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 .
Abstract. For any compact strictly pseudoconvex CR manifold M endowed with a contact form θ we obtain the Bochner type formula(involving the sublaplacian ∆ b and the pseudohermitian Ricci curvature ρ). When M is compact of CR dimension n and ρ(X, X)+ 2A(X, JX) ≥ k G θ (X, X), X ∈ H(M ), we derive the estimate −λ ≥ 2nk/(2n − 1) on each nonzero eigenvalue λ of ∆ b satisfying Eigen(∆ b ; λ) ∩ Ker(T ) = (0) where T is the characteristic direction of dθ.
We build a variational theory of geodesics of the Tanaka-Webster connection ∇ on a strictly pseudoconvex CR manifold M . Given a contact form θ on M such that (M, θ) has nonpositive pseudohermitian sectional curvature (k θ (σ) ≤ 0) we show that (M, θ) has no horizontally conjugate points. Moreover, if (M, θ) is a Sasakian manifold such that k θ (σ) ≥ k 0 > 0 then we show that the distance between any two consecutive conjugate points on a lengthy geodesic of ∇ is at most π/(2 √ k 0 ). We obtain the first and second variation formulae for the Riemannian length of a curve in M and show that in general geodesics of ∇ admitting horizontally conjugate points do not realize the Riemannian distance.
We give a description of the effect of the gravitational field by using the geodesic equation of motion with respect to a first order Finslerian approximation of the Minkowski metric. This motivates linking the physical force of gravity to the non flat nature of space in the Finslerian setting and leads to an anisotropic version of the red shift formula. We solve the linearized Finslerian field equations proposed by S.F. Rutz (Gen.
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