The Positive Mass and Isoperimetric Inequalities for Axisymmetric Black Holes in four and five dimensions

TL;DR

This paper extends positive mass and isoperimetric inequalities to axisymmetric black holes in four and five dimensions, generalizing Brill's proof and including cases with apparent horizons, with potential applications in numerical relativity.

Contribution

It generalizes Brill's proof of positive mass to higher dimensions and includes apparent horizons, establishing new inequalities for axisymmetric black holes.

Findings

Proved positive mass theorem for 3D axisymmetric initial data.

Established Riemannian Penrose inequality with apparent horizons.

Extended Brill's formula to 4+1 dimensions with axisymmetry.

Abstract

In this paper we revisit Brill's proof of positive mass for three-dimensional, time-symmetric, axisymmetric initial data and generalise his argument in various directions. In 3+1 dimensions, we include an apparent horizon in the initial data and prove the Riemannian Penrose inequality in a wide number of cases by an elementary argument. In the case of 4+1 dimensions we obtain the analogue of Brill's formula for initial data admitting a generalised form of axisymmetry. Including an apparent horizon in the initial data, the Riemannian Penrose inequality is again proved for a large class of cases. The results may have applications in numerical relativity.