Slowly rotating regular black holes with a charged thin shell

We obtain rotating solutions of regular black holes which are constructed of de Sitter spacetime with the axisymmetric stationary perturbation within the timelike charged thin shell and the Kerr-Newman geometry with sufficiently small rotation outside the shell. To treat the slowly rotating thin shell, we employ the method developed by de la Cruz and Israel. The thin shell is assumed to be composed of a dust in the zero-rotation limit and located inside the inner horizon of the black hole solution. We expand the perturbation in powers of the rotation parameter of the Kerr-Newman metric up to the second order. It is found that with the present treatment, the stress tensor of the thin shell in general has anisotropic pressure, i.e., the thin shell cannot be composed of a dust if the rotational effects are taken into account. However, the thin shell can be composed of a perfect fluid with isotropic pressure if the degrees of freedom appearing in the physically acceptable matching of the two distinct spacetimes are suitably used. We numerically investigate the rotational effects on the spherically symmetric charged regular black hole obtained by Uchikata, Yoshida, and Futamase in detail.

Figure 1

Radius of the unperturbed thin shell, $R$, given as a function of the rest mass of the unperturbed thin shell, $4\pi R^2{\sigma }_0$. Each line corresponds to the equilibrium sequence characterized by the same black hole mass, $M$, whose values are given near the corresponding line.

Figure 2

Same as Fig. 1, but for the ratio of the charge to the mass ratio of the black hole, $Q/M$.

Figure 3

Dimensionless angular velocity of the thin shell, $\overline{\mathrm{\Omega }}/\epsilon \equiv \mathrm{\Omega }/(\epsilon \sqrt{M/R^3})$, given as a function of the rest mass of the unperturbed thin shell, $4\pi R^2{\sigma }_0$. Each line corresponds to the equilibrium sequence characterized by the same black hole mass, $M$, whose values are given near the corresponding line.

Figure 4

Same as Fig. 3, but for the dimensionless angular velocity of the frame dragging in $V^{-}$, $\omega /\mathrm{\Omega }$.

Figure 5

Same as Fig. 3, but for the integration constant $C_2$.

Figure 6

Dimensionless quadrupole radial displacement of the thin shell defined in $V^{-}$, ${\xi }_2/R$, given as a function of the rest mass of the unperturbed thin shell, $4\pi R^2{\sigma }_0$. Each line corresponds to the equilibrium sequence characterized by the same black hole mass, $M$, whose values are given near the corresponding line.

Figure 7

Same as Fig. 6, but for the dimensionless quadrupole radial displacement of the thin shell defined in $V^{+}$, ${\zeta }_2/R$.

Figure 8

Same as Fig. 6, but for the eccentricity of the thin shell, $e/\epsilon$.

Figure 9

Same as Fig. 6, but for the dimensionless quadrupole energy density perturbation, $\delta {\overline{\sigma }}_2\equiv \delta {\sigma }_2/(M/4\pi R^2)$.

Figure 10

Same as Fig. 6, but for the dimensionless quadrupole pressure perturbation, $\delta {\overline{p}}_2\equiv \delta p_2/(M/4\pi R^2)$.

Figure 11

Same as Fig. 6, but for the dimensionless integration constant $D_2/M^2$.

Figure 12

Dimensionless spherically symmetric radial displacement of the thin shell defined in $V^{+}$, ${\zeta }_0/R$, given as functions of the rest mass of the unperturbed thin shell, $4\pi R^2{\sigma }_0$. Each line corresponds to the equilibrium sequence characterized by the same black hole mass, $M$, whose values are given near the corresponding line.

Figure 13

Same as Fig. 12, but for the dimensionless spherically symmetric radial displacement of the thin shell defined in $V^{-}$, ${\xi }_0/R$.

Figure 14

Same as Fig. 12, but for the dimensionless spherically symmetric energy density perturbation, $\delta {\overline{\sigma }}_0\equiv \delta {\sigma }_0/(M/4\pi R^2)$. The short unlabeled line at the left top corresponds to the result for the $M=1.9$ sequence.

Figure 15

Same as Fig. 12, but for the dimensionless spherically symmetric pressure perturbation, $\delta {\overline{p}}_0\equiv \delta p_0/(M/4\pi R^2)$.

Figure 16

Same as Fig. 12, but for the relative changes in the gravitational mass of the back hole, $\delta M/M$.

Figure 17

Same as Fig. 12, but for the integration constant $C_4$ appearing in $h_0$.

Figure 18

The dimensionless integration constant, $D_2/M^2$, the eccentricity of the thin shell, $e/\epsilon$, the dimensionless energy density perturbation, $\delta {\overline{\sigma }}_2\equiv \delta {\sigma }_2/(M/4\pi R^2)$, and the dimensionless pressure perturbation, $\delta {\overline{p}}_2\equiv \delta p_2/(M/4\pi R^2)$, given as functions of the unperturbed matching radius, $R$.

Figure 19

Same as Fig. 18, but for the relative change in the gravitational mass, $\delta M/M$, the dimensionless energy density perturbation, $\delta {\overline{\sigma }}_0\equiv \delta {\sigma }_0/(M/4\pi R^2)$, the dimensionless pressure perturbation, $\delta {\overline{p}}_0\equiv \delta p_0/(M/4\pi R^2)$, and the integral constant, $C_4$.

Figure 20

Same as Fig. 18, but for the dimensionless radial displacement of the matching radius, ${\xi }_0/R$, ${\zeta }_0/R$, ${\xi }_2/R$, and ${\zeta }_2/R$.