---
title: Rotating BTZ-like Black Holes in Modified Gravity
url: https://www.emergentmind.com/topics/rotating-btz-like-black-hole
type: topic
---

# Rotating BTZ-like Black Holes in Modified Gravity

Searching arXiv for recent and foundational papers on rotating BTZ-like black holes to ground the article in published work.
A rotating BTZ-like black hole is a \(2+1\)-dimensional stationary black-hole geometry that preserves the characteristic BTZ rotating structure—an AdS-like lapse sector, a nontrivial angular shift, and horizon-bearing circular symmetry—while incorporating deformations coming from modified matter content, Lorentz-symmetry breaking, running couplings, higher-curvature corrections, quantum backreaction, or analogue-gravity constructions. In the strict BTZ case, the geometry depends on mass, angular momentum, and a negative cosmological constant, and it provides the canonical rotating black hole of three-dimensional AdS gravity. In the broader “BTZ-like” sense, the term refers to rotating geometries that reduce to BTZ in an appropriate limit or retain its causal and kinematical architecture while modifying the radial sector, asymptotics, thermodynamics, or perturbative dynamics [2010.16183], [2302.01580].

## 1. Canonical rotating BTZ geometry

The standard rotating BTZ black hole is written in the form
\[
ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,
\]
with
\[
N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.
\]
Here \(M\) is the black-hole mass, \(J\) is the angular momentum, and \(l\) is the AdS radius, related in the paper to the cosmological constant by
\[
l=-(\Lambda)^{-1/2},
\]
to be interpreted as the usual AdS relation for \(\Lambda<0\) [2010.16183].

The horizons are located at
\[
r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},
\]
with the horizon-existence condition
\[
|J| \le Ml,
\]
and the extremal limit
\[
|J|=Ml
\]
[2010.16183]. Equivalent horizon formulas appear in other rotating BTZ treatments, including
\[
r_\pm=\ell \left[\frac{M}{2}\left\{1\pm \sqrt{1-\left(\frac{J}{M\ell}\right)^{2}\right\}\right]^{1/2}
\]
with \(\Lambda=-1/\ell^2\) [2304.12359].

A frequently used equivalent BTZ form is
\[
ds^2=-f(r)dt^2+\frac{dr^2}{g(r)}+r^2\left(d\phi-\frac{J}{2r^2}dt\right)^2,
\qquad
f(r)=g(r)= -M+\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}},
\]
which makes the angular-velocity function
\[
\Omega=\frac{J}{2r^{2}}, \qquad \Omega_H=\frac{J}{2r_+^{2}}
\]
manifest [2304.12359]. The standard thermodynamic quantities in this representation are
\[
T_0=\frac{r_+^2-r_-^2}{2\pi \ell^2 r_+}, \qquad S_{BH}=4\pi r_+,
\]
together with the first law
\[
dM = T\, dS + \Omega\, dJ
\]
[2304.12359].

This canonical geometry is the baseline against which BTZ-like rotating deformations are compared. A plausible implication is that “BTZ-like” should be reserved for rotating \(2+1\)-dimensional geometries that preserve at least the stationary circular form, an angular shift encoding frame dragging, and a horizon structure controlled by BTZ-type mass and spin parameters.

## 2. Null geodesics, photon orbits, and lensing structure

In the rotating BTZ background, null geodesics can be derived from conserved energy \(E\) and angular momentum \(L\). The first integrals quoted in the lensing analysis are
\[
\dot{t} = \dfrac{E r^{2} - L J /2}{r^{4}/l^{2}-M r^{2} + J^{2}/4},
\]
\[
\dot{\phi} = \dfrac{E J/2 - L M + L r^{2}/l^{2}}{r^{4}/l^{2}-M r^{2} + J^{2}/4},
\]
and
\[
\dot{r}^2 = \dfrac{L^{2} M - E L J}{r^{2}} + E^{2} - \dfrac{L^{2}}{l^{2}}.
\]
The radial motion factorizes as
\[
\dot{r}^2 = \left(E - V^{+}_{eff}\right)\left(E - V^{-}_{eff}\right),
\]
with
\[
V^{\pm}_{eff} = \frac{Jl}{2r^2} \pm \frac{1}{2} \sqrt{\frac{J^2 L^2}{r^4}-4\left(\frac{M L^2}{r^2}-\frac{L^2}{l^2}\right)}
\]
[2010.16183].

