From bending of light to positive mass: a non-PDE perspective

TL;DR

This paper offers a new non-PDE approach to understanding the positive mass theorem by reducing the problem to causality comparisons in compactified spacetimes, avoiding complex PDE issues.

Contribution

It introduces an intermediate diagonal metric and a causality comparison method, providing a simpler, more natural perspective on the positive mass theorem compared to traditional PDE-based proofs.

Findings

Avoids PDE complexities in positive mass proofs

Uses causality comparison in compactified spacetimes

Applicable to asymptotically flat, vacuum, stationary solutions

Abstract

Penrose et al. investigated the physical incoherence of the spacetime with negative mass via the bending of light. Precise estimates of time-delay of null geodesics were needed and played a pivotal role in their proof. In this paper, we construct an intermediate diagonal metric and make a reduction of this problem to a causality comparison in the compactified spacetimes regarding timelike connectedness near the conformal infinities. This different approach allows us to avoid encountering the difficulties and subtle issues Penrose et al. met. It provides a new, substantially simple, and physically natural non-PDE viewpoint to understand the positive mass theorem. This elementary argument modestly applies to asymptotically flat solutions which are vacuum and stationary near infinity.