Fluctuation around horizon on a Schwarzschildblack hole using the null geodesic method

Using the null geodesic method, Hawking radiation from the horizon of a Schwarzschild black hole is calculated. The thermodynamics can be built successfully on the horizon where the apparent horizon and event horizon are coincident with each other. When a relativistic perturbation is given to the horizon, the first law of thermodynamics can also be constructed at a new supersurface near the horizon successfully. The expressions of the characteristic position and temperature are consistent with the previous result while the thermodynamics was built on the event horizon in a Vaidya black hole. Therefore, the thermodynamics of a dynamical black hole should be constructed on the apparent horizon exactly, and the event horizon thermodynamics is just one of the perturbations near the apparent horizon.