The importance of general relativistic shock calculation in the light of neutron star physics

Numerical simulation of hydrodynamic equations forms the central part of solving various modern astrophysical problems. In the case of shocks, one can have either dynamical equations or jump conditions (the conservation equations without any time evolution). The solution of the jump condition in curve space-time is derived and analyzed in detail in the present work. We also derive the Taub adiabat or combustion adiabat equation from the jump condition. We have analyzed both time-like and space-like shocks in the present work. 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 relativistic case. To analyze the shock wave solution for the curved space-time, one needs the information of metric potentials describing the space-time, which for the present work is taken to be a neutron star. We assume that a shock wave is generated at the centre of the star and is propagating outward. As the shock wave is propagating outwards, it combusts nuclear matter to quark matter, and we have a combustion scenario. We find that the general relativistic treatment of shock conditions is necessary to study shocks in neutron stars so that the results are consistent with the solution of the TOV equation while calculating the maximum mass for a given equation of state. We also find that with such general relativistic treatment, the combustion process in neutron stars is always a detonation.