Homogeneous Lorentzian Structures on Some Gödel-Levichev’s Spacetimes, and Associated Reductive Decompositions

P.M. Gadea and Ana Primo Ramos

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

For the Levichev homogeneous spacetimes of type 2a on the Gödel group, the homogeneous Lorentzian structures and the associated reductive decompositions are determined.

Mathematics Subject Classification: 53C30, 14M17 Key words: homogeneous space, reductive decomposition, Lorentzian structure, Godel-Levichev spacetime, Ambrose-Singer equation

1 Introduction and preliminaries

É. Cartan gave in [2] the classical characterization of Riemannian symmetric spaces as the spaces of parallel curvature. This was extended by Ambrose and Singer, who gave in [1] a characterization for a connected, simply connected and complete Riemannian manifold to be homogeneous, in terms of a (1, 2) tensor field S, called by Tricerri and Vanhecke in [7] a homogeneous Riemannian structure, which satisfies certain equations (see (1.1) below). In [3] it is defined a homogeneous pseudo-Riemannian structure on a pseudo-Riemannian manifold (M, g) as a tensor field S of type (1, 2) such that ∇ being the Levi-Civita connection and R its curvature tensor, the connection ˜∇ = ∇ − S satisfies the Ambrose-Singer equations

˜∇g = 0, ˜∇R = 0, ˜∇S = 0. (1.1)

In [3] it is proved that if the pseudo-Riemannian manifold (M, g) is connected, simply connected and geodesically complete then it admits a homogeneous pseudo-Riemannian structure if and only if it is a reductive homogeneous pseudo-Riemannian manifold. This means that M = G/H, where G is a connected Lie group acting transitively and effectively on M as a group of isometries, H is the isotropy group at a point o ∈ M, and the Lie algebra g of G may be decomposed into a vector space direct sum of the Lie algebra h oh H and an Ad (H)-invariant subspace m, that is g = h ⊕ m, Ad (H)m ⊂ m. (If G is connected and M is simply connected then H is connected, and the latter condition is equivalent to [h, m] ⊂ m.)

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 45-51.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Let (M, g) be a connected, simply connected, and geodesically complete pseudo-Riemannian manifold, and suppose that S is a homogeneous pseudo-Riemannian structure on (M, g). We fix a point o ∈ M and put m = T_{o}(M). If R̃ is the curvature tensor of the connection ˜∇ = ∇ − S, we can consider the holonomy algebra h̃ of ˜∇ as the Lie subalgebra of “skew-symmetric” endomorphisms of (m, g_{o}) generated by the operators R̃_{ZW}, where Z, W ∈ m. Then, according to the Ambrose-Singer construction [1, 7], a Lie bracket is defined in the vector space direct sum g̃ = h̃ ⊕ m by

\begin{matrix}[U, V] = UV − V U, & U, V ∈ h̃, \\ [U, Z] = U(Z), & U ∈ h̃, Z ∈ m, \\ [Z, W] = R̃_{ZW} + S_{Z}W − S_{W} Z, & Z, W ∈ m , \end{matrix} (1.2)

and we say that (g̃, h̃) is the reductive pair associated to the homogeneous pseudo-Riemannian structure S.

Tricerri and Vanhecke [7] have classified the homogeneous Riemannian structures into eight classes, which are defined by the invariant subspaces of certain space S_{1} ⊕ S_{2} ⊕ S_{3}. In [4] a similar classification for the pseudo-Riemannian case is given. For more details see below.

On the other hand, Levichev consider in [5] the usual Gödel metric

g = −\frac{e^{−2x_{4}}}{2} dx_{1}^{2} − 2 e^{−2x_{4}} dx_{1} dx_{2} − dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2},

as a left-invariant metric on the Gödel group G, and defines several families of metrics on G, thus obtaining several types of homogeneous Lorentz spaces. The ones of type 2a are connected, simply connected, and geodesically complete. In the present note we determine the homogeneous Lorentzian structures on these homogeneous spacetimes and their type in Tricerri-Vanhecke’s classification, and the associated reductive decompositions.

2 Homogeneous Lorentzian structures

The Gödel group is the simply connected Lie group G whose Lie algebra g has four generators e_{1}, e_{2}, e_{3}, e_{4}, with the only nonvanishing bracket

[e_{4}, e_{1}] = e_{1}.

