Isotropic coordinates for Schwarzschild black hole radiation

The isotropic coordinate system of Schwarzschild spacetime has several attractive properties similar with the Painleve-Gullstrand coordinates. The purpose for us to choose the isotropic coordinates is to resolve the ambiguities of the tunneling picture in Hawking radiation. Based on energy conservation, we investigate Hawking radiation as massless particles tunneling across the event horizon of the Schwarzschild black hole in the isotropic coordinates. Because the amplitude for a black hole to emit particles is related to the amplitude for it to absorb, we must take into account the contribution of ingoing solution to the action, ImS=ImSout-ImSin. It will be shown that the imaginary part of action for ingoing particles is zero (ImSin=0) in the Painleve-Gullstrand coordinates, so the equation ImS=ImSout-ImSin is valid in both the isotropic coordinates and the Painleve-Gullstrand coordinates