--- title: 'Rotating Black Bounces: Regular Kerr Extensions' url: https://www.emergentmind.com/topics/rotating-black-bounces type: topic --- # Rotating Black Bounces: Regular Kerr Extensions Searching arXiv for recent and foundational papers on rotating black bounces. Rotating black bounces are stationary, axisymmetric deformations of Kerr in which a nonzero bounce or regularization scale removes the Kerr ring singularity and replaces it with a regular central region, a throat, or a bounce surface. In the constructions studied to date, the same line element can interpolate between a regular rotating black hole and a traversable wormhole, depending on the parameter regime. The subject now includes the rotating Simpson–Visser geometry, charged black-bounce–Kerr–Newman spacetimes, anisotropic-fluid and string-cloud variants, loop-quantum-inspired models, and more general deformations controlled by extra exponents; recent work has also shown that their superradiant response can differ sharply from Kerr, including regimes with substantially enhanced scalar-wave amplification [2109.13813; 2104.11376; 2408.08542; 2509.00249; 2307.09344; 2601.03415]. ## 1. Metric constructions and parameterizations A standard rotating black-bounce metric is obtained by introducing a bounce radius into Kerr through \[ h(r)=\sqrt{r^2+\ell^2}, \] with \[ \Sigma(r,\theta)=r^2+\ell^2+a^2\cos^2\theta,\qquad \Delta(r)=r^2+\ell^2+a^2-2M\sqrt{r^2+\ell^2}. \] In Boyer–Lindquist coordinates, the line element can be written in the Kerr-like form \[ ds^2 = -\frac{\Delta-a^2\sin^2\theta}{\Sigma}\,dt^2 -2\,\frac{2Ma h(r)\sin^2\theta}{\Sigma}\,dt\,d\phi +\frac{\Sigma}{\Delta}\,dr^2+\Sigma\,d\theta^2 +\frac{\bigl[(h(r)^2+a^2)^2-a^2\Delta\sin^2\theta\bigr]}{\Sigma}\sin^2\theta\,d\phi^2, \] where \(M\) is the ADM mass, \(a\) is the Kerr rotation parameter, and \(\ell\ge 0\) is the bounce scale. The limit \(\ell\to0\) recovers ordinary Kerr, while \(a\to0\) gives the static Simpson–Visser bounce geometry [2109.13813]. The most general rotating black-bounce family currently analyzed for superradiance introduces a regularization parameter \(p\) and two integer deformation exponents \((k,n)\). In that construction, \[ \Sigma(r,\theta)=r^2+p^2+a^2\cos^2\theta, \] and \[ \Delta(r)=r^2+p^2+a^2-\frac{2M\,(r^2+p^2)\,r^k}{(r^{2n}+p^{2n})^{(k+1)/(2n)}}. \] Here \(p\ge0\), \(k\ge0\), and \(n\ge1\). In the limit \(p\to0\), the metric exactly reduces to Kerr. Turning on \(p\) and choosing \((k,n)\neq(0,1)\) smoothly deforms the near-center geometry while preserving regularity [2601.03415]. This class has been extended in several directions. The charged black-bounce–Kerr–Newman geometry replaces \(r\) by \(z(r)=\sqrt{r^2+\ell^2}\) in the Kerr–Newman functions and introduces \[ \Delta(r)=r^2+\ell^2+a^2-2m\,z(r)+Q^2, \] with charge \(Q\) [2104.11376]. Loop-quantum-inspired versions add an LQG regularization constant \(\delta\) and modify \(\Delta(r)\) through the static seed function before rotation is introduced via the revised Newman–Janis algorithm [2408.08542]. Other families are sourced by an anisotropic fluid [2509.00249] or by a string cloud [2307.09344]. ## 2. Horizons, ergoregions, and bounce surfaces In all of these geometries, horizons are determined by the real positive roots of \(\Delta(r)=0\). For the rotating Simpson–Visser case this yields outer and inner horizons \(r_+\) and \(r_-\), while the stationary-limit surface is given by \(g_{tt}=0\), so that the ergoregion lies between \(r_+\) and \(r_{SLS}(\theta)\). The outer stationary-limit surface is pushed inward relative to Kerr as \(\ell\) increases, so the ergosphere volume shrinks as \(\ell\) increases [2109.13813]. In the \((p,k,n)\) family, the combined effect of \(p>0\) and the exponents \((k,n)\) shifts both horizons. For \(n=1\), \(r_+\) decreases monotonically with \(p\) at fixed \(a\), so the outer horizon shrinks relative to Kerr. For \(n>1\), \(r_+(p)\) can be non-monotonic, first