The group G admits a realization as R^{4} = {(x_{1}, x_{2}, x_{3}, x_{4})} with multiplication z = x · y obtained from the matrix expression

x ≡ ( \begin{matrix}e^{x_{4}} & 0 & 0 & x_{1} \\ 0 & 1 & 0 & x_{2} \\ 0 & 0 & 1 & x_{3} \\ 0 & 0 & 0 & 1 \end{matrix} ) .

The commutation relations of its Lie algebra in the system of coordinates chosen on G coincide with the brackets above.

Consider the subspaces L_{1}, L_{2}, L_{3} of g generated respectively by e_{1}; e_{2}, e_{3}; and e_{1}, e_{2}, e_{3}. Then the homogeneous Lorentz group of type 2a is defined by the conditions: L_{2}, L_{3} are timelike, and L_{1} is spacelike (for more details see [5]). Then, for each couple of real numbers p, q with 0 ≤ p < 1, q > 0, the left-invariant Lorentzian metric g_{p,q} on G obtained by left translations from the scalar product at the origin with matrix given, with respect to the above basis of g, by

⟨ , ⟩_{p,q} = ( \begin{matrix}1 & p & 0 & 0 \\ p & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & q \end{matrix} ) , (2.1)

is given by

g_{p,q} = ( \begin{matrix}e^{−2x_{4}} & e^{−2x_{4}}p & 0 & 0 \\ e^{−2x_{4}}p & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & q \end{matrix} ) .

As causal spacetimes, the Lorentz Lie groups corresponding to the Gödel group with the metric of type 2a are homogeneously globally hyperbolic, which is a strong causality condition. We recall that: A causal curve in a Lorentz manifold M is a curve whose velocity vectors are all nonspacelike; if M is globally hyperbolic then any pair of points that can be joined by a causal curve can be joined by a (longest) causal geodesic; a solvable Lorentz Lie group G is said to be homogeneously globally hyperbolic if it is globally hyperbolic and has a Cauchy surface S passing through the identity element e ∈ G and containing the center of G (for more details see [5, 6]); a Cauchy surface of a spacetime is a subset that is met exactly once by every inextendible timelike curve in the spacetime.

On account of Koszul’s formula for the Levi-Civita connection for a left-invariant metric g on a Lie group,

2g(∇_{e_{i}}e_{j}, e_{k}) = g([e_{i}, e_{j}], e_{k}) − g([e_{j}, e_{k}], e_{i}) + g([e_{k}, e_{i}], e_{j}),

we obtain that the non-null covariant derivatives between generators are

\begin{matrix}∇_{e_{1}}e_{1} = \frac{1}{q}e_{4}, ∇_{e_{1}}e_{2} = ∇_{e_{2}}e_{1} = \frac{p}{2q} e_{4}, \\ ∇_{e_{1}}e_{4} = \frac{p^{2} − 2}{2(1 − p^{2})}e_{1} + \frac{p}{2(1 − p^{2})}e_{2}, \end{matrix}

\begin{matrix}∇_{e_{2}}e_{4} = ∇_{e_{4}}e_{2} = −\frac{p}{2(1 − p^{2})}e_{1} + \frac{p^{2}}{2(1 − p^{2})}e_{2}, \\ ∇_{e_{4}}e_{1} = −\frac{p^{2}}{2(1 − p^{2})}e_{1} + \frac{p}{2(1 − p^{2})}e_{2}. \end{matrix}

So, the nonvanishing components of the curvature tensor, with the convention R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z, are, putting R_{e_{i}e_{j}}e_{k} for R(e_{i}, e_{j})e_{k},

R_{e_{1}e_{2}}e_{1} = \frac{p^{3}}{4q(1 − p^{2})}e_{1} − \frac{p^{2}}{4q(1 − p^{2})}e_{2},

R_{e_{1}e_{2}}e_{2} = \frac{p^{2}}{4q(1 − p^{2})}e_{1} − \frac{p^{3}}{4q(1 − p^{2})}e_{2},

R_{e_{1}e_{4}}e_{1} = \frac{p(2 − p)}{4q(1 − p^{2})}e_{4},  R_{e_{1}e_{4}}e_{2} = −\frac{p^{3}}{4q(1 − p^{2})}e_{4},

R_{e_{1}e_{4}}e_{4} = \frac{p^{2} − 4}{4(1 − p^{2})}e_{1} + \frac{p}{1 − p^{2}} e_{2},  R_{e_{2}e_{4}}e_{1} = −\frac{p^{3}}{4q(1 − p^{2})}e_{4},

