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

We prove the timelike Brunn-Minkowski inequality $\mathsf{T}\mathsf{B}\mathsf{M}(K,N)$ 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 $\mathsf{T}\mathsf{C}\mathsf{D}(K,N)$ condition, and the fact that $\mathsf{T}\mathsf{C}\mathsf{D}(K,N)$ spaces support the timelike Brunn-Minkowski inequality, draws an equivalence between $\mathsf{T}\mathsf{B}\mathsf{M}(K,N)$ and $\mathsf{T}\mathsf{C}\mathsf{D}(K,N)$ in the smooth setting.