Alicia Herrero
email: aherrero@mat.upv.es

Juan Antonio Morales-Lladosa

Painlevé-Gullstrand synchronizations in spherical symmetry

Keywords: numbers: 04, 20, Cv, 04, 20, -q

A Painlevé-Gullstrand synchronization is a slicing of the space-time by a family of flat spacelike 3-surfaces. For spherically symmetric space-times, we show that a Painlevé-Gullstrand synchronization only exists in the region where (dr) 2 ≤ 1, r being the curvature radius of the isometry group orbits (2-spheres). This condition says that the Misner-Sharp gravitational energy of these 2-spheres is not negative and has an intrinsic meaning in terms of the norm of the mean extrinsic curvature vector. It also provides an algebraic inequality involving the Weyl curvature scalar and the Ricci eigenvalues. We prove that the energy and momentum densities associated with the Weinberg complex of a Painlevé-Gullstrand slice vanish in these curvature coordinates, and we give a new interpretation of these slices by using semi-metric Newtonian connections. It is also outlined that, by solving the vacuum Einstein's equations in a coordinate system adapted to a Painlevé-Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.

Introduction

It is known that Painlevé [1], Gullstrand [2] and (some years later) Lemaître [3] used a non-orthogonal curvature coordinate system which allows to extend the Schwarzschild solution inside its horizon, see Eq. ( 58) below. In this coordinate system, from now on called a Painlevé-Gullstrand (PG) coordinate system, the metric is not diagonal, but asymptotically flat, and regular across the horizon, and then, everywhere nonsingular up to r = 0. Furthermore, one has a very simple spatial 3-geometry: the space-time appears foliated by a synchronization of flat instants (hereafter called PG synchronizations). ‡ Nowadays, there is an increasing interest in the study of this type of synchronizations. For instance: (i) in connection with astrophysical applications, by taking into account that the dynamics of the gravitational collapse should be pursued beyond the Schwarzschild radius in a PG coordinate system [START_REF] Eardley | Dynamical Spacetimes and Numerical Relativity[END_REF][START_REF] Papadopoulos | [END_REF]8,9], (ii) in spherically symmetric space-times (SSSTs), as a convenient initial condition preserved under time evolution [10,11], (iii) in relativistic hydrodynamics, when an effective Lorentzian metric is introduced [START_REF] Lichnerowicz | Relativistic Hydrodymamics and Magnetohydrodynamics[END_REF], (iv) in non-relativistic situations admiting a Lorentzian description, namely, in 'analog gravity models' (see, for example, [START_REF] Unruh | [END_REF]14,15,16,[START_REF] Barceló | Analogue Gravity[END_REF]18]), and also (v) in modeling the black hole geometry and its associated physics, or to describe some quantum effects by starting from a Hamiltonian formulation [19,20,21,22]. More physical issues about the use of PG coordinates and their interpretation can be found in [4,5,23,24]. For a description of the causal character of the geometric elements (coordinate lines, coordinate 2-surfaces and coordinate 3-surfaces) associated with PG coordinates see [START_REF] Lladosa | Coordinates and frames from the causal point of view A Century of Relativity Physics Proceeding[END_REF][START_REF] Lladosa | Newtonian and relativistic location systems Relativistic Astrophysics and Cosmology Proceeding of the XXX Spanish Relativity Meeting ERE-2007 ed Oscoz A[END_REF].

The existence of PG synchronizations in SSSTs has been studied in the static case considering that the induced metric has vanishing Ricci tensor [27], and some specific constructions are presented in [28]. In more general cases, this existence is usually taken for granted but, recently, a limitation to this ansatz has been pointed out [29]. However, as far as the authors are aware, a definitive interpretation of this limitation as well as the analysis of the domains where a PG synchronization exists have not been done up to now. Then, some related questions arise: does every SSST admits a region of physical interest where a synchronization by flat instants exists? and, what are the advantages of adapting coordinates to a flat spatial 3-geometry? The main contribution of this paper is to state the above limitation clearly, in a form that it is coordinate independent, and to provide its physical interpretation.

Generalized (but, in general, non-flat) PG synchronizations have been constructed in Schwarzschild [START_REF]Lake K A class of quasi-stationary regular line elements for Schwarzschild geometry[END_REF], Reissner-Nordström [29], and Kerr [START_REF] Doran | [END_REF]32] geometries, and also in non-vacuum SSSTs, where new insights in the study of gravitational collapse scenarios are achieved (by evolving an initial 3-geometry [33,34]). However, here we restrict ourselves to flat synchronizations in order to discuss their existence in SSSTs.