R_{e_{2}e_{4}}e_{2} = −\frac{p^{2}}{4q(1 − p^{2})}e_{4},  R_{e_{2}e_{4}}e_{4} = \frac{p^{2}}{4(1 − p^{2})}e_{2},

and the nonvanishing components of the Riemann-Christoffel curvature tensor, with the convention R(X, Y, Z, W) = g(R(Z, W)Y, X), putting R_{e_{i}e_{j}e_{k}e_{l}} for g(R(e_{k}, e_{l})e_{j}, e_{i}), are

\begin{matrix}R_{e_{1}e_{2}e_{1}e_{2}} = \frac{p^{2}}{4q} , & R_{e_{1}e_{4}e_{1}e_{4}} = \frac{5p^{2} − 4}{4(1 − p^{2})}, \\ R_{e_{1}e_{4}e_{2}e_{4}} = \frac{p^{3}}{4(1 − p^{2})}, & R_{e_{2}e_{4}e_{2}e_{4}} = \frac{p^{2}}{4(1 − p^{2})}. \end{matrix}

We shall now determine the homogeneous Lorentzian structures on these spaces. For this, we must solve the Ambrose-Singer equations 1.1. The first Ambrose-Singer equation amounts to S_{XY Z} = −S_{XZY} for any homogeneous pseudo-Riemannian structure S. One can write the second Ambrose-Singer equation ˜∇R = 0 as

\begin{matrix}R_{∇_{U} XY ZW} + R_{X∇_{U} Y ZW} + R_{XY ∇_{U} ZW} + R_{XY Z∇_{U} W} \\ = S_{UXR(Z,W)Y} − S_{UY R(Z,W)X} + S_{UZR(X,Y)W} − S_{UW R(X,Y)Z}. \end{matrix}

Solving, we obtain that the nonvanishing components of S are

S_{e_{1}e_{2}e_{4}} = 1 − p^{2}, S_{e_{4}e_{1}e_{2}} = \frac{p}{2},

except for S_{e_{i}e_{1}e_{4}}, i = 1, …, 4, for which we must use the third Ambrose-Singer equation. In our case, since we are considering left-invariant differential forms, the forms involved in this equation are linear combinations with constant coefficients of the basis {θ^{1}, θ^{2}, θ^{3}, θ^{4}} of left-invariant forms on G dual to the basis {e_{1}, e_{2}, e_{3}, e_{4}}. Moreover, since for a constant function f, one has ∇_{X}f = 0 and ˜∇_{X}f = 0, we also have S_{X}f = 0. Thus, the third Ambrose-Singer equation S̃ = 0 can be written as

S_{∇_{X}Y ZW} + S_{Y ∇_{X}ZW} + S_{Y Z∇_{X}W} = S_{S_{X}Y ZW} + S_{Y S_{X}ZW} + S_{Y ZS_{X}W} ,

for X, Y, Z, W ∈ g.

Solving, we obtain the nonzero components

S_{e_{1}e_{1}e_{4}} = 1, S_{e_{2}e_{1}e_{4}} = \frac{p}{2}.

Consequently, the non-null components S_{e_{i}}e_{j} are

S_{e_{1}}e_{1} = \frac{1}{q}e_{4}, S_{e_{1}}e_{2} = \frac{1 − p^{2}}{q} e_{4},

S_{e_{1}}e_{4} = \frac{−p^{3} + p − 1}{1 − p^{2}}e_{1} + \frac{p^{2} + p − 1}{1 − p^{2}}e_{2}, S_{e_{2}}e_{1} = \frac{p}{2q} e_{4},

S_{e_{4}}e_{1} = −\frac{p^{2}}{2(1 − p^{2})}e_{1} + \frac{p}{2(1 − p^{2})}e_{2}, S_{e_{4}}e_{2} = −\frac{p}{2(1 − p^{2})}e_{1} + \frac{p^{2}}{2(1 − p^{2})}e_{2},

Then, with the convention v ∧ w = v ⊗ w − w ⊗ v for the exterior product, we have proved the following

Theorem 1 The homogeneous Lorentzian structures on the Gödel-Levichev space (G, g_{p,q}) of type 2a are given by

θ^{1} ⊗ θ^{1} ∧ θ^{4} + (1 − p^{2}) θ^{1} ⊗ θ^{2} ∧ θ^{4} + \frac{p}{2} (θ^{2} ⊗ θ^{1} ∧ θ^{4} + θ^{4} ⊗ θ^{1} ∧ θ^{2}).

