New magnetized squashed black holes—thermodynamics and Hawking radiation

Abstract

We construct a new exact solution in the 5D Einstein–Maxwell-dilaton gravity describing a magnetized squashed black hole. We calculate its physical characteristics, and derive the Smarr-like relations. The Hawking radiation of the solution is also investigated by calculating the graybody factor for the emission of massless scalar particles in the low-energy regime. In distinction to the case of asymptotically flat magnetized black holes, the thermodynamical characteristics and the Hawking radiation of the solution depend on the external magnetic field, and consequently can be tuned by its variation.

Appendix

The 5D Einstein–Maxwell-dilation equations with a dilaton coupling parameter $a=\sqrt{8/3}$ possess the following more general black hole solution:

The 1-form W ^t and the metric functions W and V are given by

(46)

and the Maxwell 2-form is expressed as

$$F = dt\wedge d A_t + d\psi\wedge d A_\psi+ d \phi \wedge d A_\phi, $$

with electromagnetic potentials

(47)

The coordinates vary in the limits 0≤r<∞, 0≤θ≤π and ϕ is a periodic coordinate with period Δϕ=2π. The parameters γ, r ₀ and r ₊ take the same ranges as in Sect. 2, R _∞ is expressed as $R^{2}_{\infty}= r_{0}(r_{0} + r_{+})$, and υ is a second real parameter characterizing the electromagnetic field. In the limit υ=0 the solution reduces to the magnetized black hole which we discussed in Sect. 2.

The solution possesses coordinate singularities at θ=0 (or θ=π), which can be resolved if the coordinates t and ψ are identified with periods Δt=4πsinhγcoshυsinhυ  r ₊ and Δψ=4π. Consequently, the cross-sections at $r=t =\mathrm{const.}$, and at $r = \psi= \mathrm{const.}$ represent Hopf fibration of S ³ with S ¹ parameterized by the ψ and t coordinates, respectively. Due to the periodic identification of the time coordinate the solution contains closed time-like curves, meaning that the causality of the spacetime is violated. In the limit at υ=0, however, the fiber bundle associated with the time coordinate becomes trivial and no closed time-like curves occur.