Exact d-dimensional Bardeen-de Sitter black holes and thermodynamics Md Sabir Ali 1,* and Sushant G. Ghosh 1,2,† 1 Center for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India 2 Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa (Received 26 January 2018; published 16 October 2018) The Bardeen metric is the first spherically symmetric regular black hole solution of Einstein’ s equations coupled to nonlinear electrodynamics, which has an additional parameter (e) due to nonlinear charge apart from mass (M). We find a d-dimensional Bardeen-de Sitter black hole and analyze its horizon structure and thermodynamical properties. Interestingly, in each spacetime dimension d, there exists a critical mass parameter μ ¼ μ E , which corresponds to an extremal black hole when Cauchy and event horizons coincide, which for μ > μ E describes a nonextremal black hole with two horizons and no black hole for μ < μ E . We also find that the extremal value μ E is influenced by the spacetime dimension d. Owing to the nonlinear charge corrected metric, the thermodynamic quantities of the black holes also get modified and a Hawking-Page-like phase transition exists. The phase transition is characterized by a divergence of the heat capacity at a critical radius r þ ¼ r C þ , with the stable (unstable) branch for C e > ð<Þ0. The Hawking evaporation of black holes leads to a thermodynamically stable double-horizon black hole remnant with the vanishing temperature. DOI: 10.1103/PhysRevD.98.084025 I. INTRODUCTION The gravitational collapse of a sufficiently massive star (∼3.5 M ⊙ ) necessarily forms a spacetime singularity—this is a fact established by the famous theorem due to Hawking and Penrose [1,2]. The existence of singularity by its very definition means spacetime fails to exist and therefore signaling a breakdown of physics laws. Sakharov [3] and Gliner [4] suggest that singularities could be avoided by matter with a de Sitter core. The first regular black hole solution, based on this idea, was proposed by Bardeen [5] with horizons but no singularity. The Bardeen black hole was reinterpreted as an exact solution to Einstein equations coupled to nonlinear electrodynamics [6]. Recently, the spherically symmetric Bardeen-de Sitter black hole was derived [7] whose metric reads ds 2 ¼ −fðrÞdt 2 þ 1 fðrÞ dr 2 þ r 2 ðdθ 2 þ sin 2 θdϕ 2 Þ; ð1Þ where fðrÞ is a nonlinear metric function given by fðrÞ¼ 1 − 2mr 2 ðr 2 þ e 2 Þ 3=2 − Λr 2 3 ; r ≥ 0: Here m represents black hole mass and e is the nonlinear charge of a self-gravitating magnetic field of a nonlinear electrodynamic source. An analysis of fðrÞ¼ 0 in the absence of cosmological constant Λ reveals the existence of a critical e  such that fðrÞ has a double root if e ¼ e  , two roots if e<e  and no root if e>e  , with e  ¼ 2m=3 ffiffi 3 p . These cases, respectively, correspond to an extreme black hole with degenerate horizons, a black hole with Cauchy and event horizons, and no black hole [8]. It can be seen that the metric (1) asymptotically behaves as [7] fðrÞ ∼ 1 − 2m r þ 3me 2 r 3 − Λr 2 3 þ O  1 r 5  : ð2Þ The Bardeen-de Sitter metric, in the limit e → 0, becomes the Schwarzschild-de Sitter metric, and for small r fðrÞ ∼ 1 − Λ eff 3 r 2 ; for r ∼ 0; ð3Þ where Λ eff ¼ Λ þ 6m=e 3 . The Bardeen-de Sitter solution is regular everywhere which can be realized from the behavior of scalar invariant R ¼ R ab R ab (R ab is the Ricci tensor) and the Kretschmann invariant K ¼ R abcd R abcd (R abcd is the Riemann tensor) which are given by R ¼ 6mg 2 ð4e 2 − r 2 Þ ðr 2 þ e 2 Þ 7=2 þ 4Λ; K ¼ 12m 2 ðr 2 þ e 2 Þ 7 ½8e 8 − 4e 6 r 2 þ 47e 4 r 4 − 12e 2 r 6 þ 4r 8 þ 8e 2 Λm  4e 2 − r 2 ðe 2 þ r 2 Þ 7=2  þ 8 3 Λ 2 : ð4Þ * alimd.sabir3@gmail.com † sghosh2@jmi.ac.in, sgghosh@gmail.com PHYSICAL REVIEW D 98, 084025 (2018) 2470-0010=2018=98(8)=084025(10) 084025-1 © 2018 American Physical Society