The Diffiety Institute Preprint Series

Preprint DIPS–13 November 22, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Vacuum Einstein metrics with bidimensional Killing leaves I-Local aspects

by

G. Sparano, G. Vilasi, A. M. Vinogradov

Vacuum Einstein metrics with bidimensional Killing leaves I-Local aspects

G. Sparano, G. Vilasi, A. M. Vinogradov

Abstract. The solutions of vacuum Einstein’s field equations, for the class of Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with bidimensional leaves.

Subj. Class.: Differential Geometry, General Relativity.

Keywords: Einstein metrics, Killing vectors.

MSclassification: 53C25

Contents

1. Introduction 2 2. Metrics admitting a bidimensional Lie algebra G_{2} of Killing fields. 3 3. Killing leaves 5 4. The Ricci tensor field 7 4.1. The Ricci tensor in the case det F <0 9 4.2. The Ricci tensor field in the case F >0. 11 5. Solutions of vacuum Einstein field equations 11 5.1. Solutions of Einstein equations in the case det F <0 12 5.1.1. The case h_{22} ≠ 0 12 5.1.2. The case h_{22} = 0. 14 5.2. Solutions of Einstein equations in the case det F >0 16 5.2.1. The case h_{22} ≠ 0 16 5.2.2. The case h_{22} = 0. 18 6. The Abelian limit (s = 0) 20 7. Ricci-flat metrics admitting a 3-dimensional Killing algebra with bidimensional

leaves 21 7.1. Kil(dx^{2} + dy^{2})-invariant Ricci-flat metrics 23 7.2. so (3)-invariant Ricci-flat metrics 24 References 26

Research supported in part by the Italian Ministero dell’ Università e della Ricerca Scientifica e Tecnologica.

1. Introduction

In this paper we describe in an exact form local solutions (metrics) of the vacuum Einstein equations assuming that they admit a Lie algebra G of Killing vector fields such that:

I.: the distribution D, generated by the vector fields belonging to G, is bidimensional, II.: the distribution D^{⊥}, orthogonal to D, is completely integrable and transversal to D.

Global, in a sense, solutions of the Einstein equations constructed on the basis of the local solutions found in this paper are discussed in the subsequent one. There can occur two qualitatively different cases according to whether the dimension of G is 2 or 3. Both of them, however, have an important feature in common, which makes reasonable to study them together. Namely, all manifolds satisfying the assumptions I and II are in a sense fibered over ζ-complex curves (see section 7 and [12]).

dim G = 2 Recall that, up to isomorphisms, there are two bidimensional Lie algebras: Abelian

and non-Abelian, which in what follows will be denoted by A_{2} and G_{2} respectively.

A metric g satisfying the assumptions I and II, with G = A_{2} or G_{2}, will be called G-integrable.

The study of A_{2}-integrable metrics were started by Belinsky, Geroch, Khalatnikov, Zakharov and others [3], [4], [7]. Some remarkable properties of the reduced, accordingly with the above symmetry assumptions, vacuum Einstein equations were discovered in 1978. In particular, a suitable generalization of the Inverse Scattering Transform, allowed to integrate the equations and to obtain solitary wave solutions [4]. Some physical consequences of these reduced equations were analyzed in a number of works (see for instance [5, 2]). This paper will be devoted to the analysis of G_{2}-integrable solutions, for which some partial results can be found in [8, 1, 6].

In this case, the Killing fields ”interact” non-trivially one another (for instance, [X, Y] = Y, for a suitable choice of the basis vectors in G), while in the Abelian case these fields are absolutely free (i.e., [X, Y] = 0). Hence, it is natural to expect that the former case is more rigid, with respect to the latter, and, as such, it allows a more complete analysis. It occurs to be the case, namely, metrics in question are parametrized by solutions of a linear equation in two independent variables, which, in its turn, depends linearly on a choice of a ζ-harmonic function. Thus, this class of solutions has a ”bilinear structure” and, hence, is subjected to two superposition laws.

dim G = 3 In this case, assumption II follows automatically from I and the local structure of this class of Einstein metrics can be explicitly described. Some well known exact solutions [10], such as, for instance, that of Schwarzschild, belong to this class.

Geometrical properties of solutions described in the paper will be discussed with more details separately.

In the paper, as it is usual, everything is assumed to be of C^{∞} class and the following terminological and notational convention are adopted.

• manifolds are assumed to be connected and C^{∞}, • metric refers to a non-degenerate symmetric (0, 2) tensor field,

• k-metric refers to a metric on a k-dimensional manifold,

• the Lie algebra of all Killing fields of a metric g is denoted by Kil(g) while the term Killing algebra refers to a subalgebra of Kil(g),

• integral submanifolds of the distribution, generated by vector fields of a Killing algebra G, are called Killing leaves,

• A_{2} stands for a bidimensional Abelian Lie algebra, while G_{2} for a non-Abelian one,

• a G-integrable metric is a metric satisfying the assumptions I and II, with G = A_{2} or G_{2}.

• the elements of a matrix will be denoted with the corresponding lower case letter, for instance A = (a_{ij}).

2. Metrics admitting a bidimensional Lie algebra G_{2} of Killing fields.

For a given s ∈ ℝ, s ≠ 0, we fix a basis {e, ε} in G_{2} such that [e, ε] = sε. It is defined uniquely up to transformations of the form

e ↦ λe + με, ε ↦ λ^{−1}ε, λ, μ ∈ ℝ, λ ≠ 0.

The parameter s is introduced in order to include, into our subsequent analysis, the Abelian case (s = 0 ) as well.

In what follows, it will be useful the following general fact.

Lemma 1. Let g be a metric on a differential manifold M. If X ≠ 0 and fX, f ∈ C^{∞} (M), are two of its Killing fields, then f is constant.

Proof. The proof results from the formula

L_{fX} (g) = fL_{X} (g) + i_{X} (g) df, (1)

where the second term in the right hand side is the symmetric product of two differential 1 -forms, and i_{X} (g) the natural insertion of X in g. Indeed, L_{X} (g) = 0 and L_{fX} (g) = 0 imply, in view of relation (1), i_{X} (g) df = 0. This shows that df vanishes at those points where i_{X} (g) ≠ 0. Since g is non-degenerate, i_{X} (g) vanishes exactly at the same points where X does. Therefore, df = 0, on supp X = \overline{\{a ∈ M|X_{a} ≠ 0\}}. On the other hand, if a Killing field vanishes on an open subset of M, then, obviously, it vanishes everywhere on M. For this reason supp X is dense on M and, so, df = 0 on M.

Let g be a metric on a manifold M admitting G_{2} as a Killing algebra. Then, for the Killing vector fields X and Y corresponding, respectively, to e and ε, one has

[X, Y] = sY. (2)

Denote by D the Frobenius distribution, possibly with singularities, generated by X and Y.

Proposition 2. The distribution D is bidimensional and in a neighborhood of a nonsingular point of D there exists a local chart (x_{α}) in M such that

X = ∂_{n−1} , Y = e^{sx_{n−1}}∂_{n}.

Proof. First of all, show that dimD = 2. Indeed, in view of the above lemma if locally X = ϕY, then ϕ is constant and X and Y commute, in contradiction with Eq.(2). Thus, the vector Y_{a} and X_{a} are independent for almost all points a ∈ M, i.e. in an everywhere dense open subset M_{0} of M. Choose now a function ϕ such that the fields X and ϕY commute. In view of Eq. (2), this is equivalent to X (ϕ)+sϕ = 0. This equation admits, obviously, a solution in a neighborhood of any point a ∈ M_{0}. In a local chart (y_{μ}) in which X = \frac{∂}{∂y_{n−1}} , ϕY = \frac{∂}{∂y_{n}}, the equality X (ϕ) + sϕ = 0 looks as \frac{∂ϕ}{∂y_{n−1}} + sϕ = 0 and hence, ϕ = e^{−sy_{n−1}+λ} where the function λ does not depend on y_{n−1}. By passing now to coordinates (x_{α}) with x_{α} = y_{α}, α < n, and x_{n} = β (y_{1}, ⋯, y_{n−2}, y_{n}) one finds the desired result with β such that \frac{∂β}{∂y_{n}} = e^{−λ}. Indeed, since λ does not depend on y_{n−1}, the last equation admits a solution not depending on y_{n−1}.

Definition 1. A chart of the kind introduced in the above proposition will be called semi-adapted (with respect to X, Y).

All metrics g admitting the {X, Y } Killing algebra, i.e. such that L_{Y} (g) = L_{X} (g) = 0, are characterized by the following proposition.

Proposition 3. An n-metric g admits the vector fields X and Y as Killing fields iff in a semi-adapted chart it has the following block matrix form

M_{C} (g) = ( \begin{matrix}(g_{ij}) & (sm_{i}x_{n} + l_{i}) (−m_{i}) \\ \begin{matrix}(sm_{i}x_{n} + l_{i}) & T \\ (−m_{i}) & T \end{matrix} & \begin{matrix}s^{2}λx_{n}^{2} − 2sμx_{n} + ν & −sλx_{n} + μ \\ −sλx_{n} + μ & λ \end{matrix} \end{matrix} )

where C = {dx_{μ}} , and g_{ij}, m_{i}, l_{i}, λ, μ, ν, are functions of x_{l}, 1 ≤ l ≤ n − 2.

