Equivalence between the timelike Brunn-Minkowski inequality and timelike Bakry-Émery-Ricci lower bound on weighted globally hyperbolic spacetimes

TL;DR

This work establishes a Lorentzian curvature–dimension relation by proving that the timelike Brunn–Minkowski inequality $\mathsf{TBM}(K,N)$ yields a lower bound on the Bakry–Émery Ricci tensor $\mathrm{Ric}^{N,\mathfrak{m}}\ge K$ in weighted globally hyperbolic spacetimes. The authors develop a Lorentzian optimal transport framework, analyze infinitesimal volume distortion along transported sets, and compare optimal vs geodesic interpolations using Jacobi fields and normal coordinates. The result, together with the known equivalence between timelike Ricci lower bounds and $\mathsf{TCD}(K,N)$, places $\mathsf{TBM}(K,N)$ in correspondence with $\mathsf{TCD}(K,N)$ in the smooth setting, mirroring the Riemannian BM–CD theory. This provides a robust, transport-based characterization of curvature bounds in Lorentzian geometry, with potential extensions to non-smooth Lorentzian spaces via TBM. The work deepens the connection between geometric analysis, causal structure, and optimal transport in relativity-inspired contexts.

Abstract

We prove the timelike Brunn-Minkowski inequality $\mathsf{TBM}(K,N)$ implies a timelike lower bound on the Bakry-Émery-Ricci curvature on weighted globally hyperbolic spacetimes. This result, together with the well-known equivalence between timelike Bakry-Émery-Ricci lower bounds and the $\mathsf{TCD}(K,N)$ condition, and the fact that $\mathsf{TCD}(K,N)$ spaces support the timelike Brunn-Minkowski inequality, draws an equivalence between $\mathsf{TBM}(K,N)$ and $\mathsf{TCD}(K,N)$ in the smooth setting.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 2.1
• Definition 2.1: Timelike Brunn-Minkowski Inequality
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4
• proof