This work stands apart: there is almost nothing nearby. The closest neighbors are brief notes on modified gravity, black holes with curved momentum space, and quantum foam. All of them, in one way or another, seek to go beyond the Einsteinian picture, but the BTZ spacetime in gravity with inverse Ricci tensor had not been considered before. This is the first test of whether the three-dimensional black hole survives in a theory where the action includes not the Ricci tensor itself but its inverse.
They took the metric of the rotating BTZ hole and substituted it into the field equations for two theories: gravity with the inverse Ricci tensor (the general class f(R, A, AμνAμν)) and f(R) gravity. For the first, they computed the inverse Ricci tensor (curvature antitensor), its trace, and the quadratic invariant. Then they solved the vacuum equations and found the effective cosmological constant. After that, they integrated the geodesic equations for three cases: nonzero mass and angular momentum, zero mass and momentum, and the case M = −1, J = 0 (AdS3-like).
In gravity with the inverse Ricci tensor, the effective cosmological constant is Λm = Λ − 6α1Λ2 − 108α2Λ3 + 5β1/(4Λ) + 15β2/(8Λ2) + 7γ/(8Λ2). In f(R) gravity, the corrections come only from the α1 and α2 terms: Λm = Λ − 6α1Λ2 − 108α2Λ3. Both formulas reduce to Λ in the limit of zero coupling constants. The geodesics are fully integrated: for massive particles, hyperbolic, exponential, and trigonometric solutions are obtained depending on the relation between energy, momentum, and the cosmological constant. The effective potential changes noticeably: for nonzero coupling constants, the barrier can disappear or shift, which changes the conditions for particle capture.
The difference from neighbors lies in the object: they consider cosmological solutions or four-dimensional black holes, whereas here the three-dimensional BTZ hole is tested for the first time in a theory with the inverse Ricci tensor. This extends the class of admissible solutions and shows that modifications of gravity do not break the BTZ structure but merely renormalize the cosmological constant. For quantum gravity in 2+1 dimensions this is important: the BTZ hole is a key model for holography, and its survival in modified theories opens a path to testing the AdS/CFT correspondence beyond general relativity.
A natural next step is thermodynamics: entropy, heat capacity, and phase transitions of the BTZ hole in these theories. One could also study the stability of the solution under perturbations of the metric and scalar fields. Of particular interest is the quantization of geodesics and the spectrum of quasinormal modes, which is directly related to the holographic description.
The results will affect research on lower dimensions in quantum gravity and on testing modified theories through holography. Possible applications include condensed matter physics, where BTZ-like metrics arise in models of graphene and topological insulators.
Compute thermodynamic quantities (entropy, temperature, heat capacity) for the BTZ hole in both theories and check the first law of thermodynamics. Investigate quasinormal modes and their dependence on the coupling constants.
For gravity in 2+1 dimensions, the question of quantization and how modified theories fit into the holographic principle remains open. This work shows that the BTZ solution is stable under the addition of the inverse Ricci tensor, but it does not solve the problem of the microscopic description of entropy. The connection to the black hole information paradox is not yet clarified.
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What the work does
This study examines how modified gravity theories, namely Ricci-inverse gravity and f(R) gravity, affect the three-dimensional BTZ black hole. It is shown that both theories admit this solution, but the cosmological constant receives corrections from the coupling constants. The geodesic equations for massive and massless particles are fully integrated, with solutions expressed in terms of elementary functions, and it is demonstrated that particle trajectories change noticeably compared to general relativity.
Where its strength is
For the first time, the survival of the BTZ black hole is verified in Ricci-inverse gravity, expanding the class of admissible solutions and showing that modifications do not break the structure but merely renormalize the cosmological constant. This provides a foundation for testing holographic correspondences beyond general relativity.
Where it could go next
Worth trying
Why this may matter
The survival of the BTZ black hole in modified theories opens a path to testing holographic correspondences beyond general relativity, and the obtained exact geodesic solutions provide a basis for observational predictions.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.58, against a usual 0.64. It looks like few have drilled nearby.
The BTZ hole has no curvature singularity at the center, unlike the Schwarzschild hole. This makes it an ideal 'toy' model for testing theories of gravity.
In the series 'The Expanse', the ring gates use exotic geometry reminiscent of three-dimensional wormholes. The BTZ hole, with its nontrivial horizon topology, is a mathematical relative of such constructions.