On the space‐times admitting a synchronization of constant curvature

In fact, this is what we shall make here: the analytical integration of Einstein’s constraint and evolution equations in the complete integrability case.

The necessary and sufficient constraints that the Cauchy data of the Einstein equations must verify in order that the space‐time admits a r‐dimensional isometry group have been given by

B.

Coll

, , ();

B.

Coll

, , (). But the integrability conditions for these constraints, not yet studied, depend on the isometry group admitted by the intrinsic metric on the initial hypersurface Σ, that is, in other words, on the Σ‐Killing vector fields.

The notion of rigidity related to a synchronization has been considered by

B.

Coll

, , ();

and

C.

Bona

, , ().

A.

Krasinski

, , ().

C. B.

Collins

and

D. A.

Szafron

, , , ().

For $n>3,$ the pentadimensional case $(n = 4)$ is particularly adapted to a formulation in terms of synchronizations. Some interesting aspects of the cases $n<3$ have been pointed out recently by

S.

Giddings

,

J.

Abbott

, and

K.

Kuchar

, , ().

For our purposes, we only need that ĝ be nondegenerate. Nevertheless, in order to have a direct physical interpretation on $V4,$ we shall suppose ĝ to be also Lorentzian.

δ denotes, to within a sign, the divergence operator. In local charts $(δT̂)A = −∇ρT̂Aρ,$ A being some set of tensorial indices.

For the physical meaning of a synchronization, see B. Coll, “A relativistic notion of rigidity,” to be published.

d is the usual operator of exterior differentiation on forms, and A denotes the exterior product.

The evolution formalisms of general relativity are relative in the sense that the space‐time is replaced by a time‐dependent three‐dimensional spatial geometry, and the geometrical objects considered are defined on this evolutive geometry. The standard evolution formalism, the only one considered here, is the one in which every instant of the synchronization is characterized by its first and second fundamental forms. However, many other formalisms may be considered, in which the induced metric is replaced by the quotient metric relative to the given motion, or by a conformal one, the extrinsic curvature and the characterization of the motion being replaced by “more or less” adapted quantities. For some of these different possibilities see, for example,

L.

Bel

and

J. C.

Escard

, , , ();

J.

Stachel

, , ();

R. A.

Nelson

, , ().

B. Coll, Thése d’état, Univ. Pierre et Marie Curie, Paris, 1980.

In local charts, the interior product is $(i(n̂*)T̂)A≡n̂ρT̂ρA,$ where A represents some set of tensorial indices.

The canonical isomorphisms between tensors and cotensors defined by the metrics ĝ and g on $(Vn+1,ĝ)$ and $(EΣg),$ respectively, allow us to extend the notion of Σ characterization to the contravariant tensors of $(Vn+1,ĝ):$ the Σ characterization $E*$ of a p tensor $Ê*$ on $(Vn+1,g)$ is the p‐extensor field of $(Σ,g)$ associated by g to the Σ characterization E of the p‐cotensor field $Ê$ associated by ĝ to $Ê*.$

The direct calculation is long and tedious. A short way, based on the geometric properties of the Σ characterizations, is given in Ref. 12.

A. Lichnerowicz, Thèse d’état, Univ. de Paris, Paris, 1939.

A.

Lichnerowicz

, , ().

Y.

Choquet‐Bruhat

, , ().

Y.

Choquet‐Bruhat

, , ().

R. Arnowitt, S. Deser, and C. W. Misner, “The dynamics of General Relativity,” in Gravitation: An Introduction to Current Research (Wiley, New York, 1962).

In fact, being a tensor field on $(Vn+1,ĝ),$ $ĝI$ is a convenient one‐parameter family of induced metrics.

See, for example, A. Lichnerowicz, Théories relativistesde la gravitation et de l’électromagnetisme (Masson, Paris, 1955).

C. Bona and B. Coll, “Classification of the space‐times admitting a constant curvature synchronization,” in preparation.

In the Lorentzian case and signature $−(n−−1),$ one has $cij = −δij.$

That is, $Yi$ transforms the local representation {σ;0} of ${ΓN,w,S}$ in the local representation {σ;s} of {Γ,S}, where s is given by (39).

Of course, $a,b = 1,…,n$ and $m(ab) = −m(ba).$

For the notion of umbilical points see, for example, J. A. Schouten, Ricci‐Calculus (Springer‐Verlag, Berlin, 1954).

For the introduction of the Gauss and Codazzi tensors in $(V4,ĝ),$ see Ref. 12.

The Petrov classification [

A. Z.

Petrov

, , ()] involves only three types of space‐times. The complete one, involving the five well‐known (nontrivial) types, was obtained by L. Bel, Thèse d’état, Univ. de Paris, Paris, 1959; for this reason, we call it the Petrov‐Bel classification.

H.

Stephani

, , (). See also, D. Kramer et al., Exact Solutions of Einstein’s Field Equations (VEB Deutscher Verlag der Wissenschaften, Berlin, 1980), Chap. 32.

R.

Debever

, , ().

R.

Sachs

, , ().

We have used the signature $(+ − − −);$ thus, g is negative definite.