Abstract
We carry out the Kerr/CFT correspondence in a four-dimensional extremal rotating regular black hole with a non-linear magnetic monopole (NLMM). One problem in this study would be whether our geometry can be a solution or not. We search for the way making our rotating geometry into a solution based on the fact that the Schwarzschild regular geometry can be a solution. However, in the attempt to extend the Schwarzschild case that we can naturally consider, it turns out that it is impossible to construct a model in which our geometry can be a exact solution. We manage this problem by making use of the fact that our geometry can be a solution approximately in the whole space-time except for the black hole's core region. As a next problem, it turns out that the equation to obtain the horizon radii is given by a fifth-order equation due to the regularization effect. We overcome this problem by treating the regularization effect perturbatively. As a result, we can obtain the near-horizon extremal Kerr (NHEK) geometry with the correction of the regularization effect. Once obtaining the NHEK geometry, we can obtain the central charge and the Frolov–Thorne temperature in the dual CFT. Using these, we compute its entropy through the Cardy formula, which agrees with the one computed from the Bekenstein–Hawking entropy.
Highlights
Getting understanding for the microscopic states of the Bekenstein-Hawking entropies is one of the very important issue for us
The regular black hole geometry with a non-linear magnetic monopole (NLMM) in this study is given, and in Sec.3, considering an Einstein-nonlinear Maxwell action, we search for the way to make our rotating geometry with a NLMM into a solution. It turns that it is adding a new term vanishing at zero-rotation to the action, because the form of the geometry seems to have no space where we can add further modification for making into a solution
This would be the natural extension we can consider from the Schwarzschild case with a NLMM [46]
Summary
Getting understanding for the microscopic states of the Bekenstein-Hawking entropies is one of the very important issue for us. In Sec., the regular black hole geometry with a NLMM in this study is given, and in Sec., considering an Einstein-nonlinear Maxwell action, we search for the way to make our rotating geometry with a NLMM into a solution. It turns that it is adding a new term vanishing at zero-rotation to the action, because the form of the geometry seems to have no space where we can add further modification for making into a solution. In Appendix.A and B, a computation of the central charge based on the Lagrangian formalism [60, 61], and the expression of the Hawking temperature in our geometry are shown
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