Imagine a city drawn on a sheet of paper. The streets are the same, the intersections are in place, but the distances between blocks have suddenly changed. That is what the three-dimensional BTZ black hole looks like in modified gravity: the metric, the city map, remains the same, while the cosmological constant, which sets the scale, receives corrections.

The BTZ hole is a solution to Einstein's equations in 2+1-dimensional spacetime with a negative cosmological constant. It has no curvature singularity at the center, making it an ideal "toy" model for testing theories of gravity. The authors took the metric of a rotating BTZ hole and plugged it into the field equations of two theories: gravity with the inverse Ricci tensor (a general class of f(R, A, AμνAμν)) and f(R) gravity. For the former, they had to compute the inverse Ricci tensor, its trace, and a quadratic invariant. The result: both theories admit the BTZ solution, but the effective cosmological constant changes.

The formula for the effective cosmological constant in gravity with the inverse Ricci tensor reads:

\[\Lambda_m = \Lambda - 6\alpha_1 \Lambda^2 - 108\alpha_2 \Lambda^3 + \frac{5\beta_1}{4\Lambda} + \frac{15\beta_2}{8\Lambda^2} + \frac{7\gamma}{8\Lambda^2}.\]

Here Λ is the original cosmological constant, and α₁, α₂, β₁, β₂, γ are the coupling constants. In the limit of zero constants, the formula reduces to Λ, as expected. In f(R) gravity, only the terms with α₁ and α₂ contribute.

The authors then integrated the geodesic equations, the paths of freely falling particles, for three cases: nonzero mass and angular momentum, zero mass and angular momentum, and the case M = −1, J = 0 (of the AdS₃ type). The solutions are expressed in terms of hyperbolic, exponential, and trigonometric functions. The effective potential, which determines the barrier for particle motion, changes noticeably: with nonzero coupling constants, the barrier can disappear or shift, altering the conditions for particle capture.

This work is the first to test the survival of a three-dimensional black hole in a theory with the inverse Ricci tensor. Previously, it was unknown whether the BTZ spacetime remains a solution in such gravity and how geodesics change. Now it is shown that the metric is a vacuum solution in both theories, but with a modified cosmological constant; explicit formulas for particle trajectories are obtained. The approach is simple: plug the known metric into the new field equations, compute the inverse Ricci tensor, and integrate the equations of motion.

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. The results can be used to test modified theories of gravity in three-dimensional models and to construct holographic duals.

The city map remains the same, but the distances have changed. Now we need to understand how this affects the thermodynamics of the black hole, its stability, and its quantum properties.