Static and stationary loop quantum black bounces

Recent advances in observational technologies have revitalized interest in gravitation theory. High-precision measurements have led to the direct detection of gravitational waves by the LIGO-VIRGO-KAGRA observatories, to the imaging of supermassive black holes by the Event Horizon Telescope (EHT), and to precise determination of cosmological parameters, significantly enhancing our understanding of the universe and offering new challenges to our theories Abbott et al. (2016, 2017); Akiyama et al. (2022, 2019); Aghanim et al. (2020); Aiola et al. (2020); Gunn et al. (2006). Nonetheless, despite the success in the observational determination of model parameters, there are fundamental reasons to believe that General Relativity (GR) may not account for all the possible gravitational effects. Consequently, the search for modified theories of gravity has attracted significant attention from the theoretical physics community.

In line with this, new black hole solutions have recently been found by applying the quantization techniques of Loop Quantum Gravity to spherically symmetric solutions Kelly et al. (2020); Lewandowski et al. (2023), following the success of Loop Quantum Cosmology models Ashtekar et al. (2003, 2006a, 2006b, 2008); Assanioussi et al. (2019). To this end, an effective Hamiltonian and a corresponding metric are introduced to describe the vacuum exterior solution within the context of the Oppenheimer-Snyder (OS) collapse model, specifically in spherically symmetric spacetimes Kelly et al. (2020). A novel approach has also been proposed for the quantum OS model, which bridges the interior and exterior regions. This method incorporates LQC corrections within the interior spacetime, extending them to the exterior via specific boundary matching conditions Lewandowski et al. (2023).

In particular, in ref. Kelly et al. (2020) the authors found a line element of the form

$$ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2}-r^{2}d\Omega^{2},\;f(r)=1-\frac{2M}{r}+\frac{\alpha^{2}M^{2}}{r^{4}},$$

(1)

where the domain of the radial coordinate is bounded from below to impose a minimum radius able to cure the singularity. This minimal radius is given by

$$r>r_{b}=(\alpha^{2}M/2)^{1/3},$$

(2)

where $\alpha^{2}=16\sqrt{3}\pi\gamma^{3}\ell^{2}_{P}$, with $\ell_{P}$ being the Planck length and $\gamma$ is the Barbero-Immirzi parameter. The above cut-off is found in other models of LQG Gambini et al. (2020). One can check that the line element (1) has two horizons located approximately at

$$r_{inner}=\left(\frac{\alpha^{2}M}{2}\right)^{1/3}\!+\ \frac{1}{6}\left(\frac{\alpha^{4}}{4M}\right)^{1/3}\!\!\!\!\!\!,\quad r_{outer}=2M-\frac{\alpha^{2}}{8M}.$$

(3)

The properties of the above black hole have been studied, finding that the solution is stable under scalar and vector perturbations by computing the quasinormal modes, while the shadows were studied in Yang et al. (2023). This space-time has been generalized to the LQG-AdS case, and its thermodynamics was also studied Wang et al. (2024).

An undesired feature of the line element (1) is that the domain of the radial coordinate $r$ can be naturally extended to $r\to 0$, leading to a singular space-time. Thus, an analytical way to avoid the singularity without artificially restricting the domain of $r$ would be very valuable. A natural way out of this problem is provided by the Simpson-Visser procedure Simpson and Visser (2019), which consists on the replacement $r\to\sqrt{r^{2}+a^{2}}$, where $a$ is a regularization parameter. If we choose $a=r_{b}$, we get the desired regular solution with the same properties as the original one in central and far away regions. Furthermore, if we keep $a$ as a free parameter, we can get LQG-inspired wormholes and regular black holes by suitably tunning the value of $a$. It is in this extended scenario where we develop the main idea of this work: the Loop Quantum Black Bounce (LQBB). Our approach ensures the absence of the singularity and retains the main qualitative aspects of the LQG-inspired solution (1).