The literature distinguishes direct and retrograde photon motion by the sign of \(JL\):
- direct motion: \(JL>0\),
- retrograde motion: \(JL<0\)

[2010.16183]. The turning point of a scattering trajectory is
\[
r_{0} = \sqrt{\frac{ELJ-ML^2}{E^2-\frac{L^2}{l^2}}},
\qquad
u_0=\frac1{r_0} = \sqrt{\frac{E^{2}- \frac{L^{2}}{l^{2}}}{E L J - M L^{2}}}.
\]
The orbit equation is then expressed in terms of \(u=1/r\) and decomposed into a form involving \(C_\pm\), \(u_\pm\), \(\eta\), \(m\), \(n\), and \(k\) [2010.16183].

A central result of this analysis is that the effective potential has no minimum, hence there is no stable photon orbit near the rotating BTZ horizon; only unstable null orbits occur [2010.16183]. In the extremal case, the effective potential analysis yields the critical impact parameter
\[
b=l,
\]
with
\[
V_{eff}=b^{-2}=l^{-2},
\]
and the photon orbit is again interpreted as unstable [2010.16183].

The exact deflection angle is given by
\[
\delta = 2\int_{0}^{u_{0}} \left[ \dfrac{C_{+}} {\eta\, u _{+} \left(1- m u^{2}\right) \sqrt{1 - k u^{2}}} + \dfrac{C_{-}} {\eta\, u _{-} \left(1- n u^{2}\right) \sqrt{1 - k u^{2}}} \right] du - \pi,
\]
which can be integrated to a closed expression involving inverse hyperbolic functions and then rewritten logarithmically [2010.16183]. The principal lensing trend is that the bending angle increases as the closest approach decreases, for both direct and retrograde trajectories, and that it “slightly decreases as the value of cosmological constant increases in the negative region” [2010.16183].

These features are specific to the BTZ lensing problem in \(2+1\)-dimensional AdS gravity, but they supply a reference pattern for rotating BTZ-like geometries: frame-dragging splits direct and retrograde branches, unstable critical null orbits replace a stable photon sphere, and the AdS scale enters explicitly in both orbit structure and light bending.

## 3. Thermodynamics, quantization, and fluctuation corrections

The rotating BTZ black hole supports a standard thermodynamic description. In one formulation,
\[
T_0=\frac{r_+^2-r_-^2}{2\pi\ell^2 r_+},\qquad \Omega_H=\frac{J}{2r_+^2},\qquad S_{BH}=4\pi r_+
\]
[2304.12359]. In another convention,
\[
T_H=\frac{r_+}{2\pi l^2}-\frac{8G^2J^2}{\pi r_+^3},\qquad S_0=\frac{\pi r_+}{2G},\qquad V_r=\pi r_+^2,
\]
with thermodynamic potentials
\[
F_r=-\frac{r_+^2}{8Gl^2}+\frac{6GJ^2}{r_+^2},
\qquad
H_r=\frac{r_+^2}{8Gl^2}+\frac{2GJ^2}{r_+^2},
\qquad
U_r=\frac{2GJ^2}{r_+^2},
\qquad
G_r=\frac{6GJ^2}{r_+^2}
\]
[1912.00767].

An independent line of work quantizes the horizon area via the adiabatic invariant and Bohr–Sommerfeld quantization. For the rotating BTZ metric
\[
ds^2 = -N^2(r)\,dt^2 + N^{-2}(r)\,dr^2 + r^2\bigl(d\phi + N^\phi(r)\,dt\bigr)^2,
\]
with
\[
N^2(r) = -M + \frac{r^2}{\ell^2} + \frac{J^2}{4r^2}, \qquad N^\phi(r) = -\frac{J}{2r^2},
\]
the action variable reduces to the entropy,
\[
I_v = S_H.
\]
Bohr–Sommerfeld quantization then yields an equally spaced entropy spectrum and hence an equally spaced area spectrum, with spacing independent of the rotation parameter [1204.1786]. The horizon circumference is
\[
A_H = 2\pi r_+,
\]
and the entropy is
\[
S_H = \frac{A_H}{4\hbar G}
\]
[1204.1786]. The paper states that the area spectrum of the rotating BTZ black hole is equally spaced and irrelevant to the rotating parameter [1204.1786].

