Joan Josep Ferrando
email: joan.ferrando@uv.es

Juan Antonio Sáez
email: juan.a.saez@uv.es

An intrinsic characterization of 2 + 2 warped spacetimes

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

Introduction

The study of warped product metrics is of interest in general relativity because they include a wide variety of physically relevant solutions of Einstein equations. For instance, spherically symmetric spacetimes, Lemaître-Friedmann-Robertson-Walker universes and static metrics are warped products.

Carot and da Costa proposed [1] an invariant characterization of warped product spacetimes and a classification scheme for them. They also studied the isometry group that each class can admit. Later, the energy tensor algebraic types compatible with these classes was analyzed [2]. Recent studies about warped product spacetimes concern Ricci collineations [3], parallel transport [4] and causal structure [5].

Besides the metric tensor, the characterization conditions given in [1] involve vector fields and functions that are not given in terms of the metric tensor. Consequently, a characterization of warped spacetimes built with intrinsic (depending solely on the metric tensor) and explicit conditions is not yet known. The goal of this paper is to obtain such an intrinsic and explicit characterization.

The intrinsic labeling of metrics starts with the beginning of the Riemannian geometry [6,[START_REF] Bianchi | Lezioni di geometria differenziale[END_REF][START_REF] Cotton | [END_REF]9,10]. Cartan [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF] showed that a metric may be characterized in terms of the Riemann tensor and its covariant derivatives. Brans [START_REF] Brans | [END_REF] introduced the Cartan invariant scheme in general relativity, and Karlhede [13] developed a method to study the equivalence of two metric tensors. It is worth pointing out that the covariant obtention of the underlying geometry of the Weyl and Ricci tensors is a necessary tool to characterize spacetimes intrinsically. This study can be found in [14] for the Ricci tensor and in [15] for the Weyl tensor.

The intrinsic and explicit characterization of spacetimes is interesting not only from the conceptual point of view but also from the practical one. Indeed, it provides an algorithmic way to test if a metric tensor, given in an arbitrary coordinate system, is a specific solution of Einstein equations. This is a powerful tool for subsequent applications. Thus, intrinsic labeling of a solution can be useful in obtaining a fully algorithmic characterization of the initial data which correspond to this solution. For instance, the algorithmic characterization of the Schwarzschild initial data [16] has been achieved by applying our intrinsic and explicit labeling of the Schwarzschild geometry [17]. Recently, we have accomplished an analogous study for the Kerr black hole [18] which will undoubtedly help to obtain algorithmic Kerr initial data. Similar approaches, essentially based on the accurate inquiry of the underlying geometry of the Weyl and Ricci tensors, have allowed us to intrinsically characterize several families of spacetimes which are interesting from the physical and/or geometrical point of view [15,17,19,20,21,22,23,24,25,26,27,28].

The conditions that we can find in all the intrinsic characterizations quoted above are local restrictions on the curvature tensor. Here we also study local properties and consider locally warped products, that is, n-dimensional Riemann spaces satisfying that in a neighborhood of each point there is a coordinate system {x α } such that the metric line element can be written:

ds 2 = h 1 ij (x k )dx i dx j + e 2λ(x k ) h 2 AB (x C )dx A dx B , (1)
where i, j, k = 1, .., p, and A, B, C = p + 1, ..., n. The two factors h 1 and h 2 are p and n-p dimensional metrics, respectively, and λ is the warping factor. For short, we write g = h 1 + e 2λ h 2 and say that g is a warped metric. For a formal definition and basic properties of warped product metrics see [START_REF] Neill | Semi-Riemannian Geometry[END_REF].

In relativity we can consider 1+3 and 2+2 warped spacetimes. The 1+3 case can be characterized in terms of the kinematic properties of a unitary vector field [1]. For 2+2 warped spacetimes the known characterization is not as neat because involves two null vectors and also the warping factor λ [1,2].

Here we restrict ourselves to study the 2+2 warped spacetimes and we improve our understanding of them by giving several equivalent geometric conditions that characterize them: in terms of the projector v or of the canonical 2-form U associated with the 2-planes of the warped product, or imposing the existence of a Killing-Yano tensor subject to complementary properties.

The principal planes, and consequently the invariant tensors v and U , are closely linked with the Ricci and/or Weyl tensors and could be obtained using the covariant approach given in [14,15]. This fact enables us to give an intrinsic labeling of the the non conformally flat 2+2 warped spacetimes imposing conditions on explicit Weyl concomitants. The results that we present here will allow us to deal with an old problem stated by Takeno [START_REF] Takeno | [END_REF]: the intrinsic an explicit characterization of the spherically symmetric spacetimes. This question will be analyzed elsewhere [START_REF] Ferrando | An intrinsic characterization of spherically symmetric spacetimes[END_REF] by considering these metrics as particular 2+2 warped spacetimes.

A 2+2 warped spacetime is conformal to a 2+2 product spacetime with particular conformal factor. This fact determines the approach used here to obtain our results. In Section 2 we analyze the product metrics: we give several equivalent conditions which characterize them, we summarize known properties of their curvature tensor and we offer an algorithmic way to detect them. In Section 3 we present a similar study for the conformally 2+2 product metrics and we give an invariant classification of these spacetimes. In Section 4 we show that the 2+2 warped spacetimes correspond to two of these invariant classes, the t-warped and the s-warped metrics. As a consequence, we obtain the complementary conditions for a conformally 2+2 product metric to be a warped spacetime. Moreover we see that, for type D metrics, these conditions impose restrictions on the Weyl principal 2-form arriving thus to an intrinsic and explicit characterization of these spacetimes. The algorithmic nature of our results allows us to present a flow chart in Section 5, which distinguishes every invariant class of conformally 2+2 product metrics and, in particular, the t-warped and s-warped spacetimes. Finally, in Section 6 we comment on our main results and work underway.

In this paper we work on an oriented spacetime with a metric tensor g of signature {-, +, +, +}. The Riemann, Ricci and Weyl tensors are defined as given in [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF] and denoted, respectively, by Riem(g), R = Ric(g) and W = W (g). For the metric product of two vectors we write (x, y) = g(x, y), and we put x 2 = g(x, x). Other basic notation used in this work is summarized in Appendix A. In Appendix B we introduce basic concepts on 2+2 almost-product structures and summarize their invariant classification. Finally, Appendix C presents the covariant obtention of the canonical 2-form of a type D Weyl tensor.

2+2 product spacetimes

In a 2+2 product spacetime the metric tensor takes (locally) the expression ĝ = v + ĥ, where v and ĥ denote two 2-dimensional metrics, Lorentzian and Riemannian, respectively. We have two principal planes, a time-like one V , with projector v, and its orthogonal complement, the space-like plane H, with projector ĥ. The canonical 2-form of the 2+2 product spacetime is the volume element Û of the time-like plane, Û 2 = v.

Invariant characterization

In a 2+2 product spacetime the generalized second fundamental forms Q v of V and Q ĥ of H vanish, that is, the principal planes define an integrable and totally geodesic 2+2 almost-product structure. Then, one says that (v, ĥ) is a product structure and that ĝ is a 2+2 product metric. In Appendix B we can find the basic concepts and properties on almost-product structures. From them we can easily obtain the following characterizations of the 2+2 product metrics.

Proposition 1 A spacetime is (locally) a 2+2 product if, and only if, one of the following equivalent conditions holds:

(i) There exists a 2+2 product structure (v, ĥ):

Q v = 0, Q ĥ = 0 .
(ii) There exists a covariantly constant simple and unitary 2-form Û :

tr Û 2 = 2 , Û • * Û = 0 , ∇ Û = 0 . (2)
(iii) There exists a covariantly constant 2-projector v:

v2 = v , tr v = 2 , ∇v = 0 . (3)
(iv) There exists a covariantly constant traceless involutive 2-tensor Π:

Π2 = ĝ , tr Π = 0 , ∇ Π = 0 . (4)
(v) There exists a simple and unitary Killing-Yano tensor of order two.

The 2-form Û mentioned in (ii) is the canonical 2-form of the product structure cited in (i), and its square is the projector v cited in (iii). In this statement we can replace v by the projector ĥ on the space-like plane. The 2-tensor Π quoted in (iv) is the structure tensor Π = v -ĥ. The Killing-Yano tensor named in (v) is Û : the Killing-Yano equation implies a vanishing covariant derivative under the simple and unitary restrictions. Obviously, v = Û 2 is a Killing tensor. All the characterizations in the proposition above are invariant properties of the spacetime. Nevertheless, these conditions are not intrinsic because they involve, besides the metric tensor, other geometric quantities, namely, the 2-tensors Û , v or Π. In order to obtain a fully intrinsic characterization we must study the curvature tensor of a product metric.