To better understand the inner workings of the Simpson-Visser procedure, it will be instructive to study the effective sources that generate the line element (1) and of its transformed LQBB version. The thermodynamics of the LQBB model will also be studied and compared with the case of Schwarzschild. We will also consider the extension of the LQBB line element to the rotating case, paying special attention to the fate of the ring singularity that arises in the usual Kerr black hole.

The paper is structured as follows: In Section II, we identify the source for the exterior solution of the LQG black hole. Section III discusses the application of Simpson-Visser regularization, where we derive the quantum loop black bounce solution and examine the horizons and energy conditions. In Section IV, we determine the sources for the regularized case. Section V delves into the thermodynamics of the solutions, while Section VI extends the analysis to the rotating case and investigates the resulting horizon structure, ergospheres, regularity, and shadows. Finally, Section VII presents our conclusions.

As a warm up for what will come, we first revisit the solution of Ref. Kelly et al. (2020) by computing the effective energy sources that generate that line element, studying the corresponding energy conditions, and determining their nature in terms of fundamental fields.

The line element given in Eq. (1) can be naturally extended to the region $r\to 0$, where a singularity exists. Given the electromagnetic nature of this space-time, it is easy to see that timelike geodesics never reach to the center, $r\to 0$, as they bounce off the potential barrier at some finite radius $r>0$. Radial null geodesics, instead, do reach the center, implying the existence of incomplete geodesics and, as a consequence, of a singularity.

By applying the Simpson-Visser prescription $r\to\sqrt{r^{2}+r_{b}^{2}}$ to the metric (1), as described in Simpson and Visser (2019), a new spacetime that is well behaved on the whole real line, $r\in]-\infty,+\infty[$, can be constructed. This procedure prevents the emergence of a singularity as $r\to\sqrt{r^{2}+r_{b}^{2}}$, similar to the approach in a LQG-inspired theory. As a result, we obtain

$$ds^{2}=\left[1\!-\!\frac{2M}{\sqrt{r^{2}+r_{b}^{2}}}\!+\!\frac{\alpha^{2}M^{2}}{(r^{2}+r_{b}^{2})^{2}}\right]\!dt^{2}-\left[1\!-\!\frac{2M}{\sqrt{r^{2}+r_{b}^{2}}}+\frac{\alpha^{2}M^{2}}{(r^{2}+r_{b}^{2})^{2}}\right]^{-1}\!\!dr^{2}-(r^{2}+r_{b}^{2})d\Omega^{2}.$$

(27)

In principle, we treat $r_{b}$ as a free parameter. However, in certain cases, we define the bounce radius as a function of $\alpha$ according to Eq. (2), without any loss of generality.

In Fig. 1, we depict the metric coefficient $g_{tt}$ as a function of $r$ to verify the horizons’ structure associated with the black bounce. It can be seen that the black bounce is a black hole with zero, two, or four horizons (zero, one, or two in the $r>0$ domain). The black hole will be an extremal one, presenting two horizons (one in $r>0$), if $M=\frac{4\alpha}{3\sqrt{3}}$. For $M$ below this value, the black bounce is horizonless and becomes a traversable wormhole.

Fig. 2 illustrates the parameter space $(M,\alpha)$, highlighting the regions where wormholes and black holes can exist. The blue area indicates the presence of event horizons ($r_{h}\geq r_{b}$), corresponding to regular black holes, while the white area represents the region where LQBB is without horizons, indicating the occurrence of wormholes.