Thermal and statistical fluctuations provide a different modification. One corrected entropy formula is
\[
S=S_0-\alpha\ln(S_0T_H^2),
\]
which for rotating BTZ becomes
\[
S_c=\frac{\pi r_+}{2G} -\alpha \ln\!\left[\frac{(r_+^4-16G^2J^2l^2)^2}{8\pi G l^2 r_+^5}\right]
\]
[1912.00767]. Another treatment adopts
\[
S=S_{0}-\frac{1}{2}\ln(S_{0}T^{2})+\frac{\gamma}{S_{0}},
\]
with
\[
S_0=\frac{\pi r_+}{2}
\]
for the rotating BTZ black hole [1710.06305]. In that framework the specific heat becomes
\[
C=2\pi^{2}l^{2}T-\frac{3}{2}-\frac{\gamma}{2\pi^{2}l^{2}T},
\]
and the paper concludes that higher-order quantum corrections affect stability, especially for small black holes [1710.06305].

The thermodynamic literature therefore divides into three distinct BTZ-related themes: exact classical thermodynamics, semiclassical area quantization, and fluctuation-induced corrections. This suggests that “BTZ-like” should not be read as a purely geometric label; in current usage it often also implies a thermodynamic comparison class.

## 4. Rotating BTZ-like deformations in modified gravity and matter-coupled theories

Several exact and effective rotating BTZ-like families are now known. One prominent example is the rotating Einstein-bumblebee solution
\[
ds^2 = -f(r)\,dt^2 +\frac{1+\ell}{f(r)}\,dr^2 +r^2\left(d\phi-\frac{j}{2r^2}dt\right)^2,
\]
with
\[
f(r)=-m-(1+\ell)\Lambda_e r^2+\frac{j^2}{4r^2}.
\]
Here \(\ell=\varrho b_0^2\) is the Lorentz-violating parameter induced by a radial bumblebee vacuum expectation value, and the solution exists only for a linear functional potential of the bumblebee field [2302.01580]. The horizon radii are
\[
r_\pm = \sqrt{ \frac{1}{-(1+\ell)\Lambda_e} \left( M \pm \sqrt{M^2+(1+\ell)\Lambda_e J^2} \right) },
\]
the ergosphere radius is
\[
r_{\rm erg}=\sqrt{r_+^2+r_-^2},
\]
and the thermodynamic quantities require modified area and volume,
\[
A_\pm=2\pi\sqrt{1+\ell}\,r_\pm,\qquad
S_\pm=\frac{\pi}{2}\sqrt{1+\ell}\,r_\pm,\qquad
V_\pm=(1+\ell)\pi r_\pm^2
\]
[2302.01580]. The entropy product
\[
S_+S_-=\frac{(1+\ell)\pi^2}{-\Lambda_e}\,J^2
\]
is universal, and the dual CFT central charges satisfy
\[
c_L=c_R=\frac{12\pi\sqrt{1+\ell}}{\sqrt{-\Lambda_e}}
\]
[2302.01580].

A closely related Einstein-bumblebee family is written as
\[
ds^2 = -f(r)\, dt^2 +\frac{1+s}{f(r)}\, dr^2 + r^2 \left(d\varphi - \frac{j} {2r^2}dt\right)^2,
\qquad
f(r)=\frac{r^2}{\ell^2} - M +\frac{j^2}{4r^2},
\]
with Lorentz-symmetry-breaking parameter \(s>-1\) [2501.15759]. In this version the horizons
\[
r^{2}_{\pm} = \frac{\ell^2}{2}\left(M \pm \sqrt{M^2 -\frac{j^2}{\ell^2}}\right),
\]
the ergoregion radius
\[
r_{\rm erg}=\ell\sqrt{M},
\]
and the horizon angular velocity
\[
\Omega_H=\frac{r_-}{\ell r_+}
\]
remain BTZ-like, while the radial sector and scalar dynamics are deformed [2501.15759].

