Bekenstein-Hawking entropy quantifies the entropy of a black hole as proportional to its event horizon area.Black holes seem like bottomless pits where everything falls in and disappears. But they have a special memory — entropy, a measure of disorder. The more falls in, the larger the area of their 'surface' — the event horizon — becomes. The Bekenstein-Hawking formula links this area to entropy: it's enormous! Imagine a black hole is an ideal draft document that can't be read from the outside, but the amount written can be judged by the size of the crumpled paper.
Jacob Bekenstein realized in 1973 that without attributing entropy to black holes, the second law of thermodynamics would be violated when matter falls in. He proposed proportionality to horizon area. Stephen Hawking, initially skeptical, in 1974 used quantum field theory in curved spacetime to derive that black holes radiate and independently confirmed the 1/4 factor. Their work gave birth to modern black hole thermodynamics.
Used in astrophysics, for calculating the information capacity of black holes, in string theory, and in AdS/CFT correspondence. Applicable to any equilibrium black holes, but is semiclassical: quantum corrections become significant in the final stages of evaporation. For extremal black holes, entropy is zero, requiring separate interpretation.
Stephen Hawking initially argued with Bekenstein and even made a bet, but later helped derive the exact formula himself! The story of the two geniuses reads like a scientific detective novel.
| S | variable | Entropy of the black hole — entropy joule per kelvin |
| A | variable | Area of the event horizon — area square metre |
| k | constant | Boltzmann constant — boltzmann constant = 1.380649e-23 joule per kelvin |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| G | constant | Newtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| ħ | constant | Reduced Planck constant — reduced planck constant = 1.054571817e-34 joule second |
| S_gen | variable | Generalized entropy — entropy joule per kelvin |
| A | variable | Area of the horizon — area square metre |
| S_matter | variable | Entropy of matter outside the horizon — entropy joule per kelvin |
| G | constant | Newton's gravitational constant, determining the strength of gravity — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| S | variable | Black hole entropy — entropy joule per kelvin |
| A | variable | Black hole horizon area — area square metre |
| S_A | variable | Entanglement entropy of boundary region A — entanglement entropy |
| Area(minimal surface | variable | Area of the minimal surface in the bulk homologous to A — area square metre |
| G_N | constant | Newton's gravitational constant in the bulk — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| S | variable | Entropy of the system — entropy joule per kelvin |
| W | variable | Number of microstates — number of microstates |
| k_B | constant | Relates energy to temperature at the particle level — boltzmann constant = 1.380649e-23 joule per kelvin |
| ln | operator | Natural logarithm — natural logarithm |
| \eta | variable | Baryon to entropy ratio — baryon to entropy ratio |
| n_B | variable | Baryon number density — baryon number density per cubic metre |
| s | variable | Entropy density — entropy density joule per cubic metre per kelvin |