Before considering the specific case (27), let us consider the more general one

$$ds^{2}=f(r)dt^{2}-f(r)^{-1}dr^{2}-\Sigma^{2}(r)\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right).$$

(30)

The corresponding components of the Einstein tensor are given by:

	$\displaystyle\ {}G^{0}_{0}=-\frac{f^{\prime}(r)\Sigma^{\prime}(r)}{\Sigma(r)}-\frac{f(r)\Sigma^{\prime}(r)^{2}}{\Sigma(r)^{2}}-\frac{2f(r)\Sigma^{\prime\prime}(r)}{\Sigma(r)}+\frac{1}{\Sigma(r)^{2}},$
	$\displaystyle\ {}G^{1}_{1}=-\frac{f^{\prime}(r)\Sigma^{\prime}(r)}{\Sigma(r)}-\frac{f(r)\Sigma^{\prime}(r)^{2}}{\Sigma(r)^{2}}+\frac{1}{\Sigma(r)^{2}},$
	$\displaystyle\ {}G^{2}_{2}=G^{3}_{3}=-\frac{f^{\prime}(r)\Sigma^{\prime}(r)}{\Sigma(r)}-\frac{f^{\prime\prime}(r)}{2}-\frac{f(r)\Sigma^{\prime\prime}(r)}{\Sigma(r)}.$		(31)

The first thing we must note is that

$$G^{0}_{0}-G^{1}_{1}=-\frac{2f(r)\Sigma^{\prime\prime}(r)}{\Sigma(r)}\neq 0$$

(32)

Therefore, a NED source is not enough to generate this type of geometry. To address this, we will add a scalar field to the action (12), given by

$$S_{\phi}=\int\sqrt{-g}d^{4}x\left[\epsilon g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]\ ,$$

(33)

where $\phi$ is the scalar field, $V(\phi)$ the scalar field potential, $\epsilon=\pm 1$ depending on whether the scalar field is canonical ($+$) or phantom ($-$). Accordingly, we get the system of equations

	$\displaystyle\ {}\nabla_{\mu}\left[L_{F}F^{\mu\nu}\right]=0,\;\nabla_{\mu}^{*}F^{\mu\nu}=0,$		(34)
	$\displaystyle\ {}2\epsilon\nabla_{\mu}\nabla^{\mu}\phi=-\frac{dV(\phi)}{d\phi}\ ,$		(35)
	$\displaystyle\ {}G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa^{2}\left(T_{\mu\nu}^{\phi}+T_{\mu\nu}^{EM}\right)\ ,$		(36)

where

$$T_{\mu\nu}^{\phi}=2\epsilon\partial_{\nu}\phi\partial_{\mu}\phi-g_{\mu\nu}\big{(}\epsilon\partial^{\alpha}\phi\partial_{\alpha}\phi-V(\phi)\big{)}\$$

(37)

and $T_{\mu\nu}^{EM}$ was given previously in Eq. (15). With the same assumptions as before, (34) implies

$$F^{10}=\frac{q_{e}}{\Sigma^{2}L_{F}},\quad F_{23}=q_{m}\sin\theta\to L_{F}^{2}F=-\frac{q_{e}^{2}}{2\Sigma^{4}},F=\frac{q_{m}^{2}}{2\Sigma^{4}}.$$

(38)

By using the auxiliary field (17) we get

$$P=-\frac{q_{e}^{2}}{2\Sigma^{4}},F=\frac{q_{m}^{2}}{2\Sigma^{4}}.$$

(39)

Now, assuming that $\phi=\phi(r)$, we get for the stress-energy tensors :

	$\displaystyle\ {}{T^{\phi}}^{\mu}_{\ \nu}=\epsilon f\phi^{\prime}{}^{2}\,{\rm diag}\,(1,-1,1,1)+\delta^{\mu}_{\nu}V(\phi)\ ,$		(40)
	$\displaystyle\ {}{T^{EM}}^{\mu}_{\ \nu}=\,{\rm diag}\,\Big{(}L-\sqrt{PF},\ L-\sqrt{PF},\ L+\sqrt{PF},\ L+\sqrt{PF}\Big{)}\ .$		(41)

Note that now we have ${T}^{0}_{\ 0}\neq{T}^{1}_{\ 1},\,{T}^{2}_{\ 2}={T}^{3}_{\ 3}$, with the same symmetries as the Einstein tensor, Eq. (IV). Before considering each case separately, we will use $\Sigma^{2}=r^{2}+r_{b}^{2}$ and simplify further by finding the solution to the scalar field.