We recall some definitions and a result from Tricerri and Vanhecke [7] (see also [4]). Let E be a real vector space of dimension n endowed with an inner product ⟨ , ⟩ of signature (k, n−k). The space (E, ⟨ , ⟩) will be the model for each tangent space T_{x}M, x ∈ M, of a reductive homogeneous pseudo-Riemannian manifold of signature (k, n − k). Consider the vector space S(E) of tensors of type (0, 3) on (E, ⟨ , ⟩) satisfying the same symmetries as those of a homogeneous pseudo-Riemannian structure S, that is, S(E) = {S ∈ ⊗^{3}E^{∗} : S_{XY Z} = −S_{XZY} , X, Y, Z ∈ E}, where S_{XY Z} = ⟨S_{X}Y, Z⟩. Let

c_{12}: S(E) → V^{∗} be the map defined by c_{12}(S)(Z) = ∑_{i=1}^{n} ε_{i}S_{e_{i}e_{i}Z}, Z ∈ E, where {e_{i}}

is an orthonormal basis of E, ⟨e_{i}, e_{i}⟩ = ε_{i} = ±1. Then we have that if dim E ≥ 3, then S(E) decomposes into the orthogonal direct sum of subspaces which are invariant and irreducible under the action of the pseudo-orthogonal group O(k, n − k) : S(E) = S_{1}(E) ⊕ S_{2}(E) ⊕ S_{3}(E), where

S_{1}(E)  = { S ∈ S(E) : S_{XY Z} = ⟨X, Y ⟩ω(Z) − ⟨X, Z⟩ω(Y), ω ∈ E^{∗} },
S_{2}(E)  = { S ∈ S(E) : 𝔖_{XYZ}S_{XY Z} = 0, c_{12}(S) = 0 },
S_{3}(E)  = { S ∈ S(E) : S_{XY Z} + S_{Y XZ} = 0 }.
S_{1}(E) ⊕ S_{2}(E) = { S ∈ S(E) : 𝔖_{XYZ}S_{XY Z} = 0 },
S_{2}(E) ⊕ S_{3}(E) = { S ∈ S(E) : c_{12}(S) = 0 },
S_{1}(E) ⊕ S_{3}(E) = { S ∈ S(E) : S_{XY Z} + S_{Y XZ} = 2⟨X, Y ⟩ω(Z)
− ⟨X, Z⟩ω(Y) − ⟨Y, Z⟩ω(X), ω ∈ E^{∗} }.

In the present case we deduce

Corollary 1 The homogeneous Lorentzian structures on (G, g_{p,q}) belong to

S_{1} ⊕ S_{2} ⊕ S_{3} − {(S_{1} ⊕ S_{2}) ∪ (S_{1} ⊕ S_{3}) ∪ (S_{2} ⊕ S_{3})}.

In particular none of the associated reductive homogeneous spaces is either Lorentzian symmetric, or naturally reductive or cotorsionless.

Proof. Take the orthonormal basis

ẽ_{1} = \frac{1}{\sqrt{2(1 + p)}}(e_{1} + e_{2}), ẽ_{2} = \frac{1}{\sqrt{2(1 − p)}}(e_{1} − e_{2}), ẽ_{3} = e_{3}, ẽ_{1} = \frac{1}{\sqrt{q}} e_{4}.

As a calculation with respect to this basis shows, the condition c_{12}(S) = 0 is not satisfied. On the other hand, since for instance S_{e_{1}e_{2}e_{4}} + S_{e_{2}e_{4}e_{1}} + S_{e_{4}e_{1}e_{2}} ≠ 0, no structure belong to S_{1} ⊕ S_{2}. Moreover, since for instance S_{e_{1}e_{2}e_{4}} ≠ −S_{e_{2}e_{1}e_{4}}, no structure belong to S_{3}; not even to S_{2} ⊕ S_{3}, as the sum S_{e_{1}e_{2}e_{4}} + S_{e_{2}e_{1}e_{4}} shows. The Lorentzian symmetric spaces correspond to the class {0}, and in [4] it has been proved the equivalence of the third class with the naturally reductive spaces, and of the class S_{1} ⊕ S_{2} with the cotorsionless spaces. For more details see [4].

