Abstract
In this paper, we obtain higher dimensional topological black hole solutions of Einstein-$\Lambda$ gravity in the presence of a class of nonlinear electrodynamics. First, we calculate the conserved and thermodynamic quantities of ($n+1$)-dimensional asymptotically flat solutions and show that they satisfy the first law of thermodynamics. Also, we investigate the stability of these solutions in the (grand) canonical ensemble. Second, we endow a global rotation to the static Ricci-flat solutions and calculate the conserved quantities of solutions by using the counterterm method. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta and the electric charge, and show that these quantities satisfy the first law of thermodynamics. Then, we perform a stability analysis of the rotating solutions both in the canonical and the grand canonical ensembles.
Highlights
About 80 years ago Born and Infeld [19,20] introduced an interesting kind of nonlinear electrodynamics (NLED) in order to remove the divergence of the self-energy of a point-like charge
It is notable to mention that the exponential form of NLED does not cancel the divergency of the electric field but its singularity is much weaker than that in the Maxwell theory
In order to obtain the electric charge per unit volume Vn−1 of the black hole, we use the flux of the electric field at infinity, yielding which shows that, this kind of nonlinearity does not change the electric charge
Summary
About 80 years ago Born and Infeld [19,20] introduced an interesting kind of NLED in order to remove the divergence of the self-energy of a point-like charge. We take into account the Eq (1) as a NLED source and investigate the effects of nonlinearity on the properties of static and rotating black hole/brane solutions. In transition from the Maxwell theory to NLED, the logical decision is to consider the effects of weak nonlinearity variations, not strong effects. This means that, one can expect to obtain precise physical results with experimental agreements, provided one regards the nonlinearity as a correction to the Maxwell field. We consider the (n + 1)-dimensional topological static black hole solutions of Einstein gravity in presence of the mentioned NLED and investigate their properties.
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