Rotating Einstein-Bumblebee Black Holes

• Rotating Einstein-Bumblebee black holes are solutions in a vector-tensor gravity framework where a bumblebee field attains a nonzero vacuum expectation value, breaking local Lorentz symmetry and modifying the spacetime geometry.
• The research employs both perturbative slow-rotation techniques and Kerr-like Newman-Janis algorithms to analyze the impact of the Lorentz-violating parameter on quasinormal modes, horizon structure, and frame dragging.
• Observational implications include altered black-hole shadows, modified accretion disk dynamics, and differing thermodynamic prescriptions, underscoring the model-dependent deviations from standard Kerr black holes.

Searching arXiv for recent and foundational papers on rotating Einstein-bumblebee black holes and closely related topics. Rotating Einstein-bumblebee black holes are rotating black-hole solutions in Einstein-bumblebee gravity, a vector-tensor framework in which a bumblebee field $B_\mu$ acquires a nonzero vacuum expectation value $b_\mu$ and spontaneously breaks local Lorentz symmetry. In the rotating sector, the deformation is typically controlled by a Lorentz-violating parameter written as $\ell\equiv \varrho b^2$, $\ell\equiv \xi b^2$, or $\ell\equiv \kappa b^2$, depending on notation, and it modifies the radial geometry, frame dragging, horizon structure, perturbation spectra, shadows, accretion observables, and thermodynamics relative to Kerr, Kerr-Newman, BTZ, Myers-Perry, and their AdS analogues (Liu et al., 2022, Ding et al., 2023, Chen et al., 2 Jul 2026).

1. Einstein-bumblebee framework and the Lorentz-violating sector

In four dimensions, a representative Einstein-bumblebee action is

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$

with $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$, and spontaneous Lorentz breaking occurs when $B_\mu$ acquires a nonzero vacuum expectation value $b_\mu$ satisfying $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$. Varying with respect to $b_\mu$0 and $b_\mu$1 yields

$b_\mu$2

In the frozen limit $b_\mu$3, one obtains the effective vacuum equation

$b_\mu$4

The characteristic deformation parameter is then $b_\mu$5 in the four-dimensional radial vacuum branch, with $b_\mu$6 required in the static seed discussed in the quasinormal-mode analysis (Liu et al., 2022).

Closely related formulations appear in charged and higher-dimensional constructions. In the charged four-dimensional theory one writes

$b_\mu$7

with $b_\mu$8, while in five dimensions the action takes the form

$b_\mu$9

or the corresponding $\ell\equiv \varrho b^2$0 extension, with $\ell\equiv \varrho b^2$1 and $\ell\equiv \varrho b^2$2 (Liu et al., 2024, Chen et al., 2 Jul 2026).

A recurrent structural feature is the choice of a purely radial spacelike vacuum expectation value. In four-dimensional slow-rotation studies one uses $\ell\equiv \varrho b^2$3 or a radial-plus-small-$\ell\equiv \varrho b^2$4 deformation, while in five dimensions one takes $\ell\equiv \varrho b^2$5 with $\ell\equiv \varrho b^2$6 enforced on shell. This common choice localizes the Lorentz-violating effect in the radial sector and explains why $\ell\equiv \varrho b^2$7 typically acquires an overall factor $\ell\equiv \varrho b^2$8 or why the effective radial proper distance is rescaled (Ding et al., 2020, Chen et al., 2 Jul 2026).

2. Four-dimensional rotating geometries

The earliest explicit rotating solutions in this literature are slow-rotation metrics constructed perturbatively in the spin parameter $\ell\equiv \varrho b^2$9. To linear order in $\ell\equiv \xi b^2$0, one finds

$\ell\equiv \xi b^2$1

so that

$\ell\equiv \xi b^2$2

For a purely radial bumblebee vacuum expectation value, the slow-rotation solution exists for arbitrary $\ell\equiv \xi b^2$3. For the branch with an additional small $\ell\equiv \xi b^2$4-component of the bumblebee field, consistency of $\ell\equiv \xi b^2$5 requires $\ell\equiv \xi b^2$6, i.e. $\ell\equiv \xi b^2$7 (Ding et al., 2020).