The scalar field equation (35) yields

$$\frac{2}{\Sigma^{2}(r)}\partial_{r}\left[\Sigma^{2}(r)f(r)\partial_{r}\phi\right]=V^{\prime}(\phi)\ .$$

(42)

Furthermore, by combining again the ${0\choose 0}$ and ${1\choose 1}$ components of the Einstein field equations (14) as

$$G^{1}_{\ 1}-G^{0}_{\ 0}=\frac{2f\Sigma^{\prime\prime}}{\Sigma}=\kappa^{2}T^{1}_{\ 1}-\kappa^{2}T^{0}_{\ 0}=-2\kappa^{2}\epsilon f\phi^{\prime 2}\ ,$$

(43)

we find that the scalar field is given by

$$\phi^{\prime 2}=\frac{\Sigma^{\prime\prime}}{\kappa^{2}\Sigma}=\frac{r_{b}^{2}}{\kappa^{2}\Sigma^{4}}\to\phi(r)=\frac{1}{\kappa}\arctan{\left(\frac{r}{r_{b}}\right)},$$

(44)

and the potential $V(r)$ is

$$V(r)=-\frac{4Mr_{b}^{2}}{\kappa^{2}}\left[\frac{\alpha^{2}M}{4\left(r_{b}^{2}+r^{2}\right)^{4}}-\frac{1}{5\left(r_{b}^{2}+r^{2}\right)^{5/2}}\right]=-\frac{4Mr_{b}^{2}}{\kappa^{2}}\left[\frac{\alpha^{2}M}{4\Sigma^{8}}-\frac{1}{5\Sigma^{5}}\right].$$

(45)

In Fig. (5), the left side shows that the height of the potential barrier decreases as quantum effects increase (i.e., as $\alpha$ increases). This reduction in barrier height facilitates the tunneling of a quantum particle to another region of the universe. A similar trend is observed on the right side of the figure. Additionally, the potential at the origin results in a static equilibrium state for the scalar field when $\alpha>2^{-4/3}\approx 0.4$. It is worth mentioning that in the limit $\alpha\to 0$ one recovers the results discussed in Rodrigues and Silva (2023).

Now, the stress tensor of the scalar field simplifies to

$${T^{\phi}}^{\mu}_{\ \nu}=-(1-\frac{2M}{\Sigma}+\frac{\alpha^{2}M^{2}}{\Sigma^{4}})\frac{r_{b}^{2}}{\kappa^{2}\Sigma^{4}}\,{\rm diag}\,(1,-1,1,1)-\delta^{\mu}_{\nu}\-\frac{4Mr_{b}^{2}}{\kappa^{2}}\left(\frac{\alpha^{2}M}{4\Sigma^{8}}-\frac{1}{5\Sigma^{5}}\right)$$

(46)

With this, we will now consider the magnetic and electric cases separately.

Let’s calculate the surface gravity at the event horizon for the regular black hole case. The Killing vector, which is null at the event horizon, is $\xi^{\mu}=\partial_{t}$. This gives the following norm:

$$\xi^{\mu}\xi_{\mu}=g_{\mu\nu}\xi^{\mu}\xi^{\nu}=g_{tt}=-\left[1-\frac{2M}{\sqrt{r^{2}+r_{b}^{2}}}+\frac{\alpha^{2}M^{2}}{(r^{2}+r_{b}^{2})^{2}}\right].$$

(54)

Then, we have the following relation for the surface gravity $\kappa$ (see for instance Cembranos and Valcarcel (2017); Balakin et al. (2016)):

$$-\nabla_{\nu}g_{tt}=2\kappa\xi_{\nu},$$

(55)

from which the Hawking temperature is calculated as follows:

$$T_{H}=\frac{\kappa}{2\pi}=\frac{1}{4\pi}\left.\frac{dg_{tt}}{dr}\right|_{r=r_{h}}.$$