Ricci and Weyl tensors

The Weyl and Ricci tensor of a product metric have been studied in detail by many authors. The important fact is that they exclusively depend on the curvatures of the two 2-dimensional metrics v and ĥ. We summarize some properties that we need here.

Proposition 2 Let ĝ = v + ĥ be a 2+2 product metric, X and Y the Gauss curvature of v and ĥ, respectively, and Û such that Û 2 = v. It holds:

(i) Riem(ĝ) = -X Û ⊗ Û + Y * Û ⊗ * Û . (ii) Ric(ĝ) = Ric(v) + Ric( ĥ) = X v + Y ĥ . (iii) tr Ric(ĝ) = 2(X + Y ). (iv) W (ĝ) = ρ(3 Ŝ + Ĝ) , ρ ≡ -1 6 (X + Y ) , Ŝ ≡ Û ⊗ Û - * Û ⊗ * Û , Ĝ ≡ 1 2 ĝ ∧ ĝ .
As a direct consequence of this proposition we have:

Corollary 1 In a 2+2 product spacetime:

(i) The Weyl tensor is Petrov-Bel type D (or O, i.e. conformally flat) being ρ = -1 6 (X + Y ) the double eigenvalue. In the type D case, the Weyl principal planes are the principal planes of the product.

(ii) The Ricci tensor is Segre type [(11)( 11)] (or [(1111)], i.e. Einstein space), with eigenvalues X and Y . In the first case, the Ricci eigen-planes are the principal planes of the product.

(iii) The scalar curvature vanishes if, and only if, the spacetime is conformally flat.

Intrinsic and explicit characterization

Proposition 2 shows that the 2-form Û quoted in point (ii) of proposition 1 is the principal 2-form of a type D Weyl tensor. Moreover, the 2-tensors v and Π quoted in points (iii) and (iv) are, respectively, the projector on the time-like eigen-plane of a Ricci tensor of type [(11) [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF]] and the associated structure tensor. Consequently, these invariant characterizations become intrinsic and we can state: Proposition 3 A spacetime (which is not an Einstein space) is a 2+2 product if, and only if, its Ricci tensor is of type [(11) [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF]] and the projector on the Ricci time-like eigen-plane is covariantly constant.

Proposition 4 A non conformally flat spacetime is a 2+2 product if, and only if, it is Petrov-Bel type D and the principal 2-form of the Weyl tensor is covariantly constant.

It is worth remarking that the two propositions above cover all the metrics with non constant sectional curvature. Moreover, as a consequence of corollary 1, a product metric with constant sectional curvature is, necessarily, flat. Note that a non flat product metric admits a unique factorization. If the metric is flat, we have a factorization for every flat plane.

The intrinsic conditions in propositions 3 or 4 may be tested for a given metric tensor if we are able to impose the algebraic type and to determine the Weyl principal 2-form or the Ricci eigen-planes. Moreover, if we know a covariant expressions of these geometric elements in terms of the Weyl or Ricci tensors, we can obtain an intrinsic and explicit labeling of the 2+2 product spacetimes.

Let R ≡ Ric(ĝ) and r ≡ tr Ric(ĝ) be the Ricci tensor and the scalar curvature of the product metric ĝ, and N the Ricci trace-less part, N = R -1 4 r g. Then, R is of type [(11) [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF]] if, and only if, tr N 2 > 0 and N 2 = 1 4 tr N 2 g. Moreover, the structure tensor Π of the Ricci eigen-planes can be obtained as:

Π = 2 √ tr N 2 N , ≡ N (x, x) |N (x, x)| , (5)
where x is an arbitrary time-like vector. Then, under the above algebraic restrictions, the necessary and sufficient condition for the metric to be a 2+2 product is ∇Π = 0. Consequently, substituting expression (5) of Π, we obtain:

Theorem 1 A spacetime (which is not an Einstein space) is a 2+2 product if, and only if, the trace-less part N of the Ricci tensor, N = R -1 4 r g = 0, satisfies:

tr N 2 > 0 , N 2 = 1 4 tr N 2 g , ∇N = 1 2 d ln[tr N 2 ] ⊗ N . ( 6
)
Moreover the two metric factors can be obtained as v = 1 2 (g + Π) and ĥ = 1 2 (g -Π), where Π is given in (5).

Note that the third (differential) condition in (6) states that N is a recurrent tensor, that is, a vector field e exists such that

∇N = e ⊗ N . (7)
Moreover, under the two first (algebraic) conditions given in (6), equation ( 7) implies e = 1 2 d ln[tr N 2 ]. Consequently, we can state. Corollary 2 A spacetime (which is not an Einstein space) is a 2+2 product if, and only if, the trace-less part N of the Ricci tensor, N = R -1 4 r g = 0, is of type [(11)(11)] and recurrent.

We have shown in [17] that the Weyl tensor W is type D with real eigenvalues if, and only if, ρ ≡ -( 112 tr W 3 ) 1 3 = 0 and Ŝ2 + Ŝ = 0, where Ŝ ≡ 1 3ρ (W -ρ G). Moreover ρ is the double Weyl eigenvalue and S depends on the principal 2-form Û as Ŝ ≡ Û ⊗ Û - * Û ⊗ * Û . Then, can be easily shown that Û is covariantly constant if, and only if, ∇ Ŝ = 0. Consequently, substituting the expression of Ŝ in terms of the Weyl tensor and applying proposition 4, we obtain: Theorem 2 A non conformally flat spacetime is a 2+2 product if, and only if, the Weyl tensor satisfies:

ρ ≡ -( 1 12 tr W 3 ) 1 3 = 0 , W 2 + ρW -2ρ 2 G = 0 , ∇W = d ln |ρ| ⊗ W . ( 8
)
Moreover the two metric factors can be obtained as v = Û 2 and ĥ = ĝv, where

Û ≡ U [W ] is given in Appendix C.
Note again that the third (differential) condition in [START_REF] Cotton | [END_REF] states that W is a recurrent tensor, that is, a vector field f exists such that

∇W = f ⊗ W . (9)
Moreover, under the two first (algebraic) conditions given in [START_REF] Cotton | [END_REF], equation ( 9) implies f = d ln |ρ|. Consequently, we can state.

Corollary 3 A non conformally flat spacetime is a 2+2 product if, and only if, the Weyl tensor is Petrov-Bel type D with real eigenvalues and recurrent.

The intrinsic and explicit characterizations given in theorems 1 and 2 allow us to build an algorithm to test whether a metric tensor is a 2+2 product. Note that if the spacetime is an Einstein space, the Cotton tensor vanishes, ∇•W = 0, and then ∇U = 0 if, and only if, the Weyl eigenvalue is constant. Then we arrive to the algorithm that we present below as a flow chart.

Conformally 2+2 product spacetimes

In a conformally 2+2 product spacetime the metric tensor takes the expression g = e 2λ ĝ, where ĝ is a (locally) 2+2 product metric, ĝ = v + ĥ. The principal planes V and H have associated the projectors v = e 2λ v and h = e 2λ ĥ. Then g = v + h and Π = v -h is the structure tensor. Taking into account the relations between two conformal metrics, we obtain the change of the generalized second fundamental forms:

Lemma 1 If g = v + h = e 2λ (v +
ĥ), we have

Q v = e 2λ [Q v -v ⊗ ĥ(dλ)], Q h = e 2λ [Q ĥ -ĥ ⊗ v(dλ)] .
Note that this lemma implies the conformal invariance of the umbilical and integrable properties.

Invariant characterization

If we use the results of the previous section on product metrics and lemma above, and we take into account that vβ α = v β α , we have that the necessary and sufficient conditions for a metric g to be conformal to a 2+2 product one is that g = v + h with

Q v = -v ⊗ h(dλ), Q h = -h ⊗ v(dλ) ( 1 0 )
Moreover, Q v and Q h determine (and are determined by) the covariant derivatives of the canonical elements associated with the structure. Consequently, we obtain the following:

Proposition 5 A conformally 2+2 product spacetime is characterized by one of the following equivalent conditions:

(i) There exists a 2+2 almost-product structure (v, h) and a function λ such that

Q v = -v ⊗ h(dλ), Q h = -h ⊗ v(

dλ) .

(ii) There exists a simple and unitary 2-form U (U • * U = 0 , trU 2 = 2) such that

2 ∇U = -U ∧ a + h ∧ U (b) , d(a + b) = 0 , ( 11
)
where v ≡ U 2 , h ≡ g -v, a ≡ - * U (∇• * U ), b ≡ U (∇•U ).
(iii) There exists a traceless and involutive 2-tensor Π (trΠ = 0 , Π 2 = g) such that

2∇Π = Π 23 ∼ ⊗ Φ -g 23 ∼ ⊗ Π(Φ) , dΦ = 0 , Φ ≡ 1 2 Π(∇•Π) . ( 12
)
(iv) There exists a simple conformal Killing-Yano tensor of order two.