Proof. Indeed, the invariance with respect to X shows that the components of the metric do not depend on x_{n−1} while the invariance with respect to Y is equivalent to

∂_{n}g_{ij} = 0, ∀i, j ≤ n − 2 (3) ∂_{n}g_{n−1n−1} + sg_{nn−1} = 0 (4) ∂_{n}g_{n−1n} + sg_{nn} = 0 (5) ∂_{n}g_{nn} = 0 (6) ∂_{n}g_{in−1} + sg_{in} = 0 (7) ∂_{n}g_{in} = 0 (8)

Eq. (3) tells that, for i, j < n−1, the components g_{ij} do not depend also on x_{n}, while Eqs. (4), (5) and (6), imply that, for a, b = n − 1, n

(g_{ab}) = ( \begin{matrix}s^{2}λx_{n}^{2} − 2sμx_{n} + ν & −sλx_{n} + μ \\ −sλx_{n} + μ & λ \end{matrix} ) , (9)

where λ, μ and ν depend only on the coordinates x_{i}.

Eqs. (7) and (8) have the solution

( g_{in−1}, g_{in} ) = ( sm_{i}x_{n} + l_{i} (x_{j}) , −m_{i} (x_{j}) ) .

where l_{i} and m_{i} are arbitrary functions.

For further computations it is more convenient to work with a basis, say {e_{i}}, of vector fields invariant with respect to the Killing algebra. It is easy to see that all such fields are linear combinations of

e_{i} = ∂_{i}, e_{n−1} = ∂_{n−1} + sx_{n}∂_{n,} e_{n} = −∂_{n}. (10)

whose coefficients are G_{2}-invariant functions, i.e. not depending on x_{n−1}, x_{n} . So, the set (10) can be taken as such a basis. Obviously, the basis of differential 1 -forms Θ = {ϑ^{i}} dual to {e_{i}}

ϑ^{i} = dx_{i}, ϑ^{n−1} = dx_{n−1}, ϑ^{n} = sx_{n}dx_{n−1} − dx_{n}. (11)

is also G_{2}-invariant. The bases (10), (11) are ”slightly” non- holonomic because in the relations

[e_{μ}, e_{ν}] = C_{μν}^{α}e_{α}, dϑ^{α} = −\frac{1}{2}C_{μν}^{α}ϑ^{μ} ∧ ϑ^{ν},

all the structure constants C_{μν}^{α} are vanishing, except C_{n−1n}^{n}, which equals −s. They will be called non-holonomic semi-adapted.

The expression of the metric of proposition 3 in terms of the basis (11) is

g = g_{ij}ϑ^{i}ϑ^{j} + λ ϑ^{n}ϑ^{n} + ν ϑ^{n−1}ϑ^{n−1} − 2μϑ^{n−1}ϑ^{n} + 2l_{i} ϑ^{i}ϑ^{n−1} + 2m_{i}ϑ^{i}ϑ^{n}.

Corollary 4. An n-metric g admits the vector fields X and Y as Killing fields iff its components, in a semi-adapted non-holonomic basis Θ, do not depend on x_{n−1} and x_{n}. The matrix of g with respect to the basis Θ is

M_{Θ} (g) = ( \begin{matrix}(g_{ij}) & (l_{i}) & (m_{i}) \\ (l_{i})^{T} & ν & −μ \\ (m_{i})^{T} & −μ & λ \end{matrix} ) .

3. Killing leaves

The assumption II of the Introduction imposed on the metrics g considered in this paper allows, obviously, to construct semi-adapted charts, {x_{i}}, such that the fields e_{i} = \frac{∂}{∂x_{i}}, i = 1, .., n − 2, belong to D^{⊥}. In such a chart, called from now on, adapted, the components l_{i}’s and m_{i}’s vanish. The corresponding non-holonomic semi-adapted bases will be called non-holonomic adapted.

We will call orthogonal leaf an integral (bidimensional) submanifold of D^{⊥}. Since D^{⊥} is assumed to be transversal to D, the restriction of g to any Killing leaf, say S, is non-degenerate. So, (S, g|_{S}) is a homogeneous bidimensional Riemannian manifold. In particular, the Gauss curvature K = K (S) of the Killing leaves is constant. It can be easily computed by noticing that the matrix of the components of g|_{S} with respect to the chart x̃ = x_{n−1}|_{S} , ỹ = x_{n}|_{S} is

M_{(dx̃,dỹ)} (g|_{S}) = ( \begin{matrix}s^{2}λ̃ỹ^{2} − 2sμ̃ỹ + ν̃ & −sλ̃ỹ + μ̃ \\ −sλ̃ỹ + μ̃ & λ̃ \end{matrix} ) ,

where the symbol ”tilde” refers to the restriction to S and λ̃, μ̃, and ν̃ are constants according to proposition 3. The result is

K (S) = \frac{λ̃s^{2}}{μ̃^{2} − λ̃ν̃}, λ̃ν̃ − μ̃^{2} = M_{(dx̃,dỹ)} (g|_{S}) .

This shows that the following cases can occur for (S, g|_{S}).

1. λ̃ > 0, λ̃ν̃ − μ̃^{2} > 0: (S, g|_{S}) is a non-Euclidean plane, i.e. a bidimensional Riemannian manifold of negative constant Gauss curvature.

2. λ̃ < 0, λ̃ν̃ − μ̃^{2} > 0: (S, g|_{S}) is an ”anti” non-Euclidean plane, i.e. is endowed with the metric of the previous case multiplied by −1.

3. λ̃ν̃ − μ̃^{2} < 0: (S, g|_{S}) is any indefinite bidimensional metric of constant Gauss curvature.

Since the Killing leaves are parametrized by x_{1}, x_{2}, the function

K = K (x_{1}, .., x_{n−2}) = \frac{λs^{2}}{μ^{2} − λν}

describes the behavior of the Gauss curvature when passing from one Killing leave to another.

It is worth to note that the Killing algebra G_{2} is a subalgebra of the algebra Kil (g_{0}), g_{0} being a bidimensional metric of constant curvature (for instance, g_{0} = g|_{S}).

If g_{0} is positive (respectively, negative) definite and of positive (respectively, negative) Gauss curvature, then Kil (g_{0}) is isomorphic to so (3). But so (3) does not admit bidimensional subalgebras at all. This explains why g|_{S} cannot be a positively (respectively, negative) curved metric in the case (1) ( respectively, (2)).

Similarly, if g_{0} is a positive or negative definite flat metric, then Kil (g_{0}) admits only Abelian bidimensional subalgebras. This explains why both positive and negative definite flat metrics are absent in the above list for g|_{S}.

In all other cases, the algebra Kil (g_{0}) admits bidimensional non-Abelian subalgebras. More exactly, if g_{0} is not flat, then Kil(g_{0}) is isomorphic to so (2, 1). Let g be the

Killing form of so (2, 1). Then, the tangent planes to the isotropic cone of g exhaust the bidimensional non-Abelian Lie subalgebras of so (2, 1). If g_{0} is flat and, thus, indefinite, then any bidimensional subspace of the algebra Kil (g_{0}) different from its commutator, which is Abelian, is a non-Abelian subalgebra.

It is not difficult to describe the algebra Kil (g|_{S}) in the semi-adapted coordinates (x̃, ỹ). A direct computation shows that Kil (g_{0}) has the following basis:

X̃ = ∂x̃, Ỹ = e^{sx̃}∂ỹ, Z̃ = e^{−sx̃} [2 (sλ̃ỹ − μ̃) ∂x̃ + (s^{2}λ̃ỹ^{2} − 2sμ̃ỹ + ν̃) ∂ỹ] ,

[X̃, Ỹ] = sỸ, [X̃, Z̃] = −sZ̃, [Ỹ, Z̃] = 2sλ̃X̃,

In the case λ = 0, the metric g|_{S} is flat indefinite and it is convenient to identify (S, g|_{S}) with the standard plane (ℝ^{2}, dξ^{2} − dη^{2}), ℝ^{2} = {(ξ, η)}. To do that it is necessary to choose a bidimensional non-commutative subalgebra in Kil(dξ^{2} − dη^{2}) (they are all equivalent). For instance, by choosing Y_{0} = ∂_{ξ} + ∂_{η}, X_{0} = −η∂_{ξ} − ξ∂_{η}, we have [X_{0}, Y_{0}] = Y_{0} , X_{0}, Y_{0} ∈ Kil(dξ^{2} − dη^{2}) and, for s ≠ 0, one can identify the quadruple (S, 2 (dx̃dỹ − ỹdx̃^{2}) , X|_{S} , Y |_{S}) with (ℝ^{2}, dξ^{2} − dη^{2}, X_{0}, Y_{0} ).

The simply connected Lie group G corresponding to G is isomorphic to the group of affine transformations of ℝ. Then, both S and ℝ^{2} are diffeomorphic to G as homogeneous G−spaces and the above identification of them is an equivalence of G−spaces.

The Killing form of G determines naturally a symmetric covariant tensor field on the

G−space G which is identified with dx̃^{2} on S and with (\frac{dξ−dη}{ξ−η})^{2} on ℝ^{2}. We will continue to call it Killing form. Thus, in the above identification the metric g|_{S} for λ = 0 and s = 0 corresponds to

μ̃ (dξ^{2} − dη^{2}) + ν̃ (\frac{dξ − dη}{ξ − η} )^{2} . (12)

This representation of the metric g|_{S} will be used to describe global solutions of the Einstein equations in section 5.

4. The Ricci tensor field

In the following we will consider 4-dimensional manifolds and will use the following convention for the indices: Greek letters take values from 1 to 4; the first Latin letters take values from 3 to 4, while i, j from 1 to 2.