Einstein–Maxwell–Dilaton theory yields another exact rotating BTZ-like family. Starting from
\[
ds^2=-B(r)\,dt^2+\frac{dr^2}{B(r)}+R(r)^2\left(d\theta+u(r)\,dt\right)^2,
\]
the solution takes
\[
R(r)=b^x r^{1-x},
\qquad
u(r)=\frac{J_0\,r^{3x-2}}{3x-2},
\]
and
\[
B(r) = J_0^2\frac{ b^{2x} r^{4x-2}}{(2-3x)^2} +\frac{2r^2}{l^2(3x^2-5x+2)}\left(\frac{b}{r}\right)^{2x} +\frac{4Q_0^2}{3x-2}\left(\frac{b}{r}\right)^{-x}\ln\!\left(\frac{r}{l}\right) -M_0 r^x
\]
[2210.15704]. The \(\alpha\to0\) limit recovers the charged rotating BTZ geometry, whereas for \(\alpha\neq0\) the asymptotics cease to be AdS and the entropy becomes
\[
S=4\pi^2 b^x r_h^{1-x},
\]
which is always positive [2210.15704].

Scale-dependent gravity supplies a different class of BTZ-like deformation. Preserving the BTZ rotating ansatz,
\[
ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2\left[N(r)\,dt+d\phi\right]^2,
\]
one obtains
\[
G(r)=\frac{G_0}{1+r\epsilon},
\qquad
N(r)=-\frac{4G_0J_0}{r^2}\,Y(r),
\]
\[
f(r) = -8M_0G_0\,Y(r) + \frac{r^2}{\ell_0^2} + \frac{16G_0^2J_0^2}{r^2}Y(r)^2,
\]
with
\[
Y(r)\equiv 1-2r\epsilon+2(r\epsilon)^2\ln\!\left(1+\frac{1}{r\epsilon}\right)
\]
[1806.03024]. The running coupling produces a curvature singularity at \(r=0\), since
\[
R= -\frac{64G_0^2J_0^2}{r^3}\,\epsilon\left(1+\mathcal{O}(r)\right)
\]
[1806.03024].

Higher-curvature deformations need not preserve the full BTZ causal pattern. In three-dimensional Gauss–Bonnet gravity, the rotating solution
\[
ds^2= -f_{GB}(\Xi_{\rm eff} dt-a d\varphi)^2 +\frac{r^2}{\ell_{\rm eff}^4}(a dt-\Xi_{\rm eff}\ell_{\rm eff}^2 d\varphi)^2 +\frac{dr^2}{f_{GB}}
\]
with
\[
f_{GB}=-\frac{r^2}{2\alpha}\Bigl(1 - \sqrt{1+\frac{4\alpha}{r^2}\Bigl(\frac{r^2}{\ell^2}-m\Bigr)}\Bigr)
\]
is BTZ-like but not of constant curvature [2005.13732]. It possesses an ergoregion and outer horizon but does not have an inner horizon [2005.13732]. This is a direct counterexample to the common misconception that every rotating BTZ-like geometry necessarily retains the BTZ pair of horizons.

## 5. Quantum, semiclassical, and perturbative rotating BTZ-like black holes

Quantum or semiclassical corrections generate further rotating BTZ-like spacetimes. The rotating quantum BTZ black hole, or qBTZ black hole, is a geometry that captures the exact backreaction of strongly coupled quantum conformal fields [2405.04597]. In the brane-induced description,
\[
g_{\hat r \hat r}=\frac{1}{H(r)},
\]
with horizon condition
\[
H(r)=0,
\]
or equivalently
\[
Q(r)=r^2H(r)= \frac{r^4}{\ell_3^2} -8{\cal G}_3Mr^2 +(4{\cal G}_3J)^2 -\mu\ell(1-\tilde a^2)^{3/2}\Delta^3\sqrt{r^2-r_S^2}
\]
[2405.04597]. The extremality condition is
\[
Q(H)=0,\qquad Q'(H)=0,
\]
and the paper shows that an extremal rotating qBTZ black hole cannot be overspun by test-particle capture: for a particle with maximal allowed angular momentum, the first-order change satisfies
\[
\delta H=0
\]
[2405.04597].