In proposition above, ∧ and 23

∼

⊗ denote, respectively, the anti-symmetrization and symmetrization in the second and third indexes of the tensorial product (see notation in Appendix A).

As happens in the 2+2 product case, all these geometrical elements are closely related to each other. The 2-tensor Π cited in (iii) is the structure tensor of the almostproduct structure quoted in (i), and the 2-form U in (ii) is the associated canonical 2-form. In (ii) we can substitute the time-like 2-form U by the space-like one * U , 2 ∇ * U = - * U ∧ b + v ∧ * U (a) . Moreover Φ = a + b = -2 dλ. The conformal Killing-Yano tensor cited in (iv) is e λ U , and the proof of this point follows by considering the projections of the conformal Killing-Yano equation on the planes V and H. The traceless part of the square of this conformal Killing-Yano tensor is the conformal Killing tensor e 2λ Π.

Carot and Tupper [START_REF] Carot | [END_REF] have classified the conformally 2+2 product spacetimes according to their conformal algebra. Moreover, they give an invariant characterization which imposes conditions on two null vectors and also involves the conformal factor. However, the invariant characterizations given in proposition 5 involve geometric elements linked with the curvature tensor. Then, we can obtain an intrinsic and explicit characterization of a conformally 2+2 product metric if we know its Ricci and Weyl tensors.

Ricci and Weyl tensors

The Weyl tensor W α βµν is conformal invariant. Then only the Weyl eigenvalue and the Ricci tensor change. The following proposition summarizes these known results. Proposition 6 Let g = e 2λ (v + ĥ) be a conformally 2+2 product metric and let us denote ∇ the covariant derivative associated to the metric ĝ = v + ĥ, and X, Y the Gauss curvature of v and ĥ. Then, (i) Ric(g) = X v + Y ĥ -2 ∇dλdλ ⊗ dλ -∆λ + 2ĝ(dλ, dλ) ĝ

(ii) The Weyl tensor is type D(or O) with double eigenvalue ρ ≡ -1 6 e -2λ (X + Y ). In the type D case, the canonical 2-form U , U 2 = v, is the principal 2-form of the Weyl tensor which takes the expression:

W = ρ(3S + G) , S ≡ U ⊗ U - * U ⊗ * U , G ≡ 1 2 g ∧ g . (13)

Intrinsic and explicit characterization

Proposition 6 shows that the algebraic type of the Ricci tensor of a conformally 2+2 metric strongly depends on the Hessian of the conformal factor λ. Thus, we have no restrictions on the Segre type and the Ricci tensor is, generically, algebraically arbitrary. On the other hand, the Weyl tensor is, necessarily, type D or O. In the conformally flat case we can only detect the principal structure if one considers particular conformal factors that restrict the Ricci tensor type. But we do not consider here these particular situations. Nevertheless, now we acquire an intrinsic characterization of the non conformally flat metrics conformal to a 2+2 product because of the 2-form U quoted in point (ii) of proposition 5 is the Weyl principal 2-form. Proposition 7 A non conformally flat spacetime is conformal to a 2+2 product if, and only if, it is Petrov-Bel type D and the Weyl principal 2-form satisfies [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF].

The differential conditions [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF] can be tested if we know the Weyl principal 2-form U . In making more explicit these conditions we can consider in a first step the Weyl concomitant S ≡ 1 3ρ (W -ρ G) which depends on U as S = U ⊗ U - * U ⊗ * U . Then, a straightforward calculation allows us to write [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF] in terms of S. Moreover, taking into account the characterization of the type D Weyl tensors with real eigenvalues [17], one arrives to: Theorem 3 A non conformally flat spacetime is conformal to a 2+2 product if, and only if, the Weyl tensor satisfies:

ρ ≡ -( 1 12 tr W 3 ) 1 3 = 0 , S 2 + S = 0 , S ≡ 1 3ρ (W -ρ G); ( 14
) 2 ∇•S + 3 S(∇•S) -g ∧ Φ = 0 , dΦ = 0 , Φ ≡ tr[S(∇•S)] . ( 15
)
Moreover the two metric factors can be obtained as v = U 2 and h = g -v, where U ≡ U [W ] is given in Appendix C, and the conformal factor λ is a function satisfying -2dλ = Φ.

The notation used in this theorem is explained in detail in Appendix A. Note that ∇•S and Φ are, respectively, a vector valued 2-form and a 1-form. Finally, if one introduces the 1-form ω = 3ρ(3ρΦ -2dρ) and substitutes the expression of S in terms of the Weyl tensor in the theorem above, one obtains: Theorem 4 A non conformally flat spacetime is conformal to a 2+2 product if, and only if, the Weyl tensor satisfies:

ρ ≡ -( 1 12 tr W 3 ) 1 3 = 0 , W 2 + ρW -2ρ 2 G = 0 ; ( 16
)
ρ ∇•W + W (∇•W ) - 1 3 g ∧ ω = 0 , d 1 ρ 2 ω = 0 , ω ≡ tr [W (∇•W )] . ( 17
)
Note that when the Cotton tensor vanishes, ∇•W = 0, the differential conditions (17) identically hold. Consequently, we recover a result given in [23]: every type D metric with vanishing Cotton tensor is conformal to a product metric.

Invariant classification of conformally 2+2 product metrics

When a family of spacetimes has an outlined 2+2 almost-product structure we can classify them by considering the invariant classification of this structure (see Appendix B). For example, elsewhere [23] we have classified type D metrics attending to the differential properties of the Weyl principal structure, and we have shown that only 16 of the 64 possible classes are compatible with the vacuum condition.

The principal structure of a conformally 2+2 product spacetime is umbilical and integrable. Consequently, following the notation given in Appendix A, necessarily the structure is of type ( 00• 00• ), that is, only the classes ( 001 001 ), ( 001 000 ), ( 000 001 ) and ( 000 000 ) are possible. For short we will write ( r s ) to indicate the conformally 2+2 product metrics of class ( 00r 00s ):

Definition 1 We say that a conformally 2+2 product metric is of: class ( 0 0 ) if both principal planes are minimal (product metrics). class ( 0 1 ) if the time-like principal plane is minimal and the space-like one is not. class ( 1 0 ) if the space-like principal plane is minimal and the time-like one is not. class ( 1 1 ) if none of the principal planes is minimal.

The classification given in the definition above is based on the minimal character of the principal planes and imposes differential conditions on the canonical 2-form U . Nevertheless, every class can also be characterized in terms of the gradient of the conformal factor. Indeed, as a consequence of proposition 5, and taking into account that -2 dλ = a + b, we obtain: Proposition 8 For a conformally 2+2 product metric g = e 2λ (v + ĥ), let U be the canonical 2-form and a ≡ - * U (∇• * U ), b ≡ U (∇•U ). Then, the metric is:

class ( 0 0 ) iff a = b = 0 iff λ is constant (dλ = 0). class ( 0 1
) iff a = 0, b = 0 iff dλ lies on the time-like principal plane (dλ = 0, * U (dλ) = 0). class ( 1 0 ) iff a = 0, b = 0 iff dλ lies on the space-like principal plane (dλ = 0, U (dλ) = 0). class ( 1 1 ) iff a = 0, b = 0 iff dλ does not lie on a principal plane (U (dλ) = 0, * U (dλ) = 0).

2+2 warped spacetimes

We can consider two different types of 2+2 warped spacetimes depending on the Lorentzian or Riemannian character of the warped factor h 2 (see expression (1)). In the first case the warped plane is time-like, the metric tensor takes the expression g = e 2λ v + h, with v(dλ) = 0, and we say that the spacetime is t-warped. In the second case the warped plane is space-like, the metric tensor takes the expression g = v + e 2λ ĥ, with ĥ(dλ) = 0, and we say that the spacetime is s-warped. Here, v and v denote two 2-dimensional Lorentzian metrics, and h and ĥ denotes two 2-dimensional Riemannian metrics. In [2] two subclasses of s-warped metrics were considered depending on the null or non null character of dλ. But in our study this distinction plays no role.

If for a 2+2 t-warped (respectively, s-warped) metric we define ĥ = e -2λ h (respectively, v = e -2λ v), we obtain that a 2+2 warped metric is conformal to a 2+2 product metric, g = e 2λ (v + ĥ), with v(dλ) = 0 (respectively, ĥ(dλ) = 0). Then, from proposition 8, we easily obtain: Proposition 9 The 2+2 t-warped (respectively, s-warped) spacetimes are the conformally 2+2 product spacetimes of class ( 1 0 ) (respectively, ( 0 1 )).