growing then decreasing, and admits ultra-spinning \((a>M)\) solutions. Larger \(k\) tends to shrink the horizon radii for fixed \((p,n)\). Once \(r_+\) is found, the horizon angular velocity and surface gravity are \[ \Omega_H=\frac{a}{r_+^2+a^2+p^2},\qquad \kappa=\frac{1}{2}\,\frac{d\Delta}{dr}\Big|_{r_+}\Big/(r_+^2+a^2+p^2), \] with Killing generator \(\chi=\partial_t+\Omega_H\partial_\phi\) [2601.03415]. A recurring feature is the transition from black hole to wormhole. For \(\ell\) large compared to \(M\), the horizons disappear altogether and one smoothly interpolates to a traversable-wormhole geometry with a throat at \(r=0\) [2109.13813]. In the charged case, two distinct horizons require \[ m^2\ge a^2+Q^2,\qquad \ell\le m-\sqrt{m^2-a^2-Q^2}, \] while sufficiently large \(\ell\) or \(a^2+Q^2>m^2\) yields no horizons and a traversable wormhole [2104.11376]. The bounce surface itself acquires a nontrivial geometry once rotation is turned on. In the loop-quantum-inspired model, the throat at \(r=0\) becomes an ellipsoid of revolution, \[ \frac{x^2+y^2}{\ell^2+a^2}+\frac{z^2}{\ell^2}=1, \] with semi-axes \(\sqrt{\ell^2+a^2}\) in the equatorial plane and \(\ell\) along the spin axis. This makes explicit that rotation does not restore the Kerr ring singularity; instead, the spherical bounce becomes an ellipsoid [2408.08542]. ## 3. Regularity, effective matter sources, and energy conditions The defining structural feature of a rotating black bounce is the removal of the Kerr ring singularity. In the rotating Simpson–Visser geometry, \[ \Sigma=r^2+\ell^2+a^2\cos^2\theta \ge \ell^2>0 \] for \(\ell>0\), so there is no curvature singularity in the manifold, and all curvature invariants remain finite for all finite \((r,\theta)\) [2109.13813]. The charged generalization similarly satisfies \(\rho^2\ge \ell^2>0\) everywhere, so all curvature invariants remain finite [2104.11376]. In the LQG-inspired rotating bounce, the Ricci and Kretschmann scalars have denominators built from powers of \(\rho^2=r^2+\ell^2+a^2\cos^2\theta\), and therefore no divergence occurs anywhere [2408.08542]. Regularity does not imply a vacuum solution. These metrics are generally supported by an effective stress–energy tensor obtained from Einstein’s equations. In the rotating Simpson–Visser spacetime one finds diagonal \(T^{\hat\mu}{}_{\hat\nu}=\mathrm{diag}(-\rho,P_r,P_\theta,P_\phi)\), with \[ \rho+P_r<0,\qquad \rho+P_\theta<0,\qquad \rho<0 \] in regions near the throat or inner region, so the null, weak, and strong energy conditions are all violated there [2109.13813]. In the charged black-bounce–Kerr–Newman case, outside any horizon, \[ \varepsilon+p_r=-\frac{\ell^2\,\Delta}{8\pi\,\rho^6}<0, \] so the null energy condition is generically violated in the bounce region, though on the horizon \(\varepsilon+p_r=0\) [2104.11376]. The matter interpretation depends on the model. In the charged case, the source can be split into a Maxwell part plus either charged dust or an anisotropic fluid with non-zero shear [2104.11376]. In the anisotropic-fluid black-bounce family, the source is specified by \[ T_{\mu\nu}=(\rho+p_t)u_\mu u_\nu+p_t g_{\mu\nu}+(p_r-p_t)v_\mu v_\nu, \qquad p_r=-\rho,\quad p_t=\omega\,\rho, \] with density \(\rho(r)=\rho_0(r^2+\ell^2)^{-(1+\omega)}\), and this fluid is responsible for the non-singular behavior of all curvature invariants [2509.00249]. A common misconception is that singularity resolution by itself places rotating black bounces close to Kerr in all dynamical respects. The literature instead shows that regularity is typically accompanied by nontrivial matter content and energy-condition violation, often localized near the bounce, though in some models the violation becomes mild far from the central region or decreases as the LQG parameter grows [2408.08542]. ## 4. Separability, geodesics, and optical observables Several rotating black-bounce metrics preserve enough hidden symmetry to allow separation