The paper is structured as follows. Section 2 is devoted to introduce some general formulae for the induced geometry on space-like hypersurfaces and surfaces in SSSTs. In Sec. 3 the condition for the existence of a PG synchronization in SSSTs is analyzed and physically interpreted. In Sec. 4 we write the components of the Weinberg pseudotensor [START_REF] Weinberg | Gravitation and Cosmology[END_REF] with respect to a PG synchronization, and we prove that the energy and momentum densities of each PG slice vanish. In Sec. 5 we consider the semi-metric connection (see [START_REF] Bel | Rigid motion invariance of Newtonian and Einstein's theories of General Relativity Recent developments in gravitation[END_REF]) associated with a spherically symmetric metric expressed in PG coordinates and provide new insights on the Newtonian interpretation of the properties exhibited by the Schwarzschild field in these coordinates. Sec. 6 deals with the 3 + 1 decomposition coordinate character of the 'Schwarzschild singularity' obtaining an extended metric form for the Schwarzschild solution. In fact, the main motivation in papers [1, 2, 3] is rather different. For historical remarks about this subject, and some physical interpretations see [4,5].

of the Einstein equations with respect to a PG synchronization. By integration of the vacuum equations one recovers the extended Schwarzschild metric in PG coordinates, including the region inside the horizon. Finally, in Sec. 7 we discuss the role that our results can play for a better understanding of the geometry and physics in SSSTs. Some preliminary results of this work were presented at the Spanish Relativity meeting ERE-2009 [START_REF] Herrero | Flat synchronizations in spherically symmetric spacetimes[END_REF].

Let us precise the used notation. The curvature tensor R k lij of a symmetric connection ∇ is defined according with the identity ∇ i ∇ j ξ k -∇ j ∇ i ξ k = R k lij ξ l for the vector field ξ, and R ij ≡ R k ikj is the Ricci tensor. We take natural units in which c = G = 1 and the Einstein constant is κ = 8π. We say that {t, r, θ, ϕ} is a curvature coordinate system if for constant t and r the line element is dl 2 = r 2 dΩ 2 (with dΩ 2 ≡ dθ 2 + sin 2 θ dϕ 2 the metric of the unit sphere). In these coordinates, the metric form is, in general, non-diagonal. When, in addition, the 3-surfaces defined by t = constant are flat, the curvature coordinate system is called a PG coordinate system. From now on, it will be understood that the title of the sections and the results of this work always concern space-times with spherical symmetry. The agreement for space-time signature is (-+ ++).

Some geometrical relations

In this section we present the geometric background needed in the following sections: expressions for the Ricci tensor and the extrinsic curvature of a spherically symmetric synchronization, as well as, the mean curvature vector and the Gauss identity for the 2-spheres of a SSST. Of course, this material is not new and it may be bypassed or used as a glossary of formulae, which are conveniently referred throughout the text of the remaining sections. For an account on 2+2 warped space-times properties allowing to intrinsically characterize SSST see [START_REF] Ferrando | An intrinsic characterization of 2+2 warped spacetimes[END_REF][START_REF] Ferrando | An intrinsic characterization of spherically symmetric spacetimes[END_REF].

Let (V 4 , g) be a SSST, and let us consider a canonical coordinate system {T, R, θ, ϕ} of V 4 , which is adapted to the symmetries of the metric g. Then we may express the metric in the following general form [START_REF] Synge | Relativity: The General Theory[END_REF][START_REF] Takeno | The Theory of Spherically Symmetric Space-Times[END_REF][START_REF] Plebański | An Introduction to General Relativity and Cosmology[END_REF]

g = A dT ⊗ dT + B dR ⊗ dR + C(dT ⊗ dR + dR ⊗ dT ) + D σ, (1)
with A(T, R), B(T, R), C(T, R) verifying the condition δ ≡ AB -C 2 < 0, D(T, R) = 0, and σ ≡ dθ ⊗ dθ + sin 2 θdϕ ⊗ dϕ being the metric on the unit 2-sphere.

