Regular black holes: Electrically charged solutions, Reissner-Nordström outside a de Sitter core

To have the correct picture of a black hole as a whole, it is of crucial importance to understand its interior. The singularities that lurk inside the horizon of the usual Kerr-Newman family of black hole solutions signal an endpoint to the physical laws and, as such, should be substituted in one way or another. A proposal that has been around for sometime is to replace the singular region of the spacetime by a region containing some form of matter or false vacuum configuration that can also cohabit with the black hole interior. Black holes without singularities are called regular black holes. In the present work, regular black hole solutions are found within general relativity coupled to Maxwell’s electromagnetism and charged matter. We show that there are objects which correspond to regular charged black holes, whose interior region is de Sitter, whose exterior region is Reissner-Nordström, and the boundary between both regions is made of an electrically charged spherically symmetric coat. There are several types of solutions: regular nonextremal black holes with a null matter boundary, regular nonextremal black holes with a timelike matter boundary, regular extremal black holes with a timelike matter boundary, and regular overcharged stars with a timelike matter boundary. The main physical and geometrical properties of such charged regular solutions are analyzed.

Figure 1

A plot of the several radii as a function of $\alpha =r_0/q$. The minimum value of $\alpha$ is $\sqrt{3}/3$, and the vertical dashed-dotted line is at $\alpha =2/3$.

Figure 2

A plot of $B(r)$ for four different $\alpha \equiv \frac{r_0}{q}$: $\alpha =\sqrt{3}/3$ (solid line), $\alpha =\sqrt{3}/3+.1$ (dashed line), $\alpha =2/3$ (dash-dotted line), and $\alpha =2/3+.1$ (dotted line).

Figure 3

The Carter-Penrose diagram in the possible four distinct situations, three for regular black holes and one for a star: (i) The regular nonextremal black hole with a lightlike matter boundary, where $r_0=r_{-}$. (ii) The regular nonextremal black hole with a timelike matter boundary, where $r_0<r_{-}$. (iii) The regular extremal black hole with a timelike matter boundary, where $r_0<r_{-}=r_{+}$. (iv) The regular overcharged star with a timelike matter boundary, where there is no $r_{\pm }$. In all cases, $r_0$ represents the surface matching an interior de Sitter spacetime region to an exterior Reissner-Nordström region. Note the regular black hole constraint $r_0\le r_{-}$.