Mathematical Structure of Charged-Particle Dynamics in Curved Spacetime

How does the geometry of spacetime enter the equation of motion of a charged particle? We study a test charge obeying the covariant Lorentz force law \(u^\nu\nabla_\nu u^\mu = eF^\mu{}_\nu u^\nu\), \(e = q/m\), and state every assumption on the metric and the field explicitly. Geometry acts through the connection, through the metric that defines measured energies and fields, and through isometries. We prove that every Killing vector preserving \(F\) yields a gauge-invariant constant of motion and that Killing observers measure \(d(N\gamma)/d\tau = eNE_\mu u^\mu\), so the magnetic field they see never does work. In the Reissner–Nordström spacetime a charge can hover only if \(eQ > M\), at a radius given in closed form, and every sub-extremal equilibrium is unstable with Killing-time rate \(\kappa_+ (r_+/r_{\mathrm{eq}})^2\). The ISCO condition is a polynomial in \(e^2\); for \(|e| = 1\) it gives \(r_{\mathrm{ISCO}} = 3M + \sqrt{9M^2 - 6Q^2}\) for both signs of the charge. For a Schwarzschild black hole in Wald's uniform magnetic field we solve for the angular velocity of circular orbits and prove that only anti-Larmor orbits exist inside the photon sphere. All results are verified symbolically and numerically.