Exact radial solution in 2+1 gravity with a real scalar field

TL;DR

This paper explores circularly symmetric static spacetimes in 2+1 dimensions, deriving an exact scalar field solution that exhibits unique properties, including a flat spatial metric and asymptotic AdS behavior, highlighting differences from higher-dimensional cases.

Contribution

It provides a new derivation of 2+1 AdS space and presents a unique exact scalar field solution specific to 2+1 dimensions, not extendable to 3+1 dimensions.

Findings

Derived a simple form of 2+1 AdS space-time

Presented an exact scalar field solution with asymptotic AdS behavior

Highlighted differences between 2+1 and 3+1 dimensional gravitating scalar solutions

Abstract

In this paper we give some general considerations about circularly symmetric, static space-times in 2+1 dimensions, focusing first on the surprising (at the time) existence of the BTZ black hole solution. We show that BTZ black holes and Schwarzschild black holes in 3+1 dimensions originate from different definitions of a black hole. There are two by-products of this general discussion: (i) we give a new and simple derivation of 2+1 dimensional Anti-de Sitter (AdS) space-time; (ii) we present an exact solution to 2+1 dimensional gravity coupled to a self-interacting real scalar field. The spatial part of the metric of this solution is flat but the temporal part behaves asymptotically like AdS space-time. The scalar field has logarithmic behavior as one would expect for a massless scalar field in flat space-time. The solution can be compared to gravitating scalar field solutions in 3+1 dimensions but with certain oddities connected with the 2+1 dimensional character of the space-time. The solution is unique to 2+1 dimensions; it does not carry over to 3+1 dimensions.