A spacetime route to positive mass

Brien Nolan

Brien Nolan is a Senior Lecturer in the School of Mathematical Sciences, Dublin City University

This paper provides an important, unexpected and very satisfying route to positivity of mass in General Relativity. It shows positivity of the Trautman-Bondi mass in a way that avoids both the heavy differential geometric machinery of the work of Schoen and Yau, and the spinorial methods used by Witten and by Ludvigsen and Vickers in their positive mass results.

The approach is a purely spacetime one, relying on (essentially) the tools that the pioneers of gravitational radiation theory used in the early 1960’s, albeit with the deep insights provided by the recent work of Choquet-Bruhat, Chrusciel and Martin-Garcia. To my mind, the key is that the same philosophy is at work here as in the papers of Bondi and colleagues: focus on null cones and apply the Einstein equations. It is of great interest to see that this approach continues to yield insights into General Relativity.

Although we know from the result of Ludvigsen and Vickers that the Trautman-Bondi mass is indeed positive, it is very difficult to see how this must be true solely on the basis of the relevant definition – pick out a coefficient in the asymptotic expansion of a certain metric function written in a particular coordinate system. The simple and elegant proof in this paper shows how this quantity, when written in terms of coordinates adapted to solving the Einstein equations on a null cone, equates to the integral of a positive term, and so is positive.

Read the full article in Classical and Quantum Gravity:
The mass of light-cones
Piotr T Chruściel and Tim-Torben Paetz
Class. Quantum Grav. 31 102001

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