Gravitational lensing by a Lorentz-violating black hole

TL;DR

This paper develops a rigorous strong-field gravitational lensing framework for a Lorentz-violating (LV) black hole in metric-affine bumblebee gravity, deriving an analytic deflection-angle formula $a(b) = - \tilde{a} \ln\left(\frac{b}{b_c}-1\right) + \tilde{b}$ and identifying the LV-dependent coefficients $b_c$, $\tilde{a}$, and $\tilde{b}$. The LV black hole is cast into a Schwarzschild-like form with LV corrections entering through the parameter $X$, yielding a deficit solid angle and LV contributions to the deflection, including an explicit $\alpha_X(b)$ term. Lensing observables such as the image positions $\theta_n$, magnifications $\mu_n$, and Bozza observables $\theta_\infty$, $s$, and $\tilde{r}$ are derived and related to the strong-field coefficients, enabling potential inference of $X$ from data. The framework is applied to Sagittarius A* as a concrete test case, showing that $\theta_\infty \approx 26.55\ \mu\text{as}$ with LV corrections entering at $\mathcal{O}(X)$, and that the observables $s$ and $\tilde{r}$ respond predictably to changes in $X$. The results provide a pathway to constrain LV physics with high-resolution relativistic-lensing observations and motivate extensions to other LV black hole solutions.

Abstract

In this work, we study the gravitational lensing by a Lorentz-violating (LV) black hole inspired by the recent contribution [1]. Explicitly, we concentrate on a specific application: we perform the computation of gravitational lensing effects under the strong field limit. In particular, we analytically derive the deflection angle so that the lens equation can also be addressed. This methodological approach yields physically measurable outcomes, including the determination of relativistic image positions and their corresponding magnifications. As an application of this methodology, we consider the gravitational lensing by Sagittarius A${}^*$ and obtain the corresponding observables expressed as functions of the LV parameter.

Figures (7)

• Figure 1: The deflection angle as a function of $X$ for different values of $b$ and $M$.
• Figure 2: The LV contribution of the deflection $\alpha(b)$ a function of $b$ for different values of $X$. In this case, the GR represents the Schwarzschild solution of the general relativity
• Figure 3: The LV contribution of the deflection $\alpha_{X}(b)$ a function of $b$.
• Figure 4: The LV contribution of the deflection $\alpha_{X}(b)$ a function of $X$ for different values of $b$.
• Figure 5: The emitted light from the source, denoted as $S$ (red point), undergoes deflection on its path toward the observer, identified as $O$ (purple point), influenced by the presence of the compact object located at point $L$ (orange point); $I$ (blue point) represents the image seen from the observer $O$; $D_{OL}$ signifies the distance between the lens $L$ and the observer $O$, while $D_{LS}$ denotes the distance between the projection of the source in relation to the optical axis and the lens.