Entropy of the Schwarzschild Black Hole in the Painlevé and the Lemaitre Coordinates

In the Painlevé and the Lemaitre coordinates, the statistical-mechanical entropies of the Schwarzschild black hole arising from the quantum scalar field are investigated by using the ’t Hooft’s brick wall model. At first sight, it seems that the results would be different from that in the standard Schwarzschild coordinate since both the Painlevé and the Lemaitre spacetimes do not possess the event horizon obviously. However, we prove that the entropies in these coordinates are exactly equivalent to that in the Schwarzschild coordinate.