Abstract
In this Letter 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.
Highlights
The study of general relativity in dimensions larger than the 3 + 1 that are observed has a history almost as long as general relativity itself
Given the requirement that the local presence of a mass-energy source was required to curve space–time it was a surprise that it was possible to construct a black hole space–time [11] (a review of (2 + 1)-dimensional BTZ black holes is given in [12]). This was accomplished by considering a point mass source in a space–time with a negative cosmological constant, Λ < 0 – i.e. Anti-de Sitter (AdS) space–time
In this Letter we have found an r-symmetric, exact, analytical solution for a real scalar field coupled to gravity in 2 + 1 dimensions
Summary
Article history: Received 24 January 2013 Accepted 7 March 2013 Available online 14 March 2013 Editor: M. In this Letter 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. 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 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
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