Invariant characterization

As a consequence of proposition 9 we can characterize the warped spacetimes by adding the conditions that label classes ( 1 0 ) or ( 0 1 ) (see proposition 8) to the conditions given in section 4 that characterize the metrics conformal to 2+2 product. Indeed, a simple reasoning leads to: Proposition 10 A 2+2 t-warped spacetime is characterized by one of the following equivalent conditions: (i) There exists a 2+2 almost product structure (v, h) and a function λ such that

Q v = -v ⊗ dλ, Q h = 0.
(ii) There exists a simple, time-like and unitary 2-form U (U • * U = 0 , tr U 2 = 2) such that

2 ∇U = -U ∧ a , da = 0 , a≡ - * U (∇• * U ) = 0 . (18)
(iii) There exists a time-like 2-projector v (v 2 = v , tr v = 2 , v(x, x) < 0 , where x in an arbitrary time-like vector) such that

2 ∇v = v 23 ∼ ⊗ a , da = 0 , a≡ ∇•v = 0 . (19)
(iv) There exists a simple and time-like Killing-Yano tensor of order two.

Proposition 11 A 2+2 s-warped spacetime is characterized by one of the following equivalent conditions:

(i) There exists a 2+2 almost product structure (v, h) and a function λ such that

Q v = 0, Q h = -h ⊗ dλ.
(ii) There exists a simple, time-like and unitary 2-form U (U • * U = 0 , tr U 2 = 2) such that

2 ∇ * U = - * U ∧ b , db = 0 , b ≡ U (∇•U ) = 0 . (20)
(iii) There exists a space-like 2-projector h (h 2 = h , tr h = 2 , h(x, x) ≥ 0 , where x in an arbitrary time-like vector) such that

2 ∇h = h 23 ∼ ⊗ b , db = 0 , b ≡ ∇•h = 0 . (21)
(iv) There exists a simple and space-like Killing-Yano tensor of order two.

All the geometrical elements in the proposition 10 (respectively, 11) above are closely related to each other. The 2-form U of (ii) is the canonical 2-form of the structure cited in (i), and the 2-tensor in (iii) is v = U 2 (respectively, h = g -v). Moreover, a = -2dλ (respectively, b = -2dλ). The Killing-Yano tensor cited in (iv) is e λ U (respectively, e λ * U ). The square of this Killing-Yano tensor is the Killing tensor e 2λ v (respectively, e 2λ h).

Ricci and Weyl tensors

Now, to obtain an intrinsic and explicit characterization from the above invariant properties, we analyze the Ricci and Weyl tensors of a 2+2 warped metric.

A 2+2 warped spacetime is a conformally 2+2 product with particular (non constant) conformal factor λ. Then, the Ricci tensor is given in proposition 6. Now, for a t-warped (respectively, s-warped) spacetime we have U (dλ) = 0 (respectively, * U (dλ) = 0) and, consequently, ∇dλdλ ⊗ dλ is a 2-tensor of the space-like plane H (respectively, time-like plane V ). Then, we have the following known result (see, for example, [2]):

Proposition 12 In a 2+2 t-warped (respectively, s-warped) spacetime the time-like principal plane (respectively, space-like plane) is an eigen-plane of the Ricci tensor.

When the Ricci eigen-plane quoted in the proposition above corresponds to a double eigenvalue (Segre types [ (11)11] or [11(11)]), the projectors v or h can be covariantly obtained from the Ricci tensor and, then, the invariant characterizations given in point (iii) of propositions 10 or 11 become intrinsic and explicit. Nevertheless, there are Segre types admitting a time-like or a space-like eigen-plane that are more degenerate. In these cases the intrinsic labeling of the 2+2 warped spacetimes using the Ricci tensor requires a more detailed analysis that will be considered elsewhere [START_REF] Ferrando | On the conformally flat warped spacetimes: an intrinsic approach[END_REF].

On the other hand, as also stated in proposition 6, the metric is Petrov-Bel type D (or O). In the type D case the Weyl tensor takes the expression (13) where ρ ≡ -1 6 e -2λ (X + Y ) is the double eigenvalue and U is the Weyl principal 2-form which satisfies U 2 = v. Consequently, for non conformally flat 2+2 warped spacetimes we can obtain an intrinsic labeling. To distinguish between t-warped and s-warped spacetimes we must add another condition. Under the assumption * S(Φ, Φ) = 0 we have that U (Φ) = 0 or * U (Φ) = 0. Then, the indefinite quadratic form Q = S(Φ; Φ) = U (Φ) ⊗ U (Φ) - * U (Φ) ⊗ * U (Φ) becomes semi-definite: negative for t-warped case and positive for s-warped spaces. Moreover, the sign of a semi-definite quadratic form is the sign of its trace with whatever elliptic metric associated with g, 2x ⊗ x + g, x being a unitary time-like vector. Thus, we have the following result. Proposition 14 A non conformally flat 2+2 warped metric is: (i) t-warped if, and only if, Q = S(Φ; Φ) is semi-definite negative, 2Q(x, x) + tr Q < 0.

(ii) s-warped if, and only

if, Q = S(Φ; Φ) is semi-definite positive, 2Q(x, x) + tr Q > 0.
where S and Φ are given in proposition 13 and x is an arbitrary unitary time-like vector.

The two propositions above and theorem 3 that characterizes the conformally 2+2 product metrics lead to: Theorem 5 For a non conformally flat metric g let ρ, S and Φ be the Weyl concomitants

ρ ≡ -( 1 12 tr W 3 ) 1 3 = 0 , S ≡ 1 3ρ (W -ρG) , Φ ≡ tr[S(∇•S)] . (22)
Then, g is a 2+2 warped metric if, and only if, it satisfies:

S 2 + S = 0 , 2 ∇•S + 3 S(∇•S) -g ∧ Φ = 0 , dΦ = 0 , * S(Φ; Φ) = 0. ( 23
)
In addition, the metric is t-warped (respectively, s-warped), if, and only if, 2S(Φ, x; Φ, x) -Φ 2 < 0 (respectively, 2S(Φ, x; Φ, x) -Φ 2 > 0), where x is an arbitrary unitary time-like vector.

A summary in algorithmic form

Theorems 3 and 5 unable us to perform an algorithm which distinguishes, between all the non conformally flat metrics, the four classes of conformally 2+2 product spacetimes: ( 0 0 ) (product), ( 1 0 ) (t-warped), ( 0 1 ) (s-warped) and ( 11 ) (regular conformally product). The flow chart (see below) shows the role played by every condition in these theorems. The input data are the Weyl concomitants ρ = ρ(g), S = S(g) and Φ = Φ(g) defined in (22).

Discussion and work in progress

In this work we have acquired an intrinsic and explicit labeling of the non conformally flat 2+2 warped spacetimes (theorem 5). This result is based on the invariant characterization of these spacetimes in terms of a time-like and unitary 2-form U which result to be the canonical 2-form of the (type D) Weyl tensor. In contrast, the invariant characterization given previously [1,2] imposes conditions on two null vectors and it also involves the warping factor λ.

In [2] the warped spacetimes were classified taking into account the projection of the gradient of the warped factor on two null directions. Here we have shown that t-warped and s-warped spacetimes correspond to two classes of conformally product metrics when one applies a general classification of the 2+2 almost-product structures.

In the non conformally flat case we have labeled with intrinsic and explicit conditions (theorems 2, 3 and 5) the four compatible classes of this classification (product, t-warped, s-warped and regular conformally product). Thus, we were able to build an algorithm to distinguish them that we have presented as a flow chart.

Conformally flat product metrics have been fully studied here (theorem 1). Nevertheless, the intrinsic characterization of the 2+2 warped conformally flat spacetimes requires a more detailed analysis of the different Ricci algebraic types that will be considered elsewhere [START_REF] Ferrando | On the conformally flat warped spacetimes: an intrinsic approach[END_REF].

We have also shown in this work that the four classes quoted above can also be characterized in terms of a Killing-Yano or a conformal Killing-Yano tensor (see propositions 1, 5, 10 and 11). The existence of quadratic first integrals of the geodesic equation in these spacetimes has been shown in [2]. But here we have shown that, the existence of an algebraically special Killing-Yano or conformal Killing Yano-tensor is a necessary and sufficient condition for a metric to be of a fixed class.

It is worth remarking that if a metric tensor, given in arbitrary coordinates, satisfies our characterization theorems, from the Riemann tensor we can obtain the geometric elements that appear in its canonical form, namely, the two factors v and h and the conformal (or warping) factor λ (see theorem 3).

Finally, we wish to comment that our results on s-warped metrics pave the way to acquiring the intrinsic labeling of the spherically symmetric spacetimes, a question that is considered elsewhere [START_REF] Ferrando | An intrinsic characterization of spherically symmetric spacetimes[END_REF].

4. 3 .