Ricci tensor

The induced metric γ on the 3-surfaces T = constant, is written as

γ = BdR ⊗ dR + Dσ, (2)
with B = 0. The Ricci tensor of γ, Ric(γ), is given by

Ric(γ) = B 2 R - B D F dR ⊗ dR + D 4 R + F 2 σ ( 3
)
where

F = 1 - (∂ R D) 2 4BD (4)
and R ≡ R(γ), the scalar curvature of γ, results

R = 2 D 1 + ∂ R B∂ R D 2B 2 + (∂ R D) 2 4BD - ∂ 2 R D B =        2F D + 4∂ R F ∂ R D if ∂ R D = 0, 2 D if ∂ R D = 0. (5)
Notice that the 3-surfaces T = constant are conformally flat § but, in general, they are not flat. From Eqs. ( 3), ( 4) and ( 5), we see that γ is a flat metric if, and only if, F = 0, that is

Ric(γ) = 0 ⇐⇒ 4BD = (∂ R D) 2 . ( 6
)

Extrinsic curvature

The extrinsic curvature of the slicing T = constant is defined as K = -1 2 L n γ where L n is the Lie derivative along the unit normal vector n,

n = 1 α ∂ ∂T - C B ∂ ∂R (7)
with

α 2 = - δ B
. From (2), we obtain

K = ΨB dR ⊗ dR + ΦD σ (8)
where

Ψ = K R R = 1 2Bα 2∂ R C - C B ∂ R B -∂ T B (9)
Φ = K θ θ = K ϕ ϕ = 1 2Dα C B ∂ R D -∂ T D (10)
are the eigenvalues of K. Developing the Lie derivative of K with respect to the shift vector β = C B ∂ ∂R we arrive to the expression

L β K = B C ∂ R C B 2 ΨB dR ⊗ dR + C B ∂ θ (ΦD) σ, (11)
which will be needed in Sec. 6 to split the Einstein evolution equations with respect to a PG synchronization. § Notice that the Cotton tensor of γ, C ijk (γ) ≡ D i Q jk -D j Q ik , (D i is the covariant derivative with the Levi-Civita connection of γ, and

Q ij ≡ R ij -R 4 γ ij ) identically vanishes, C ijk (γ) = 0
. This is the result that one could expect ought to the algebraic properties of the Cotton tensor and the assumed spherical symmetry. This means that γ is a conformally flat metric. Then, γ may always be written in isotropic form by making a coordinate change on each 3-surface T = constant. An interesting summary on the Cotton tensor properties is given in [START_REF] García | [END_REF].

Mean curvature vector

The mean curvature vector H of a 2-sphere S defined by constant T and R is given by

H = - 1 Dδ (B∂ T D -C∂ R D) ∂ ∂T + (A∂ R D -C∂ T D) ∂ ∂R . (12)
This expression directly follows by taking the trace (with respect to the induced metric Dσ) of the extrinsic curvature tensor K of each S, which is defined by

K(e a , e b ) ≡ -(∇ ea e b ) ⊥ = -Γ T ab ∂ ∂T + Γ R ab ∂ ∂R (13)
where the minus sign is taken as a matter of convention. ∇ is the Levi-Civita connection of g, {e i ≡ ∂ ∂x i } is a coordinate basis of S, i = a, b = θ, ϕ, and the symbol ⊥ stands for the projection on S ⊥ : the time-like 2-surface orthogonal to S. For a detailed study of K with applications in relativity see, for example, [44,45,46]. Then, it results

K = 1 2 Dσ ⊗ H. ( 14
)
The one form Γ metrically equivalent to H, Γ α = g αβ H β , is written as

Γ = - 1 D (∂ T DdT + ∂ R DdR) = -d ln D, (15)
which can be also obtained from the general expressions presented in [47].

Gauss identity

Given a space-like 2-surface Σ of a space-time (V 4 , g), the Gauss identity provides a scalar relation involving the background geometry and the intrinsic and extrinsic properties of Σ, and it may be expressed as

R(h) = 2 3 R(g) + g(H, H) + 2tr(K l × K k ) + 2Ric(l, k) -2W (l, k, l, k) (16)
where R(h) is the scalar curvature of the induced metric h on Σ, R(g), Ric and W are, respectively, the scalar curvature, the Ricci and the Weyl tensor of g, and l and k are two independent future pointing null vectors normal to Σ satisfying g(l, k) = -1. Now, H is the mean curvature of Σ, that is H = trK, with K the extrinsic curvature tensor of Σ (as defined by the first equality of [START_REF] Unruh | [END_REF], taking e a and e b tangent to Σ); K l and K k are, respectively, the second fundamental forms of Σ relative to l and k, that is, K l (e a , e b ) = g(l, K(e a , e b )), and similarly for K k . The trace is taken with respect to h, that is,

tr(K l × K k ) = h ad h bc (K l ) ab (K k ) cd .
The Gauss identity is usually given in terms of the Riemann curvature tensor R αβγδ (see, for instance [45,46]) from which the expression (16) follows by taking into account the algebraic decomposition:

