[Paper Review] The gravitational bending angle by static and spherically symmetric black holes in bumblebee gravity

Í. D. D. Carvalho, G. Alencar+2 more|arXiv (Cornell University)|Mar 5, 2021

Relativity and Gravitational Theory4 citations

TL;DR

This paper computes the gravitational bending angle of massive particles and light in bumblebee gravity, a Lorentz symmetry-breaking theory, using the Ishihara method adapted to the Jacobi metric for massive particles. It derives deflection angles for two static, spherically symmetric black hole solutions—Bertolami-Páramos (asymptotically flat) and Maluf-Neves (non-asymptotically flat)—showing that Lorentz symmetry breaking modifies the deflection beyond general relativity, with explicit analytical expressions in finite-distance and weak-field limits.

ABSTRACT

This work investigates the influence of the Lorentz symmetry breaking in the bending angle of massive particles and light for bumblebee black hole solutions. The solutions analyzed break the Lorentz symmetry due to a non-zero vacuum expectation value of the bumblebee field. We use the Ishihara method, which allows us to study the bending angle of light for finite distances, and it is applicable to non-asymptotically flat spacetimes when considering the receiver viewpoint. In order to analyze the deflection of massive particles, we systematize the Ishihara method for its application in the Jacobi metric. This systematization allows the study of the deflection angle of massive particles using the Gauss-Bonnet theorem. We consider two backgrounds: the first was found by Bertolami et al. and is asymptotically flat. The second was found recently by Maluf et al. and is not asymptotically flat due to an effective cosmological constant.

Motivation & Objective

• To investigate the influence of Lorentz symmetry breaking on the gravitational bending angle of massive particles and light in bumblebee gravity.
• To extend the Ishihara method, originally for light deflection in non-asymptotically flat spacetimes, to massive particles via the Jacobi metric.
• To analyze deflection angles for two distinct bumblebee black hole solutions: one asymptotically flat (Bertolami-Páramos) and one with an effective cosmological constant (Maluf-Neves).
• To derive analytical expressions for the deflection angle in finite-distance configurations and in limiting cases (e.g., weak field, vanishing bumblebee field).

Proposed method

• Adapts the Ishihara method to massive particle trajectories by reformulating the problem in the Jacobi metric, enabling application of the Gauss-Bonnet theorem.
• Applies the Gauss-Bonnet theorem to compute the deflection angle using the Riemannian geometry of spatial sections in static, spherically symmetric spacetimes.
• Uses the receiver viewpoint to handle non-asymptotically flat spacetimes, avoiding divergences when source or receiver is not at the horizon.
• Derives deflection angle expressions for two bumblebee black hole solutions: one from Bertolami et al. and one from Maluf et al., with distinct asymptotic behaviors.
• Performs analytical expansions in the weak-field limit (b >> M) and in the limit of vanishing bumblebee field (λ = 1), recovering known solutions like Schwarzschild in the latter.
• Considers three limiting cases: (a) light deflection (v = 1), (b) distant source and receiver (b u_R, b u_S → 0), and (c) vanishing bumblebee field (λ = 1).

Experimental results

Research questions

• RQ1 How does Lorentz symmetry breaking, induced by a non-zero vacuum expectation value of the bumblebee field, affect the deflection angle of massive particles in static, spherically symmetric black hole spacetimes?
• RQ2 Can the Ishihara method be systematically extended to massive particles using the Jacobi metric, and does it yield consistent results in non-asymptotically flat spacetimes?
• RQ3 What are the deflection angle expressions for the Bertolami-Páramos and Maluf-Neves bumblebee black hole solutions in finite-distance and weak-field regimes?
• RQ4 How do the deflection angles reduce to known results (e.g., Schwarzschild or general relativity) in the limit of vanishing bumblebee field (λ = 1)?
• RQ5 What are the implications of the effective cosmological constant in the Maluf-Neves solution for the deflection of massive particles and light?

Key findings

• The deflection angle of massive particles in the Bertolami-Páramos background is derived in equation (25) for finite distances, with explicit expressions in limiting cases.
• In the weak-field limit (b u_R, b u_S → 0), the deflection angle reduces to equation (27), which depends on the bumblebee field parameter λ and particle velocity v.
• When the bumblebee field vanishes (λ = 1), the deflection angle in the Bertolami background reduces to the Schwarzschild-like form, as shown in equation (28).
• For the Maluf-Neves background, the deflection angle is given by equation (31), which includes an effective cosmological constant Λ_e = κσ/ξ, modifying the deflection beyond general relativity.
• In the light-deflection limit (v = 1), the deflection angle for the Maluf-Neves background becomes equation (32), showing dependence on λ and the effective cosmological constant.
• The apparent divergences in equations (33) and (34) are only problematic if the source or receiver is located at the horizon, confirming the robustness of the method for finite-distance configurations.