 3 Intrinsic and explicit characterization Propositions 8 and 9 imply that the 2+2 t-warped (respectively, s-warped) spacetimes are the conformally 2+2 product ones subject to the complementary restriction b = v(Φ) = 0 (respectively, a = h(Φ) = 0), a condition which is equivalent to U (Φ) = 0 (respectively, * U (Φ) = 0). On the other hand, for the type D metrics (W = 0) the double 2-form S given in (13) satisfies * S ≡ U ∼ ⊗ * U , and consequently, * S(Φ; Φ) = U (Φ) ∼ ⊗ * U (Φ) . Thus, we arrive to the following: Proposition 13 A (non conformally flat) conformally 2+2 product is a 2+2 warped spacetime if, and only if, the Weyl tensor satisfies * S(Φ; Φ) = 0 where S ≡ 1 3ρ (W -ρG) and Φ ≡ tr[S(∇•S)] = 0. Note that we have * W = ρ(3 * S -η), and, consequently, we can substitute condition * S(Φ; Φ) = 0 in proposition above by * W (Φ; Φ) = 0.

S

  (∇•S)-g ∧ Φ =0 S 2 +S = 0 dΦ = 0 * S(Φ; Φ) = 0 2S(Φ, x; Φ, x) -Φ 2 > 0

Acknowledgments

This work has been supported by the Spanish Ministerio de Ciencia e Innovación, MICIN-FEDER project FIS2009-07705.

(c) Action on a vector x, as an endomorphism A(x), and as a quadratic form A(x, x):

A(x) α = A α β x β , A(x, x) = A αβ x α x β . (A.3) (d) Exterior product as double 1-forms: A ∧ B is the double 2-form,

(e) Exterior product with a vector x as double 1-forms: A ∧ x is the vector-valued 2-form,

(f) Symmetrized tensorial product with a vector x as double 1-forms:

(ii) Products and other formulas involving double 2-forms P and Q:

(a) Composition as endomorphisms of the 2-forms space: P • Q,

(b) Square and trace as an endomorphism:

(c) Action on a 2-form X, as an endomorphism P (X), and as a quadratic form P (X, X), P (X) αβ = 1 2 P µν αβ X µν , P(X, X) =

Appendix B. 2+2 almost-product structures

The generalized second fundamental form Q v of a non-null p-plane V (set of vector fields generated by p independent vector fields) is the (2,1)-tensor:

where v is the projector on V and h = g -v. Let us consider the invariant decomposition of Q v into its antisymmetric part A v and its symmetric part

v is a traceless tensor:

The p-plane V is a foliation if, and only if, A v = 0, and, similarly, V is said to be minimal, umbilical or geodesic if tr S v = 0, S T v = 0 or S v = 0, respectively. On the spacetime, a 2+2 almost-product structure is defined by a time-like plane V and its space-like orthogonal complement H. Let v and h = g -v be the respective projectors and let Π = v -h be the structure tensor. A 2+2 spacetime structure is also determined by the canonical unitary 2-form U , volume element of the time-like plane V . Then, the respective projectors are v = U 2 and h = -( * U ) 2 .

The 2+2 almost-product structures can be classified by taking into account the invariant decomposition of the covariant derivative of the structure tensor Π or, equivalently, according to the foliation, minimal or umbilical character of each plane [23,25]. Every principal 2-plane can be subject or not to three properties, so 2 6 = 64 classes can be considered.

Definition 2 Taking into account the foliation, minimal or umbilical character of each 2-plane we distinguish 64 different classes of 2+2 almost-product structures.

We denote the classes as ( p q r lmn ), where the superscripts p, q, r take the value 0 if the time-like principal plane is, respectively, a foliation, a minimal or an umbilical plane, and they take the value 1 otherwise. In the same way, the subscripts l, m, n collect the foliation, minimal or umbilical nature of the space-like plane.

The most degenerated class that we can consider is ( 000 000 ) which corresponds to a product structure, and the most regular one is ( 111 111 ) which means that neither V nor H are foliation, minimal or umbilical planes. We will put a dot in place of a fixed script (1 or 0) to indicate the set of metrics that cover both possibilities. So, for example, the metrics of type ( 111 11 • ) are the union of the classes ( 111 111 ) and ( 111 110 ); or a metric is of type

We will say that a structure is integrable when both, V and H are a foliation, that is, of type ( 0 • • 0 • • ). We will say that the structure is minimal (respectively, umbilical) if both, V and H are minimal (respectively, umbilical), that is, of type (

Appendix C. Covariant obtenion of the canonical 2-form U of a type D Weyl tensor with real eigenvalues

The covariant determination of the canonical elements of the Weyl tensor for all the Petrov-Bel types was presented in [15] using the self-dual formalism. In a more recent paper [18] we have also given the expression of the canonical 2-form U of a type D Weyl tensor using real formalism. Here we particularize it for the case of real eigenvalues:

Lemma 2 For a Petrov-Bel type D Weyl tensor with real eigenvalue ρ ≡ -( 1 12 tr W 3 ) 1 3 , the canonical 2-form U can be obtained as:

where Z is an arbitrary 2-form and


Joan Josep Ferrando
email: joan.ferrando@uv.es

Juan Antonio Sáez
email: juan.a.saez@uv.es

An intrinsic characterization of spherically symmetric spacetimes

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

We give the necessary and sufficient (local) conditions for a metric tensor to be a non conformally flat spherically symmetric solution. These conditions exclusively involve explicit concomitants of the Riemann tensor. As a direct application we obtain the ideal labeling of the Schwarzschild, Reissner-Nordström and Lemaître-Tolman-Bondi solutions.

Introduction

The spherically symmetric solutions of the Einstein equation have played an essential role in further development of the General Relativity theory. The simplicity of their geometry has allowed analytic developments in modeling several astrophysical scenarios: the Schwarzschild and Reissner-Nordström solutions as the gravitational field exterior to a compact object, the geodesic motion in the Schwarzschild geometry explaining the perihelion advance and the bending of light, the relativistic equations of stellar structure in spherically symmetric perfect fluid solutions, the Friedmann-Lemaître-Robertson-Walker cosmological models, among others (see, e.g., [START_REF] Plebański | An introduction to General Relativity and Cosmology[END_REF]).

The spherically symmetric solutions continue being a subject of study and, moreover, interest in them is increasing nowadays. For instance, in numerical relativity they are an important tool in testing fully general-relativistic time-dependent numerical codes which evolve matter in strong gravitational fields (see, e.g., [START_REF] Romero | [END_REF] an references therein). Nevertheless, despite the extensive literature on spherically symmetric solutions, a fully algorithmic characterization of these spacetimes has not been accomplished. The goal of this paper is to attain such a characterization.

The intrinsic characterization of spherically symmetric spacetimes is an old problem established and partially solved by Takeno in 1952 [3] (see also [START_REF] Takeno | The Theory of Spherically Symmetric Space-Times[END_REF]). In the abstract he advanced: "In this note a theory is developed concerning the problem of determining whether or not a space-time defined by a line element arbitrarily given in any coordinate system is spherically symmetric". And in the introduction Takeno adds: "If the answer of this problem is given by the tensor equations to be satisfied by the curvature tensor specially when the equations contain no tensor other than the metric tensor, the metric volume element and the curvature tensor, we may say that the problem is solved in the most desirable form". Finally, in his conclusions he says:"...a theory concerning the discrimination of the spherically symmetric spacetimes has been constructed. Although it is not of the ideal form,...". Given Takeno's work and his clear ideas two questions naturally arise: Is it possible to solve the problem of characterizing the spherically symmetric spacetimes in "the ideal form"? Why doesn't Takeno solve the problem in this "most desirable form"?

The answer to the first question is affirmative as a consequence of a result by Cartan [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF] who showed that each Riemannian geometry may be characterized in terms of the Riemann tensor and its covariant derivatives. Nevertheless, the labeling of metrics in the "ideal form" starts with the beginning of the Riemannian geometry. The characterization of locally flat Riemann spaces [START_REF] Riemann | [END_REF], Riemann spaces with a maximal group of isometries [START_REF] Bianchi | Lezioni di geometria differenziale[END_REF], and locally conformally flat Riemann spaces [START_REF] Cotton | [END_REF] [9] [10] was achieved before 1921. In all these historic results, the conditions involve explicit concomitants of the curvature tensor (Riemann, Weyl and Cotton tensors). Then, they are intrinsic (depend solely on the metric tensor g) and, besides, the explicit expression of these concomitants in terms of g is known. We say that a characterization with these two properties, intrinsic and explicit, is an ideal characterization. ‡

It is worth pointing out that an understanding of the algebraic structure of the Weyl and Ricci tensors and the covariant determination of their underlying geometry are necessary tools to characterize spacetimes in an ideal form. But this comprehension of the Riemann tensor had not been attained in the epoch of the Takeno's aforementioned papers, which explains why he could not achieve the characterization of the spherically symmetric spacetimes in an ideal form.

