Rényi's entropy on Lorentzian spaces. Timelike curvature-dimension conditions

Braun, Mathias

doi:10.48550/arxiv.2206.13005

Cited by 1 publication

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“…4.1] for every p ∈ (0, 1), cf. Theorem 1.1, and in fact • the stronger TCD p (K, n) property [1,Def. 3.3] for every p ∈ (0, 1) if g is at least C 1,1 , cf.…”

Section: Introductionmentioning

confidence: 91%

“…Section 2.1 -physically, one should always think of (M, g) to solve the Einstein equation with given cosmological constant Λ ∈ R and energy-momentum tensor T . If g is smooth 1 , [26,29] and later [1] showed convexity properties of certain entropy functionals with respect to the volume measure vol g along "chronological" geodesics in P(M) to characterize the condition Ric g ≥ K in all timelike directions.…”

Section: Introductionmentioning

confidence: 99%

“…Here, P(M) is the space of Borel probability measures on M. These discoveries have lead to the synthetic theory of TCD and TMCP spaces 2 via the Boltzmann entropy [3] and later via the Rényi entropy [1] in the framework of measured Lorentzian spaces [3,20], i.e. natural generalizations of spacetimes.…”

Section: Introductionmentioning

confidence: 99%

“…3.35,Thm. 4.20], yet the approach in [1] yields stronger geometric properties a priori, as made precise below.…”

Section: Introductionmentioning

confidence: 99%

“…Thence, we get timelike geometric inequalities in Corollary 1.5, Corollary 1.6, and Corollary 1.7. Notably, these are obtained in sharp form even though the regularity of g might be below C 1,1 , where g might admit timelike branching [3,Def. 1.10], and localization [3,Ch.…”

Section: Introductionmentioning

confidence: 99%

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Timelike Ricci bounds for low regularity spacetimes by optimal transport

Braun¹,

Calisti²

2022

Preprint

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We prove that a globally hyperbolic smooth spacetime endowed with a C 1 -Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measurecontraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption.If the metric is even C 1,1 , in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics. Contents 1. Introduction 2. Spacetimes of low regularity 2.1. Terminology 2.2. C 1 -spacetimes as Lorentzian geodesic spaces 2.3. Approximation 2.4. Lorentzian optimal transport 3. Proofs of the main results 3.1

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“…4.1] for every p ∈ (0, 1), cf. Theorem 1.1, and in fact • the stronger TCD p (K, n) property [1,Def. 3.3] for every p ∈ (0, 1) if g is at least C 1,1 , cf.…”

Section: Introductionmentioning

confidence: 91%