Lensing Fingerprints of Scalar Hair and Broken Symmetries via Three Semi-Analytical Methods

TL;DR

This paper derives the weak-field light deflection in scalar hairy and Lorentz-violating black hole spacetimes using three semi-analytical methods, demonstrating their agreement and efficiency for gravitational lensing calculations.

Contribution

It introduces three independent semi-analytical techniques for calculating light deflection in complex black hole spacetimes, showing their consistency and computational advantages.

Findings

All three methods agree exactly in results.

The methods are computationally efficient for lensing calculations.

The techniques are applicable to spherically symmetric spacetimes.

Abstract

We present a detailed derivation of the weak-field deflection of light in the scalar hairy black hole spacetime with scalar charge $Q_s$ and bumblebee gravity black hole spacetime, characterized by a Lorentz-violating parameter $\ell$. Starting from the metric, we derive the photon orbit equation up to second order in the small parameters $M/b$, $Q_s$ or $\ell$. Then we derive the weak-field deflection angle via three independent semi-analytical techniques- (1) Homotopy-Perturbation Method, (2) Variational Iteration Method, (3) Single-Impulse (Transverse-Kick) Method-are each applied in full detail. We demonstrate exact agreement among these approaches and discuss their relative computational efficiencies, thereby offering diverse tools for lensing calculations in spherically symmetric spacetimes.