R αβγδ = W αβγδ + g α[γ R δ]β -g β[γ R δ]α - R 3 g α[γ g δ]β
where the bracket stands for antisymmetrization of index pairs,

T [αβ] ≡ 1 2 (T αβ -T βα ).
For Σ = S, a 2-sphere (orbit) in a SSST, the Gauss identity may be written as

ρ = 2λ + 1 3 (4µ -µ 1 -µ 2 ) + 1 2 H 2 . ( 17
)
with ρ = 2/D the scalar curvature of Dσ, λ the simple eigenvalue ¶ of W (provided that W = 0), and {µ, µ 1 , µ 2 } stand for the Ricci eigenvalues. More precisely, the tangent space to S is the eigenplane associated with the eigenvalue µ. The eigenvalues µ 1 and µ 2 may be real (simple or multiple) or complex, and have associated the invariant 2-plane tangent to S ⊥ . Here, we have considered the algebraic classification of Churchill-Plebański for a symmetric 2-tensor in Lorentzian geometry, and its peculiarities in spherical symmetry (see [48], and also [49] for an intrinsic approach). We have taken into account that R(g) = 2µ + µ 1 + µ 2 , and 2Ric(l, k) = -(µ 1 + µ 2 ). In addition (14), the second fundamental forms K l and K k are both proportional to h = Dσ.

H 2 ≡ g(H, H) = -4tr(K l × K k ) because, according to

Painlevé-Gullstrand slicings

In this section we find the condition ensuring the existence of a PG-synchronization in spherical symmetry, and we discuss its invariant meaning in terms of the eigenvalues of the Weyl and Ricci tensors. Also, using a radial curvature coordinate r, we provide a physical interpretation of this condition in terms of the Misner-Sharp gravitational energy of a 2-sphere of radius r. As commented below, this result shows an interesting interconnection between the study and classification of trapped surfaces (see [50]) and the existence of flat slicings.

Existence condition

Let us start from the general metric form (1). Exploring the gauge freedom to make coordinate transformations of the form T = T (t, r), R = R(t, r), we look for a function t(T, R) whose level hypersurfaces t = constant are Euclidean, i. e., the induced metric is positive and flat. Under such a transformation, the metric (1) is expressed as

ds 2 = ξ 2 dt 2 + χ 2 dr 2 + 2ξ • χ dt dr + D(t, r)dΩ 2 , (18)
with D(t, r) ≡ D(T (t, r), R(t, r)), and the vector fields ξ and χ defined by

ξ ≡ Ṫ ∂ ∂T + Ṙ ∂ ∂R , χ ≡ T ∂ ∂T + R ∂ ∂R . (19)
Over-dot and prime stand for partial derivative with respect t and r, respectively, and J ≡ Ṫ R -T Ṙ = 0 assures coordinate regularity. The scalar products ξ 2 ≡ g(ξ, ξ), χ 2 ≡ g(χ, χ) and ξ • χ ≡ g(ξ, χ) can be written as

δ(dD) 2 Ṫ 2 + 2 Ḋ Z Ṫ + B Ḋ2 = (∂ R D) 2 ξ 2 , ( 20
) ¶ One has 1 2 W µναβ (l α k β -l β k α ) = λ(l µ k ν -l ν k µ ), and then, W (l, k, l, k) ≡ W µναβ l µ k ν l α k β = -λ. δ(dD) 2 T 2 + 2D Z T + BD 2 = (∂ R D) 2 χ 2 , ( 21
) (∂ R D) 2 ξ • χ = δ(dD) 2 Ṫ T + Z (D Ṫ + ḊT ) + B ḊD , (22)
where we have used the relations Ḋ = Ṫ ∂ T D + Ṙ∂ R D and D = T ∂ T D + R ∂ R D to substitute Ṙ and R in terms of Ṫ and T . We have denoted Z ≡ C∂ R D -B∂ T D. Then, due to the Lorentzian character of the metric (δ < 0), Eqs. ( 20) and ( 21) lead to real values for Ṫ and T if, and only if, the inequalities

ξ 2 (dD) 2 ≤ Ḋ2 , χ 2 (dD) 2 ≤ D 2 (23)
are satisfied. Now, looking for a flat synchronization, we have that the induced metric on the 3-surfaces t = constant is flat if, and only if,

4 D χ 2 = D 2 (24)
according to Eq. ( 6). Consequently, in the case of a flat synchronization, the second inequality in ( 23) is equivalent to

(d √ D) 2 ≤ 1. ( 25
)
So, under the assumed spherical symmetry, Eq. ( 25) provides the necessary and sufficient condition to be fulfilled for the existence of a flat slicing. The first inequality in (23) guarantees that the slices are space-like, that is, that the slicing is a PG synchronization.