Let g be a G_{2}-integrable 4-metric. The results of the previous sections allow to choose a non-holonomic adapted basis Θ such that the matrix M_{Θ} (g) associated to g is of the form

M_{Θ} (g) = ( \begin{matrix}F & 0 \\ 0 & H \end{matrix} ) (13)

where F and H are 2 × 2 matrices whose elements depend only on x_{1} and x_{2}. We will distinguish two cases according to whether F, i.e., the matrix associated to the metric restricted to D^{⊥}, has negative or positive determinant.

• det F < 0. In this case, owing to the bidimensionality of D^{⊥}, and the independence of F on x_{3} and x_{4}, the coordinates x_{1} and x_{2}, can be further specified to be characteristic coordinates on any integral submanifold of D^{⊥}, so that, without changing the properties of M_{Θ} (g) in (13), F takes the following form

F = ( \begin{matrix}0 & f \\ f & 0 \end{matrix} ) .

• det F > 0. Similarly, in this case, in some isothermal coordinates, the matrix F gets the form

F = ( \begin{matrix}f & 0 \\ 0 & f \end{matrix} )

Thus, we have:

Proposition 5. A 4-metric g, is G_{2}-integrable iff there exists a non-holonomic adapted basis Θ such that the matrix M_{Θ} (g) of g takes one of the following block forms, according to whether det F < 0 or det F > 0 .

M_{Θ} (g) = ( \begin{matrix}\begin{matrix}0 & f \\ f & 0 \end{matrix} & 0 \\ 0 & H \end{matrix} ) , M_{Θ} (g) = ( \begin{matrix}\begin{matrix}f & 0 \\ 0 & f \end{matrix} & 0 \\ 0 & H \end{matrix} )

H = ( \begin{matrix}ν & −μ \\ −μ & λ \end{matrix} )

λ, μ, ν being arbitrary functions of x_{i}. In the corresponding adapted holonomic basis C = {dx^{μ}} we have

M_{C} (g) = ( \begin{matrix}\begin{matrix}0 & f \\ f & 0 \end{matrix} & 0 \\ 0 & \overline{H} \end{matrix} ) , M_{C} (g) = ( \begin{matrix}\begin{matrix}f & 0 \\ 0 & f \end{matrix} & 0 \\ 0 & \overline{H} \end{matrix} )

where

\overline{H} = ( \begin{matrix}s^{2}λx_{4}^{2} − 2sμx_{4} + ν & −sλx_{4} + μ \\ −sλx_{4} + μ & λ \end{matrix} ) .

It is worth to observe that det \overline{H} = det H = λν − μ^{2} is a functions of x_{i}’s only.

In the following sections the explicit expressions of the components R_{μν} of the Ricci tensor field in terms of the function f and of the elements h_{ab} of the matrix H in the adapted non-holonomic basis of corollary 5 are found.

Recall that

R_{μν} = R_{μ νβ}^{β} = e_{[ν} (γ_{β]μ}^{β}) + γ_{[ν ρ}^{β}γ_{β]μ}^{ρ} − C_{νβ}^{ρ}γ_{ρμ}^{β}.

with the Christoffel symbols

γ_{μν}^{α} = \frac{1}{2}g^{ασ} (−e_{σ} (g_{μν}) + e_{μ} (g_{σν}) + e_{ν} (g_{σμ}))

− \frac{1}{2} (C_{νμ}^{α} + g^{ρα}g_{σμ}C_{νρ}^{σ} + g^{ρα}g_{σν}C_{μρ}^{σ}) .

It is easy too see that the γ_{μν}^{α}’s and R_{μν}’s are first order polynomials in s and it is convenient to single out their constant terms Γ_{μν}^{α} and S_{μν}, respectively. More exactly, one has:

γ_{μ} = Γ_{μ} + Λ_{μ} = \frac{1}{2}g^{−1}G_{μ} + Λ_{μ}

where γ_{μ}, Γ_{μ}, Λ_{μ},G_{μ} are matrices whose elements γ_{μν}^{α}, Γ_{μν}^{α}, Λ_{μν}^{α}, G_{μαν}, are defined by

Γ_{μν}^{α} = \frac{1}{2}g^{ασ} (−e_{σ} (g_{μν}) + e_{μ} (g_{σν}) + e_{ν} (g_{σμ}))
(14)  Λ_{μν}^{α} = −\frac{1}{2} (C_{νμ}^{α} + g^{ρα}g_{σμ}C_{νρ}^{σ} + g^{ρα}g_{σν}C_{μρ}^{σ})
G_{μσν} = −e_{σ} (g_{μν}) + e_{μ} (g_{σν}) + e_{ν} (g_{σν}) .

R_{μν} = S_{μν} + T_{μν}

where

S_{μν} = e_{[ν} Γ_{β]μ}^{β} + Γ_{[ν ρ}^{β}Γ_{β]μ}^{ρ}
T_{μν} = e_{[ν} Λ_{β]μ}^{β} + (Γ_{[ν} Λ_{β]})_{μ}^{β} + (Λ_{[ν} Γ_{β]})_{μ}^{β} + (Λ_{[ν} Λ_{β]})_{μ}^{β} − C_{νβ}^{ρ}γ_{ρμ}^{β}.

Now we pass to the calculation of the Ricci tensor.

4.1. The Ricci tensor in the case det F <0. Note that for s = 0 the adapted nonholonomic basis becomes holonomic and coincides with the one used in [4]. This is why the expressions for S_{μν} given below coincide with the expressions for the components of the Ricci tensor found in [4]. Observe also that only the fields e_{1} and e_{2} give nontrivial contributions to expressions (14) for the Γ_{μν}^{α}’s and all components Λ_{μν}^{α}, except possibly Λ_{ba}^{c}, vanish.

• Components R_{ij}:

Let us note that

T_{ij} = Λ_{βρ}^{β}Γ_{ji}^{ρ} − C_{jβ}^{ρ}γ_{ρi}^{β} = 0,

this is due to the fact that C_{jβ}^{ρ} = 0, the components Λ_{μν}^{α} with an index equal to 1 or 2 vanish and Γ_{ij}^{a} = 0. So,

R_{ij} = S_{ij} = e_{[j} (Γ_{β]i}^{β}) + Γ_{[j ρ}^{β}Γ_{β]i}^{ρ}.

The first term of this expression gives,

e_{[j} (Γ_{β]i}^{β}) = ∂_{j}∂_{i} (ln |f|) − δ_{ij}∂_{i}^{2} (ln |f|) + ∂_{j}∂_{i} (ln α) ,

where α = \sqrt{|det H|} and the second term gives

Γ_{[j ρ}^{β}Γ_{β]i}^{ρ} = tr (Γ_{j}Γ_{i}) − (Γ_{β}Γ_{j})_{i}^{β} =

\frac{1}{4}tr [H^{−1}∂_{j} (H) H^{−1}∂_{i} (H)] + \frac{(∂_{i}f)^{2}}{f^{2}} δ_{ij} + δ_{ij} \frac{(∂_{i}f)^{2}}{f^{2}} +

+ δ_{ij}∂_{i} (ln |f|) ∂_{i} (ln α) .

Finally, one has

R_{ij} = ∂_{j}∂_{i} (ln |f|) − δ_{ij}∂_{i}^{2} (ln |f|) + ∂_{j}∂_{i} (ln α) +

\frac{1}{4}tr [H^{−1}∂_{j} (H) H^{−1}∂_{i} (H)] − δ_{ij}∂_{i} (ln |f|) ∂_{i} (ln α) .

• Components R_{ab} = S_{ab} + T_{ab}: For what concerns S_{ab}, it is more convenient to use the following expression

S_{ab} = \frac{1}{\sqrt{|det g|}}∂_{ρ} (\sqrt{|det g|}Γ_{ab}^{ρ}) − ∂_{a}∂_{b} (ln \sqrt{|det g|}) − Γ_{ρa}^{β}Γ_{βb}^{ρ}

taking into account that |det g| ≡ |det F| |det H| = f^{2}α^{2} and α = \sqrt{|det H|}. The result is

(S_{ab}) = \frac{1}{2fα}H [(αH^{−1}∂_{1} (H))_{,2} + (αH^{−1}∂_{2} (H))_{,1}] .

For T_{ab} one finds

T_{ab} = e_{[b} (Λ_{β]a}^{β}) + (Γ_{[b} Λ_{β]})_{a}^{β} + (Λ_{[b} Γ_{β]})_{a}^{β} + (Λ_{[b} Λ_{β]})_{a}^{β} − C_{bβ}^{ρ}γ_{ρa}^{β}
= −C_{bβ}^{ρ}γ_{ρa}^{β},

so that

(T_{ab}) = s^{2}h_{22} (det H^{−1}) H.

and

(R_{ab}) = \frac{1}{2fα}H [(αH^{−1}∂_{2} (H))_{,1} + (αH^{−1}∂_{1} (H))_{,2} + \frac{2s^{2}}{α} fh_{22}1_{2}] ,

where 1_{2} stands for the unit (2 × 2)- matrix.

