Deriving the expression for Hawking temperature involves combining principles from general relativity and thermodynamics. Here's a high-level overview of the derivation:
Consider a black hole described by the laws of general relativity. In particular, focus on the properties of its event horizon, which is the boundary beyond which nothing can escape the gravitational pull of the black hole.
Apply the laws of classical thermodynamics to the black hole. Treat the event horizon as a thermodynamic system and assign it properties such as entropy and temperature.
Utilize the concept of the Bekenstein-Hawking entropy. In 1973, Jacob Bekenstein proposed that black holes possess entropy proportional to their event horizon area. Stephen Hawking subsequently showed that black holes should emit thermal radiation, now known as Hawking radiation, as a consequence of quantum effects near the event horizon.
Apply the principles of thermodynamics to the process of black hole evaporation through Hawking radiation. The key idea is that the total entropy of a closed system should not decrease. As a black hole emits radiation, its mass decreases, leading to a decrease in its entropy. To account for this, the radiation itself must possess entropy, and the black hole's entropy is considered to reside on its event horizon.
Use the first law of thermodynamics to relate changes in the black hole's mass, area, and entropy. The first law states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In the case of black hole thermodynamics, the internal energy corresponds to the black hole's mass, and the work done is associated with changes in the black hole's area.
Combine the first law of thermodynamics with the Bekenstein-Hawking entropy expression and perform calculations involving the black hole's surface gravity to derive an expression for the Hawking temperature. The surface gravity is related to the black hole's mass and area.
The detailed mathematical derivation involves intricate calculations and considerations of quantum field theory in curved spacetime. It is beyond the scope of a brief explanation. However, the key steps mentioned above provide an outline of the procedure for deriving the Hawking temperature using principles from general relativity and thermodynamics.