Polytropic stars in three-dimensional spacetime

Abstract

We investigate three-dimensional perfect fluid stars with polytropic equation of state, matched to the exterior three-dimensional black hole geometry of Bañados, Teitelboim and Zanelli. A new class of exact solutions for a generic polytropic index is found and analysed.In the past decade, lower-dimensional theories of gravity have been intensively investigated, primarily because of the insights they can provide into conceptual issues arising in four-dimensional general relativity. The interest in lower-dimensional theories of gravity has dramatically increased with the discovery [1,2] that three-dimensional general relativity with a negative cosmological constant admits a black hole solution (called BTZ), which, apart from being asymptotically anti-de Sitter, exhibits features closely related to the Schwarzschild and Kerr solutions. Further generalizations have shown that three-dimensional theories of gravity have a very rich structure of black hole solutions [3][4][5][6][7][8][9][10][11].The BTZ black hole metric can be matched to a three-dimensional perfect fluid star. Cruz e Zanelli [12], studying the stability of these stars for the case of a generic equation of state p = p(ρ), concluded that they can undergo gravitational collapse into a black hole. For the particular case of pressureless dust, Mann and Ross [13] have investigated the circunstances under which a BTZ black hole can be formed from gravitational collapse, concluding that it occurs naturally, in a way that parallels the Oppenheimer-Snyder collapse in four dimensions.In this letter, we consider three-dimensional perfect fluid stars in hydrostatic equilibrium, matched to the BTZ black hole exterior geometry. The equation of state relating the pressure and the energy density is assumed to be of the polytropic type,where n is the polytropic index and K is the polytropic constant. This equation of state encompasses a wide range of physically interesting cases, namely constant energy density for n = 0, nonrelativistic degenerate fermions for n = 1, pure radiation for n = ∞ and K = 1/2, and stiff matter for n = ∞ and K = 1. For constant energy density an exact solution is known [12]. Here, exact solutions for a generic polytropic index are found and analysed. Note that in four spacetime dimensions analytical solutions for polytropic stars are known just for a few particular values of the polytropic index and only in the Newtonian approximation. For the three-dimensional spherically symmetric, static metric,the Einstein equations with a negative cosmological constant Λ < 0 in the presence of a perfect fluid with energy density ρ and pressure p related by a polytropic equation of state,

Curvature invariants and lower dimensional black hole horizons

“…Interests in lower-dimensional black holes is motivated from the existence of physical systems whose motion is known to be confined in lower dimensions, like cosmic strings and domain walls which arise in the Polyakov model [73,74] (see also [75] for a review about applications of BTZ black holes in the light of this connection). The existence of hydrodynamic equilibrium has led to the proposal of stellar models in (1+2) gravity [76][77][78]. Distinguishing a lower-dimenional black hole from a lower-dimensional star is important in laboratory settings which investigate Bose-Einstein condensate [79], and the optical properties of graphene [80] through analogue sonic BTZ black holes.…”

Static circularly symmetric perfect fluid solutions with an exterior BTZ metric

“…Cornish et al [2] found an exact solution for a 2 + 1 dimensional star with a polytropic equation of state, and a flat exterior spacetime. Sá [12] consider the same equation of state but in an (anti)-de Sitter background, so the exterior correspond to a BTZ spacetime. In 3 + 1 dimension the situation is very different and over one hundred solution have been found.…”

Spinor solutions of a Chern–Simons model for the superconformal algebra

“…Three-dimensional theories of gravity have a very rich structure of black hole solutions with matter: polytropic stars from [4], rigidly rotating perfect fluid stars [5], low mass strange stars based on the Heintzmann ansatz [6], circular thin shells in 2+1 F(R) gravity [7], exact solutions in (2 + 1)-dimensional anti-de Sitter space-time admitting a linear or non-linear equation of state [8] and references therein.…”