A separate four-dimensional line of work uses a Kerr-like metric in Boyer-Lindquist coordinates,

$\ell\equiv \xi b^2$8

with

$\ell\equiv \xi b^2$9

In that form the horizons satisfy

$\ell\equiv \kappa b^2$0

and the static-limit surface is

$\ell\equiv \kappa b^2$1

The extremal-spin bound becomes $\ell\equiv \kappa b^2$2 (Ding et al., 2019). A related rotating-black-hole construction writes the Kerr-like metric with

$\ell\equiv \kappa b^2$3

so that the horizon radius is

$\ell\equiv \kappa b^2$4

again implying $\ell\equiv \kappa b^2$5. In this model, if $\ell\equiv \kappa b^2$6, the extremal spin is $\ell\equiv \kappa b^2$7, and an extremal rotating Einstein-bumblebee black hole can occur with $\ell\equiv \kappa b^2$8 (Islam et al., 2024).

Charged rotating generalizations are Kerr-Newman-like. To linear order in $\ell\equiv \kappa b^2$9, one uses

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$0

with

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$1

The electromagnetic potential is

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$2

and the bumblebee field remains purely radial,

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$3

The horizons are

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$4

the ergosurface obeys $$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$5, and the horizon angular velocity is

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$6

with

$$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$7

The net effect of $$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$8 is described in that work as a rescaling $$S \;=\;\int\!d^4x\,\sqrt{-g}\Bigl[\tfrac{1}{16\pi}\bigl(R+\varrho\,B^\mu B^\nu R_{\mu\nu}\bigr) -\tfrac14 B_{\mu\nu}B^{\mu\nu}-V(B^\mu B_\mu\pm b^2)\Bigr]\,,$$9, a dressed charge term $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$0, and a prefactor $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$1 in $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$2 (Liu et al., 2024).

The status of exact four-dimensional rotation has remained technically nonuniform. A 2020 slow-rotation study stated that “Till now there seems to be no full rotating black hole solution, so one can't use the Newman-Janis algorithm to generate a rotating solution in Einstein-bumblebee theory” (Ding et al., 2020). Later work nevertheless presented Kerr-like rotating geometries obtained by a modified Newman-Janis algorithm (Islam et al., 2024), and a 2025 quasinormal-mode study explicitly compared two Lorentz-violating rotating backgrounds: one constructed via the Newman-Janis algorithm and another obtained by direct series expansion of the field equations (Deng et al., 18 Jul 2025). This suggests that the rotating sector is best understood as a family of competing constructions rather than a single universally accepted metric.

3. Perturbations and quasinormal spectra in four dimensions

For the slowly rotating Einstein-bumblebee black hole, scalar, vector, and odd-parity axial gravitational perturbations can be reduced by the standard separation

$B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$3

to a Schrödinger-like ordinary differential equation,

$B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$4

where, to first order in the rotation parameter $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$5,

$B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$6

The effective potentials were obtained explicitly for scalar, axial electromagnetic, and axial gravitational sectors, and quasinormal-mode boundary conditions were imposed as purely ingoing waves at the horizon and purely outgoing waves at infinity,

$B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$7

with $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$8 and $B_{\mu\nu}=\nabla_\mu B_\nu-\nabla_\nu B_\mu$9 (Liu et al., 2022).

Two numerical schemes were used in that analysis: the matrix method and the continued-fraction method. In the matrix method, $B_\mu$0 is mapped to $B_\mu$1 by $B_\mu$2, the asymptotic factors are extracted, and the problem becomes a generalized eigenvalue equation $B_\mu$3. In the continued-fraction method, $B_\mu$4 is expanded as $B_\mu$5, leading to a three-term recursion for spin $B_\mu$6 or a six-term recursion for spin $B_\mu$7, and the quasinormal frequencies are found by root-finding on the resulting continued fraction. For $B_\mu$8, $B_\mu$9, and $b_\mu$0, the two methods agree to better than $b_\mu$1 (Liu et al., 2022).