of the Hamilton–Jacobi and Klein–Gordon equations. In the charged black-bounce–Kerr–Newman spacetime, although the geometry fails to be Petrov type D and admits no principal Killing–Yano tensor, it does admit a rank-2 Killing tensor \(K_{\mu\nu}\) satisfying \(\nabla_{(\lambda}K_{\mu\nu)}=0\). This yields a Carter-like constant \[ \mathcal Q=K_{\mu\nu}u^\mu u^\nu-(L_z-aE)^2, \] and both Hamilton–Jacobi and Klein–Gordon equations separate because \[ [\Box,\nabla_\mu K^{\mu\nu}\nabla_\nu]=0 \] has been verified explicitly [2104.11376]. The anisotropic-fluid rotating black bounce also admits separation for null geodesics. With the ansatz \[ S=-Et+L_z\phi+S_r(r)+S_\theta(\theta), \] one obtains a Carter constant \(K\), impact parameters \(\xi=L_z/E\) and \(\eta=K/E^2\), and radial and polar potentials \(\mathcal R(r)\) and \(\Theta(\theta)\). The shadow boundary is determined by unstable spherical photon orbits satisfying \[ \mathcal R(r)=0,\qquad \frac{d\mathcal R}{dr}=0,\qquad \frac{d^2\mathcal R}{dr^2}>0. \] For horizonless cases there is an additional contribution from rays whose radial turning point approaches the throat \(r=0\), producing a circular branch; the final shadow is the inner envelope of the photon-sphere branch and the throat circle [2509.00249]. This produces a broader optical taxonomy than in Kerr. The anisotropic-fluid model exhibits nearly Kerr-like D-shaped shadows for small spin, cusped triangular outlines in certain wormhole-near cases, horizontally elongated shadows in other regimes, and mixed silhouettes with one contribution from the photon sphere and one from the throat [2509.00249]. In the string-cloud variant, the shadow is very sensitive to the string-cloud parameter \(L\): increasing \(L\) can significantly increase the boundary of the shadow, enlarge the overall size \(R_s\), and reduce the distortion \(\delta_s\), whereas varying the bounce parameter \(\ell\) has only a very weak effect on \(R_s\) and \(\delta_s\) [2307.09344]. In the loop-quantum model, increasing the LQG parameter \(\delta\) reduces both the apparent radius and the distortion of the shadow, consistent with a smaller photon region [2408.08542]. These results indicate that the observational appearance of rotating black bounces is not controlled by a single “regularization scale” alone. Instead, the shadow may be governed by a competition among spin, matter profile, and throat structure, and in horizonless cases the throat can close the contour directly. ## 5. Superradiant scattering and amplification The most detailed dynamical analysis to date concerns superradiant scattering of massless scalar waves off rotating black-bounce black holes. For a test scalar field \(\Phi\), stationarity and axisymmetry allow \[ \Phi(t,r,\theta,\phi)=e^{-i\omega t+im\phi}\,R_{\omega\ell m}(r)\,S_{\omega\ell m}(\theta), \] with the angular equation reducing to the usual spheroidal harmonics problem and the radial function satisfying \[ \Delta\,\frac{d}{dr}\Bigl(\Delta\,\frac{dR}{dr}\Bigr) +\Bigl[-\omega^2(r^2+p^2+a^2)^2+2am\omega(r^2+p^2)+m^2a^2-\lambda_{\omega\ell m}\Delta\Bigr]R=0. \] Near the outer horizon the solution is purely ingoing, while at spatial infinity it has incident and reflected components with amplitudes \(\mathcal I\) and \(\mathcal R\). Superradiant amplification occurs if and only if \[ 0<\omega0\) precisely in the superradiant regime [2601.03415]. For the dominant massless scalar mode \((\ell,m)=(1,1)\), the maximal amplification \(\overline Z=\max_\omega Z(\omega)\) and peak frequency \(\overline\omega\) have been mapped over the \((a/M,p/M)\) plane for all \((k,n)\) with \(0\le k\le3\) and \(1\le n\le3\). The resulting parameter dependence is sharply structured. For \(n=1\), including the rotating Simpson–Visser and Bardeen-like cases, \(\overline Z\) is always maximized in the Kerr limit \(p=0\), so the regularization can only suppress superradiance. For \(n>1\), there are regions where \(\overline