Geometric interpretation

In terms of the scalar curvature ρ = 2/D of the metric DdΩ 2 , the above condition (25) may be expressed as follows

(dρ) 2 ≤ 2ρ 3 , (26)
which involves the sole invariant ρ. On the other hand, according to Eq. ( 15), H 2 = Γ 2 = (d ln D) 2 , and then Eq. ( 25) gives an upper bound for the norm of the mean extrinsic curvature H of the group orbits,

H 2 ≤ 4 D = 2ρ. ( 27
)
Moreover, from the Gauss relation [START_REF] Barceló | Analogue Gravity[END_REF], we arrive to the following result.

Proposition 1 In a spherically symmetric space-time the following conditions are equivalent.

(i) There exists a Painlevé-Gullstrand synchronization.

(ii) (dρ) 2 ≤ 2ρ 3 , where ρ is the scalar curvature of the 2-spheres.

(iii) H 2 ≤ 2ρ, where H is the mean curvature vector of the 2-spheres.

(iv) µ 1 + µ 2 -4µ ≤ 6λ, where µ 1 , µ 2 , and µ (double) are the Ricci eigenvalues, and λ is the simple eigenvalue of the Weyl tensor, or λ = 0 when the space-time is conformally flat.

Notice that this is a geometric result, which will be physically interpreted in the next subsection. Taking into account the Einstein equations, in the above item (iv) the Ricci eigenvalues, {µ 1 , µ 2 , µ} may by substituted by the corresponding energy tensor eigenvalues, {e 1 , e 2 , e}, giving

e 1 + e 2 -e ≤ 3λ. ( 28
)

Physical interpretation

By definition, see [START_REF] Takeno | The Theory of Spherically Symmetric Space-Times[END_REF][START_REF] Synge | Relativity: The General Theory[END_REF], r is a coordinate of curvature for the spherically symmetric metric form (18) if D(t, r) = r 2 , so that D = 2r and Ḋ = 0. Then, [START_REF] Lladosa | Coordinates and frames from the causal point of view A Century of Relativity Physics Proceeding[END_REF] says that (dr) 2 ≤ 1, and taking into account that the Misner-Sharp gravitational energy E of a 2-sphere of radius r is expressed as (see [51,50])

E = r 2 1 -(dr) 2 , ( 29
)
we arrive to the following result.

Proposition 2 Any spherically symmetric space-time admits a Painlevé-Gullstrand synchronization in the region where the Misner-Sharp gravitational energy is nonnegative, E ≥ 0.

The Misner-Sharp energy has been painstakingly analyzed in [50], providing useful criteria to study trapped surfaces. The main novelty here has been to relate this concept and the existence of PG synchronizations. + Moreover, the flatness condition (24) implies that χ 2 = 1, and the metric ( 18) is written as

ds 2 = ξ 2 dt 2 + 2ξ • χ dt dr + dr 2 + r 2 dΩ 2 . ( 30
)
Then, accordingly to ( 20) and ( 22), the following relations must occur

A(t, r) ≡ ξ 2 = J 2 δ(dr) 2 B(t, r) ≡ ξ • χ = εJ δ[(dr) 2 -1] (31)
where ε = ±1. So, the real function B exists in the region where (dr) 2 ≤ 1, and we have the following result.

Proposition 3 Let r be the radius of curvature of the orbits (2-spheres) of the isometry group of a spherically symmetric space-time with metric g. In the region defined by the condition

(dr) 2 ≡ g µν ∂ µ r∂ ν r ≤ 1, ( 32
)
the Misner-Sharp energy is not negative and a curvature coordinate system {t, r, θ, ϕ} exists in which the metric line element may be written as

ds 2 = A(t, r) dt 2 + 2B(t, r) dt dr + dr 2 + r 2 dΩ 2 , ( 33
)
where dΩ 2 = dθ 2 + sin 2 θ dϕ 2 .

+ Marc Mars inspired us in the obtaining of this relation.

