Construction of regular black holes in general relativity

We present a general procedure for constructing exact black hole solutions with electric or magnetic charges in general relativity coupled to a nonlinear electrodynamics. We obtain a variety of two-parameter family spherically symmetric black hole solutions. In particular, the singularity at the center of the space-time can be canceled in the parameter space and the black hole solutions become regular everywhere in space-time. We study the global properties of the solutions and derive the first law of thermodynamics. We also generalize the procedure to include a cosmological constant and construct regular black hole solutions that are asymptotic to anti–de Sitter space-time.

Figure 1

The metric function $f(r)$ for a Bardeen class solution with zero Schwarzschild mass. Along the vertical axis, the value of $\mu$ decreases from top to bottom. For solid lines $\mu =5$, 3 and for dashed lines $\mu =2$, 1. Some parameters have been set as $\alpha =1/2$, $q=1$. The dotted line corresponds to a Schwarzschild black hole with $M=2$.