Black Hole Thermodynamics

Von Neumann Entropy $-\mathrm{Tr}[\rho \log \rho ]$ is time independent under the unitary evolution of Quantum Mechanics.

Jacob Bekenstein: wanted to generalise Entropy such that it wouldn’t decrease around a Black hole. Stephen Hawking had proven that the area of a black hole horizon can only increase. Bekenstein proposed that entropy is proportional to this area. This would be $S=\frac{A}{4G\hslash }$. For a Schwarzschild blackhole this becomes $S=\frac{4\pi G{M}^2}{\hslash }$ .

We generally have $dE=TdS$ . Stephen Hawking discovered that black holes have an internal temperatura, proportional to $\hslash$. Hawking observed this by plotting the behavior of light using a Penrose diagram in the presence of a black hole.

Keywords

• Penrose diagram ( time vertically, space horizontally ), are a possible visualization of Minkowski space
• Cauchy hypersurface
• Geodesic
• Antiperiodicity is the hallmark of a thermal correlation function
• Entanglement entropy
• Schwarzschild solution
• Coordinate singularity
• Einstein-Rosen bridge
• Infrared cutoff