Uniqueness of the static vacuum asymptotically flat spacetimes with massive particle spheres

TL;DR

This paper proves that a four-dimensional static vacuum asymptotically flat spacetime with a massive particle sphere must be isometric to Schwarzschild spacetime, extending uniqueness theorems to include massive particle spheres.

Contribution

It establishes a new uniqueness theorem for static vacuum spacetimes with massive particle spheres, showing they are uniquely Schwarzschild, and introduces a spacetime foliation by such spheres.

Findings

Spacetime with a massive particle sphere is isometric to Schwarzschild.

A single massive particle sphere suffices for uniqueness.

Existence of a foliation by massive particle spheres across energies.

Abstract

In this paper, we establish that a four-dimensional static vacuum asymptotically flat spacetime containing a massive particle sphere is isometric to the Schwarzschild spacetime. Our results expand upon existing uniqueness theorems for static vacuum asymptotically flat spacetimes, which focus on scenarios featuring event horizons or photon spheres. Similarly to the uniqueness theorems concerning photon spheres or event horizons, only a single massive particle sphere is sufficient to obtain a unique solution. However, in contrast to previous theorems, our result leads to the existence of an entire spacetime foliation sliced by a set of massive particle spheres spanning various energies.