Comparison theory in Lorentzian and Riemannian geometry

Ehrlich, Paul E.; Kim, Seon-Bu

doi:10.1016/j.na.2008.10.022

Cited by 4 publications

(9 citation statements)

References 20 publications

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“…where we have also extended the integral to the full line, using the fact that | f | 2 is real and even, and used Parseval's theorem in the final step. As we have argued, this expression can be used -at least for sufficiently small support width -as a replacement for the integral in (20). We can say a bit more about the magnitude of this term in relation to the others, because smooth functions f , compactly supported in an interval I of length τ, obey the Rayleigh-Ritz inequality…”

Section: Quantum Strong Energy Inequalitymentioning

confidence: 99%

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A semiclassical singularity theorem

Fewster,

Kontou

2021

Preprint

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Quantum fields do not satisfy the pointwise energy conditions that are assumed in the original singularity theorems of Penrose and Hawking. Accordingly, semiclassical quantum gravity lies outside their scope. Although a number of singularity theorems have been derived under weakened energy conditions, none is directly derived from quantum field theory. Here, we employ a quantum energy inequality satisfied by the quantized minimally coupled linear scalar field to derive a singularity theorem valid in semiclassical gravity. By considering a toy cosmological model, we show that our result predicts timelike geodesic incompleteness on plausible timescales with reasonable conditions at a spacelike Cauchy surface.

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Section: Quantum Strong Energy Inequalitymentioning

confidence: 99%

“…An interesting account of the interplay between Riccati and index-form methods in a slightly different context can be found in Ref. [20].…”

Section: Introductionmentioning

confidence: 99%

A semiclassical singularity theorem

Fewster,

Kontou

2021

Preprint

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“…We assume, for simplicity, that the eigenvalues are smooth. 5 Finally, note that, since s · S(s) → Id as s → 0, the eigenvalues satisfy the asymptotic condition that s · λ i (s) → 1 as s → 0 for i = 1, . .…”

Section: Equation (33)mentioning

confidence: 99%

Areas and Volumes for Null Cones

Grant

2011

Ann. Henri Poincaré

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Abstract. Motivated by recent work of Chruściel,, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volume monotonicity theorem and Günther's volume comparison theorem. We briefly discuss how these estimates may be used to control the null second fundamental form of slices of the null cone in Ricci-flat Lorentzian four-manifolds with null curvature bounded above.

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“…We now abstractly study some properties of solutions of the Riccati equation (5). Let E be an n-dimensional, real vector space with positive-definite inner product ·, · .…”

Section: Riccati Comparisonmentioning

confidence: 99%

“…For example, the interplay between Riccati techniques and index techniques in both fields has been emphasised by Ehrlich [5]. The combination of completeness (which guarantees minimizing geodesics) and curvature conditions (which, via Riccati techniques, imply that geodesics have conjugate points) are the key ingredients in the proof of, for example, Myers's theorem in Riemannian geometry.…”

Section: Introductionmentioning

confidence: 99%