• Components R_{ai}: In this case,

S_{ai} = e_{[i} (Γ_{β]a}^{β}) + Γ_{[i ρ}^{β}Γ_{β]a}^{ρ} = 0,

Indeed, the first term vanishes since Γ_{i}’s are diagonal and Γ_{a} are anti-diagonal. The second term also vanishes since the matrices Γ_{i}Γ_{j} are diagonal while Γ_{i}Γ_{b} or Γ_{b}Γ_{i} anti-diagonal. Thus,

R_{ai} = T_{ai}

and

T_{ai} = e_{i} (Λ_{ba}^{b}) + (Γ_{[i} Λ_{β]})_{a}^{β} + (Λ_{[i} Γ_{β]})_{a}^{β} + (Λ_{[b} Λ_{β]})_{a}^{β} − C_{iβ}^{ρ}γ_{ρa}^{β}
= (Γ_{i}Λ_{b})_{a}^{b} − (Λ_{b}Γ_{i})_{a}^{b}

or, equivalently,

( \begin{matrix}T_{3i} \\ T_{4i} \end{matrix} ) = s ( \begin{matrix}(H^{−1}∂_{i} (H))_{2}^{2} − (H^{−1}∂_{i} (H))_{1}^{1} \\ −2 (H^{−1}∂_{i} (H))_{2}^{1} \end{matrix} ) .

So, the final result is

(R_{i3}, R_{i4}) = s ((H^{−1}∂_{i} (H))_{2}^{2} − (H^{−1}∂_{i} (H))_{1}^{1} , −2 (H^{−1}∂_{i} (H))_{2}^{1}) .

The above calculations are summarized in the following proposition

Proposition 6. Let g be a G_{2}-integrable 4-metric. If det F < 0, then the components of the Ricci tensor in a non-holonomic adapted basis are

(R_{ab}) = \frac{H}{2fα} [(αH^{−1}∂_{1} (H))_{,2} + (αH^{−1}∂_{2} (h))_{,1} + \frac{2s^{2}}{α} fh_{22}1_{2}]
R_{12} = ∂_{1}∂_{2} (ln |f| + ln α) + \frac{1}{4}tr [H^{−1}∂_{1} (H) H^{−1}∂_{2} (H)]
R_{ii} = −∂_{i} (ln α) ∂_{i} (ln |f|) + ∂_{i}^{2} (ln α) + \frac{1}{4}tr [H^{−1}∂_{i} (H) H^{−1}∂_{i} (H)]
( \begin{matrix}R_{i3} \\ R_{i4} \end{matrix} ) = s ( \begin{matrix}(H^{−1}∂_{1} (H))_{2}^{2} − (H^{−1}∂_{1} (H))_{1}^{1} & −2 (H^{−1}∂_{1} (H))_{2}^{1} \\ (H^{−1}∂_{2} (H))_{2}^{2} − (H^{−1}∂_{2} (H))_{1}^{1} & −2 (H^{−1}∂_{2} (H))_{2}^{1} \end{matrix} )

with α = \sqrt{|det H|} .

Remark 1. Note that for s = 0 the above expressions for the components of the Ricci tensor field coincide with the ones given in [4]. In particular, the components R_{ai} vanish identically.

4.2. The Ricci tensor field in the case F >0. We use again the adapted nonholonomic basis Θ described in proposition 2, so that the matrix of g is

M_{Θ} (g) = ( \begin{matrix}\begin{matrix}2f & 0 \\ 0 & 2f \end{matrix} & 0 \\ 0 & H \end{matrix} ) = ( \begin{matrix}F & 0 \\ 0 & H \end{matrix} ) .

In this case essentially the same computation as before gives the following result.

Proposition 7. Let g be a G_{∈}-integrable 4-metric. If det F > 0, then the components of the Ricci tensor in a non-holonomic adapted basis are

(R_{ia}) = s ( \begin{matrix}(H^{−1}∂_{1} (H))_{2}^{2} − (H^{−1}∂_{1} (H))_{1}^{1} & −2 (H^{−1}∂_{1} (H))_{2}^{1} \\ (H^{−1}∂_{2} (H))_{2}^{2} − (H^{−1}∂_{2} (H))_{1}^{1} & −2 (H^{−1}∂_{2} (H))_{2}^{1} \end{matrix} );
(R_{ab}) = \frac{H}{2fα} [\frac{1}{2} [(αH^{−1}∂_{1} (H))_{,1} + (αH^{−1}∂_{2} (Hh))_{,2}] + \frac{2s^{2}}{α} fh_{22}1_{2}];
R_{11} = \frac{1}{2} [△ (ln α ln |f|) + \frac{1}{2}tr (H^{−1}∂_{1}H)^{2} − \frac{α,_{1}}{α} ∂_{1} (ln |f|)] +

+ \frac{1}{2} [\frac{α,_{2}}{α} ∂_{2} (ln |f|) + ∂_{1} (\frac{α,_{1}}{α} ) − ∂_{2} (\frac{α,_{2}}{α} )];

R_{22} = \frac{1}{2} [△ (ln α ln |f|) + \frac{1}{2}tr ( H^{−1}∂_{2}H)^{2} + \frac{α,_{1}}{α} ∂_{1} (ln |f|)] +

− \frac{1}{2} [\frac{α,_{2}}{α} ∂_{2} (ln |f|) − ∂_{1} (\frac{α,_{1}}{α} ) + ∂_{2} (\frac{α,_{2}}{α} )];

R_{12} = \frac{1}{2} [−\frac{α,_{1}}{α} ∂_{2} (ln |f|) − \frac{α,_{2}}{α} ∂_{1} (ln |f|) + 2∂_{1}∂_{2} (ln α)] +

+ \frac{1}{4}tr [ H^{−1}∂_{1} (H) H^{−1}∂_{2} (H)];

with

△ = \frac{∂^{2}}{∂x_{1}^{2}} + \frac{∂^{2}}{∂x_{2}^{2}} .

Remark 2. Also in this case the components R_{ai} vanish identically for s = 0.

5. Solutions of vacuum Einstein field equations

In this section we will limit ourselves to discuss only the general form of local solutions of vacuum Einstein equations

R_{μν} = 0

for G_{2}-integrable normal (see after) metrics.

Let us consider separately the cases characterized by det F < 0 and det F > 0 .

5.1. Solutions of Einstein equations in the case det F <0. Note that, for s = 0 (Abelian case) the equations R_{ai} = 0 become identities, while for s ≠ 0 they impose the following strong conditions on the metric:

\{ \begin{matrix}(H^{−1}∂_{i} (H))_{2}^{2} = (H^{−1}∂_{i} (H))_{1}^{1} \\ (H^{−1}∂_{i} (H))_{2}^{1} = 0 \end{matrix} . (15)

The two cases h_{22} ≠ 0 and h_{22} = 0 are qualitatively different and will be discussed separately.

5.1.1. The case h_{22} ≠ 0. In this case equations (15) imply that (H^{−1}∂_{i} (H))_{1}^{2} = 0 for any symmetric (2 × 2)-matrix H. This means that H^{−1}∂_{i} (H) is a scalar matrix, i.e.,

∂_{1} (H) = φH, ∂_{2} (H) = ψH

for some functions φ = φ (x_{i}), ψ = ψ (x_{i}).

The compatibility condition ∂_{2} (φ) = ∂_{1} (ψ) for the above system, implies the existence (locally) of a function γ (x_{i}) such that φ = ∂_{1} (γ), ψ = ∂_{2} (γ) . The function γ can be chosen in such a way that H = e^{γ}M, M being a constant symmetric (2 × 2)-matrix such that det M = ±1. Thus,

α = e^{γ}.

Then the equations R_{ab} = 0 can be written as

α,_{12} +s^{2}fm_{22} = 0, (16)

or

f = cα,_{12} ,

α,_{i} ≡ ∂_{i} (α) , α,_{ij} ≡ ∂_{i}∂_{j} (α), and

c = −\frac{1}{s^{2}m_{22}} .

This brings Einstein equations to the form

(17)  H = e^{γ}M = αM
(18)  f = cα,_{12}
(19)  ∂_{i} (ln |f|) = ∂_{i} (ln \frac{|α,_{i} |}{\sqrt{α}})
(20)  ∂_{1}∂_{2} (ln |f|) = −\frac{1}{α}α,_{12} +\frac{1}{2α^{2}} α,_{1} α,_{2} .

For the two possible values of the index i Eq. (19) gives

(21)  f = H (x_{2}) ∂_{1} (\sqrt{α})

= K (x_{1}) ∂_{2} (\sqrt{α}) (22)

where H and K are arbitrary functions, or, equivalently,

H∂_{1}α = K∂_{2}α. (23)

From Eq. (18) one gets

α,_{1} = \frac{1}{c}K (\sqrt{α} − A) , α,_{2} = \frac{1}{c}H (\sqrt{α} − A) ,

where A is a constant, or, equivalently,

dα = \frac{1}{c} (\sqrt{α} − A) (Kdx_{1} + Hdx_{2}) .

By setting β^{2} = α the above equation integrates to the equality

β + A ln |β − A| = F (x_{1}) + G (x_{2}) ,

with F (x_{1}) ≡ \frac{1}{2c} ∫ Kdx_{1}, G (x_{2}) ≡ \frac{1}{2c} ∫ Hdx_{2}. The above equation will be called the tortoise equation. Finally, the remaining Einstein equations show Eq. (20) to be an identity.

By summing up we give the components of the metric in the basis C = {dz_{1}, dz_{2}, dx, dy} with z_{1} = \frac{1}{2} (x_{1} + x_{2}) , z_{2} = \frac{1}{2} (x_{1} − x_{2}) , x = x_{3}, y = x_{4}, where the x_{μ}’s are the adapted coordinates mentioned in proposition 4.

