Supersymmetry and Positive Energy in Classical and Quantum Two-Dimensional Dilaton Gravity

Abstract

An $N = 1$ supersymmetric version of two dimensional dilaton gravity coupled to matter is considered. It is shown that the linear dilaton vacuum spontaneously breaks half the supersymmetries, leaving broken a linear combination of left and right supersymmetries which squares to time translations. Supersymmetry suggests a spinorial expression for the ADM energy $M$, as found by Witten in four-dimensional general relativity. Using this expression it is proven that ${M}$ is non-negative for smooth initial data asymptotic (in both directions) to the linear dilaton vacuum, provided that the (not necessarily supersymmetric) matter stress tensor obeys the dominant energy condition. A {\it quantum} positive energy theorem is also proven for the semiclassical large-$N$ equations, despite the indefiniteness of the quantum stress tensor. For black hole spacetimes, it is shown that $M$ is bounded from below by $e^{- 2 \phi_H}$, where $\phi_H$ is the value of the dilaton at the apparent horizon, provided only that the stress tensor is positive outside the apparent horizon. This is the two-dimensional analogue of an unproven conjecture due to Penrose. Finally, supersymmetry is used to prove positive energy theorems for a large class of generalizations of dilaton gravity which arise in consideration of the quantum theory.