Shock waves in (1+1-dimensional) curved space-time

Shock jump conditions are widely used to solve various astrophysical problems. From the hydrodynamic equation, we derive the jump condition and the Taub adiabat equation in curve space-time for both time-like and space-like shocks. We find that the change in entropy for the weak shocks for curved space-time is small, similar to that for flat space-time. We also find that for general relativistic space-like shocks, the Chapman-Jouguet point does not necessarily correspond to the sonic point for downstream matter, unlike the special relativistic case. To analyse the shock wave solution for the curved space-time, one needs the information of metric potentials describing the space-time, which is assumed to be a neutron star for the present work. Assuming a shock wave is generated at the star's centre, and as it propagates outward, it combusts nuclear matter to quark matter. We find that the general relativistic treatment of shock conditions is necessary to study shocks in neutron stars so that the results are consistent. We also find that with such general relativistic treatment, the combustion process in neutron stars is always a detonation.