Asymptotic Schwarzschild solutions in $f(R)$ gravity and their observable effects on the photon sphere of black holes

TL;DR

The paper investigates static, spherically symmetric solutions in $f(R)$ gravity that asymptotically approach Schwarzschild GR and analyzes their strong-field lensing signatures. By focusing on the quadratic truncation $f(R)=R+aR^2$, it derives asymptotic solutions for large $r$ and then numerically integrates inward to obtain the photon-sphere radius $r_P$, capture parameter $b_c$, and a new observable, the photon-sphere width $oldsymbol{ m oldsymbol{\delta_P}}$. It finds two branches: an exponentially decaying exterior for $a>0$ and an oscillatory exterior for $a<0$, with both yielding a smaller horizon than GR and a reduced $b_c$, while $oldsymbol{ m oldsymbol{ ho_{oldsymbol{ m \delta_P}}}}$ can increase by up to $ oughly 20 ext{\%}$ for realistic parameter choices. These results imply that even modest $f(R)$ deviations from GR can leave imprints in strong-field observables that future high-resolution black-hole imaging could constrain; the study also discusses limitations of the truncated model and outlines pathways to incorporate rotation and higher-order curvature terms. Overall, the work provides a concrete framework to connect asymptotically Schwarzschild $f(R)$ spacetimes with measurable photon-ring features in strong gravity regimes.

Abstract

We investigate asymptotic Schwarzschild exterior solutions in the context of modified gravity theories, specifically within the framework of $f(R)$ gravity, where the asymptotic behavior recovers the standard Schwarzschild solution of General Relativity. Unlike previous studies that rely mainly on analytical approximations, our approach combines asymptotic analysis with numerical integration of the underlying differential equations. Using these solutions, we analyze strong lensing effects to obtain the photon sphere radius and the corresponding capture parameter. Considering rings produced by total reflection, we define the photon sphere width as the difference between the first total reflection and the capture parameter; and study how it is modified in the $f(R)$ scenario. Our results show that the photon sphere width increases in the presence of $f(R)$-type modifications, indicating deviations from GR that could be observable in the strong-field regime.

Figures (13)

• Figure 1: From left to right and upper to lower: numerical and asymptotic solutions for $U(x)$, $P(x)$, $m(x)$; and numerical solutions for metric functions $A(x)$ and $B(x)$ with the corresponding functions for the SW case $A_{SW}(x)$ and $B_{SW}(x)$. We can see that result are consistent with the asymptotic behavior. At lower radii both metric functions tend to zero and the function $P(x)$ diverges. In the left lower panel we also plot the behavior for low $x$ which is $m(x\to 0) = x - 1$. These results are obtained using $\alpha = 1$ and $P_{\alpha} = 1$.
• Figure 2: $m(x)$ for different values of $P_{\alpha}$ and with $\alpha = 1$ fixed. As we can see, all curves tends to $x-1$ when $x \to 0$. Increasing $P_{\alpha}$ makes the maximum of $m(x)$ bigger at the same radii.
• Figure 3: $m(x)$ for different values of $\alpha$ and with $P_{\alpha} = 0.5$ fixed. As we can see, all curves tends to $x-1$ when $x \to 0$. Increasing $\alpha$ makes the maximum of $m(x)$ bigger and moves it to larger radius values.
• Figure 4: From left to right and upper to lower: numerical and asymptotic solutions for $U(x)$, $P(x)$, $m(x)$; and numerical solutions for metric functions $A(x)$ and $B(x)$ with the corresponding functions for the SW case $A_{SW}(x)$ and $B_{SW}(x)$. We can see that the numerical results are consistent with the asymptotic behavior. The metric functions at low radii are similar to SW with some oscillating pattern and with an event horizon lower than $x=1$. These results are obtained using $|\alpha| = 1$ and $P_{\alpha} = 0.1$.
• Figure 5: $B(x)$ for different values of $P_{\alpha}$ and with $|\alpha| = 1$ fixed. As we can see, bigger values of $P_{\alpha}$ make $B(x)$ greater than the SW solution at the same radii.