Black hole based general relativistic limit of f(R) theory of gravity

TL;DR

This work studies a Kerr-Scalaron metric, an exact stationary, axisymmetric vacuum solution in $f(R)$ gravity, to test deviations from general relativity in the Galactic Center. By analyzing photon orbits and the resulting black hole shadow, the authors connect the scalaron mass $M_{\psi}$ to observable features such as shadow size, displacement, and quadrupole deviation, and show that $M_{\psi} \sim 10^{-16}$ eV yields Kerr-like behavior while satisfying the Cassini $γ$ constraint. They further relate $M_{\psi}$ to curvature and potential scales via gravitational identifiers $(κ,φ)$ inferred from S-star orbits, obtaining consistent masses around $10^{-17}$–$10^{-16}$ eV, thereby supporting a general relativistic limit of $f(R)$ gravity at horizon scales. The results suggest the GR limit of $f(R)$ gravity is scale-invariant and that GC shadow analyses, stellar dynamics, and solar-system tests provide a complementary cross-scale probe of scalaron gravity with potential implications for gravitational-wave observations.

Abstract

The Galactic Center black hole environment gives us new opportunity to test deviation from General Relativity and black hole physics. In this work we analytically generate the shape of the Galactic Center black hole by using a recently developed exact stationary, axisymmetric and vacuum solution of $f(R)$ gravity theory. By using scalaron mass as a free parameter we find that the shadow shape along with displacement and asymmetry is sensitive to the scalaron mass, even after keeping the black hole spin low. We recognize scalaron mass which is compatible with Kerr like quadrupole moment and hence black hole "no-hair" theorem. The same mass scale is found to reproduce the PPN parameter ($γ$) constrained in the weak field limit of the solar system. Gravitational identifiers, the Kretschmann scalar ($κ$) and gravitational potential ($φ$) have been used to infer scalaron masses in the regime of S-stars which are found to be consistent with the limits obtained using shadow scales. We ensure that $f(R)$ gravity scalaron has an appropriate general relativistic limit in the horizon scale of the black hole. We also identify the possibility of scale invariance of the general relativistic limit.

Figures (5)

• Figure 1: The shadow of Kerr-scalaron black hole as observed by a distant observer at $\theta_0 = \pi/2$ for spin 0.4 (left) and 0.99(right).Here KS denotes Kerr-scalaron.
• Figure 2: Variation of displacement (a) and asymmetry (b) with inclination angle for a fixed value of black hole spin and various scalaron masses.
• Figure 3: Variation of PPN parameter $\gamma$ with length scale for various scalaron masses.
• Figure 4: Scalaron mass realized in other astrophysical tests, including planetary perihelia shift, EHT's measurement of deviation parameter for shadow, S2's periapsis shift and this study.
• Figure 5: Variation of $M_{\psi}$ for identifiers $\kappa$ and $\phi$. The $\kappa$ and $\phi$ values for S-stars have been adopted from Table 3 of 2015ApJ...802...63B.