We prove that a globally hyperbolic smooth spacetime endowed with a
$\smash{\mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from
below in all timelike directions, in a distributional sense, obeys the timelike
measure-contraction 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 $\smash{\mathrm{C}^{1,1}}$, in fact the stronger
timelike curvature-dimension condition holds. In this regularity, we also
obtain uniqueness of chronological optimal couplings and chronological

Source link


Leave A Reply