The Lorentzian character of the metric impose that the functions A and B must satisfy the sole restriction A < B 2 , which is implied by [START_REF] Doran | [END_REF].

The Misner-Sharp energy is a geometric invariant that may be physically interpreted as an effective gravitational energy whose origin is the interaction between the energetic content and its associated field (see [50]). Given that the intrinsic and extrinsic scalar curvatures of the 2-spheres are ρ = 2/r 2 and H 2 = 4(d ln r) 2 , and according with (29) one has the invariant expression

E = 1 √ 2ρ 1 - 1 2ρ H 2 . (34)
Finally, notice that Eq. ( 27) does not constraint the causal character of the mean curvature vector H, which might be time-like, light-like or space-like. This is a remarkable property, because a 2-sphere is said to be trapped, marginal or untrapped if H is, respectively, time-like, light-like or space-like (see e. g. [45,46,50,52]).

Energy and momenta densities of a Painlevé-Gullstrand slice

In this section we establish the following result.

Proposition 4

In any spherically symmetric space-time, the Weinberg energy and momenta densities vanish for every Painlevé-Gullstrand synchronization.

Of course, to find a coordinate system in which the Weinberg densities vanish is not a surprising property, due to the non-tensorial character of them. However, the novelty here is to show that, for every SSST, such a vanishing property occurs in PG coordinates.

In order to proof the above result, let us consider the metric (33) written in a quasi-Minkowskian form, that is g = η + h with η the Minkowski metric, h 00 = 1 + A, h 0i = Bx i /r, and h ij = 0.

We start from the expression of the Weinberg pseudo-tensor [START_REF] Weinberg | Gravitation and Cosmology[END_REF],

2Q i0λ = ∂h µ µ ∂x 0 η iλ - ∂h µ µ ∂x i η 0λ - ∂h µ0 ∂x µ η iλ + ∂h µi ∂x µ η 0λ + ∂h 0λ ∂x i - ∂h iλ ∂x 0 ,
where Latin and Greek indexes go from 1 to 3 and from 0 to 3, respectively, and all indexes are raised and lowered with the flat metric η. In this case, it results Q i00 = 0 (according with [53]) and

2Q i0j = B r + B δ ij + B r -B x i r x j r (35)
The derivative of this expression leads to

2 ∂Q i0j ∂x k = B r - B r 2 + B δ ij x k r + B r 2 - B r δ ik x j r + δ jk x i r + 3 B r -3 B r 2 -B x i r x j r x k r .
and by contraction of the indexes, directly follows that ∂Q i0j ∂x i = 0. Then, the fourmomentum density vanishes,

τ 0λ ≡ - 1 8πG
∂Q i0λ ∂x i = 0 [START_REF] Bel | Rigid motion invariance of Newtonian and Einstein's theories of General Relativity Recent developments in gravitation[END_REF] and hence, the angular momentum densities j iλ = x i τ 0λ -x λ τ 0i also vanish, according with the announced conclusion.

For the special case of the Schwarzschild geometry, the vanishing of the energy density may be intuitively understood invoking the Einstein equivalence principle. Taking = 1 in the extended form (58) of the Schwarzschild metric, t represents the proper time of a radial geodesic observer which initially stays, in r = ∞, at rest with respect to a static observer. Locally, such an observer does not feel any gravitational effect.

Painlevé-Gullstrand slicings semi-metric connections

In the eighties, Bel proposed an extended Newtonian theory of gravitation based on a semi-metric connection associated with an observer congruence and a flat spatial 3-metric [START_REF] Bel | Rigid motion invariance of Newtonian and Einstein's theories of General Relativity Recent developments in gravitation[END_REF]. In a space-time, with metric g µν , which admits a spatially flat slicing given by the coordinate hypersurfaces x 0 = constant, the connection coefficients of the aforementioned semi-metric connection are written as [START_REF] Bel | Rigid motion invariance of Newtonian and Einstein's theories of General Relativity Recent developments in gravitation[END_REF],

Λ k = -Γ k 00 = 1 2 δ ki (∂ i g 00 -2∂ 0 g 0i ) (37)
Ω k j = -2Γ k 0j = δ ki (∂ i g 0j -2∂ j g 0i ). (38)
Consequently, a SSST metric admits a Newtonian interpretation when it is written in PG coordinates and it is considered in the above context. In fact, taking into account the expression (33) of the metric, we have g 00 = A, g 0i = Bx i /r, and then

∂ i g 0j = B r δ ij + B - B r x i r x j r = ∂ j g 0i .
Then, the connection coefficients result

Λ k = 1 2 A -2 Ḃ x k r (39)
Ω k j = 0, [START_REF] Synge | Relativity: The General Theory[END_REF] which means that, in the region of a SSST where a PG synchronization exist, the gravitational field may be interpreted as an inertial field of radial accelerations and vanishing rotation.

