On limit behavior of quasi-local mass for ellipsoids at spatial infinity

TL;DR

This paper investigates the behavior of different quasi-local mass definitions for ellipsoids at spatial infinity in asymptotically flat spacetimes, revealing that Hayward mass converges to a finite value while Hawking mass diverges.

Contribution

It demonstrates that Hayward mass remains finite and positive at infinity for ellipsoids, and establishes a positive mass theorem in this context.

Findings

Hawking mass tends to -infinity for these ellipsoids.

Hayward mass converges to a finite, positive value.

Numerical simulations show Hayward mass increases monotonically near infinity.

Abstract

We discuss the spatial limit of the quasi-local mass for certain ellipsoids in an asymptotically flat static spherically symmetric spacetime. These ellipsoids are not nearly round but they are of interest as an admissible parametrized foliation defining the Arnowitt-Deser-Misner (ADM) mass. The Hawking mass of this family of ellipsoids tends to $-\infty$. In contrast, we show that the Hayward mass converges to a finite value. Moreover, a positive mass type theorem is established. The limit of the mass has a uniform positive lower bound no matter how oblate these ellipsoids are. This result could be extended for asymptotically Schwarzschild manifolds. And numerical simulation in the Schwarzschild spacetime illustrates that the Hayward mass is monotonically increasing near infinity.