Comparison theorems for causal diamonds

TL;DR

This paper establishes geometric inequalities for causal diamonds in curved spacetimes, revealing monotonic properties of the red-shift factor and generalizing classical inequalities to higher dimensions and non-spherical cases.

Contribution

It introduces new inequalities for causal diamonds involving the red-shift factor, generalizes Bishop's inequality without Ricci positivity, and explores dimensional differences in volume behaviors.

Findings

Red-shift factor is monotonic and maximal at the center in spherically symmetric cases.

Generalized Bishop's inequality without assuming Ricci tensor positivity.

Volume behavior differs in four dimensions versus higher dimensions near infinity.

Abstract

We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the centre. As a byproduct of our discussion we re-derive Bishop's inequality without assuming the positivity of the spatial Ricci tensor. We then generalize our considerations to arbitrary, static and not necessarily spherically symmetric, asymptotically flat spacetimes. In the case of spacetimes with a horizon our generalization involves the so-called {\it domain of dependence}. The respective volume, expressed in terms of the duration measured by a distant observer compared with the volume of the domain in Minkowski spacetime, exhibits behaviours which differ if $d=4$ or $d>4$. This peculiarity of four dimensions is due to the logarithmic subleading term in the asymptotic expansion of the metric near infinity. In terms of the invariant duration measured by a co-moving observer associated with the diamond we establish an inequality which is universal for all $d$. We suggest some possible applications of our results including the comparison theorems for entanglement entropy, causal set theory and fundamental limits on computation.