Mixed gravitational field equations on globally hyperbolic spacetimes

Abstract

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting T (M) = D ⊕ R∇T of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43-50). We give sufficient geometric conditions (e.g. T is isoparametric and D is totally umbilical) for the existence of exact solutions −β dT ⊗ dT + g to mixed field equations in free space. We linearize and solve the mixed field equations Ric D (g) μν − ρ D (g) g μν = 0 for empty space, where ρ D (g) is the mixed scalar curvature of foliated spacetime (M, D) (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g = g 0 + γ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (R 4 \ R, g ) to order O( ). We derive the equations of motion of a material particle in the gravitational field g μν governed by the mixed field equations Ric D (g) μν − ρ D (g) ω μ ω ν − g μν = 2πκc −2 T μν − 1 2 T g μν . In the weak field ( 1) and low velocity ( v /c 1) limit, the motion equations are d 2 r/dt 2 = ∇φ + F, where φ = ( /2)c 2 γ 00 .

Einstein-Hilbert type action on spacetimes

“…and which governs the interaction of the "gravitational field" with the given field representing the matter. Note that, see [1],…”

“…The mixed Einstein-Hilbert action has been introduced in [1] as analog of the Einstein-Hilbert action, with the scalar curvature replaced by S mix :…”

“…The integral is taken over M if it converges; otherwise, one integrates over an arbitrarily large, relatively compact domain Ω containing supports of variations g t with g 0 = g. The physical meaning of (0.1) has been discussed in [1] for the case of a globally hyperbolic spacetime (M 4 , g) when D = Span(N ) and hence S mix = g(N, N ) Ric N,N , and the following have been obtained: the Euler-Lagrange equations, see (3.12), called the mixed gravitational field equations; sufficient conditions for existence of solutions for empty space, the conservation law analogous to conservation law of stress-energy tensor in relativity; equations of motion of a particle on isoparametric foliations in the "mixed gravitational field"; the linearized mixed field equations about the Minkowski metric; the value of coupling constant a in the weak field and low velocity limit. In this paper, we explore (0.1) for any spacetime, the obtained mixed gravitational field equations generalize the result of [1]. In fact, we work in arbitrary number of dimensions of a pseudo-Riemannian manifold endowed with a distribution, and also generalize certain results of [2,8], where the particular case of variations of metric (called adapted variations) has been examined.…”

“…In Proposition 2.2 we observe that the replacement of S mix by Scal D in (0.1) leads to the same Euler-Lagrange equations (0.2). In Section 3 we explore Ric D for spacetimes, see (3.6); for integrable D we compare the tensor, see (3.10), with its initial prototype defined in [1] for globally hyperbolic spacetimes.…”

Variations of the total mixed scalar curvature of a distribution

“…Since we consider also non-compact manifolds, we assume that the total mixed scalar curvature is in fact an integral of the mixed scalar curvature over a sufficiently large, relatively compact set. When viewed as a functional of the metric, the total mixed scalar curvature may be considered an analogue of the Einstein-Hilbert action [1].…”