The Geometry Study Of Photon Surface And The Evolutions Of The Innermost Stable Circular Orbits In Dynamical Spacetimes

Black holes are significant celestial bodies in our universe,and they had been predicted in general relativity a long time ago.As a macroscopic object with quantum effects,it is important to confirm its existence from both classical and quantum perspectives.The first image of a black hole taken by the Event Horizon Telescope was published in 2019.We can clearly see a halo surrounding a shaded region.This shaded region is called the black hole shadow,and the halo is the accretion disk of the black hole.The studies of black hole shadow and accretion disk are important to get the information of the black holes.On the one hand,we study the photon sphere and photon surface which are closely connected with the black hole shadow.Based on the geometry of the codimension-2 surface in general spherically symmetric spacetime,we give a quasi-local definition of a photon sphere as well as a photon surface.The new definition effectively excludes the photon surface in spacetime without gravity which is allowed in the definition given by Claudel,Virbhadra and Ellis.And for the system which we are interested in,we give the same results as Claudel,Virbhadra and Ellis.The application of the definition to the Lemaitre-Tolman-Bondi(LTB)model of gravitational collapse reduces to a second order differential equation problem.Crucially,we find that the energy balance on the boundary of the dust ball can provide one of the appropriate boundary conditions to this equation.Based on this key investigation,we find an analytic photon surface solution in the Oppenheimer-Snyder(OS)model and reasonable numerical solutions for the marginally bounded collapse in the LTB model.Interestingly,in the OS model,we find that the time difference between the occurrence of the photon surface and the event horizon is mainly determined by the total mass of the system but not the size or the strength of the gravitational field of the system.On the other hand,we studied the innermost stable circular orbit(ISCO)which is the boundary of the accretion disk.At first,we reviewed the two different methods to obtain the ISCO in Schwarzschild spacetime,i.e,extremum method and effective potential method.Further,we showed that the extremum method is also applicable in general static and stationary spacetimes,and demonstrated it is equivalent to the effective potential method in general static and stationary spacetimes.Then,we generalize the extremum method to general dynamical spacetimes.From this generalization,we studied the evolutions of the ISCOs in Vaidya spacetime,Vaidya-Anti de Sitter spacetime and the slow rotation limit of Kerr-Vaidya spacetime.We find that the evolution behavior of the ISCO in the dynamical spacetime is similar to the evolution behavior of a photon sphere which is an unstable circular orbit.