The third order correction on Hawking radiation and entropy conservation during black hole evaporation process

Using Parikh–Wilczek tunneling framework, we calculate the tunneling rate from a Schwarzschild black hole under the third order WKB approximation, and then obtain the expressions for emission spectrum and black hole entropy to the third order correction. The entropy contains four terms including the Bekenstein–Hawking entropy, the logarithmic term, the inverse area term, and the square of inverse area term. In addition, we analyse the correlation between sequential emissions under this approximation. It is shown that the entropy is conserved during the process of black hole evaporation, which consists with the request of quantum mechanics and implies the information is conserved during this process. We also compare the above result with that of pure thermal spectrum case, and find that the non-thermal correction played an important role.