Generating rotating regular black hole solutions without complexification

We drop the complexification procedure from the Newman-Janis algorithm and introduce more physical arguments and symmetry properties, and we show how one can generate regular and singular rotating black hole and non-black-hole solutions in Boyer-Lindquist coordinates. We focus on generic rotating regular black holes and show that they are regular on the Kerr-like ring, but physical entities are undefined there. We show that rotating regular black holes have much smaller electric charges, and, with increasing charge, they turn into regular non-black-hole solutions well before their Kerr-Newman counterparts become naked singularities. No causality violations occur in the region inside a rotating regular black hole. The separability of the Hamilton-Jacobi equation for neutral particles is also carried out in the generic case, and the innermost boundaries of circular orbits for particles are briefly discussed. Other, but special, properties pertaining to the rotating regular counterpart of the Ayón-Beato–García regular static black hole are also investigated.

Figure 1

(a) Using different horizontal and vertical scales, we show in the $(a^2,M^2)$ plane the extremality condition. Continuous plot: Rotating regular black hole (24) with $F$ given by (25). The black hole region is above this curved line. The curve itself represents an extremal black hole, and the region beneath it represents regular non-black-hole solutions. Dashed plot: Rotating Kerr-Newman black hole. The Kerr-Newman black hole region is above this straight line of the equation $M^2/Q^2=a^2/Q^2+1$. Notice that the region between the two plots corresponds to $Q^2<M^2$, which is within the black hole region for the Kerr-Newman solution but within the non-black-hole region ($\forall a^2\ge 0$) for the rotating regular black hole (24). This is a parametric plot of $1/(2s{)}^2$ versus $u^2$ (see Appendix pp2). (b) The common radius ${r_{\text{ext}}}^2$ of the merging horizons versus $a^2$. For $a^2=0$, we have ${r_{\text{ext}}}^2/Q^2\simeq 2.51155$, yielding $r_{\text{ext}}/|Q|\simeq 1.58$, as found in Ref. [17]. This is a parametric plot of $t-1$ versus $u^2$ (see Appendix pp2).

Figure 2

For all the plots, we took $M^2=16$, $Q^2=1$, and $a^2=1$, corresponding, according to Fig. 1, to the black hole region for the Kerr, the Kerr-Newman, and the rotating regular black hole solutions (24), with $F$ given by (25). (a) Implicit plot of $\mathrm{\Sigma }(\sin \theta ,r)=0$, where $\mathrm{\Sigma }=(r^2+a^2{)}^2-a^2\mathrm{\Delta }{\sin }^2\theta$ and $\mathrm{\Delta }=r^{2}F+a^2$, for the Kerr black hole (dashed line, $F=1-2M/r$), the Kerr-Newman black hole (continuous line, $F=1-2M/r+Q^2/r^2$), and the rotating regular black hole solution (24), with $F$ given by (25) (the point $\sin \theta =1$ and $r=0$). Causality violations and CTCs occur on the right of each curve, where $g_{\varphi \varphi }>0$. The Kerr black hole has CTCs for $r<0$ only while the Kerr-Newman one has CTCs for both signs of $r$. For the rotating regular black hole solution (24) with $F$ given by (25), no causality violations or CTCs occur, since $g_{\varphi \varphi }<0$ [except at the point ($\sin \theta =1$ and $r=0$), where $\mathrm{\Sigma }=0$ and $g_{\varphi \varphi }$ is undefined]. (b) Implicit plot of $\mathrm{\Sigma }(\sin \theta ,r^2)=0$ for the rotating regular black hole solution (24), with $F$ given by (25). The plot confirms that solutions to $\mathrm{\Sigma }(\sin \theta ,r^2)=0$ for $\sin \theta \ne 1$ are such that $r^2<0$.