Nonexistence of marginally trapped surfaces and geons in 2+1 gravity

TL;DR

This paper proves that under certain energy and boundary conditions, 2+1 gravity cannot contain nontrivial topologies or marginally outer trapped surfaces, implying trivial topology for such initial data sets.

Contribution

It establishes the nonexistence of geons and MOTSs in 2+1 gravity under broad conditions, strengthening previous results and extending implications to quantum gravity.

Findings

Any 2+1 initial data set with mbda  0 and mild asymptotic conditions has trivial topology.

Such data sets cannot contain marginally outer trapped surfaces.

Results apply to classical and quantum 2+1 gravity, reinforcing the nonexistence of nontrivial topologies.

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 \Lambda \geq 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.