The algebraic classification of the space-time symmetric tensors was given by Churchill [11], and since the sixties a lot of work has been devoted to studying the Ricci tensor from an algebraic point of view (see, e.g., the paper by Plebański [12]). However, the general covariant method to determine the Ricci eigenvectors and their causal character was not published until 1992 [13].

The pioneering papers by Petrov [14] and Bel [15] studied the algebraic classification of the Weyl tensor. Since then several algorithms have been proposed to determine the Petrov-Bel type. However, the general covariant expressions that give the underlying geometry of the Weyl tensor for every Petrov-Bel type were not published until 2001 ‡ The use of the appellation IDEAL (as an acronym) is used by B. Coll and collaborators long ago. This acronym seems to be adequate if the conditions involved, as well as being Intrinsic and Explicit, are Deductive (no inference process is necessary) and ALgorithmic (a flow chart with a finite number of steps can be built). [16].

The accurate analysis of the underlying geometry of the Weyl and Ricci tensors has allowed us to characterize in an ideal form several families of spacetimes which are interesting from the physical and/or geometrical point of view. We can quote the characterizations of the Schwarzschild [17], Reissner-Nordström [18] and Kerr [19] black holes, among others (see references in [20]).

As Takeno claimed, the ideal characterization of spacetimes provides an algorithmic way to test if a metric tensor, given in an arbitrary coordinate system, is a specific solution of Einstein equations. This can be useful to check wether a new solution is an already known one. Also, it can be of interest in obtaining a fully algorithmic characterization of the initial data which correspond to such a solution. For instance, our ideal labeling of the Schwarzschild geometry [17] has allowed us to achieve the algorithmic characterization of the Schwarzschild initial data [21].

A spherically symmetric spacetime is a 2+2 warped product and this quality determines the algebraic properties of the curvature tensor of a metric with spherical symmetry. The approach that we present here is based on this fact. We start from the ideal characterization of the 2+2 warped spacetimes recently obtained [20], and we look for the complementary conditions on the curvature tensor which lead to spherical symmetry. Section 2 is devoted to the study of the s-warped metrics, the class of the 2+2 warped spacetimes containing the spherically symmetric ones: we present their ideal labeling and summarize some properties of their curvature tensor.

In Section 3 we analyze the complementary constraints that the spherical symmetry imposes and, when the Weyl tensor does not vanish, we write these new conditions in terms of explicit concomitant of the Weyl and Ricci tensors. Thus, we obtain our main result: the ideal characterization of the non conformally flat spherically symmetric spacetimes. Further, we give the algorithm to determine the canonical elements of a spherically symmetric spacetime from the Riemann tensor. An alternative characterization which applies for a wide range of Ricci algebraic types is also offered.

In Section 4 we show that our result can be useful to label specific spherically symmetric solutions in an ideal form. As a consequence of our main theorem we recover the already known characterizations of the Schwarzschild and Reissner-Nordström spacetimes [17] [18], and we obtain the ideal labeling of the Lemaître-Tolman-Bondi solution.

In this paper we work on an oriented spacetime with a metric tensor g of signature {-, +, +, +}. The Riemann, Weyl and Ricci tensors and the scalar curvature are defined as given in [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF] and denoted, respectively, by Riem(g), W = W(g), R = Ric(g) and r. For the metric product of two vectors we write (x, y) = g(x, y), and we put x 2 = g(x, x). The symbols ∇, ∇• and ∆ denote, respectively, the covariant derivative, the divergence operator and the Laplacian operator. Other basic notation used in this work is summarized in Appendix A.

s-warped spacetimes

A classification scheme for the warped product spacetimes was proposed by Carot and da Costa [START_REF] Carot | [END_REF] and the isometry group that each class can admit was also studied. The energy tensor algebraic types compatible with these classes were later analyzed [24].

In a recent paper [20] we have given an invariant classification of the spacetimes conformal to a 2+2 product and, for the non conformally flat case, we have characterized each class in an ideal form. The 2+2 warped metrics correspond to two of these invariant classes, the t-warped and the s-warped spacetimes.

In the s-warped spacetimes the warped plane is space-like, and the metric tensor takes the expression [20]:

g = v + e 2λ ĥ , ĥ(dλ) = 0 , ( 1
)
where λ is the warping factor and where v and ĥ denote two 2-dimensional metrics, Lorentzian and Riemannian, respectively. A s-warped spacetime has two principal planes, the time-like V with projector v and the space-like H with projector h = e 2λ ĥ. The pair (v, h) defines a 2+2 almostproduct structure with structure tensor Π = v -h. A 2+2 spacetime structure is also determined by the canonical time-like unitary 2-form U , volume element of the plane V, U 2 = v.

Invariant characterization of the s-warped spacetimes

The properties of the principal structure (v, h) offer an invariant characterization of the s-warped metrics [20]. More precisely, the structure is umbilical and integrable, the plane V is minimal, and the trace of the second fundamental form of the plane H is a closed 1-form. These differential constraints can be explicitly written in terms of either the canonical 2-form U or the projector h and, consequently, one obtains an invariant labeling of the s-warped metrics [20]: Proposition 1 A 2+2 s-warped spacetime is characterized by one of the following equivalent conditions: (i) There exists a simple, time-like and unitary 2-form

U (U • * U = 0 , tr U 2 = 2) such that 2 ∇ * U = - * U ∧ b , db = 0 , b ≡ U (∇•U ) = 0 . (2)
(ii) There exists a space-like 2-projector h (h 2 = h , tr h = 2 , h(x, x) ≥ 0 , where x in an arbitrary time-like vector) such that

2 ∇h = h 23 ∼ ⊗ b , db = 0 , b ≡ ∇•h = 0 . (3)
Moreover, U is the canonical 2-form and h is the projector on the space-like plane. The warping factor λ satisfies 2 dλ = -b.

Ricci and Weyl tensors

If we define v = e -2λ v, we obtain that a s-warped metric can be written as conformal to a 2+2 product metric, g = e 2λ (v + ĥ), with ĥ(dλ) = 0. Then, taking into account how the Weyl and Ricci tensors change by a conformal transformation, we have [20]:

Proposition 2 Let g = e 2λ (v + ĥ) be a s-warped metric and let X, Y be the Gauss curvatures of v and ĥ. Then, (i) The Weyl tensor is type D (or O) with double eigenvalue ρ ≡ -1 6 e -2λ (X + Y ). In the type D case, the canonical 2-form U , U 2 = v, is the principal 2-form of the Weyl tensor which takes the expression:

W = ρ(3S + G) , S ≡ U ⊗ U - * U ⊗ * U , G ≡ 1 2 g ∧ g . (4)
(ii) The space-like principal plane is an eigen-plane of the Ricci tensor R, that is, R • h = µ h where µ is the associated eigenvalue. The Ricci tensor takes the expression:

R = X v + Y ĥ -2∇dλ -2dλ ⊗ dλ -[∆λ -2g(dλ, dλ)] g . (5)
Note that the gradient of the warping factor is given by the vector b defined in proposition 1, 2 dλ = -b. Then, if we take the trace in expression (5) and we consider the expression of the Weyl eigen-value given in point (i) of proposition 2, we get:

4 ρ + r 3 = ∇ • b + 1 2 b 2 , ( 6
)
where r is the scalar curvature, ρ the double Weyl eigenvalue and b ≡ ∇ • h, h being the projector on the space-like plane.

On the other hand, we can obtain the Gauss curvature of the metric h, κ = e -2λ Y , as follows. Note that h(b) = 0 and, consequently, h αβ ∇ α b β = -b 2 . Then, if we take the trace with the projector h in expression [START_REF] Cartan | Leçons sur le Géométrie des Espaces de Riemann[END_REF], we obtain:

κ = e -2λ Y = µ - 1 2 ∇ • b , (7)
where µ is the Ricci eigenvalue associated with the eigen-plane H, 2µ = h αβ R αβ . Moreover, if we remove ∇ • b from equations ( 6) and ( 7), we can obtain an alternative expression for the Gauss curvature κ:

κ = -2 ρ - 1 6 r + µ + 1 4 b 2 . ( 8
)

Ideal characterization of the non conformally flat s-warped metrics

As a consequence of proposition 2, in the non conformally flat case the Weyl tensor is type D and the canonical 2-form U can be obtained from the Weyl tensor. Then, the invariant characterization given in proposition 1 leads to the following characterization of the s-warped metrics [20]:

Proposition 3 For a non conformally flat metric g let ρ, S and Φ be the Weyl concomitants

ρ ≡ - 1 12 tr W 3 1 3 = 0 , S ≡ 1 3ρ (W -ρG) , Φ ≡ tr[S(∇•S)] . (9)
Then, g is a 2+2 s-warped metric if, and only if, it satisfies:

S 2 + S = 0 , (10) 2
∇•S + 3 S(∇•S) -g ∧ Φ = 0 , dΦ = 0 , ( 11
) * S(Φ; Φ) = 0, 2 S(Φ, x; Φ, x) -Φ 2 > 0 , ( 12
)
where x is an arbitrary unitary time-like vector. Moreover the warping factor λ satisfies Φ = -2 dλ.