In particular, for the case of the Schwarzschild metric, we have A = -1 -2m r , B = 2m r , and then the vector component of the connection reduces to

Λ = - m r 2 e r , (41)
where e r is the unit vector in the radial direction. The above expression [START_REF] Takeno | The Theory of Spherically Symmetric Space-Times[END_REF] gives the acceleration of a unit mass particle radially falling in the Newtonian field of a mass m. Similar Newtonian interpretations have been considered from a different point of view (see, for example, [4,5,23]).

Painlevé-Gullstrand slicings and Einstein equations

In General Relativity, when dealing with the evolution (or 3 + 1) formalism (see [START_REF] Coll | Isometries et conditions initials en Relativité Général Thèse d' état[END_REF][START_REF] Bona | [END_REF], and [START_REF]Gourgoulhon É 3+1 formalism and bases of Numerical Relativity[END_REF] for a recent review) one introduces a vorticity free observer n, n 2 = -1, and Einstein equations are decomposed in the following set of constraint equations (κ is the Einstein constant),

R(γ) + (trK) 2 -trK 2 = 2 κ τ (42) ∇ • (K -trK γ) = κ q (43)
and this other system of evolution equations

∂ t γ = -2αK + L β γ (44)
∂ t K = -∇∇α-κα[Π+ 1 2 (τ -p)γ]+α[Ric(γ)+trK K-2K 2 ]+L β K . (45)
Here, γ and K are, respectively, the metric and the extrinsic curvature of the spacelike slices whose normal vector is n; ∇ is the Levi-Civita connection of γ, and the Ricci tensor and scalar curvature of γ are denoted by Ric(γ) and R(γ), respectively; the trace operator associated with γ is denoted by tr, so that, (∇ • K) a ≡ (tr∇K) a ≡ γ ij ∇ i K ja is the divergence of K with respect to γ. In the usual evolution formalism notation, n is written as n = α -1 ( ∂ ∂t -β), where α is the lapse function and β is the shift vector. The energy content T ≡ {τ, q, p, Π} has been decomposed relatively to n, that is

T = τ n ⊗ n + n ⊗ q + q ⊗ n + Π + pγ, (46)
with τ ≡ T (n, n), q ≡ -⊥T (n, •), p and Π being the energy density, the energy flux, the mean pressure and the traceless anisotropic pressure as measured by n, respectively; ⊥ is the projector on the 3-space orthogonal to n associated with the 3-metric γ ≡ g + n ⊗ n.

Spherical symmetry

In the case of a SSST, using the expression (8) of the extrinsic curvature, the constraint equations ( 42) and ( 43) are equivalent to

Φ(Φ + 2Ψ) = κτ - R 2 (47) 2∂ R Φ + ∂ R D D (Φ -Ψ) = -κq R (48)

Discussion

In this work we have analyzed the existence of flat synchronizations in SSSTs. The condition (27) provides an upper bound for the norm of the mean extrinsic curvature vector of the isometry group orbits which, using the Gauss identity, may be expressed in terms of curvature invariants. Moreover, the associated flat slices have vanishing Weinberg energy and momentum densities. We have seen that any spherically symmetric metric admits a Newtonian interpretation in the context of the Bel extended Newtonian theory of gravitation. Our study offers a new perspective about the meaning of the Painlevé-Gullstrand coordinates. This study applies for any SSST in the region where these coordinates exist. In this region, the gradient of the radial Painlevé-Gullstrand coordinate r may be space-like, light-like, or time-like, according to the condition (dr) 2 ≤ 1, which means that the Misner-Sharp gravitational energy of a sphere of radius r is non-negative. This condition may be tested for any SSST, starting from the general metric form (1). For instance, it occurs elsewhere in the Schwarzschild geometry, as it has been pointed out at the end of Sec. 6. Moreover, one has that (dr) 2 ≤ 1 everywhere for any Robertson-Walker metric with an energetic content which satisfies the usual energy conditions. In fact, if we put in (1) A = -1, B = a 2 (t)/(1 + k 4 r 2 ) 2 (with k = 1, 0, -1 the universe curvature index), C = 0 and D = r 2 B, we obtain that, in this case, ( 27) is equivalent to k + ȧ2 ≥ 0, which means that the proper energy density of the cosmological fluid is non negative. Consequently, any Robertson-Walker space-time that satisfies this energy condition admits a PG synchronization. This property is also obtained directly from the inequality (28). In this case, τ = -e 1 and p = e 2 = e are, respectively, the energy density and the pressure of the cosmological fluid, and λ = 0, because the Robertson-Walker metric is conformally flat. We leave for a future work the obtaining of the Painlevé-Gullstrand form of these Robertson-Walker cosmological models.

