
Geometric diagram of a black hole in units of entropy.

Geometric diagram of a black hole in units of entropy.

In cosmological thermodynamics, black hole entropy or “Bekenstein-Hawking entropy” is the entropy of a black hole, hypothesized to measurable according to the surface area of the black hole; which, according to Stephen Hawking’s 1970 area theorem, increases for any given process. The equation for black hole entropy is typically given by the formula:

$S_{BH} = \frac{kA}{4\ell_{\mathrm{P}}^2}$

where k is the Boltzmann constant, A is the surface area of the black hole, and lP is the Planck length.

Tea cups, entropy, and black holes
In circa 1971, American theoretical physicist John Wheeler commented to Mexican-born Jewish physicist Jacob Bekenstein, a student of his at the time, that black holes seem to flout the second law of thermodynamics. Supposedly, Wheeler had been bothered by the notion that black holes provide for a loophole in the second law of thermodynamics in that they conceal actions associated with entropy increase. In recollection, Wheeler comments:

“I told him [Bekenstein] of the concern I always feel when a hot cup of tea exchanges energy with a cold cup of tea … by allowing that transfer of heat I do not alter the energy of the universe, but I do increase its microscopic disorder, its information loss, its entropy. The entropy of the world always increases in an irreversible process like that. The consequences of my crime, Jacob, echo down to the end of time. But if a black hole swims by, and I drop the teacup into it, I conceal from all the world the evidence of my crime.”

Bekenstein pondered the conundrum, and came back several months later with an idea of how to resolve it: [4]

“You don’t destroy entropy when you drop those teacups into the black hole,” as Wheeler recalls the conversation, “the entropy already has entropy, and you only increase it!”

Bekenstein gave a picture in which black holes grow in size as their entropy increases. [3]

In short, in circa 1972 Bekenstein had showed that black holes should have a well-defined entropy. [1] Bekenstein built on the previous 1970 work of Stephen Hawking, in particular his “law of areas” for black holes (resembling the second law), showing that the area of a black hole called an event horizon always increases in all black hole processes, to argue that this area is a representative measure of the entropy of a black hole. [2] Hawking later came across this work and had objections or reservations about it and began to marinade on the issue. He eventually arrived at a remedy to the issue by contemplating the logic that if a black hole rotated it might then emit particles, thus having a temperature and an entropy. [3]

In 1999, American theoretical physicist Brian Greene argued that string theory has most impressively flexed its muscles on solving the issue of black hole entropy. [2]

References
1. Baeyer, Hans Christian von. (2004). Information - the New Language of Science. Cambridge, (pgs. 205-11). Massachusetts: Harvard University Press.
2. Green, Brian. (1999). The Elegant Universe (section: Black Hole Entropy, pgs. 333-40). Vintage Books.
3. Ferris, Timothy. (1998). The Whole Shebang: a State-of-the-Universe(s) Report (pgs. 90-97, esp. 92). Simon and Schuster.
4. Wheeler, John A. and Ford, Kenneth W. (2000). Geons, Black Holes, and Quantum Foam (pg. 314). W.W. Norton and Co.

External links
● Black hole entropy – String Theory Wiki.
● Bekenstein-Hawking entropy – Scholarpedia.