Relativistic oblique shocks with ordered or random magnetic fields: tangential field governs

Relativistic magnetohydrodynamic shocks are efficient particle accelerators, often invoked in the models of gamma-ray bursts (GRBs) and shock-powered fast radio bursts (FRBs). Most theoretical studies assume a perpendicular shock with an ordered magnetic field perpendicular to the shock normal. However, the degree of magnetization $\sigma$ and the magnetic field geometry in shock-powered GRB/FRB scenarios are still poorly constrained by observations. Analogous to the magnetization $\sigma$ associated with the total field strength, we define a tangential magnetization $\sigma_\perp$ associated with the tangential field component. We explore the jump conditions of magnetized relativistic shocks, either with an ordered field of arbitrary inclination angle or with a random field of arbitrary anisotropy. In either case, we find that the jump conditions of relativistic shocks are governed by the tangential magnetization $\sigma_\perp$ instead of the total magnetization $\sigma$, insensitive to the inclination angles or the anisotropy of the pre-shock magnetic field. The approximated analytical solution developed in this work could serve as a quick check for numerical simulations and apply to theoretical studies of GRBs/FRBs with a more general field geometry.