Quasi-local mass at the null infinity of the Vaidya spacetime

TL;DR

This paper investigates the Wang-Yau quasi-local mass at null infinity in Vaidya spacetime, confirming positivity and deriving explicit formulas related to the mass aspect function, thus providing a local perspective on mass loss.

Contribution

It explicitly solves the optimal embedding equation for the Wang-Yau quasi-local mass at null infinity in Vaidya spacetime and expresses the mass in terms of the mass aspect function.

Findings

Confirmed positivity of quasi-local mass in Vaidya spacetime

Derived explicit formula for quasi-local mass at null infinity

Connected quasi-local mass to the mass aspect function

Abstract

There are two important statements regarding the Trautman-Bondi mass [1,8,5] at null infinity: one is the positivity [7,6], and the other is the Bondi mass loss formula [1], which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.