Metric-affine modified gravity produces a slowly rotating charged BTZ-like family in Palatini Chern–Simons gravity. Starting from the charged non-rotating BTZ background
\[
ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\phi^2,
\qquad
f(r)=-M-\Lambda r^2-2\kappa^2 Q_1^2\ln\frac{r}{r_0},
\]
rotation is introduced perturbatively as
\[
ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\phi^2+2\epsilon r^2\omega(r)\,dt\,d\phi
\]
together with an induced magnetic field [2603.03436]. The asymptotic consistency condition
\[
Q_2=-\frac{\delta\Lambda Q_1}{2}
\]
ensures that
\[
\omega(r)\sim -\frac{J_F}{\epsilon r^2},
\]
so the geometry decays to BTZ-like form at large radius, while the horizon angular momentum and magnetic field remain finite [2603.03436].

A different analytic continuation leads to the double Wick rotated rotating BTZ black hole, treated as a quotient geometry. The Euclidean rotating BTZ metric
\[
ds^2 = l^2\left[ \frac{(r^2-r_+^2)(r^2-r_-^2)}{r^2}d\tau_E^2 +\frac{r^2\,dr^2}{(r^2-r_+^2)(r^2-r_-^2)} +r^2\left(dx+i\frac{r_+r_-}{r^2}d\tau_E\right)^2 \right]
\]
is double Wick rotated into a geometry that becomes equivalent to an ordinary rotating BTZ black hole after exchanging parameters and periodicities [2604.15720]. The paper states
\[
O^{\mathrm{DWR}(r_+,\tilde r_-)}=O^{\mathrm{BTZ}(\tilde r_-,r_+)},
\]
and uses this identification to reproduce geometric entropy and time-like entanglement entropy [2604.15720].

These examples indicate that the modern BTZ-like literature spans exact quantum backreaction, parity-violating first-order deformations, and analytic continuations of the BTZ quotient structure. A plausible implication is that “BTZ-like” now functions as a family resemblance term rather than a single metric class.

## 6. Perturbations, quasinormal modes, clouds, and analogue realizations

Rotating BTZ-like black holes are a major testing ground for perturbation theory. In Einstein-bumblebee gravity, the rotating BTZ-like metric
\[
ds^{2}=-f(r')dt^{2}+\frac{(s+1)}{f(r')}dr'^{2}+r'^2\left(d\theta-\frac{j}{2r'^2}dt\right)^2,
\qquad
f(r')=\frac{r'^2}{l^2}-M+\frac{j^2}{4r'^2}
\]
has horizon radii
\[
r'^2_{\pm}=\frac{1}{2}\left(Ml^2\pm l\sqrt{M^2l^2-j^2}\right)
\]
that do not depend on the Lorentz-breaking parameter \(s\) [2302.05861]. Scalar perturbations yield exact right- and left-moving frequencies,
\[
\omega_R=-\frac{m}{l} -i\frac{2(\sqrt{r_+}+\sqrt{r_-})}{l^2\sqrt{1+s}}
\left[ n+\frac{1}{2}+\frac{1}{2}\sqrt{1+l^2(1+s)\mu_0^2} \right],
\]
\[
\omega_L=\frac{m}{l} -i\frac{2(\sqrt{r_+}-\sqrt{r_-})}{l^2\sqrt{1+s}}
\left[ n+\frac{1}{2}+\frac{1}{2}\sqrt{1+l^2(1+s)\mu_0^2} \right],
\]
and the Lorentz-breaking parameter affects only the imaginary parts, not the real parts [2302.05861].