(56)

As the expression to find the horizon radius from $g_{tt}=0$ is very involved, we write the value of the black hole mass as a function of the horizon radius, given by

$$M=\frac{(r_{h}^{2}+r_{b}^{2})^{2}-\sqrt{\left(r_{b}^{2}+r_{h}^{2}\right)^{3}\left(r_{b}^{2}+r_{h}^{2}-\alpha^{2}\right)}}{\alpha^{2}\sqrt{r_{b}^{2}+r_{h}^{2}}}.$$

(57)

Hence, it follows that the temperature of Hawking radiation for our regular black hole is given by

$$T_{H}=\frac{r_{h}}{2\pi\alpha^{2}}\left[1-\sqrt{1-\frac{\alpha^{2}}{r_{b}^{2}+r_{h}^{2}}}\right]\left[2\sqrt{1-\frac{\alpha^{2}}{r_{b}^{2}+r_{h}^{2}}}-1\right].$$

(58)

The above expression shows that the temperature is null for

$$r_{h}=\frac{1}{\sqrt{3}}\sqrt{4\alpha^{2}-3r_{b}^{2}}$$

(59)

For a radius fewer than this, the temperature becomes negative. Therefore, we have a remnant unless $\alpha>\sqrt{3}r_{b}/2$, namely if the LQG parameter is higher than the regularization one. In other words, the spacetime quantum effects favour the formation of remnants. Below we will analyze if this is a stable point. Fig. 6 clearly reveals this behavior. In the left panel, we have the Hawking temperature as a function of the event horizon radius, $r_{h}$, for a loop quantum black bounce, according to Eq. (58). The quantum parameter $\alpha$ is varied across different values, and for $r_{b}=0.5$, the critical value is $\alpha_{c}\approx 0.43$. In the right panel, we depict the variation of the maxima of the Hawking temperature with $\alpha$, for different bounce parameters, $r_{b}$. Observe that these maxima decrease with increasing of both the parameters.

Note that the expression for the Hawking temperature can also be written as

$$T_{H}=\frac{1}{4\pi r_{h}}\left[1-\frac{r_{b}^{2}}{r_{h}^{2}}-\frac{3\alpha^{2}}{4r_{h}^{2}}+\frac{3r_{b}^{2}\alpha^{2}}{2r_{h}^{4}}+\mathcal{O}(\alpha^{4},r_{b}^{4})\right],$$

(60)

where the multiplicative factor is just the Hawking temperature for the Schwarzschild black hole, which is obtained when $(\alpha,r_{b})\to 0$.

Another thermodynamic quantity of interest is the entropy, $S$, which we calculate via

$$dS=\frac{dM}{T_{H}}.$$

(61)

Derivating Eq. (57) relative to $r_{h}$, dividing by the expression for the Hawking temperature (58) and integrating again, we arrive at

$$S=\pi\left[r_{h}\sqrt{r_{b}^{2}+r_{h}^{2}-\alpha^{2}}+\left(\alpha^{2}+r_{b}^{2}\right)\log\left(\sqrt{r_{b}^{2}+r_{h}^{2}-\alpha^{2}}+r_{h}\right)\right].$$

(62)

The first thing we can see is that at the radius of null temperature, given by Eq. (59), we get

$$S=\pi\frac{\alpha}{3}\left[\sqrt{4\alpha^{2}-3r_{b}^{2}}+4\alpha\log\left(\frac{2\alpha+\sqrt{4\alpha^{2}-3r_{b}^{2}}}{\sqrt{3}}\right)\right].$$

(63)

