Spherically symmetric black holes with electric and magnetic charge in extended gravity: physical properties, causal structure, and stability analysis in Einstein’s and Jordan’s frames
E. Elizalde, G. G. L. Nashed, S. Nojiri, S. D. Odintsov
Abstract
Abstract Novel static black hole solutions with electric and magnetic charges are derived for the class of modified gravities: $$f({{{\mathcal {R}}}})={{{\mathcal {R}}}}+2\beta \sqrt{{{\mathcal {R}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mi>β</mml:mi><mml:msqrt><mml:mi>R</mml:mi></mml:msqrt></mml:mrow></mml:math> , with or without a cosmological constant. The new black holes behave asymptotically as flat or (A)dS space-times with a dynamical value of the Ricci scalar given by $$R=\frac{1}{r^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow></mml:math> and $$R=\frac{8r^2\Lambda +1}{r^2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>8</mml:mn><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>Λ</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow></mml:math> , respectively. They are characterized by three parameters, namely their mass and electric and magnetic charges, and constitute black hole solutions different from those in Einstein’s general relativity. Their singularities are studied by obtaining the Kretschmann scalar and Ricci tensor, which shows a dependence on the parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> that is not permitted to be zero. A conformal transformation is used to display the black holes in Einstein’s frame and check if its physical behavior is changed w.r.t. the Jordan one. To this end, thermodynamical quantities, as the entropy, Hawking temperature, quasi-local energy, and the Gibbs free energy are calculated to investigate the thermal stability of the solutions. Also, the casual structure of the new black holes is studied, and a stability analysis is performed in both frames using the odd perturbations technique and the study of the geodesic deviation. It is concluded that, generically, there is coincidence of the physical properties of the novel black holes in both frames, although this turns not to be the case for the Hawking temperature.