Non-singular rotating black hole with a time delay in the center

Abstract

As proposed by Bambi and Modesto, rotating non-singular black holes can be constructed via the Newman–Janis algorithm. Here we show that if one starts with a modified Hayward black hole with a time delay in the centre, the algorithm succeeds in producing a rotating metric, but curvature divergences reappear. To preserve finiteness, the time delay must be introduced directly at the level of the non-singular rotating metric. This is possible thanks to the deformation of the inner stationarity limit surface caused by the regularisation, and in more than one way. We outline three different possibilities, distinguished by the angular velocity of the event horizon. Along the way, we provide additional results on the Bambi–Modesto rotating Hayward metric, such as the structure of the regularisation occurring at the centre, the behaviour of the quantum gravity scale alike an electric charge in decreasing the angular momentum of the extremal black hole configuration, or details on the deformation of the ergosphere.

Appendices

A. Diagonalising $T^{\mu \nu }$

The co-rotating tetrad $e^I_\mu$ of [42] in the Bambi–Modesto metric has norms given by

$$\begin{aligned} g^{\mu \nu }e^I_\mu e^J_\nu ={\left\{ \begin{array}{ll} (-\mathrm{Sign}(\tilde{\Delta }),\mathrm{Sign}(\tilde{\Delta }),1,1) &{} \mathrm{if\;} I=J\\ 0 &{} \mathrm{if\;} I\ne J \end{array}\right. } \end{aligned}$$

(33)

The corresponding Einstein tensor defines a non-diagonal energy-momentum tensor of the form

$$\begin{aligned} T^{IJ} = \left( \begin{array}{cccc} i&{}\quad 0&{}\quad 0&{}\quad j\quad \\ 0&{}\quad k&{}\quad 0&{}\quad 0\quad \\ 0&{}\quad 0&{}\quad l&{}\quad 0\quad \\ j&{}\quad 0&{}\quad 0&{}\quad n\quad \end{array}\right) . \end{aligned}$$

(34)

We then ask whether it is possible to diagonalise this tensor with a Lorentz transformation. To that end, we can concentrate on the 2-by-2 block with $I=0,3$. The transformation will have to be either a simple rotation, when $\mathrm{Sign}(\tilde{\Delta })=-1$, between the event and the Cauchy horizons, or a boost, when $\mathrm{Sign}(\tilde{\Delta })=+1$ in the rest of the spacetime. Consider first the latter case. The most general $(1+1)$ boost reads

$$\begin{aligned} \Lambda ^I{}_{J}=\left( \begin{array}{cc} \cosh \eta &{}\quad \sinh \eta \\ \sinh \eta &{}\quad \cosh \eta \\ \end{array}\right) \,. \end{aligned}$$

(35)

The condition $D^{03}=0$, where $D^{IJ}=\Lambda ^I{}_{K} T^{KL}\Lambda ^J{}_{L}$, reduces to

$$\begin{aligned} \tanh \eta = -\frac{2j}{i+m} \end{aligned}$$

(36)

and implies

$$\begin{aligned} \left| -\frac{2j}{i+m}\right| <1 \end{aligned}$$

(37)

in order for the transformation to be valid, which in turn imposes conditions on the parameters of the metric. In between the horizons, since the transformation is a rotation, this problem does not arise because condition (36) is replaced by

$$\begin{aligned} \tan \eta = -\frac{2j}{m-i} \end{aligned}$$

(38)

which can be always satisfied. When condition (37) is not satisfied – notice that this may also happen locally, as the coefficients in (37) are spacetime functions –, it is not possible to diagonalize the energy momentum tensor starting from the co-rotating tetrad. Therefore, we need to start from scratch to find the good orthonormal basis. This is generally a not easy task and we do not address this problem here.

When a time delay is introduced, for cases (ii) and (iii) is still possible to define the co-rotating tetrad with the norms given in Eq. (33). In these cases, however, the transformed energy-momentum tensor $T^{IJ}$ has two non-null components out of the diagonal, namely $T^{03}$ and $T^{12}$. To diagonalize it, therefore, we need now to combine a boost and a rotation for the two different 2-by-2 blocks according to the sign of $\tilde{\Delta }$. This introduces the same condition (37) when $\mathrm{Sign}(\tilde{\Delta })=-1$, together with a similar one for the 1–2 block of the tensor in the opposite case. The additional condition imposes strong limitations on the procedure, so that for the same range of parameters we now typically have finite regions where the energy-momentum tensor is not diagonalisable with the co-rotating tetrad. This shows up in the void zones in the numerical plots. The void regions can be either in between the horizons or outside, depending on the value of the parameters.