The principal four-dimensional quasinormal-mode trends are field dependent. For both scalar and vector fields, the Lorentz-violating parameter significantly affects the imaginary part of the frequency while having a relatively smaller impact on the real part. In the explored slow-rotation regime, $b_\mu$2 increases weakly with $b_\mu$3 and $b_\mu$4, while $b_\mu$5 decreases as $b_\mu$6 or $b_\mu$7 grow, so the modes decay more slowly. For scalar $b_\mu$8, second-order rotational corrections improve accuracy up to $b_\mu$9. For axial gravitational perturbations, increasing $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$0 mimics the effect of increasing $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$1, creating a near-degeneracy that hampers distinguishing rotation from Lorentz violation by ringdown alone. All computed modes satisfy $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$2, so no instability was found in the slow-rotation, small-$\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$3 regime (Liu et al., 2022).

A later study considered a massive scalar field in two slowly rotating Einstein-bumblebee backgrounds, one Newman-Janis generated and one obtained by direct second-order expansion. In that work the master radial equation was written as

$\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$4

with different explicit potentials $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$5 and $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$6, and the boundary conditions were imposed through an ansatz containing $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$7 and $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$8. The continued-fraction method and the matrix method were again used. That analysis reported that increasing $\langle B^\mu\rangle\langle B_\mu\rangle=\pm b^2$9 raises both $b_\mu$00 and $b_\mu$01, that negative $b_\mu$02 reduces the influence of the field mass parameter $b_\mu$03, and that second-order terms in rotation lift the first-order $b_\mu$04-degeneracy and generate richer splittings. It also described a compressed or expanded spectral “cube” depending on the sign of $b_\mu$05 and on the choice of rotating background (Deng et al., 18 Jul 2025).

Taken together, these results show that quasinormal-mode systematics are not fully universal across the rotating Einstein-bumblebee literature. The dependence on $b_\mu$06 can differ between massless and massive perturbations, between slow-rotation metrics derived from field equations and Kerr-like metrics generated algorithmically, and between scalar, vector, and axial gravitational sectors. A plausible implication is that Lorentz-violation phenomenology in ringdown is strongly model dependent even within Einstein-bumblebee gravity.

4. Geodesics, shadows, accretion disks, and quasi-periodic oscillations

Rotating Einstein-bumblebee geometries have been studied extensively through null and timelike geodesics. In the slowly rotating charged Kerr-Newman-like solution, the Hamilton-Jacobi equation separates with

$b_\mu$07

and the conserved impact parameters are $b_\mu$08 and $b_\mu$09. The radial and polar equations retain the Carter-like structure, and the shadow boundary in celestial coordinates for an observer at infinity with inclination $b_\mu$10 is

$b_\mu$11

For this charged rotating family, numerical plots show that the shadow reference-circle radius $b_\mu$12 decreases as the Lorentz-violating parameter and charge parameter increase, while the distortion parameter $b_\mu$13 increases with both. The same work states that for fixed spin and charge the shadow becomes slightly smaller and more deformed for positive $b_\mu$14 (Liu et al., 2024).

A distinct rotating-black-hole construction, denoted RBHBG, yields a different shadow trend. In that model, the critical impact parameters are

$b_\mu$15

the photon radii satisfy

$b_\mu$16

and the shadow observables $b_\mu$17, $b_\mu$18, area $b_\mu$19, and oblateness $b_\mu$20 are used for inference. In that setting, increasing $b_\mu$21 enlarges the shadow radius $b_\mu$22 and area $b_\mu$23, produces more significant deformation, and decreases the event-horizon area; positive $b_\mu$24 also makes the shadow more flattened. Comparison with Event Horizon Telescope measurements was used to place example bounds such as $b_\mu$25 for $b_\mu$26 from M87*, and the paper summarizes the observational consistency as rough bounds $b_\mu$27–$b_\mu$28 for astrophysical spins $b_\mu$29 (Islam et al., 2024).

