The evolutions of the innermost stable circular orbits in dynamical spacetimes

Abstract

In this paper, we studied the evolutions of the innermost stable circular orbits (ISCOs) in dynamical spacetimes. At first, we reviewed the method to obtain the ISCO in Schwarzschild spacetime by varying its conserved orbital angular momentum. Then, we demonstrated this method is equivalent to the effective potential method in general static and stationary spacetimes. Unlike the effective potential method, which depends on the presence of the conserved orbital energy, this method requires the existence of conserved orbital angular momentum in spacetime. So it can be easily generalized to the dynamical spacetimes where there exists conserved orbital angular momentum. From this generalization, we studied the evolutions of the ISCOs in Vaidya spacetime, Vaidya-AdS spacetime and the slow rotation limit of Kerr–Vaidya spacetime. The results given by these examples are all reasonable and can be compared with the evolutions of the photon spheres in dynamical spacetimes.

Quasi-local studies of the particle surfaces and their stability in general spacetimes

“…This is the evolution equation of the timelike circular orbits in general spherical symmetric spacetime and it is consistent with the result in [43].…”

“…which is consistent with the result in [43]. For a more general situation, it is not hard to show definition 2 will give the same result as [43].…”

“…Combining eq. ( 6.3), (6.4), (6.14), (6.13) and (6.15), we have This is the evolution equation of the timelike circular orbits at the equatorial plane of the Kerr-Vaidya spacetime in the slow rotation limit, and it is consistent with the result in [43].…”

“…which is consistent with the result in [43]. To get the evolution equation of the ISCO in Vaidya spacetime, one needs to require v c ∇ c (K ab k a k b ) = 0.…”

“…But, one can still use definition 2 to classify the stability of the circular orbits. One can use the condition that v c ∇ c (K ab k a k b ) = 0 to get the evolution equation of the ISCO in general spacetime, and the result is consistent with [43].…”