Deflection of charged signals in a dipole magnetic field in Schwarzschild background using Gauss-Bonnet theorem

TL;DR

This paper calculates how charged particles are deflected by a dipole magnetic field in Schwarzschild spacetime using the Gauss-Bonnet theorem, revealing magnetic effects on gravitational lensing and potential for magnetic field measurement.

Contribution

It introduces a novel application of the Gauss-Bonnet theorem with Finsler geometry to compute charged particle deflection in Schwarzschild background, including higher-order effects.

Findings

Magnetic dipole affects deflection angles based on particle charge and rotation direction.

Deflection angles up to fourth order are derived.

Magnetic field influence can be used to constrain dipole properties.

Abstract

This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positively charged signal when its rotation angular momentum is parallel (or antiparallel) to the magnetic field. It is argued that the difference in the deflection angles of different rotation directions can be viewed as a Finslerian effect of the non-reversibility of the Finsler metric. The similarity of the deflection angle in this case with that for the Kerr spacetime allows us to directly use the gravitational lensing results in the latter case. The dependence of the apparent angles on the magnetic field suggests that by measuring these angles the magnetic dipole might be constrained.