The notation used in this proposition is explained in detail in Appendix A. Each of the conditions has a specific geometric interpretation. The algebraic condition (10) means that the Weyl tensor is of type D with real eigenvalues. The two differential conditions (11) ensure that the metric is conformal to a 2+2 product metric with a principal structure aligned with the Weyl tensor and, moreover, Φ = -2 dλ. The differential conditions (12) mean that Φ lies on the time-like principal plane or, equivalently, that h(dλ) = 0. This proposition will be the starting point in obtaining the characterization of the spherically symmetric spacetimes.

Characterizing spherically symmetric spacetimes

In a canonical coordinate system (t, r, θ, ϕ), the metric line element of a spherically symmetric spacetime takes the expression [START_REF] Takeno | The Theory of Spherically Symmetric Space-Times[END_REF] [1]:

ds 2 = A dt 2 + 2 C dt dr + B dr 2 + e 2λ dΩ 2 , (13)
where dΩ 2 = dθ 2 + sin 2 dϕ 2 is the metric of the 2-sphere and A, B, C and λ are smooth functions of t and r.

In the coordinate expression (13) we can identify the canonical form of a s-warped metric. More precisely, we have: Lemma 1 A spacetime is spherically symmetric if, and only if, the metric tensor can be written as:

g = v + e 2λ σ , σ(dλ) = 0 , ( 14
)
where v is an arbitrary 2-dimensional Lorentzian metric, and σ is a 2-dimensional Riemannian metric with a Gauss curvature equal to one.

Invariant characterization of the spherically symmetric spacetimes

In the previous section we have characterized the s-warped spacetimes. Now we must add new constraints in order to impose that ĥ = σ, σ being a metric with Gauss curvature Y = 1. As a consequence of [START_REF] Bianchi | Lezioni di geometria differenziale[END_REF], this condition holds if, and only, if, the Gauss curvature of h is κ = e -2λ > 0.

On the other hand, as h(dλ) = 0, the condition Y = 1 implies h(dκ) = 0. And, conversely, as Y is the curvature of the metric ĥ and κ = e -2λ Y , conditions h(dκ) = 0 and κ > 0 imply that Y is a positive constant and, consequently, we can add an additive constant to λ in order to obtain a metric σ with Gauss curvature Y = 1. Therefore, a s-product spacetime is spherically symmetric if, and only if, the Gauss curvature κ of the metric h of the space-like principal plane satisfies:

κ ≡ κ(h) > 0 , h(dκ) = 0 . ( 15
)
Then, if we take into account proposition 1, we obtain an invariant characterization of the metrics with spherical symmetry:

Proposition 4 A spherically symmetric spacetime is characterized by one of the following equivalent conditions:

(i) There exists a simple, time-like and unitary 2-form U which satisfies (2) and ( 15), where κ(h) is the Gauss curvature of the metric h = g -U 2 .

(ii) There exists a space-like 2-projector h which satisfies (3) and (15), where κ(h) is the Gauss curvature of h.

Moreover, U is the canonical 2-form and h is the projector on the 2-spheres. The warping factor λ can be obtained as λ = -1 2 ln κ, and σ = κ h is the metric of the 2-sphere.

Ideal characterization of the non conformally flat spherically symmetric spacetimes

Proposition 4 characterizes the spherically symmetric spacetimes whatever their Riemann algebraic type. In order to obtain an intrinsic labeling we must give either the canonical 2-form U or the projector h in terms the of the Weyl or Ricci tensors. The use of the Ricci tensor requires the analysis of several algebraic types and will be considered elsewhere [START_REF] Ferrando | Characterizing spherically symmetric spacetimes with the Ricci tensor[END_REF]. Here we use the Weyl tensor. When the spacetime is not conformally flat (W = 0), the Weyl tensor is of type D and, as stated in proposition 2, the canonical 2-form U is aligned with the Weyl tensor. The explicit expression U = U [W ] of this Weyl concomitant is given in Appendix B. Then, as a consequence of proposition 4, if we impose conditions (2) and ( 15) to the 2-form U = U [W ] we get an intrinsic labeling of the non conformally flat spherically symmetric spacetimes. Nevertheless, a more accurate analysis of these constraints leads to more explicit conditions in the Weyl tensor. Indeed, in [20] we have shown that conditions (2) are equivalent to equations (10), (11) and ( 12) which characterize the s-warped metrics in an ideal form (see proposition 3). Thus, at this point we have: Lemma 2 A non conformally flat spacetime is spherically symmetric if, and only if, the metric tensor satisfies the intrinsic and explicit conditions (10), ( 11) and (12), and the invariant conditions (15). Now we analyze the conditions (15) in order to complete an ideal characterization of the spherical symmetry.

As commented after proposition 3, constraint (12) states that h(Φ) = 0. Moreover, under this condition, the Weyl concomitant Φ defined in (9) equals the vector b defined in proposition 1. Then, proposition 4 implies that for spherically symmetric spacetimes the following identities hold:

Φ = -2 dλ = b = d ln κ . (16)
Note that the necessary conditions h(Φ) = 0 and Φ = d ln κ imply that dΦ = 0 and h(dκ) = 0. Consequently, we have:

Lemma 3
In the set of constraints (10), ( 11), ( 12) and (15) characterizing the spherically symmetric spacetimes we can substitute conditions dΦ = 0 and h(dκ) = 0, by the condition Φ = d ln κ.

Finally, let us analyze expression (8) of the curvature κ. The scalars ρ and r, and the vector b = Φ are explicit concomitants of the Weyl and Ricci tensors. How can the Ricci eigenvalue µ be explicitly obtained? It satisfies 2µ = h αβ R αβ , where the projector h could be obtained from the Weyl concomitant U = U [W ] given in lemma 7 as h = g -U 2 . Nevertheless, we try to avoid the use of this expression of U in our characterization theorem because it involves an arbitrary 2-form Z which could be annoying in subsequent applications. Now, we get an alternative expression for µ whatever the Ricci algebraic type by using a mixed concomitant of the Ricci and Weyl tensors as follows.

The generic expression of a Ricci tensor which has H as an eigen-plane with associated eigenvalue µ is:

R = A + µh , (17)
where A is a symmetric tensor on V, A • h = 0. Then, from the expression (4) of S, we obtain:

B ≡ R -S[R] - 1 2 rg = β Π , β ≡ 1 2 r -2µ . (18)
Thus, we have µ = 1 4 r -1 2 β, where β = 0 when B = 0, and when B = 0, β can be computed as

β = 2 √ tr B 2 , = - B(x, x) |B(x, x)| , (19)
x being an arbitrary unitary time-like vector (note that the structure tensor Π = v -h satisfies Π(x, x) < 0 for every time-like vector x). This expression of β allows us to obtain an explicit expression of the Ricci eigenvalue µ and, substituting in [START_REF] Cotton | [END_REF], of the curvature κ.

All these considerations and lemma 3 lead to the following ideal characterization.

Theorem 1 Let W = W(g) and R = Ric(g) be the Weyl and Ricci tensors of a non conformally flat metric g, and let ρ, S, Φ, r, B and κ be the Riemann concomitants

ρ ≡ - 1 12 tr W 3 1 3 = 0 , S ≡ 1 3ρ (W -ρG) , Φ ≡ tr[S(∇•S)] , (20)
r ≡ tr R , B ≡ R -S[R] - 1 2 rg , κ ≡ -2 ρ + 1 12 r + 1 4 Φ 2 - 4 √ tr B 2 . ( 21
)
where we put either = 0 if B = 0 or = -B(x,x) |B(x,x)| if B = 0, x being an arbitrary unitary time-like vector. Then, g is spherically symmetric if, and only if, it satisfies:

S 2 + S = 0 , ( 22
) 2 ∇•S + 3 S(∇•S) -g ∧ Φ = 0 , ( 23
) * S(Φ; Φ) = 0, 2 S(Φ, x; Φ, x) -Φ 2 > 0 , (24)
κ > 0 , Φ = d ln κ , ( 25
)
where x is an arbitrary unitary time-like vector.

A metric tensor g given in any coordinate system which satisfies the conditions of theorem 1 is a spherically symmetric solution and, as a consequence of lemma 1, it admits the canonical form (14). Can the canonical elements v, σ and λ be obtained from the Riemann tensor?

As stated in the last sentence of proposition 4, the warping factor can be obtained from the Gauss curvature κ as λ = -1 2 ln κ, and σ = κh. Moreover h can be obtained as h = g -v, and v = U 2 where U is the Weyl principal 2-form. Thus, we can state: Proposition 5 A non conformally flat metric g with spherical symmetry admits the canonical form:

g = v + 1 κ σ , v = U 2 , (26)
where κ and U are, respectively, the Riemann concomitants defined in (21) and (B.1).

