Generalized Vaidya spacetime: horizons, conformal symmetries, surface gravity and diagonalization

In this paper, the different properties of generalized Vaidya spacetime are considered. We define the location of horizons. We show that the apparent horizon can contain the event horizon. The locations of all types of horizons are compared with ones in the usual Vaidya spacetime. We investigate the timelike geodesics in this spacetime. New corrections to Schwarzschild and Vaidya cases appear and we give conditions when these corrections are not negligible. Also, we consider the conformal Killing vector and transform the metric to conformally-static coordinates. We introduce a new constant of motion along null and timelike geodesics, which is generated by a homothetic Killing vector. The conformally-static coordinates allow diagonalizing of the generalized Vaidya spacetime. The surface gravity has been calculated for the dust and stiff fluid cases.