Abstract
We studied the collapse solution to a singular isothermal sphere with consideration of relativistic collapse solution, and extended the collapse solutions with critical points into the shock wave solution. Apart from its Newtonian counterpart, the general relativistic shock wave is generated by the change of space-time after the collapse of the sphere begins. The shock wave propagate outward at a super sonic speed, accompanied by the collapse of the sphere inside and the formation of a central black hole. In our calculation, the shock wave solutions are capped by the isothermal sound speed. The solution only exists for the sound speed ranged at 17.32% 45.83% of speed of light. We postulate this shock wave can release the gravitational binding energy of the central black hole by the ratio of 3% 15% of its rest mass. A simple estimation shows that the released energy is about 1054erg for a black hole with mass MBH ∼ 10M☉, which correspond to the energy scale of those most energetic gamma-ray bursts in the sky.