Proposition 8. Any G_{2}-integrable 4-metric g satisfying the vacuum Einstein equations, and such that det F < 0 and h_{22} ≠ 0, has in the adapted coordinate (z_{1}, z_{2}, x, y) the following matrix form

M_{C} (g) = ( \begin{matrix}\begin{matrix}2f & 0 \\ 0 & −2f \end{matrix} & 0 \\ 0 & β^{2} ( \begin{matrix}s^{2}ky^{2} − 2sly + m & −sky + l \\ −sky + l & k \end{matrix} ) \end{matrix} )

where • k, l, m, are arbitrary constants such that km − l^{2} = ±1, k ≠ 0, •

f = −\frac{1}{4s^{2}k} (\frac{∂^{2}}{∂z_{1}^{2}} − \frac{∂^{2}}{∂z_{2}^{2}} ) β^{2}, (24)

• β is a solution of the tortoise equation

β + A ln |β − A| = F (z_{1} + z_{2}) + G (z_{1} − z_{2}) , (25)

A, F, G being an arbitrary constant and arbitrary functions respectively.

Remark 3. As it will be clarified in [12], the tortoise equation(25) leads to a deeper understanding of the so called Regge-Wheeler tortoise coordinate, which, apart from constant terms, is defined as its left hand side.

Remark 4. Concerning the signature of the metric and the character of the Killing fields, we observe that: If det M = 1 (see Eq. (17)), then H is either positive or negative definite according to the sign of k and g (Y, Y), g (X, X) have the same sign as k. The signature of g is equal to ±2, so that these metrics are of interest for general relativity; If det M = −1, then H is indefinite, g (Y, Y) has again the same sign as k while the sign of g (X, X) varies depending on the values of y. The signature of g in this case is equal to 0.

By using the results of section 3, we have:

Corollary 9. The metric g of the above proposition admits an additional Killing field

Z = e^{−sx} [2 (sky − l) ∂_{x} + (s^{2}ky^{2} − 2sly + m) ∂_{y}] ,

which generates together with X = ∂_{x} and Y = e^{sx}∂_{y} a 3-dimensional Lie algebra isomorphic to so (2, 1) (assuming that s ≠ 0):

[X, Y] = sY, [X, Z] = −sZ, [Y, Z] = 2skX

5.1.2. The case h_{22} = 0. Now, Eqs. (15) are identically satisfied, while the remaining Einstein equations become

(αH^{−1}∂_{1}H) ,_{2} + (αH^{−1}∂_{2}H) ,_{1} = 0 (26)

(27) ∂_{1}∂_{2} (ln |f| + ln α) + \frac{1}{4}tr [H^{−1}∂_{1} (H) H^{−1}∂_{2} (H)] = 0

(28) −∂_{i} ln |α| ∂_{i} ln |f| + ∂_{i}^{2} ln α + \frac{1}{4}tr [ H^{−1}∂_{i} (H) H^{−1}∂_{i} (H)] = 0.

In terms of the components μ and ν of H they reduce to

α,_{12} = 0 (29a) (αw_{,1})_{,2} + (αw_{,2})_{,1} = 0 (29b)

∂_{1}∂_{2} (ln |f|) = \frac{α,_{2} α,_{1}}{2α^{2}} (29c)

α,_{i} ∂_{i} (ln |f|) = α,_{ii} −\frac{α,_{i}^{2}}{2α} , (29d)

with α = \sqrt{|det H|} = |μ| and w = \frac{ν}{α}.

The general solution of Eq. (29a) is

α = F (x_{1}) + G (x_{2}) ,

F and G being arbitrary functions such that α is positive.

The general solution of Eq. (29c) is

f = ±α^{− \frac{1}{2}} e^{P (x_{1})+Q(x_{2})}

where P and Q are arbitrary functions.

Now equation (29d) takes the form

P′ (x_{1}) α,_{1} = α,_{11}
Q′ (x_{2}) α,_{2} = α,_{22}

and are resolved as

F = C_{1} ∫ e^{P} dx_{1} + D_{1} G = C_{2} ∫ e^{Q}dx_{2} + D_{2}.

Thus as the final result we see that the general solution of the differential system (29a), (29c), (29d) is given by

α = C_{1} ∫ e^{P} dx_{1} + C_{2} ∫ e^{Q}dx_{2} + C
f = ±α^{− \frac{1}{2}} e^{P (x_{1})+Q(x_{2})}

where C, C_{1}, C_{2}, are arbitrary constants such that α is positive.

Eq. (29b) is a linear second order partial differential equation and can be studied by standard methods. We postpone this problem to a further publication.

As in proposition 8 we summarize the obtained results by giving the components of g in the frame C = \{dz_{1}, dz_{2}, dx, dy\} where z_{1} = \frac{1}{2} (x_{1} + x_{2}) , z_{2} = \frac{1}{2} (x_{1} − x_{2}) , x = x_{3}, y = x_{4}, and x_{μ}’ s are the adapted coordinates introduced in proposition 4.

Proposition 10. Any G_{2}-integrable 4-metric g satisfying the vacuum Einstein equations and such that det F < 0 and h_{22} = 0, has the following matrix form in the adapted coordinates (z_{1}, z_{2}, x, y),

M_{C} (g) = ( \begin{matrix}\begin{matrix}2f & 0 \\ 0 & −2f \end{matrix} & 0 \\ 0 & μ ( \begin{matrix}−2sy + w & 1 \\ 1 & 0 \end{matrix} ) \end{matrix} )

where •

(30)  μ = C_{1}F (z_{1} + z_{2}) + C_{2}G (z_{1} − z_{2}) + C
(31)  f = |μ|^{− \frac{1}{2}} F′G′

F, G and C, C_{1}, C_{2} being arbitrary functions and arbitrary constants respectively, such that μ and f are everywhere nonvanishing;

• w is an arbitrary solution of the equation

μ (\frac{∂^{2}}{∂z_{1}^{2}} − \frac{∂^{2}}{∂z_{2}^{2}} ) w + \frac{∂μ}{∂z_{1}} \frac{∂w}{∂z_{1}} − \frac{∂μ}{∂z_{2}} \frac{∂w}{∂z_{2}} = 0.

In this case, det H < 0 and the metric g has signature equal to 0. The Killing field Y is isotropic, while the sign of g (X, X) varies as a function of y. The curvature K of the Killing leaves vanishes.

Remark 5. In contrast with the case h_{22} ≠ 0 (see 5.1.1) an additional Killing field, say Z, tangent to the Killing leaves and independent on X and Y exists only if w is a constant, say w_{0}. In such a case

Z = e^{−sx} [−2∂_{x} + (−2sy + w_{0}) ∂_{y}] ,

and generates together to X = ∂_{x} and Y = e^{sx}∂_{y} a 3-dimensional Lie algebra isomorphic to Kil (dx^{2} − dy^{2}):

[X, Y] = sY, [X, Z] = −sZ, [Y, Z] = 0

A canonical form for Eq. (29b) may be obtained by passing to coordinates

ξ = F (x_{1}) , η = G (x_{2})

in which Eq. (29b) becomes

2 (ξ + η) \frac{∂^{2}w̃}{∂ξ∂η} + \frac{∂w̃}{∂ξ} + \frac{∂w̃}{∂η} = 0,

with w̃ (ξ, η) ≡ w (F^{−1} (ξ) , G^{−1} (η)), or, alternatively,

\frac{∂^{2}Z}{∂ξ∂η} + \frac{1}{4 (ξ + η)^{2}} Z = 0, Z = \sqrt{ξ + η}w̃.

Its geometrical interpretation is given in [12].

5.2. Solutions of Einstein equations in the case det F >0. As before, the equations R_{ai} = 0 are satisfied trivially if s = 0 while for s ≠ 0 they coincide with (15):

\{ \begin{matrix}(H^{−1}∂_{i} (H))_{2}^{2} = (H^{−1}∂_{i} (H))_{1}^{1} \\ (H^{−1}∂_{i} (H))_{3}^{2} = 0 \end{matrix} . (32)

Again it is convenient to treat separately the cases h_{22} ≠ 0 and h_{22} = 0.

5.2.1. The case h_{22} ≠ 0. As in sec. 5.1.1, equations R_{ia} = 0 are solved as

H = e^{γ}M .

M being a constant symmetric (2 × 2)-matrix such that det M = ±1 and α = e^{γ}. Because of the non-degeneracy of g the first derivatives of α are non-vanishing, so that Einstein equations can be brought to the following form

H = αM , (33)

(\frac{△ (α)}{4f} + s^{2}m_{22}) M = 0, (34)

△ (ln α |f|) − \frac{1}{αf} (α,_{1} f,_{1} −α,_{2} f,_{2} ) + \frac{(α,_{2} )^{2}}{α^{2}} + \frac{α,_{11} −α,_{22}}{α} = 0, (35)

△ (ln α |f|) + \frac{1}{αf} (α,_{1} f,_{1} −α,_{2} f,_{2} ) + \frac{(α,_{1} )^{2}}{α^{2}} − \frac{α,_{11} −α,_{22}}{α} = 0, (36)

\frac{1}{2αf} (α,_{1} f,_{2} +α,_{2} f,_{1} ) + \frac{α,_{2} α,_{1}}{2α^{2}} − \frac{α,_{12}}{α} = 0. (37)

In its turn the last system is equivalent to

H = αM

f = \frac{c}{4}△α

∂_{1} [ln |f| − \frac{1}{2} (ln α + ln \frac{|∇ (α)|^{2}}{α^{2}} )] = −ϑ_{2}

∂_{2} [ln |f| − \frac{1}{2} (ln α + ln \frac{|∇ (α)|^{2}}{α^{2}} )] = ϑ_{1}

where c = −\frac{1}{s^{2}m_{22}} and ϑ_{1} and ϑ_{2} are the partial derivatives of

ϑ = arctan \frac{α,_{2}}{α,_{1}} .