On the other characterizations based more closely on the Ricci tensor

Theorem 1 and proposition 5 apply when the Weyl tensor does not vanish whatever the Ricci algebraic type. Under any algebraic constraint of the Ricci tensor, alternative statements can be deduced. Elsewhere [START_REF] Ferrando | Characterizing spherically symmetric spacetimes with the Ricci tensor[END_REF] we analyze in detail other characterization theorems based more closely on the Ricci tensor. Here, we illustrate these approaches by imposing an algebraic condition which is valid for a wide range of Ricci types and which avoids the use of the canonical 2-form U in obtaining the canonical elements. Let us suppose that the 2-tensor B defined in (18) does not vanish. Then β = 0 and the structure tensor Π can be obtained as Π = 1 β B. Thus, taking into account that condition (24) is equivalent to h(Φ) = 0, we can state:

Then, the necessary and sufficient conditions for g to be the Schwarzschild metric are:

Ric(g) = 0 , ρ = 0 , S 2 + S = 0 , (28) κ
≡ 1 9 (d ln ρ) 2 -2ρ > 0 , (29)
* S(dρ; dρ) = 0, 2 S(dρ, x; dρ, x) -(dρ) 2 > 0 ,

where x is an arbitrary unitary time-like vector.

Ideal characterization of the Reissner-Nordström solution

As a consequence of the generalized Birkhoff theorem (see [START_REF] Bona | [END_REF] and references therein), the only spherically symmetric non null Einstein-Maxwell solution is the Reissner-Nordström spacetime. The spherical symmetry implies that the electromagnetic field is aligned with the Weyl tensor and moreover, as the principal structure is integrable, the differential Rainich condition identically holds [30] [31]. Then, we only have to add the algebraic Rainich conditions. Consequently, we have:

Lemma 5 A metric tensor g is the Reissner-Nordström solution if, and only if, it satisfies the conditions of theorem 1 and:

tr R = 0 , R 2 = χ 2 g , R(x, x) ≥ 0 , (31)
where x is an arbitrary unitary time-like vector and χ ≡ -

1 2 √ tr R 2 .
Again, the conditions in the above lemma can be simplified. As commented above, the spherical symmetry implies the alignment of the Ricci and Weyl tensors, an algebraic condition which is equivalent to:

S[R] + R = 0 . ( 32
)
Under this constraint, the second Rainich condition in (31) is a consequence of the type D condition [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF]. Moreover, in [31] we have shown that the algebraic constraints [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF] and (32) imply the differential condition [START_REF] Carot | [END_REF] under the complementary algebraic restriction 9ρ 2 = χ 2 .

On the other hand, for aligned non null Einstein-Maxwell solutions, the Bianchi identities imply (when the Weyl tensor has real eigenvalues) [31]: dχ = 2 χΦ , 2 dρ = 3 ρΦ + χΠ(Φ) .

The second of these equations implies, under the algebraic condition 9ρ 2 = χ 2 , that h(Φ) = 0 if, and only if, h(dρ) = 0. Thus, we can substitute Φ by dρ in equation (24). Moreover, in this case µ = -χ, and from (33) we have Φ = 2 3 d ln(ρ + χ), and we obtain that the curvature κ given in (21) becomes κ = 1 9 (d ln ρ) 2 -2 ρ -χ. Thus, we recover the following ideal characterization of the Reissner-Nordström geometry [18]: Theorem 3 Let W = W(g) and R = Ric(g) be the Weyl and Ricci tensors of a metric g, and let ρ, S and χ be the Riemann concomitants ρ ≡ -1 12 tr W 3 (34)

2 √

 2 ρG) , χ ≡ -1 tr R 2 .

Acknowledgments

We thank the referee for the accurate reading of the paper. This work has been supported by the Spanish Ministerio de Ciencia e Innovación, MICIN-FEDER project FIS2009-07705.

Proposition 6 Let g be a non conformally flat metric such that the Riemann concomitant B defined in (18) does not vanish. Then, the metric is spherically symmetric if, and only if, it satisfies [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF], [START_REF] Carot | [END_REF], (25) and B(Φ) = βΦ , where β is given in (19). Then, the metric g admits the canonical form [START_REF] Ferrando | Characterizing spherically symmetric spacetimes with the Ricci tensor[END_REF], where κ is defined in (21) and v is given by v = 1 2 (g + 1 β B) .

Labeling specific spherically symmetric solutions

The ideal characterization given in theorem 1 allows us to distinguish whether a non conformally flat metric given in any coordinate system is spherically symmetric. Moreover, this result could easily help us to obtain the ideal labeling of specific spherically symmetric solutions. In this section we illustrate this fact by recovering the already known characterizations of the Schwarzschild [17] and Reissner-Nordström [18] geometries, and by obtaining the ideal characterization of the Lemaître-Tolman-Bondi solution.

In the two last cases, with a non vanishing Ricci tensor, an alternative characterization based more closely on Ricci concomitants could be acquired. Nevertheless, here we apply our above theorem 1 and proposition 6 based essentially on the Weyl tensor. This way, the characterization theorem for the Reissner-Nordström admits the Schwarzschild solution as a limiting case.

Ideal characterization of the Schwarzschild solution

The Jebsen-Birkhoff theorem [START_REF] Jebsen | [END_REF] [START_REF] Birkhoff | Relativity and Modern Physics[END_REF] states that the Schwarzschild spacetime is the only spherically symmetric vacuum solution. Consequently, we have: Lemma 4 A metric tensor g is the Schwarzschild solution if, and only if, it satisfies R ≡ Ric(g) = 0 and the conditions of theorem 1.

Let us see how the conditions in the above lemma can be simplified. If we replace S in terms of the Weyl tensor, condition ( 23) is equivalent to

On the other hand, ω and Φ are related as Φ = 1 9ρ 2 ω + 2 3 d ln ρ. Then, under the vacuum condition, R = 0, the Cotton tensor vanishes and then ∇ • W = 0. Consequently, [START_REF] Carot | [END_REF] identically holds and Φ = 2 3 d ln ρ. Then, the second condition in (25) also holds and, moreover, the curvature κ takes the expression κ = 1 9 (d ln ρ) 2 -2ρ. Finally, we can substitute Φ by dρ in equation (24). Thus, we recover the following ideal characterization of the Schwarzschild geometry [17]:

Theorem 2 Let W = W(g) and R = Ric(g) be the Weyl and Ricci tensors of a metric g, and let ρ and S be the Weyl concomitants

Then, the necessary and sufficient conditions for g to be the Reissner-Nordström metric are:

where x is an arbitrary unitary time-like vector.

If we make χ = 0 in the above theorem we recover the conditions of theorem 2 which characterize the Schwarzschild solution. Thus, theorem 3 includes the vacuum case.

Ideal characterization of the Lemaître-Tolman-Bondi solution

The Lemaître-Tolman-Bondi metrics (see, for example [START_REF] Stephani | Exact Solutions of Einstein's Field Equations[END_REF]) are the spherically symmetric dust solutions such that the gradient of the warped factor λ is not collinear with the velocity of the fluid. These constraints and the energy condition can be explicitly written in terms of the Ricci tensor. Then we obtain:

Lemma 6 A metric tensor g is the Lemaître-Tolman-Bondi solution if, and only if, it satisfies the conditions of theorem 1 and:

where T ≡ R -1 2 rg. We can simplify the conditions in this lemma considering the alternative characterization given in proposition 6 since now B = -r 2 Π = 0. Then β = -r 2 , and the curvature κ takes the expression κ = -2 ρ + 1 4 Φ 2 + 1 3 r. Thus, we obtain the following ideal characterization of the Lemaître-Tolman-Bondi geometry: Theorem 4 Let W = W(g) and R = Ric(g) be the Weyl and Ricci tensors of a metric g, and let ρ, S, Φ, r, B and κ be the Riemann concomitants

Then, the necessary and sufficient conditions for g to be the Lemaître-Tolman-Bondi metric are:

(b) Square and trace as an endomorphism:

(c) Action on a vector x, as an endomorphism A(x), and as a quadratic form A(x, x):

(e) Exterior product with a vector x as double 1-forms: A ∧ x is the vector-valued 2-form,

(f) Symmetrized tensorial product with a vector x as double 1-forms: A The explicit expression of the canonical 2-form U of a type D Weyl tensor using real formalism has been given in a recent paper [19]. Here we particularize it for the case of real eigenvalues:

Lemma 7 For a Petrov-Bel type D Weyl tensor with real eigenvalue ρ ≡ -( 1 12 tr W 3 ) 1 3 , the canonical 2-form U can be obtained as:

where Z is an arbitrary 2-form and