Finally, we have presented an improved proof of the Jebsen-Birkhoff theorem by expressing and solving the vacuum Einstein Equations in PG-coordinates. So, the extended Painlevé-Gullstrand-Lemaître metric form of the Schwarzschild solution is directly obtained.

Acknowledgments

We appreciate valuable aids provided by the tandem Joan J. Ferrando-Juan A. Sáez during the development of this paper, and the orientational comment by Marc Mars during the presentation of this subject at the Spanish Relativity Meeting ERE-2009. This work has been supported by the Spanish Ministerio de Ciencia e Innovación MICIN-FEDER project FIS2009-07705.

where q R is now the radial component of the energy flux. For the evolution equation (45), taking into account the expression (11), we have Π ϕϕ = Π θθ sin 2 θ and

For the metric (1), Eqs. ( 47), ( 48), ( 49) and ( 50) are the 3 + 1 splitting of the Einstein equations with respect to a vorticity free observer. The proper space of such an observer is Euclidean if, and only if, F = 0, and then R = 0. The integration of these equations for simple energetic contents (for instance, a dust model) should provide the corresponding metric form in PG coordinates. In the next section, the vacuum case is considered: the extended form of Schwarzschild solution in PG coordinates is obtained from the sole consideration of the field equations.

Schwarzschild vacuum solution

The extended Painlevé-Gullstrand-Lemaître metric form of the Schwarzschild solution may be obtained assuming spherical symmetry and the existence of a flat synchronization, Ric(γ) = 0, and then, solving the vacuum Einstein equations in a coordinate system adapted to such a synchronization. So, let us take τ = p = Π RR = Π θθ = F = R = 0. Then, for the metric expression (33), the lapse function is given by α 2 = B 2 -A and the constraint equations ( 47) and (48) result

When Φ = 0, taking into account also the evolution equations, we recover the Minkowski space-time. In the generic case, Φ = -2Ψ = 0, Eq. ( 53) leads to

with f (t) an arbitrary function. Substituting Eq. ( 54) in the momentum constraint (52), it reduces to α = 0. Consequently, the lapse is a function of the sole variable t, α(t), and we can take α = 1 by re-scaling the coordinate t. Then, we have

Next, the evolution equations ( 49) and ( 50) are written as

Given that Φ = -2Ψ, these last equations are equivalent to

By using the expressions ( 54) and ( 55), the equation ( 56) leads to f (t) = constant and the equation ( 57) is identically satisfied. Finally, by taking f = √ 2m, we obtain

which is the extended form of the Schwarzschild solution obtained by Painlevé, Gullstrand and Lemaître [1,2,3]. The positive parameter m is the Schwarzschild energy. The sign provides two coordinate branches for the solution, the Kruskal-Szekeres black and white hole regions being described by the above metric with = 1 and = -1, respectively, see [21]. Note that the r coordinate can take any positive value, 0 < r < +∞. In fact, from (58), we have (dr) 2 = g rr = 1 -2m r < 1, and the domain of a Painlevé-Gullstrand chart extends for every value of r = 0. Notice that (29) implies that E = m, which provides the physical interpretation of the parameter m as an effective energy [50]. Moreover, writing dt = dt S + 2m r (1 -2m r ) -1 dr, one recovers the usual Schwarzschild metric form

where t S is the coordinate time of the static observer (-∞ < t S < ∞) and the rank of the r coordinate is restricted to be r > 2m. According to the Jebsen-Birkhoff theorem (see [START_REF] Johansen | [END_REF] an references therein), we recover the Schwarzschild metric as the sole spherically symmetric solution of the vacuum Einstein equations. Other derivations of the Schwarzschild solution providing improvements of the original proof of the Jebsen-Birkhoff theorem have been achieved by solving the field equations in null coordinates (see [50,58] and reference therein). From the conceptual point of view, any of these derivations make unnecessary to get a coordinate transformation allowing to extend the domain of Schwarzschild chart from the outer to the inner horizon regions.