These equations show that ϑ and ln \frac{\sqrt{α}|f|}{|∇(α)|} are conjugated harmonic functions so that the above system can be brought to the form:

△ (ϑ) = 0

\frac{α,_{2}}{α,_{1}} = tan ϑ (38)

ln \frac{\sqrt{α} |△ (α)|}{|∇ (α)|} = Φ (39)

f = \frac{c}{4}△ (α) (40)

H = αM

where Φ is a harmonic function conjugated to ϑ, that is a primitive of the exact differential 1-form ω = ϑ_{1}dx_{2} − ϑ_{2}dx_{1}. Now one can easily check that the above system is reduced to the tortoise equation (see sec. 5.1.1)

β + A ln |β − A| = Ψ

where β^{2} = α, Ψ is an arbitrary harmonic function and A is an arbitrary constant. The functions ϑ and Φ are given, respectively, by

ϑ = arctan \frac{Ψ,_{2}}{Ψ,_{1}}
Φ = ln |∇ (Ψ)| .

By summing up we give the components of the metric in terms of the adapted holonomic frame C = {dx_{1}, dx_{2}, dx, dy} with x = x_{3}, y = x_{4}, the x_{μ}’s being the adapted coordinates introduced in proposition 4.

Proposition 11. Any G_{2}-integrable 4-metric g satisfying the vacuum Einstein equations, and such that det F > 0 and h_{22} ≠ 0, has the following matrix form in the adapted coordinates (x_{μ})

M_{C} (g) = ( \begin{matrix}\begin{matrix}2f & 0 \\ 0 & 2f \end{matrix} & 0 \\ 0 & β^{2} ( \begin{matrix}s^{2}ky^{2} − 2sly + m & −sky + l \\ −sky + l & k \end{matrix} ) \end{matrix} )

where • k, l, m, are arbitrary constants such that km − l^{2} = ±1, k ≠ 0, •

f = −\frac{1}{4s^{2}k} △ (β^{2}) , (41)

• β is a solution of the tortoise equation

β + A ln |β − A| = Ψ, (42)

such that △β^{2} ≡ (\frac{∂^{2}}{∂x_{1}^{2}} + \frac{∂^{2}}{∂x_{2}^{2}} ) β^{2} is everywhere nonvanishing, A and Ψ being an arbitrary constant and an arbitrary harmonic function.

Remark 6. Concerning the signature of g and the character of the Killing fields, we remark that: If det M = 1 (see Eq. (33)), then H is either positive or negative definite according to the sign of k as well as g (Y, Y), and g (X, X). Since the sign of the constant c is opposite to the one of k, the signature of g is always equal to 0. If det M = −1, then H is indefinite, g (Y, Y) has the same sign as k while the sign of g (X, X) varies with as a function of y. The signature of g is equal to ±2, so that these metrics are of interest for General Relativity.

Moreover, as in sec. 5.1.1 we have:

Corollary 12. The metric of the above proposition admits a third Killing field

Z = αe^{−sx_{2}} [(m − sky) ∂_{x} + (s^{2}ky^{2} − 2smy + l) ∂_{y}] ,

which together with X and Y generate a 3-dimensional Lie algebra isomorphic to so (2, 1)

[X, Y] = sZ, [X, Z] = −sZ, [Y, Z] = −2skX.

5.2.2. The case h_{22} = 0. In this case the equations R_{ia} = 0 are satisfied automatically while the matrix H has the form

H = ( \begin{matrix}ν & μ \\ μ & 0 \end{matrix} ) ,

and α = |μ|. The remaining Einstein equations reduce now to

△ (α) = 0 (43) (α∂_{1}w)_{,1} + (α∂_{2}w)_{,2} = 0 (44)

△ (ln |f|) = \frac{1}{2} [(\frac{α,_{1}}{α} )^{2} + (\frac{α,_{2}}{α} )^{2}] (45)

α,_{1} ∂_{1} (ln |f|) − α,_{2} ∂_{2} (ln |f|) = α,_{11} −α,_{22} −\frac{α,_{1}^{2} −α,_{2}^{2}}{2α} (46)

α,_{2} ∂_{1} (ln |f|) + α,_{1} ∂_{2} (ln |f|) = 2α,_{12} −\frac{α,_{2} α,_{1}}{α} . (47)

where △ = \frac{∂^{2}}{∂x_{1}^{2}} + \frac{∂^{2}}{∂x_{2}^{2}} and w = \frac{ν}{α}. If α is a solution of Eq. (43), i.e., a harmonic function, then the general solution of Eq. (45) is

f = ±α^{− \frac{1}{2}} e^{ψ}

ψ being a harmonic function. Substituting this expression in Eqs. (46), (47) one gets

α,_{1} ψ,_{1} −α,_{2} ψ,_{2} = 2α,_{11}
α,_{2} ψ,_{1} +α,_{1} ψ,_{2} = 2α,_{12}

the last relations are locally equivalent to

|∇ (α)|^{2} = ce^{ψ}

c being a constant. Therefore,

f = ±\begin{matrix}|∇ (ϕ)| & 2 \\ √ & \overline{|Dϕ + B|} \end{matrix} (48)

where α = |μ| = |Dϕ + B|, A and B are constants and ϕ a harmonic function such that α is nonvanishing. The equation (44) is a linear second order partial differential equation and can be analyzed with standard methods.

Thus, as the final result we have:

Proposition 13. Any G_{2}-integrable 4-metric g satisfying the vacuum Einstein equations, and such that det F > 0 and h_{22} = 0, has the following matrix form in the adapted coordinates (x_{1}, x_{2}, x.y)

M_{C} (g) = ( \begin{matrix}ϵ\begin{matrix}|∇(ϕ)| & ^{2} \\ √ & \overline{|Dϕ+B|} \end{matrix}1_{2} & 0 \\ 0 & (Dϕ + B) ( \begin{matrix}−2sy + w & 1 \\ 1 & 0 \end{matrix} ) \end{matrix} ) .

where ε = ±1, ϕ is a harmonic function, D and B are constants such that μ = Dϕ + B is everywhere nonvanishing and w is a solution of the equation

(μw,_{1} )_{,1} + (μw,_{2} )_{,2} = 0.

In the considered case det H is negative and the signature of g is equal to ±2. The Killing vector field Y is isotropic while the sign of g (X, X) varies as a function of y. The Gauss curvature K of the Killing leaves vanishes.

Remark 7. According to section 3, an additional Killing field, say Z, tangent to the Killing leaves and independent of X and Y, exists iff w is a constant, say w_{0}. In such a case it is given by

Z = e^{−sx} [−2∂_{x} + (−2sy + w_{0}) ∂_{y}] ,

which generates together with X = ∂_{x} and Y = e^{sx}∂_{y} a 3-dimensional Lie algebra isomorphic to Kil (dx^{2} − dy^{2}):

[X, Y] = sY, [X, Z] = −sZ, [Y, Z] = 0.

A canonical form for the equation (44) can be found by introducing new coordinates, namely ξ and η, by

ξ = α + α̃, η = α − α̃.

in which Eq. (44) becomes

(ξ + η) (\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) \binom{}{w} + \frac{∂w̃}{∂ξ} + \frac{∂w̃}{∂η} = 0,

with w̃ (ξ, η) ≡ w (x_{1} (ξ, η) , x_{2} (ξ, η)), or, alternatively,

(\frac{∂^{2}}{∂ξ^{2}} + \frac{∂^{2}}{∂η^{2}} ) (Z) + \frac{1}{2 (ξ + η)^{2}} Z = 0.

with

Z = \sqrt{ξ + η}w̃.

For its geometrical meaning see [12].

6. The Abelian limit (s = 0)

The solutions of the Einstein equations found in the previous section allow one to get exact solutions of the Belinsky-Zahkarov case just by passing to the ”Abelian limit” s = 0. Since the Abelian case was extensively studied (see, for instance, [3, 7, 4]) we shall limit ourself here simply to describe these solutions. In what follows we use the adapted coordinates to which the propositions refer and consider separately the cases h_{22} ≠ 0 and h_{22} = 0.

The case h_{22} ≠ 0. With this assumption Eqs. (15) and, which is the same (32) play the role of an ”ansatz” when passing to the Abelian limit: So, in that case as in sec. 5.1.1 and sec. 5.1.2 one sees that H = αM, M being a constant unimodular matrix.

• If det F < 0 then Eq. (16) becomes

α,_{12} = 0

and the remaining Einstein equations coincide with equations (29c) and (29d) as they appeared when analyzing the non-abelian situation assuming that h_{22} = 0 and det F < 0 (see sec. 5.1.2). Thus, the same procedure leads us to the following result:

M_{C} (g) = ( \begin{matrix}\begin{matrix}2f & 0 \\ 0 & −2f \end{matrix} & 0 \\ 0 & αM \end{matrix} ) , (49)

where α and f are given by

α = C_{1}F (z_{1} + z_{2}) + C_{2}G (z_{1} − z_{2}) + C (50)

f = \frac{F′G′}{\sqrt{|α|}} (51)

F and G being arbitrary functions, C, C_{1}, C_{2}, arbitrary constants such that α and f are everywhere nonvanishing;

• If det F > 0, then by referring to Eqs. (43)-(47) one finds that

M_{C} (g) = ( \begin{matrix}ϵ\begin{matrix}|∇(ϕ)| & ^{2} \\ √ & \overline{|Dϕ+B|} \end{matrix}1_{2} & 0 \\ 0 & (Dϕ + B) M \end{matrix} ) .

where ϵ = ±1, ϕ is a harmonic function, and D and B are constants such that Dϕ + B is everywhere nonvanishing and M is as above.

