ADM, Bondi mass, and energy conservation in two-dimensional dilaton gravities

TL;DR

This paper constructs a stress-energy pseudotensor in two-dimensional dilaton gravity theories, deriving expressions for ADM and Bondi masses, and explores energy conservation under specific boundary conditions.

Contribution

It introduces a pseudotensor formalism to define ADM and Bondi masses in 2D dilaton gravities, generalizing previous mass definitions and analyzing energy conservation.

Findings

Derived the ADM mass expression from the pseudotensor.

Defined the Bondi mass using the pseudotensor formalism.

Showed energy conservation holds under certain boundary conditions.

Abstract

We show how a stress-energy pseudotensor can be constructed in two-dimensional dilatonic gravity theories (classical, CGHS and RST) and derive the expression for the ADM mass in these theories from it. We define the Bondi mass for these theories by using the pseudotensor formalism. The resulting expression is the generalization of the expression for the ADM mass. The boundary condition needed for the energy conservation is also investigated. It is shown that under appropriate boundary conditions, our definition of the Bondi mass is exactly the ADM mass minus the matter radiation energy at null infinity.