A later study extends this Einstein-bumblebee analysis to scalar, fermionic, and vector perturbations. In the coordinate system
\[
ds^2=-\sinh^2\rho\,(dx^+)^2+(1+\ell)\,d\rho^2+\cosh^2\rho\,(dx^-)^2,
\]
the exact spectra take the CFT form
\[
\omega_L = k - 4\pi i T_L(n + h_L),\qquad \omega_R = -k - 4\pi i T_R(n + h_R),
\]
with
\[
T_L=\frac{r_+ - r_-}{2\pi l^2\sqrt{1+\ell}},\qquad
T_R=\frac{r_+ + r_-}{2\pi l^2\sqrt{1+\ell}}
\]
[2603.25470]. The paper emphasizes that the real parts remain exactly BTZ-like, while the Lorentz-symmetry-breaking parameter modifies the damping rates and conformal weights [2603.25470].

Stationary scalar clouds provide another diagnostic. For the Einstein-bumblebee metric
\[
ds^2 = -f(r) dt^2 +\frac{1+s}{f(r)} dr^2 + r^2 \left(d\varphi - \frac{j}{2r^2}dt\right)^2,
\]
the scalar ansatz
\[
\Phi=e^{-i\omega t+ik\varphi}\phi(r)
\]
reduces to a hypergeometric problem after
\[
z=\frac{r^2-r_+^2}{r^2-r_-^2}
\]
[2501.15759]. Under Robin boundary conditions
\[
\phi=\cos(\zeta)\phi^{(D)}+\sin(\zeta)\phi^{(N)},
\]
stationary clouds occur at the synchronization threshold
\[
\omega=\omega_c=k\Omega_H
\]
[2501.15759]. The paper finds only fundamental stationary scalar clouds, \(n=0\), and reports degenerate cloud-existence lines for different \((k,s)\) values [2501.15759].

The Kalb–Ramond BTZ-like black hole changes the perturbation problem qualitatively. With
\[
A(\tilde r)= - M+\frac{\tilde r^2}{(1+\ell)\lambda^2}+\frac{j^2}{4(1+\ell)\tilde r^2},
\qquad
K(\tilde r)=-\frac{j}{2\tilde r^{2}},
\]
the scalar radial equation reduces to a general Heun equation after
\[
z=\frac{r-r_+}{r-r_-}
\]
[2511.00784]. Robin boundary conditions are imposed via
\[
\phi^{(\mathcal R)}(z)=\sin(\xi)\phi^{(\mathcal N)}(z)+\cos(\xi)\phi^{(\mathcal D)}(z),
\]
and the quasinormal frequencies are determined numerically by the Wronskian condition
\[
W(1/2)=0
\]
[2511.00784]. The paper states that only the fundamental left-branch mode becomes unstable and that the Kalb–Ramond parameter shifts the threshold and the range of Robin coupling where superradiance occurs [2511.00784].

Beyond gravitational theories, rotating BTZ-like structure has even been realized in analogue gravity. In a photon-fluid model, the acoustic metric
\[
ds^{2}=\left(\frac{\rho_0}{c_s}\right)^2 \left[ -(c_s^2-v_t^2)dt^2 -2v_r\,dr\,dt -2v_\theta r\,d\theta\,dt +dr^2+(r\,d\theta)^2 \right]
\]
is matched, up to a conformal factor, to the rotating BTZ geometry by choosing
\[
v_r=c_s\sqrt{1-\left(-M+\frac{r^2}{l^2}+\frac{J^2}{4r^2}\right)},
\qquad
v_\theta=c_s\frac{J}{2r}
\]
[2303.15661]. A realizable optical vortex ansatz produces a radial flow that crosses the sound speed twice, thereby reproducing both inner and outer horizons, a defining BTZ feature absent from earlier optical rotating analogues [2303.15661].

Taken together, these studies show that rotating BTZ-like black holes are now used as laboratories for exact QNMs, AdS boundary-condition effects, superradiance, stationary clouds, and even experimental analogue horizons. This suggests that the phrase names not merely a geometry but a research program centered on \(2+1\)-dimensional rotating AdS black-hole kinematics under controlled deformations.

Source: https://www.emergentmind.com/topics/rotating-btz-like-black-hole