The case h_{22} = 0. With this assumption the Abelian limit is, obviously, obtained from the corresponding non-Abelian result (propositions 10 and 13) just by putting s = 0. Namely:

• If det F < 0, then (proposition 10)

M_{C} (g) = ( \begin{matrix}\begin{matrix}−2f & 0 \\ 0 & 2f \end{matrix} & 0 \\ 0 & μ ( \begin{matrix}w & 1 \\ 1 & 0 \end{matrix} ) \end{matrix} ) , (52)

where

μ = C_{1}F (z_{1} + z_{2}) + C_{2}G (z_{1} − z_{2}) + C (53)

f = |μ|^{− \frac{1}{2}} F′G′, (54)

F, G and C, C_{1}, C_{2}, being arbitrary functions and constants, respectively, such that μ and f be everywhere nonvanishing while w is an arbitrary solution of the equation

(μw,_{1} )_{,2} + (μw,_{2} )_{,1} = 0.

• If det F > 0, then (proposition 13)

M_{C} (g) = ( \begin{matrix}ϵ\begin{matrix}|∇(ϕ)| & ^{2} \\ √ & \overline{|Dϕ+B|} \end{matrix}1_{2} & 0 \\ 0 & (Dϕ + B) ( \begin{matrix}w & 1 \\ 1 & 0 \end{matrix} ) \end{matrix} ) . (55)

where ϵ = ±1, ϕ is a harmonic function, D and B are arbitrary constants such that μ = Dϕ + B is everywhere nonvanishing and w is an arbitrary solution of the equation

(μw,_{1} )_{,1} + (μw,_{2} )_{,2} = 0.

Remark 8. It is worth to note that in the Abelian case the Gauss curvature of the Killing leaves is equal to zero.

7. Ricci-flat metrics admitting a 3-dimensional Killing algebra with bidimensional leaves

Let g be a metric and G be one of its Killing algebras. In what follows, the Killing algebra G will be called normal if the restrictions of g to its Killing leaves are nondegenerate.

Obviously, a normal Killing algebra G is isomorphic to a subalgebra of Kil(g|_{S}) where S is a generic Killing leaf of G. Thus, when dim G = 3 and the Killing leaves are bidimensional, G = Kil (g|_{S}). As it is easy to see, in this situation there are exactly five options for Kil(g|_{S}) and, therefore, for G. Namely, they are:

so (2, 1) ,  Kil (dx^{2} − dy^{2}) ,  so (3) ,  Kil (dx^{2} + dy^{2}) ,  A_{3}, (56)

where A_{3} is a 3-dimensional Abelian Lie algebra. Since the Lie algebra A_{3} belongs to the case treated in [4] it will not be considered in the following.

Only two of these algebras, namely so (2, 1) and Kil(dx^{2} − dy^{2}), possess a noncommutative bidimensional subalgebra. Thus, one may expect that the corresponding Ricci flat 4-metrics are among the solutions described in section 5. It will be shown below that this is in fact true and that they belong to one of the cases h_{22} ≠ 0, or h_{22} = 0 with w fixed to be constant (see section 5).

As for the algebra Kil(dx^{2} + dy^{2}), it has only a bidimensional commutative subalgebra and we shall see that the corresponding Ricci-flat 4-metrics are among the solutions described in the previous section 6 (the Abelian limit with h_{22} ≠ 0).

The following assertion generalizes lemma 1 (section 2).

Lemma 14. Let X_{1}, X_{2} and f_{1}X_{1} + f_{2}X_{2}, f_{1}, f_{2} ∈ C^{∞} (M) be Killing fields of a metric (M, g). Then, supposing that X_{1} and X_{2} are independent, either f_{1} and f_{2} are functionally independent, or f_{1} and f_{2} are constant.

Proof. It results from relation (1) taking into account L_{X_{1}} (g) = L_{X_{2}} (g) = 0 that

0 = L_{f_{1}X_{1}+f_{2}X_{2}} (g) = i_{X_{1}} (g) df_{1} + i_{X_{2}} (g) df_{2}. (57)

Assuming, say, that f_{2} = φ (f_{1}) we see that

0 = L_{f_{1}X_{1}+f_{2}X_{2}} (g) = (i_{X_{1}} (g) + φ′i_{X_{2}} (g)) df_{1} = i_{X_{1}+φ′X_{2}} (g) df_{1},

If df_{1} ≠ 0, then the last equality implies, obviously, i_{X_{1}+φ′X_{2}} (g) = 0. In that case, X_{1} + φ′X_{2} = 0 due to the non-degeneracy of g in contradiction with the assumed independence of X_{1} and X_{2}. If on the contrary df_{1} = 0, then df_{2} = 0 and the second alternative takes place.

Note that it cannot happen that on a connected manifold M the first alternative takes place in U_{1} ⊂ M and the second one in U_{2} ⊂ M if ⋂_{i} U_{i} ≠ ∅. It results from the fact that if a Killing field vanishes on an open subset of M, then it vanishes everywhere.

Corollary 15. If \overline{G} is a 3-dimensional Killing algebra having bidimensional Killing leaves and the fields X_{1}, X_{2} , X_{3} generate it as a linear space, then almost everywhere X_{3} = f_{1}X_{1} + f_{2}X_{2} and f_{1} and f_{2} are functionally independent.

Proof. The fields X_{1} and X_{2} are independent according to lemma 1. So they generate almost everywhere, say in U, the tangent spaces to the Killing leaves. Thus, X_{3} = f_{1}X_{1} + f_{2}X_{2}, f_{i} ∈ C^{∞} (U). The possibility that f_{1} and f_{2} are constant offered by lemma 1 cannot occur in this context since X_{1}, X_{2} and X_{3} are supposed to be linearly independent.

Proposition 16. Any Killing algebra from the list (56) having bidimensional Killing leaves is normal. Moreover, the distribution D^{⊥} orthogonal to its Killing leaves is integrable.

Proof. Below the notation of corollary 15 is used. Since df_{1}and df_{2} are almost everywhere point-wise independent and one can deduce easily from (57) that

i_{X_{1}} (g) = λdf_{2}, i_{X_{2}} (g) = −λdf_{1}, (58)

being g nondegenerate, λ is almost everywhere non-vanishing.

Let now Y be an almost everywhere different from zero vector field. Then the equality

i_{Y} (i_{X_{1}} (g) df_{1} + i_{X_{2}} (g) df_{2}) = 0,

which is an obvious consequence of (57), is equivalent to

g (X_{1}, Y) df_{1} + g (X_{2}, Y) df_{2} = −Y (f_{1}) i_{X_{1}} (g) − Y (f_{2}) i_{X_{2}} (g) .

In view of (58) it gives

g (X_{1}, Y) df_{1} + g (X_{2}, Y) df_{2} = −λY (f_{1}) df_{2} + λY (f_{2}) df_{1},

so that

g (X_{1}, Y) = λY (f_{2}) , g (X_{2}, Y) = −λY (f_{1}) .

Hence Y (f_{1}) = Y (f_{2}) = 0 iff g (X_{1}, Y) = g (X_{2}, Y) = 0, i.e., such fields Y are orthogonal to the Killing leaves and vice versa. If Y is tangent to the Killing leaves, then

Y (f_{1}) = Y (f_{2}) = 0 ⟺ Y = 0,

since by the above corollary applied to the case M = S, df_{i}|_{S} is nondegenerate for a generic Killing leaf S. This proves that the fields Y such that Y (f_{1}) = Y (f_{2}) = 0 are transversal to the Killing leaves and that g|_{S} is non-degenerate for a generic Killing leaf S. Thus G is normal.

Finally note that the distribution D̃ spanned by the vector fields Y such that Y (f_{1}) = Y (f_{2}) = 0 is of co-dimension 2 since df_{1} and df_{2} are independent almost everywhere. Being both transversal and orthogonal to the Killing leaves, D̃ coincides with D^{⊥} by a dimension argument.

Corollary 17. The solutions found in section 5 exhaust all local Ricci-flat 4 -metrics admitting a Killing algebra isomorphic to so (2, 1) or to Kil (dx^{2} − dy^{2}).

Proof. As we already noticed, the first two algebras possess non-Abelian bidimensional subalgebras and according to the previous proposition the distribution D^{⊥} orthogonal to Killing leaves is transversal to them and integrable.

7.1. Kil(dx^{2} + dy^{2})-invariant Ricci-flat metrics. As it has been already noticed, the algebra Kil(dx^{2} + dy^{2}) has a bidimensional commutative subalgebra. We shall see that the corresponding Ricci-flat 4 -metrics are among the solutions of previous section 6 (the Abelian limit with h_{22} ≠ 0).

First, let G be a Killing algebra isomorphic to Kil(dx^{2} + dy^{2}) and let X_{i}, i = 1, 2, 3, be its standard basis, i.e.,

[X_{1}, X_{2}] = 0, [X_{1}, X_{3}] = X_{2}, [X_{2}, X_{3}] = −X_{1}.

