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

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.

Novel rotating hairy black hole in (2+1)-dimensions

“…However, due to the complicated form (12) of the function f (r), we are unable to find its zeros r = r ± analytically for generic choices of parameters. The only qualitative property we know about r ± is that r − < r min , r + > r min , and that r + is the event horizon for the non-extremal black hole.…”

“…The study of hairy extensions of the BTZ black hole has a history which is almost as long as the age of BTZ black hole itself. To this date, many explicit hairy extensions of BTZ black hole has been found, and in most cases the extra hair is a scalar field which couples to gravity either minimally [7,8,9,10,11,12] or non-minimally [13,14], the scalar may or may not couple to itself via a scalar potential, and the hairy black holes can be static, charged or rotating, see [15] for more references on related studies. Charged static solutions are relatively easy to find, however it is very difficult to find charged rotating solutions due to the complexities involved in solving the Maxwell field equation in rotating background.…”

$(2+1)$-dimensional charged black holes with scalar hair in Einstein-Power-Maxwell Theory

“…On the other hand, with a negative cosmological constant in the action, the (2 + 1)-dimensional black holes with the minimal [42][43][44] or non-minimal [45,46] scalar fields have been constructed in the Einstein's gravity. Furthermore, the charged [47], rotating [48][49][50][51], charged rotating [52] and Einstein-Born-Infeld [53] black holes with non-minimally scalar hair and rotating black hole dressed with minimal scalar field hair [54] in the (2 + 1) dimensional Einstein's gravity, and black hole dressed by a (non)minimally scalar field in New Massive Gravity [55] have been discussed.…”