We examine a stationary BTZ spacetime in the context of modified gravity featuring an inverse Ricci tensor (where the function f depends on the Ricci scalar, anti-curvature, and the square of the anti-tensor). It is demonstrated that BTZ is an exact solution in this theory, with the cosmological constant altered by coupling constants. f(R) gravity is also analyzed. A complete integration of the geodesic equations for BTZ in inverse-Ricci gravity is carried out; the solutions are expressed through elementary functions. Geodesics are classified for three scenarios: (i) M≠0, J≠0; (ii) massless BTZ (M=0, J=0); (iii) AdS₃ type (M=-1, J=0). The findings are compared with general relativity predictions.
Spacetime in Einstein's theory behaves like a rubber sheet: massive bodies bend it. But the accelerated expansion of the Universe hints that the sheet may have unusual properties. To test new rules of gravity, physicists use simplified models — for example, black holes in two-dimensional worlds. One such "toy" is the BTZ black hole: it lives on a plane but keeps the main features of the real thing.
In a modified theory of gravity, such a hole also appears, but the parameter controlling the galaxies flying apart becomes different. Particles and light move along new trajectories that were calculated precisely — like a ball rolling on a curved sheet. Comparison with the Einstein case revealed subtle differences. Perhaps one day they will make it possible to do without hypothetical dark energy to explain the acceleration of the cosmos.
🎯 In a two-dimensional universe, a black hole is not a sphere but a circle, and its event horizon is a simple line. Yet the formulas for an infalling observer are almost the same as in the three-dimensional case.
🎬 Physicists often construct worlds with fewer dimensions — like in Edwin Abbott's novel "Flatland", where flat beings are unaware of height. The BTZ black hole is their version of a black hole.