = 0.

Corollary 20.4. Suppose that σ = σ(t) and ρ = ρ(t) such that the biconformally deformed metric (20.5) satisfies the Einstein equation (17.3) with flow vector field u = ρ ∂ t , magnetic constant μ, and electromagnetic field (20.4) having zero mixed part. Let k ∈ R be as in the above theorem. If the Hubble condition (20.9)

Proof. These equations are obtained by evaluating equations (20.10) and (20.11) at t = t 0 .

The following lemma will be used to prove two of the last results in this chapter (Theorems 20.7 and 20.8), which highlight the relevance of the biconformal deformation approach of this work.

Proof. This result follows from Lemmas 1.4, 1.3, and 6.3, and the observation that

Theorem 1.6. Let X = a X a e a be a vector field on M . If X is horizontal, then

If X is vertical, then

The second equation in the theorem shows that the vertical distribution is conformal by definition ( [4] Definition 2.5.7): for a vertical vector field X and horizontal vector fields Y, Z on M ,

To obtain a formula for the change in the mixed part of Ricci curvature under conformal deformation of the metric (Corollary 15.17), we need the following lemma.

Lemma 1.7. Let X be a vector field on M . If X is horizontal, then the mixed part of L X g is given by

Proof. We first do some rearrangement: for vector fields Y, Z,

If X is horizontal, then by Lemma 10.1

APPENDIX B

Alternative computation of Ricci curvature (See the note at the beginning of Appendix A.)

Isothermal coordinates on a surface

Let (S, g S ) be an oriented (and time-oriented) Riemannian or Lorentzian surface, and let ψ = (x 1 , x 2 ) : S → ψ(S ) (S an open subset of S) be an isothermal coordinate system on S with respect to which

where τ is a smooth positive-valued function on S and 58

The Christoffel symbols Γ k ij of g S with respect to the coordinate system (x 1 , x 2 ) are given by

Let a i = τ ∂ i . The connection 1-form

of the orthonormal frame field {a 1 , a 2 } is given by

58 If g S is Lorentzian, then we start with a null coordinate system (ξ 1 , ξ 2 ) on S (see, for example, [4] Proposition 14. 1.18). With respect to this coordinate system, g S = a -1 dξ 1 dξ 2 for a nonvanishing smooth real-valued function a. If a > 0, then set

If a < 0, then set

119

FRW cosmological models

(We mostly follow [29] Chapter 12 here.) In this appendix, we write down the elements of a FRW cosmological model, which models the evolution of the universe. This model has both geometric and physical elements, and these two are related by the Einstein equation. We end this appendix with a quick discussion of the Hubble condition and of endpoint singularities.

Elements of a FRW cosmological model

First, the geometry of the model. Let (P, g P ) be a 3-dimensional Riemannian manifold of constant curvature k. In view of a theorem by Hopf (see, for example, [29] Corollary 8.25), we take P to be one of the prototypes H 3 , R 3 , or S 3 with respective curvatures -1, 0, 1. Let I ⊆ R be an open interval, equipped with the metric -dt 2 inherited from R 1 1 , and let f : I → (0, +∞) be a smooth function. The warped product M (k, I, f ) := P f ×I with metric g = f (t)

Now, the physics. Let ρ, p : P × I → R be smooth functions. Then (u, ρ, p) is a perfect fluid with energy density ρ, pressure p, and energy-momentum tensor T matter = T fluid = (ρ + p)u 2 + pg.

123

To improve the readability of calculations, we make the following assignments:

and

These sums will be used below:

and

4 , noting for the transition from (6.3a) to (6.3b) that

Hence,

To complete the description of B 0 , we also calculate

Collecting the values of B 0 on frame field elements, we get

For B * 0 , we have that for vector fields X, Y

, where we use (6.1) and (6.2) in the transition to the second line.

For C 0 and C * 0 ,

Projection from Schwarzschild spacetime onto S 2

Let (M, g 0 ) be exterior (interior, respectively) Schwarzschild spacetime with Schwarzschild mass M ∈ (0, +∞) and let (N, h) be S 2 . Let ϕ be the canonical projection from M onto S 2 . In spherical coordinates with r ∈ (2M, +∞) (r ∈ (0, 2M), respectively), the metrics g 0 and h, but for two poles of S 2 , are given by

respectively, where f (r) = 1 -2Mr -1 . From these we see that ϕ is a semi-conformal submersion with dilation

so that (M, g 0 ) satisfies the Einstein equation. Abstract : The domain of this thesis concerns Lorentzian and Riemannian geometry in dimension 4 with the objective of resolving Einstein's equations by an innovative method of biconformal deformation. This method consists of taking a base space-time, for example the Minkowski space or a Friedmann-Robertson-Walker (FRW) cosmological model, endowed with a conformal foliation by Lorentzian surfaces, and to deform the metric by factors σ and ρ normal and tangent resp. to the foliation. If σ ≡ ρ, the deformation is conformal. It turns out that biconformal deformations are optimal for controlling the Ricci curvature; conformal being better adapted to scalar curvature (Yamabe problem).

The method adapts to the Riemannian case, this time by taking a 4-dimensional Riemannian manifold endowed with conformal foliation by any surfaces. Among the significant results, we find a characterization of the solutions to Einstein's equations by deformation of the Minkowski metric with an electromagnetic field, with a perfect fluid field, or without a field. Concrete examples are constructed. The notion of Hubble condition is introduced in order to give a physical criterion for realistic biconformal deformations of FRW models. In the Riemannian framework, we construct a family of complete Einstein metrics with "ends" R 2 .