Upper bounds on the force function in spatially regular self-gravitating matter configurations

Abstract

We use the non-linearly coupled Einstein-matter field equations to prove four theorems that bound from above the dimensionless force function ${\cal F}=4πr^2\cdot p(r)$ in spatially regular curved spacetimes of spherically symmetric self-gravitating matter configurations [here $p(r)$ is the radially-dependent pressure inside the spatially regular matter configurations]. In particular, for generic (not necessarily isotropic) matter configurations it is proved that: (i) ${\cal F}\leq 2$ for matter fields that satisfy the dominant energy condition, and (ii) ${\cal F}\leq 1$ for matter fields with a non-positive energy-momentum trace. In addition, for self-gravitating isotropic matter configurations we derive the stronger upper bounds: (iii) ${\cal F}\leq 1$ for matter fields that satisfy the dominant energy condition, and (iv) ${\cal F}\leq 1/2$ for matter fields with a non-positive energy-momentum trace. Our analytically derived results are in accord with the spirit of the maximum force conjecture in general relativity.