Various methods of calculating hawking radiation and holography

Abstract

There is a number of different methods to calculate Hawking radiation and Hawking temperature. Each method has its own slightly different interpretation of the emitted radiation. However, every method yields the same expression for the Hawking temperature under some approximation limits (T=ℏc^3/8πGMk_B in 4 dimensions). In this work, we will explore four of these methods, starting with the oldest derivation of Hawking radiation i.e., Unruh effect where the ground state or vacuum state of an observer free falling into a black hole becomes a radiation ensemble at fixed temperature for a distant observer. Hawking radiation is then being considered as a radiation composed of virtual particle escaping from vicinity of the horizon region via the quantum tunneling effect. Then, we derive Hawking radiation by utilizing various forms of uncertainty principle. A momentum uncertainty of virtual particles emerging near the horizon gives us the energy of the emitted particle which leads to the temperature of the radiation emitted from the black hole. Additionally, implications of various mass scales to the black hole evaporation process and the possibility of black hole stops radiating and becomes a remnant are being considered. The entropy of black hole remnant is found to be proportional to the surface area of the horizon in unit of the Planck area, a characteristic of holography. This statement holds even when the uncertainty relation is modified to the Minimum Length Uncertainty Relations (MLURs) and extended to arbitrary non-compact D-dimension. However, the entropy of black hole subjugated by MLURs possesses holography only at large mass and remnant limit. Lastly, Hawking radiation will be formulated as a cancellation term to the Einstein anomaly preserving the consistency of quantum field theory around the horizon region. We will also briefly discuss the potential solutions of information loss paradox from each method.