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

Timelike Ricci bounds for low regularity spacetimes by optimal transport

“…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 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.…”

“…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.…”

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

“…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.…”