Therefore, as expected, we have a finite entropy as a remnant. Expanding this in $\alpha$ and $r_{b}$ we have

$$S=\pi r_{h}^{2}\left[1+\frac{r_{b}^{2}}{2r_{h}^{2}}+\frac{\alpha^{2}r_{b}^{2}}{4r_{h}^{4}}-\frac{\alpha^{2}}{2r_{h}^{2}}\right]+\pi\left(r_{b}^{2}+\alpha^{2}\right)\log(2r_{h})+\mathcal{O}(\alpha^{4},r_{b}^{4}),$$

(64)

where the first multiplicative factor represents the expression for the entropy of a Schwarzschild black hole. Note that the logarithmic corrections to the entropy arise solely due to the quantum nature of spacetime, within the framework of LQG that we are considering.

In order to analyze the phase structure of the black holes under consideration, we will calculate the thermal capacity at constant volume, $C_{v}$, and the Gibbs free energy, $G$, given respectively by

$$C_{v}=\frac{dM}{dT_{H}}=\frac{dM/dr_{h}}{dT_{H}/dr_{h}};\text{ }G=M-TS.$$

(65)

In Fig. 7, we depict $C_{v}$ (left panel) and $G$ (right panel) as functions of the horizon radius $r_{h}$, which serves as an order parameter of the system, for $r_{b}=0.5$. The behavior of $C_{v}$ reveals local phase transitions of first order, which are characterized by discontinuities where $C_{v}$ shifts from positive (indicating local stability) to negative (local instability) values. Notably, for $\alpha=0.5$, we observe two first-order phase transitions and one second-order phase transition, marked by a smooth change in the sign of $C_{v}$. This complex behavior is due to the influence of the LQG parameter $\alpha$, which alters the thermodynamic landscape, leading to such multiple phase transitions. In fact, there exists a specific range of $\alpha$ values where these phenomena are observed, underscoring the critical impact of ($\alpha$) on the phase structure of the system.

Regarding the Gibbs free energy, the plot in the right panel of Fig. 7 reveals the absence of phase transitions. However, the system is globally thermodynamically unstable due to the positive values of $G$ for any $r_{h}$. Notably, at a critical value around $\alpha=0.5$, the plot reveals two local minima in the Gibbs free energy. This indicates the presence of two stable black hole phases with different sizes. On the other hand, on evaluating the dependence of $G$ on $T_{H}$, Fig. 8 illustrates the coexistence of two distinct thermodynamic phases. At low temperatures, different values of $\alpha$ and $r_{b}$ lead to distinct phases, with the differences becoming less pronounced as $\alpha,r_{b}$ increases. At high temperatures, the curves converge, indicating a universal phase at lower critical temperatures for increasing ($\alpha$, $r_{b}$).

Applying the revised Newman-Janis algorithm Azreg-Aïnou (2014); Azreg-Ainou (2014) to the metric (27), we obtain the Kerr-like LQG-inspired black bounce solution

$$\!\!ds^{2}=\frac{\Psi}{\rho^{2}}\left\{\frac{\Delta}{\rho^{2}}(dt-a\sin^{2}{\theta}d\phi)^{2}-\frac{\rho^{2}}{\Delta}dr^{2}-\rho^{2}d\theta^{2}-\frac{\sin^{2}{\theta}}{\rho^{2}}\left[adt-\left(r^{2}+r_{b}^{2}+a^{2}\right)d\phi\right]^{2}\right\},$$

(66)

where

	$\displaystyle\Delta$	$\displaystyle=$	$\displaystyle(r^{2}+r_{b}^{2})\left[1-\frac{2M}{\sqrt{r^{2}+r_{b}^{2}}}+\frac{\alpha^{2}M^{2}}{(r^{2}+r_{b}^{2})^{2}}\right]+a^{2},\text{ and}$		(67)
	$\displaystyle\rho^{2}$	$\displaystyle=$	$\displaystyle r^{2}+r_{b}^{2}+a^{2}\cos^{2}{\theta}.$		(68)

The function $\Psi=\Psi(r,\theta,a)$ must satisfy $G_{r\theta}=0$ in addition to Einstein’s equations Azreg-Aïnou (2014). Considering our solution, we find that $\Psi=\rho^{2}$.

