From the Riccati inequality to the Raychaudhuri equation

Ehrlich, Paul E.; Kim, Seon-Bu

doi:10.1090/conm/170/01745

Cited by 9 publications

(12 citation statements)

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“…where t ∈ I = [0, t (H )) and t (H ) is the first focal value of H along γ , is called a normal geodesic variation of γ along the hypersurface H [2,3]. For each fixed q ∈ H , let γ q be the geodesic given by γ q (t) = φ(t, q) and define φ t : H → M by φ t (q) = φ(t, q) for q ∈ H such that maximal geodesic joining q and γ q (t) = φ(t, q) has the Lorentzian distance t.…”

Section: Preliminariesmentioning

confidence: 99%

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The upperbound of the volume expansion rate in a Lorentzian manifold

Kim

2009

Gen Relativ Gravit

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Section: Preliminariesmentioning

confidence: 99%

“…So if −s M + s(H t ) ≤ −Ric(γ , γ ) under Ric(γ , γ ) ≤ 0, then we get the upperbound (8) less than (2). In particular, let H be a spacelike hypersurface in R 3 1 , then we get trivially (7) and (8).…”

mentioning

confidence: 94%

The upperbound of the volume expansion rate in a Lorentzian manifold

Kim

2009

Gen Relativ Gravit

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“…The following important result on the existence of conjugate points does not follow from the previous theorem. The proof can be found in Hawking (1966a), Hawking and Ellis (1973, Proposition 4.4.2,5), Ehrlich and Kim (1994), Beem et al (1996) and Tong (2009).…”

Section: Proofmentioning

confidence: 92%

Lorentzian causality theory

Minguzzi

2019

Living Rev Relativ

157

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I review Lorentzian causality theory paying particular attention to the optimality and generality of the presented results. I include complete proofs of some foundational results that are otherwise difficult to find in the literature (e.g. equivalence of some Lorentzian length definitions, upper semi-continuity of the length functional, corner regularization, etc.). The paper is almost self-contained thanks to a systematic logical exposition of the many different topics that compose the theory. It contains new results on classical concepts such as maximizing curves, achronal sets, edges, horismos, domains of dependence, Lorentzian distance. The treatment of causally pathological spacetimes requires the development of some new versatile causality notions, among which I found particularly convenient to introduce: biviability, chronal equivalence, araying sets, and causal versions of horismos and trapped sets. Their usefulness becomes apparent in the treatment of the classical singularity theorems, which is here considerably expanded in the exploration of some variations and alternatives.

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“…Compare Theorem 4·1 with [7] where it is assumed that there are no conjugate points along a fixed null geodesic, and the conclusion only concerns that geodesic. Compare Theorem 4·1 with [7] where it is assumed that there are no conjugate points along a fixed null geodesic, and the conclusion only concerns that geodesic.…”

Section: Integral Inequalities and Their Applicationsmentioning

confidence: 99%

Lorentzian manifolds with no null conjugate points

Gutiérrez¹,

Palomo²,

Romero

2004

Math. Proc. Camb. Phil. Soc.

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An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian n( 3)-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian 2−torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.

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From the Riccati inequality to the Raychaudhuri equation

Cited by 9 publications

References 0 publications

The upperbound of the volume expansion rate in a Lorentzian manifold

The upperbound of the volume expansion rate in a Lorentzian manifold

Lorentzian causality theory

Lorentzian manifolds with no null conjugate points

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