The thin-disk literature examines equatorial circular geodesics through the orbital frequency $b_\mu$30, specific energy $b_\mu$31, specific angular momentum $b_\mu$32, and the marginally stable orbit defined by

$b_\mu$33

The radiative flux is written in Page-Thorne form,

$b_\mu$34

with $b_\mu$35, and the efficiency is $b_\mu$36. In the Schwarzschild-like case $b_\mu$37, the ISCO remains at $b_\mu$38 and $b_\mu$39 is independent of $b_\mu$40, but positive $b_\mu$41 suppresses the overall flux and temperature. For prograde rotating disks, positive $b_\mu$42 decreases $b_\mu$43, increases the flux peak and temperature at small radii, and raises the efficiency above the Kerr value; negative $b_\mu$44 does the opposite (Ding et al., 2019).

Quasi-periodic oscillations have been modeled through the relativistic precession framework. For the rotating Einstein-bumblebee metric, the azimuthal frequency is

$b_\mu$45

while the periastron and nodal precession frequencies are $b_\mu$46 and $b_\mu$47. In the rotating case, increasing the Lorentz-symmetry-breaking parameter raises the periastron and nodal precession frequencies but decreases the azimuthal frequency; in the nonrotating case, the nodal precession frequency vanishes for arbitrary Lorentz-violating parameter. Fits to GRO J1655-40, XTE J1550-564, and GRS 1915+105 yielded best-fit values of $b_\mu$48 that include negative and positive central values, but $b_\mu$49 remained inside the $b_\mu$50 region for all three sources, so no statistically significant deviation from general relativity was found (Wang et al., 2021).

The shadow and accretion results are therefore not uniform across solution families. In one rotating charged background the shadow radius decreases with $b_\mu$51, whereas in another Kerr-like rotating background the shadow radius increases with $b_\mu$52. This suggests that observational signatures of Lorentz violation depend not only on the coupling parameter but also on the specific rotating metric, the presence of charge, and the method by which the rotating solution is generated.

5. Three-dimensional rotating BTZ-like black holes

Einstein-bumblebee gravity also admits exact rotating BTZ-like black holes in three dimensions. One construction uses

$b_\mu$53

with a purely radial spacelike bumblebee field $b_\mu$54. In this family the outer and inner horizons are

$b_\mu$55

and the ergosurface is $b_\mu$56. Because of the nonminimal coupling $b_\mu$57, that work argues that horizon area and volume must be redefined as

$b_\mu$58

so that an entropy-area relation $b_\mu$59, a first law, a Smarr formula, and a universal entropy product

$b_\mu$60

can be maintained. The dual CFT central charges were then extracted as

$b_\mu$61

through the thermodynamic method (Ding et al., 2023).

A second BTZ-like rotating solution, used in later quasinormal-mode and scalar-cloud analyses, has

$b_\mu$62

with $b_\mu$63. In this formulation,

$b_\mu$64

so the horizon radii are independent of $b_\mu$65. The angular velocity is

$b_\mu$66

and the Hawking temperature is

$b_\mu$67

with

$b_\mu$68

In that work the entropy is written as $b_\mu$69, unaffected by $b_\mu$70 (Quan et al., 26 Mar 2026). A scalar-cloud study based on the same BTZ-like geometry instead states that the effective Newton constant is shifted to $b_\mu$71 and therefore

$b_\mu$72

while the coordinate positions of $b_\mu$73 remain unchanged and the temperature scales as $b_\mu$74 (Quan et al., 27 Jan 2025). These differing thermodynamic prescriptions show that entropy in the three-dimensional Lorentz-violating sector is not settled uniformly across the literature.

