Geometric and topological corrections to Schwarzschild black hole
Rocco D’Agostino, Orlando Luongo, Stefano Mancini
Abstract
Abstract In this paper, we compute departures in the black hole thermodynamics induced by either geometric or topological corrections to general relativity. Specifically, we analyze the spherically symmetric spacetime solutions of two modified gravity scenarios with Lagrangians $$\mathcal {L}\sim R^{1+\epsilon }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and $$\mathcal {L}\sim R+\epsilon \, \mathcal {G}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>∼</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mspace/> <mml:msup> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , where $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is the Euler density in four dimensions, while $$ 0<\epsilon \ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> measures the perturbation around the Hilbert–Einstein action. Accordingly, we find the expressions of the Bekenstein–Hawking entropy by the Penrose formula, and the black hole temperature and horizon of the obtained solutions. We then investigate the heat capacities in terms of the free parameters of the theories under study. In doing so, we show that healing the problem of negative heat capacities can be possible under particular choices of the free constants, albeit with limitations on the masses allowed for the black hole solutions.