Precision tests of analytical tail-term approximations for radiation reaction in Schwarzschild spacetime
We investigate the consistency and precision of approximate analytical expressions for the electromagnetic self-force acting on a charged particle in Schwarzschild spacetime endowed with weak electromagnetic fields. A fundamental requirement of relativistic particle dynamics is the preservation of the four-velocity normalization ($u^μu_μ=-1$), which implies that the total self-force must remain orthogonal to the particle's four-velocity. We introduce a covariant diagnostic based on the orthogonality condition ($u_μF^μ_{\text{tail}}=0$), which provides a quantitative measure of the internal consistency of approximate tail-term models used in radiation-reaction calculations. We apply this diagnostic to two widely used analytical approximations for the electromagnetic tail force: the conservative component derived by Smith and Will and the dissipative component derived by Gal'tsov. The analysis is performed for several physical configurations, including pure Schwarzschild spacetime, a weakly electrically charged Schwarzschild black hole, and a Schwarzschild black hole immersed in a weak external magnetic field. We find that the conservative Smith--Will term alone leads to small but measurable deviations from the orthogonality condition, while inclusion of the dissipative Gal'tsov contribution suppresses these deviations by many orders of magnitude. For realistic radiation-reaction parameters, the violation becomes extremely small. The proposed orthogonality diagnostic offers a simple and covariant tool for validating approximate self-force models in curved spacetime and may be useful for future studies of radiation-reaction dynamics near compact objects.