Z\) exceeds the Kerr maximum \(\overline Z_{\rm Kerr}^{\rm max}\simeq0.003674\). In particular, \[ (k,n)=(0,2):\quad \overline Z_{\rm max}\simeq0.00524 \] at \((a/M,p/M)\simeq(1.115,0.53)\), corresponding to a \(142\%\) enhancement, and \[ (k,n)=(0,3):\quad \overline Z_{\rm max}\simeq0.00727 \] at \((1.176,0.57)\), corresponding to a \(198\%\) enhancement [2601.03415]. The deformation exponents control this behavior systematically. Increasing \(n\) shifts the peak amplification to larger spins and larger \(p\), and raises \(\overline Z\). At fixed \(n>1\), increasing \(k\) from 0 to 3 reduces the enhancement; for \(n=3\), moving from \(k=0\) to \(k=3\) lowers the relative amplification from \(198\%\) to \(144\%\). The corresponding peak frequencies \(M\overline\omega\) range roughly from \(0.38\) in the Kerr-like region up to \(0.44\) for the most superradiant \((k,n)=(0,3)\) configuration [2601.03415]. A plausible implication is that rotating black bounces cannot be treated as uniformly “less superradiant” than Kerr. The parameter \(n\) can instead generate a steeper regular deformation that enhances energy extraction by low-frequency waves, suggesting potentially stronger superradiant instabilities for massive bosonic fields, modified bosonic-cloud phenomenology, and altered ringdown or Hawking spectra through changes in \(\Omega_H\) and \(\kappa\) [2601.03415]. ## 6. Extensions, horizon fragility, and related geometries The rotating black-bounce program now includes several non-equivalent extensions of the Kerr paradigm. The charged black-bounce–Kerr–Newman solution is a regular black-hole mimicker that smoothly interpolates between black holes and traversable wormholes while preserving a Carter-like constant without the full Killing tower [2104.11376]. The rotating black-bounce string regularizes the rotating black string in \(3+1\) dimensions by forcing a bounce on the radial coordinate, with a throat at \(r=0\) of minimal circumference \(2\pi a\); depending on parameters, it yields a regular black-bounce string, a one-way bounce, or a traversable cylindrical wormhole [2307.07404]. A separate line of inquiry concerns the robustness of the horizon. In the rotating Black–Bounce family with deformation exponents \((k,n)\), test-particle absorption and scalar-field scattering have been used to analyze horizon destruction. The horizon condition is governed by \[ a^2+m^2\le \Gamma^2=M^2\beta^2, \] and the study reports that under extreme or near-extreme conditions the event horizon can potentially be destroyed after the absorption of particle energy and angular momentum, as well as by the scattering of scalar fields. It further finds that as the parameter \(m\) increases, the event horizon becomes more susceptible to destruction after the injection of test particles or the scattering of scalar fields [2308.12150]. This issue is conceptually significant because rotating black bounces are nonsingular. The usual weak cosmic censorship question is therefore reframed: the problem is not exposure of a curvature singularity, but loss of a regular horizon and transition into a horizonless bounce or wormhole domain. The literature does not present this as a generic property of every rotating black-bounce model, but it does show that regularity alone does not enforce Kerr-like censorship behavior [2308.12150]. Taken together, these results place rotating black bounces at the intersection of regular-black-hole phenomenology, wormhole physics, hidden-symmetry methods, and strong-gravity observables. The existing constructions show a common core—Kerr-like rotation with a nonzero bounce scale and finite curvature—while the detailed phenomenology depends strongly on how the Kerr mass term is regularized, what matter sector is assumed, and whether additional deformations such as charge, anisotropic stress, string clouds, or LQG corrections are present. Source: https://www.emergentmind.com/topics/rotating-black-bounces