3 Associated reductive decompositions

Consider now the Ambrose-Singer connection ˜∇ = ∇−S. Then, the non-null covariant derivatives between generators are

˜∇_{e_{1}}e_{2} = \frac{2p^{2} + p − 2}{2q},  ˜∇_{e_{1}}e_{4} = \frac{p(2p^{2} + p − 2)}{2(1 − p^{2})}e_{1} − \frac{2p^{2} + p − 2}{2(1 − p^{2})}e_{2},

and, as a calculation shows, the only nonvanishing curvature operator is

R̃_{e_{1}e_{4}} ≡ ((2p^{2} + p − 2) \begin{matrix}0 & 0 & 0 & \frac{p}{2(1 − p^{2})} \\ 0 & 0 & 0 & −\frac{1}{2(1 − p^{2})} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{1}{2q} & 0 & 0 \end{matrix} ) ,

According to Ambrose-Singer’s Theorem on holonomy, the algebra of holonomy of a connection is generated by the curvature operators. In the present case, the holonomy algebra h̃ has the only generator V = R̃_{e_{1}e_{4}}. Putting m for g, and taking T = V + e_{1} we have

Theorem 2 The reductive pairs (g̃, h̃) associated to the reductive decompositions g̃ = h̃ ⊕ m corresponding to the homogeneous Lorentzian structures on (G, g_{p,q}) given in Theorem 1, are given in terms of the basis {e_{1}, e_{2}, e_{3}, e_{4}, T } by the (nonvanishing) Lie brackets

\begin{matrix}[T, e_{4}] = 2e_{1} − T, [e_{1}, e_{2}] = −\frac{2p^{2} + p − 2}{2q}e_{4}, \\ [e_{1}, e_{4}] = T − \frac{2p^{3} − 3p^{2} − 2p + 4}{2(1 − p^{2})}e_{1} + \frac{2p^{2} + p − 2}{2(1 − p^{2})}e_{2}. \end{matrix}

Proof. On account of the expressions (1.2), we obtain that

\begin{matrix}[V, e_{2}] = \frac{2p^{2} + p − 2}{2q}e_{4}, [V, e_{4}] = \frac{p(2p^{2} + p − 2)}{2(1 − p^{2})}e_{1} − \frac{2p^{2} + p − 2}{2(1 − p^{2})}e_{2}, \\ [e_{1}, e_{2}] = −\frac{2p^{2} + p − 2}{2q}e_{4}, [e_{1}, e_{4}] = V − \frac{2p^{3} − p^{2} − 2p + 2}{2(1 − p^{2})}e_{1} + \frac{2p^{2} + p − 2}{2(1 − p^{2})}e_{2}. \end{matrix}

Then, making the change T = V + e_{1} we conclude. Acknowledgements. Partially supported by DGI, Spain, under grant no. BFM2002-00141.

References

[1] W. Ambrose and I.M. Singer, On homogeneous Riemannian manifolds, Duke Math. J. 25 (1958) 647–669.

[2] É. Cartan, Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France 54 (1926) 214–264.

[3] P.M. Gadea and J.A. Oubiña, Homogeneous pseudo-Riemannian structures and homogeneous almost para-Hermitian structures, Houston J. Math. 18 (1992) 449– 465.

[4] P.M. Gadea and J.A. Oubiña, Reductive homogeneous pseudo-Riemannian manifolds, Monatsh. Math. 124 (1997) 17–34.

[5] A.V. Levichev, Causal structure of left-invariant Lorentz metrics on the group M_{2} ⊗ R^{2}, Siberian Math. J. 31 (1990) 607–614.

[6] A.V. Levichev, Methods of investigation of the causal structure of homogeneous Lorentz manifolds, Siberian Math. J. 31 (1990) 395–408.

[7] F. Tricerri and L. Vanhecke, Homogeneous Structures on Riemannian Manifolds, London Math. Soc. Lect. Notes Ser. 83, Cambridge Univ. Press, Cambridge, U.K., 1983.

IMAFF, CSIC, Serrano 144, 28006 Madrid,

email: pmgadea@iec.csic.es

Department of Mathematics, University of Salamanca, Pl. Merced 1–4, 37008 Salamanca, Spain, email:ana_{−}primo@hotmail.com