B. Coefficients of the $\mathcal K$ expansion

We report in this Appendix various coefficients of power series used in the main text. For the expansion (29), we have

$$\begin{aligned} c_1=\frac{1}{a^4\,G(0)}\Big [ 8G(0)\Big (2 \sqrt{G(0)} - 1 - G(0)\Big ) + a^2 G'(0)^2 \Big ], \end{aligned}$$

and the other two, $c_2$ and $c_3$, are functions of G(0) and its derivatives up to the second order, too long to be written here. However, once the solution for $c_1=c_2=0$ is plugged into the equation $c_3=0$, one gets the following relatively simple equation for G(0),

$$\begin{aligned}&G(0) \big ( 100\,\sqrt{G(0)} - 21 \big )\nonumber \\&\quad -\, G(0)^2 \big (51\, G(0) - 44\,\sqrt{G(0)} - 86 \big )\nonumber \\&\quad +\, \Gamma \; \big (1-G(0)\big )=0, \end{aligned}$$

(39)

where the function $\Gamma$ is defined by

$$\begin{aligned} \Gamma ^2:= 32\, G(0)^2\,(\sqrt{G(0)}-1)^2(55\,G(0)+10\sqrt{G(0)}-9)\;. \end{aligned}$$

The only positive solution of Eq. (39) is $G(0)=1$.

On the other hand, when the time delay is introduced after the NJ procedure, non-trivial solutions exist. The expansion still has the structure (29), but with different coefficients. Both cases (i) and (ii) give the same result,

$$\begin{aligned} \begin{aligned}&c_1 = 2\frac{G'(0){}^2}{a^2G^2(0)}, \\&c_2 = \frac{(G'(0){}^2-2G(0) G''(0))^2}{4G(0)^4}\\&c_3 = \frac{2}{3} c_2 - 4 \frac{G'(0){}^2}{15 a^2 G(0)^2} \end{aligned} \end{aligned}$$

which can be all made to vanish for $G'(0)=G''(0)=0$, while keeping an arbitrary $G(0)\ne 0$. While the coefficients are slightly different in case (iii), their vanishing leads to the same condition.

Finally, the coefficients of the equatorial expansion (31) are much longer. Explicitly for case (ii), the first three read

$$\begin{aligned} d_6= & {} \frac{2a^4 G'(0)^2}{G(0)^2}, \\ d_5= & {} \frac{a^4 G'(0)}{G(0)^3}(2G(0)G''(0)-3G'(0)^2), \\ d_4= & {} \frac{a^2}{4G(0)^4L^2}\bigg [16 a^2G(0)^2G'(0)^2 \\&+\, L^2 \Big [4a^2G(0)^2G''(0)^2 + G'(0)^2\Big (13a^2G'(0)^2 \\&-\,16a^2G(0)G''(0)+8G(0)^2\Big )\Big ]\bigg ]. \end{aligned}$$

These vanish iff the first two derivatives vanish. With this condition, $d_3\equiv 0$ and

$$\begin{aligned} d_2 = \frac{G^{(3)}(0)^2}{2G(0)^2}, \end{aligned}$$

which gives the additional condition $G^{(3)}(0)=0$ for the third derivative. Finally, the three conditions all together make $d_1$ vanish.

C. Divergence for $\gamma \ne \delta$

In Sect. 4 and the previous Appendix, we showed that introducing a time delay in the rotating case by applying the NJ algorithm to the metric of [20] unavoidably leads to a divergent $\mathcal K$, for any function G(r) and for a complexification such that M(r) and G(r) are unchanged, as choosing $\gamma =\delta$ in the Bambi–Modesto prescription. When $\gamma \ne \delta$, the dependence on r and $\theta$ is too complicated to be handled explicitly, and we can not derive a power series expansion near zero. However, the divergence can be established with an indirect argument. In fact, notice that at the equator, where the divergence lurks, the metric for $\gamma \ne \delta$ coincides with the metric for $\gamma =\delta$. Furthermore, also the first derivatives coincide, and most of the second derivatives. The only terms that differ are second derivatives in $\theta$ of three metric components. Explicitly,

$$\begin{aligned} \partial ^2_\theta (g-\bar{g}) \big |_{\theta =\frac{\pi }{2}}= (\gamma - \delta ) \frac{8 a^2 m^2L^2 G(r)}{(4mL^2 + r^3)^2}\times {\left\{ \begin{array}{ll} -1 &{} (uu)\\ a &{} (u\phi )\\ -a^2 &{} (\phi \phi )\\ \end{array}\right. } \end{aligned}$$

where g is the metric for $\gamma \ne \delta$, while $\bar{g}$ is the one with $\gamma = \delta$.

Hence, the Riemann tensors evaluated at the equator only differ in the terms of type $R^{\mu }{}_{\theta \theta \nu }$, and the difference of the Kretschmann invariants is

$$\begin{aligned} (\mathcal{K} - \bar{\mathcal{K}})\big |_{\theta =\frac{\pi }{2}} = 8 R^\mu {}_{\theta \theta \nu } A_{\mu }{}^{\nu } + 2 A^{\mu \nu } A_{\mu \nu }, \end{aligned}$$

(40)

where

$$\begin{aligned} A^\mu {}_\nu := \frac{1}{2}\,\bar{g}^{\mu \lambda }\,\partial _\theta ^2(g_{\lambda \nu }-\bar{g}_{\lambda \nu })\big |_{\theta =\frac{\pi }{2}}. \end{aligned}$$

(41)

The latter quantity is zero except for the u and $\phi$ components and, more importantly, it is finite. Equation (40), therefore, tells us that the divergence of one implies the divergence of the other.