The Entropy Formula
The entropy \(S_{BH}\) of a black hole is given by the Bekenstein-Hawking formula: \[S_{BH} = \frac{k_B c^3 A}{4 G \hbar}\] where \(k_B\) is Boltzmann's constant, \(c\) is the speed of light, \(A\) is the horizon area, \(G\) is the gravitational constant, and \(\hbar\) is the reduced Planck constant.
Quantum Gravity Connection
This formula is profound because it combines constants from five different areas of physics: thermodynamics (\(k_B\)), relativity (\(c\), \(G\)), and quantum mechanics (\(\hbar\)). It suggests that black hole entropy is fundamentally quantum mechanical in origin, despite being derived from classical geometry.
Magnitude of Black Hole Entropy
For a solar-mass black hole, the entropy is approximately \(10^{77} k_B\). This is vastly larger than the entropy of the star that formed it (about \(10^{57} k_B\)). Most of the information about the matter that fell in is encoded on the horizon surface.