Bulletin of Taras Shevchenko National University of Kyiv. Astronomy, no. 71, p. 30-33 (2025)
SPECIAL SPHERICALLY SYMMETRIC SOLUTIONS IN f(R) GRAVITY
Valery ZHDANOV, DSc (Phys. & Math.), Prof.
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Abstract
Background. f (R) gravity is a natural generalization of General Relativity, where the Lagrangian in the Einstein-Hilbert action is replaced by a more complicated function. This theory is extensively used in considerations of the early inflation, late inflation and in the dark matter models. Variation of the action in this theory leads to the fourth order equations with respect to the physical metric (Jordan frame). We study the conformally transformed metric (Einstein frame) in a static spherically symmetric (SSS) case. Methods. We use the well-known transformation, which allows to reduce the problem to usual Einstein equations with an additional scalar field dubbed scalaron. The focus is on the SSS solutions in case of large dimensionless parameter, Mμ→∞, where Mis the configuration mass and μis the scalaron mass in geometrized units. Results. A class of scalaron potentials is identified that covers well-known f(R) models and permits a straightforward analytical description of SSS objects in the region where the solutions are significantly different from the Schwarzschild case for Mμ>>1. A representation of the structure equations is found that enables numerical integration. The results may be useful for testing numerical algorithms in case of sufficiently large Mμ. Conclusions. Approximate analytic relations are found for the SSS solutions of the f (R) gravity in the Einstein frame on a finite interval of radial variable in the limit Mμ→∞ under special assumption on the initial data. The results show universal properties of solutions for a wide class of scalaron potentials.
Key words
Relativistic objects, naked singularities, modified gravity.
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