With the notation of corollary 15, let X_{3} = f_{1}X_{1} + f_{2}X_{2}. Then

X_{2} = [X_{1}, X_{3}]
= [X_{1}, f_{1}X_{1} + f_{2}X_{2}]
= X_{1} (f_{1}) X_{1} + X_{1} (f_{2}) X_{2}

and

X_{1} = [X_{3}, X_{2}]
= [f_{1}X_{1} + f_{2}X_{2}, X_{2}]
= −X_{2} (f_{1}) X_{1} − X_{2} (f_{2}) X_{2},

so that, for the independence (section 2, lemma 1) of X_{1} and X_{2}, implies that we have

X_{1} (f_{1}) = 0, X_{1} (f_{2}) = 1
X_{2} (f_{1}) = −1, X_{2} (f_{2}) = 0.

Joining to f_{1}, f_{2} a couple of independent functions z_{1}, z_{2} such that X_{i} (z_{j}) = 0, ∀i, j, one gets a local chart on M. Taking into account the above relations and passing to the standard coordinate notation x = f_{1}, y = f_{2}, we see that in the chart (x, y, z_{1}, z_{2})

X_{1} = ∂_{y}, X_{2} = −∂_{x}, X_{3} = x∂_{y} − y∂_{x}.

Introducing on S polar coordinates (r, φ), i.e., x = r cos φ, y = r sin φ, the above fields read as

X_{1} = sin φ∂_{r} + \frac{cos φ}{r}∂_{φ}, X_{2} = cos φ∂_{r} + \frac{sin φ}{r}∂_{φ}, X_{3} = ∂_{φ}.

Then, in view of proposition 16, a direct computation similar to the one of section 2 shows that any G-invariant metric has in the adapted local chart (z_{1}, z_{2}, r, φ) the form

g = 2f (dz_{1}^{2} + εdz_{2}^{2}) + μ (z_{1}, z_{2}) [dr^{2} + r^{2}dφ^{2}] ,

and, therefore, belongs to the class of metrics considered in section 6 with definite H and h_{22} ≠ 0.

Thus, we have:

Corollary 18. The solutions found in section 6exhaust all local Ricci-flat 4-metrics admitting a Killing algebra isomorphic to Kil (dx^{2} + dy^{2}).

7.2. so (3)-invariant Ricci-flat metrics. The above results lead to expect that Ricciflat 4-metrics admitting a Killing algebra isomorphic to so (3) with 2-dimensional leaves can be described essentially in the same way as it was done in section 5 with respect to those admitting a Killing algebra isomorphic to so (2, 1). The details are as follows

First, let G be a Killing algebra isomorphic to so (3) and let X_{i}, i = 1, 2, 3, be its standard basis, i.e.

[X_{1}, X_{2}] = X_{3}, [X_{2}, X_{3}] = X_{1}, [X_{3}, X_{1}] = X_{2}.

In the notation of corollary 15 let X_{3} = f_{1}X_{1} + f_{2}X_{2}. Then

X_{1} = [X_{2}, X_{3}] = [X_{2}, f_{1}X_{1} + f_{2}X_{2}]
= X_{2} (f_{1}) X_{1} + f_{1} [X_{2}, X_{1}] + X_{2} (f_{2}) X_{2}
= (X_{2} (f_{1}) − f_{1}^{2}) X_{1} + (X_{2} (f_{2}) − f_{1}f_{2}) X_{2}.

Since X_{1} and X_{2} are independent (lemma 1)

X_{2} (f_{1}) − f_{1}^{2} = 1, X_{2} (f_{2}) − f_{1}f_{2} = 0. (59)

Similarly, from the relation [X_{3}, X_{1}] = X_{2} one finds

X_{1} (f_{1}) + f_{1}f_{2} = 0, X_{1} (f_{2}) + f_{2}^{2} = −1. (60)

Joining to f_{1}, f_{2} a couple of independent functions z_{1}, z_{2} such that X_{i} (z_{j}) = 0, ∀i, j, one gets a local chart on M. Taking into account relations (59) and (60) and passing to the standard coordinate notation x = f_{1}, y = f_{2}, we see that in the chart (x, y, z_{1}, z_{2})

X_{1} = −xy∂_{x} − (1 + y^{2}) ∂_{y}, X_{2} = (x^{2} + 1) ∂_{x} + xy∂_{y}, X_{3} = y∂_{x} − x∂_{y}.

In the geographic coordinates (r, φ) , i.e., x = tan ϑ cos φ, y = tan ϑ sin φ, the above fields read as

X_{1} = −\frac{cos φ}{tan ϑ}∂_{φ} − sin φ∂_{ϑ}, X_{2} = −\frac{sin φ}{tan ϑ}∂_{φ} + cos φ∂_{ϑ}, X_{3} = −∂_{φ}.

Then, in view of proposition 16, a direct computation similar to the one of section 2 shows that any G-invariant metric has in the adapted local chart (z_{1}, z_{2}, ϑ, φ) the form

g = f (dz_{1}^{2} + εdz_{2}^{2}) + α (z_{1}, z_{2}) [dϑ^{2} + sin^{2} ϑdφ^{2}] (61)

The Ricci tensor of the above metric can be easily computed as in section 4 and the corresponding Einstein equations lead to the same equations for f and α ≡ r^{2} as already found in section 5 in the case h_{22} ≠ 0. Namely,

f = −\frac{1}{2} (\frac{∂^{2}}{∂z_{1}^{2}} + ε\frac{∂^{2}}{∂z_{2}^{2}} ) (r^{2}) , (62)

r + A ln |r − A| = u, (63)

with ε = ±1, A being an arbitrary constant and u being an arbitrary function satisfying the equation

(\frac{∂^{2}}{∂z_{1}^{2}} + ε\frac{∂^{2}}{∂z_{2}^{2}} ) (u) = 0.

Additionally, f is required to be nonvanishing.

Remark 9. In the case ε = −1,these solutions are locally diffeomorphic to the Schwarzschild solution. This will be discussed in [12].

Below, the graph of the left hand side of the equation (63) is reported for the values A = 2 and A = −2.

u = r + 2 ln |r − 2|  u = r − 2 ln |r + 2|

One can see that for A ≠ 0 there exactly three possibilities for r = r (u) that correspond to the intervals of monotonicity of u (r). For instance, for A > 0 these are ]−∞, 0[, ]0, A[, and ]A, ∞[. In these regions the corresponding metric (61) is regular and has some singularities along the curves r = 0 and r − A = 0.

Some geometrical peculiarities of the obtained local solutions show how to match them together in order to get global nonextendible Einstein metrics. To this purpose, in [12] a formalism is developed which allows to construct, starting from known solutions, ”new” global ones and to describe their singularities as well. For instance, by extracting the square root of the Schwarzschild solution, one easily finds an Einstein metric which describes parallel universes. Other examples which illustrate some aspects of our approach can be found in [12]. We stress that it generalizes naturally to some other situations as, for instance, cosmological Einstein metrics satisfying assumptions I and II (work in progress).

Acknowledgments Two of the authors (G.S. and G.V.) wish to thank G. Bimonte, B. Dubrovin and

G.Marmo for interesting discussions.

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[3] V.A.Belinsky and I.M.Khalatnikov, General solution of the gravitational equations with a physical singularities, Sov. Phys. Jetp vol. 30, 6 (1970)

[4] V.A.Belinsky and V.E.Zakharov, Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions, Sov. Phys. Jetp vol. 48, 6 (1978)

[5] V.A.Belinsky and V.E.Zakharov, Stationary gravitational solitons with axial symmetry Sov. Phys. Jetp vol. 50, 1 (1979)

[6] F. J. Chinea, New first integral for twisting type-N vacuum gravitational fields with two noncommuting Killing vectors, Class. Quantum Grav. vol. 15 (1998)367-371

[7] R. Geroch, A method for generating new solutions of Einstein’s equation. II, J.Math.Phys. vol. 13, 3 (1972).

[8] M. Hallisoy, Studies in space-times admitting two spacelike Killing vectors, J.Math.Phys. vol. 29, 2 (1988).

[9] A. S. Mishenko and A. T. Fomenko, Generalized Liouville Method of Integration of Hamiltonian Systems, Funct. Anal. Appl. 12, 2 (1978)113-121, and references therein. A. T. Fomenko, Symplectic Geometry, Gordon and Breach 1995.

[10] A. Z. Petrov, Einstein spaces, ( Pergamon Press, New York 1969)

D. Kramer, H. Stephani, E. Herlt, M. MacCallum, Exact solutions of Einstein field equations, Cambridge University Press 1980.

[11] G. Sparano and G. Vilasi, Noncommutative Integrability and Recursion Operators, J. Geom. Phys. (2000) to appear

[12] G. Sparano, G. Vilasi and A. M. Vinogradov, Vacuum Einstein metrics with bidimensional Killing leaves, II-Global aspects, DIPS 14/2001, to appear in Differential Geometry and its applications.

[13] A. M. Vinogradov, Geometric singularities of solutions of partial differential equations, in Proc. Conf. on Differential Geometry and its Applications (Reidel, Dordrecht 1987)

[14] R. Wald, General Relativity, University of Chicago Press (Chicago, 1984)

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Matematica e Informatica, Università di Salerno, Italy.

E-mail address: sparano@unisa.it

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Scienze Fisiche E.R.Caianiello, Università di Salerno, Italy.

E-mail address: vilasi@sa.infn.it

Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, Italy., Dipartimento di Matematica e Informatica, Università di Salerno, Italy.

E-mail address: vinograd@unisa.it