The horizons are determined by the condition $\Delta=0$. Fig. 9 illustrates the scenarios where there are zero, one, or two horizons within the domain $r\geq 0$.

In Fig. (10), we depict the profiles of the ergospheres, obtained by solving $g_{tt}=0$. On the left side, we depict the structure of the black bounce, with the internal and external horizons and ergospheres; on the right side, we present the external ergospheres while varying $\alpha$ and fixing $a$, including the Kerr black hole (for which $\alpha=0$). Note that the size of the ergospheres decreases with an increase in the LQG parameter $\alpha$. This reduction occurs because higher values of $\alpha$ introduce significant quantum corrections, shrinking and deforming the region where particles are forced to co-rotate with the black hole.

Concerning the curvature of the rotating LQBB spacetime, it can be shown that the Ricci scalar $R$ is generally given by $R=\frac{N}{D}$, where $N$ depends on the metric functions appearing in (66) and their derivatives up to second order, while $D=2\rho^{6}\Psi^{3}$ Azreg-Aïnou (2014). Therefore, the rotating LQBB spacetime would exhibit a singularity provided $r^{2}+r_{b}^{2}+a^{2}\cos^{2}\theta=0$. However, this quantity cannot vanish because $r_{b}\neq 0$. To better understand the geometry of this region, let us consider the line element of Eq.(66) in the region $r=0$, where it can be recast as

$\displaystyle ds^{2}=dt^{2}-\frac{\left(2Mr_{b}-\alpha^{2}M^{2}/r_{b}\right)}{r_{b}^{2}+a^{2}\cos^{2}\theta}\left(dt-a\sin^{2}\theta d\phi\right)^{2}-(r_{b}^{2}+a^{2}\cos^{2}\theta)d\theta^{2}-(r_{b}^{2}+a^{2})\sin^{2}\theta d\phi^{2}$

(69)

For the sake of comparison, one can consider the usual Kerr scenario, the limit $\alpha\to 0$ and $r_{b}\to 0$, for which the above line element boils down to

$\displaystyle ds^{2}=dt^{2}-a^{2}\cos^{2}\theta d\theta^{2}-a^{2}\sin^{2}\theta d\phi^{2}=dt^{2}-d\tilde{r}^{2}-\tilde{r}^{2}d\phi^{2}\ ,$

(70)

where we have defined $\tilde{r}=a\cos\theta$, such that $|\tilde{r}|\in[0,a]$. Thus, this line element represents the geometry of a flat $2+1$ dimensional circle of radius $a$ in Minkowski space-time in polar coordinates. The problem comes when we consider the case $\theta=\pi/2$ (equatorial plane of the Kerr geometry), because there $\tilde{r}=0$ represents a circumference of vanishing radius (at constant time) on which the Kerr curvature scalars diverge. In our case, the last two terms of the line element (69) represent the geometry of a rotating ellipsoid of radius $r_{b}$ in the $z$ direction, and of radius $\sqrt{r_{b}^{2}+a^{2}}$ on the $x-y$ plane. As one can see, this ellipsoid degenerates into two circles when $r_{b}\to 0$ (one of those circles corresponds to the interval $\tilde{r}\in[0,a]$, where $z>0$, while the other is in $\tilde{r}\in[-a,0]$, where $z<0$). This ellipsoid represents the transit (or bounce) region, with the parameter $a$ controlling its deformation from a sphere in the rotating case. As mentioned before, the curvature invariants are regular on this ellipsoid, including the equatorial circumference $\theta=\pi/2$. Further details on the properties of this ellipsoid and its corresponding Kerr-Schild coordinates will be given elsewhere.