The quasinormal-mode problem in the BTZ-like background is unusually tractable. For scalar, fermionic, and vector perturbations, exact left- and right-moving quasinormal frequencies were obtained. Their real parts remain equal to the BTZ values, $b_\mu$75, while the imaginary parts carry a universal factor $b_\mu$76, so larger positive $b_\mu$77 produces slower damping. The stated exceptions are the fundamental vector modes with $b_\mu$78 in the left sector and $b_\mu$79 in the right sector, where the imaginary part becomes independent of $b_\mu$80 (Quan et al., 26 Mar 2026). For the scalar case, the quasinormal frequencies take the form

$b_\mu$81

$b_\mu$82

and the Lorentz-violation parameter affects only the imaginary parts while enhancing the left and right conformal weights $b_\mu$83 and $b_\mu$84 of the dual operators (Chen et al., 2023).

Stationary scalar clouds add another layer of structure. Imposing Robin boundary conditions at the AdS boundary, the scalar-cloud analysis found only fundamental stationary clouds, $b_\mu$85, and reported that the Lorentz-symmetry-breaking parameter $b_\mu$86 and the angular quantum number $b_\mu$87 play opposite roles in determining the clouds, indicating degenerate scalar clouds. It also states that $b_\mu$88 does not change the superradiance condition and that superradiant instabilities appear only for the fundamental modes (Quan et al., 27 Jan 2025).

6. Five-dimensional equal-spin solutions, Kerr/CFT, and unresolved issues

Five-dimensional Einstein-bumblebee gravity admits rotating black holes with equal angular momenta. In cohomogeneity-one coordinates $b_\mu$89 with $b_\mu$90, the metric ansatz is

$b_\mu$91

and with a purely radial spacelike bumblebee expectation value one finds

$b_\mu$92

$b_\mu$93

The horizon radius $b_\mu$94 is the largest real root of $b_\mu$95, and in the $b_\mu$96 case

$b_\mu$97

In this class of solutions the Lorentz-violating parameter enters thermodynamic quantities as an overall rescaling of the radial sector (Chen et al., 2 Jul 2026).

A notable feature of the five-dimensional theory is the coexistence of two thermodynamic formalisms. The Wald covariant-phase-space method gives

$b_\mu$98

while Komar integrals give

$b_\mu$99

The two charge definitions differ by the overall factor

$\ell\equiv \varrho b^2$00

and the entropies differ by

$\ell\equiv \varrho b^2$01

This entropy mismatch is one of the clearest thermodynamic subtleties presently identified in rotating Einstein-bumblebee gravity (Chen et al., 2 Jul 2026).

The extremal near-horizon geometry supports a Kerr/CFT analysis. After the standard scaling limit, the near-horizon metric takes the warped-$\ell\equiv \varrho b^2$02 form, with all warp factors carrying a common factor $\ell\equiv \varrho b^2$03 in front of the $\ell\equiv \varrho b^2$04 directions. Imposing Kerr/CFT boundary conditions yields a Virasoro algebra with central charge

$\ell\equiv \varrho b^2$05

while the left-moving Frolov-Thorne temperature remains

$\ell\equiv \varrho b^2$06

The Cardy entropy is then

$\ell\equiv \varrho b^2$07

so the microscopic entropy reproduces the Komar/Bekenstein-Hawking entropy rather than the Wald entropy (Chen et al., 2 Jul 2026).

Several unresolved issues run through the broader rotating Einstein-bumblebee literature. In four-dimensional ringdown, increasing $\ell\equiv \varrho b^2$08 can mimic increasing the rotation parameter in the axial gravitational sector, so ringdown alone may not disentangle Lorentz violation from spin; the proposed remedy is to combine multiple harmonic indices or use independent spin measurements (Liu et al., 2022). In the five-dimensional theory, a recent proposal cited in the thermodynamic analysis is that gravitational-wave modes probe an effective radial metric and a modified surface gravity, and that using the corresponding effective temperature in the Wald formula brings $\ell\equiv \varrho b^2$09, although that proposal is explicitly described as an open question (Chen et al., 2 Jul 2026). More generally, the coexistence of slow-rotation solutions derived directly from field equations, Kerr-like constructions obtained by Newman-Janis methods, and nonidentical thermodynamic prescriptions indicates that rotating Einstein-bumblebee black holes constitute a mathematically rich but still internally nonunique sector of Lorentz-violating gravity.