Nonexistence of Marginally Trapped Surfaces and Geons in 2 + 1 Gravity

Abstract

We use existence results for Jang’s equation and marginally outer trapped surfaces (MOTSs) in 2 + 1 gravity to obtain nonexistence of geons in 2 + 1 gravity. In particular, our results show that any 2 + 1 initial data set, which obeys the dominant energy condition with cosmological constant Λ ≥ 0 and which satisfies a mild asymptotic condition, must have trivial topology. Moreover, any data set obeying these conditions cannot contain a MOTS. The asymptotic condition involves a cutoff at a finite boundary at which a null mean convexity condition is assumed to hold; this null mean convexity condition is satisfied by all the standard asymptotic boundary conditions. The results presented here strengthen various aspects of previous related results in the literature. These results not only have implications for classical 2 + 1 gravity but also apply to quantum 2 + 1 gravity when formulated using Witten’s solution space quantization.