192
12 Unsupervised Deep Learning
Column: Black Holes and Information
In Chap. 1 we explained that Maxwell’s demon made an entropy decrease by log 2.
Here, as a final column of this book, we shall talk about the introduction of black
hole entropy by J. Bekenstein and S. Hawking. Prior to their proposal, it was
pointed out that there was a similarity of the law of increasing entropy to a black
hole growing endlessly by swallowing gas around it. Bekenstein assumed that the
entropy S is given as
S = f (A) ,
(12.29)
where the area of the black hole horizon is A, and attempted to determine the
function f . The key in his idea is “What is the minimum amount of change in
S?” He assumed the answer as follows:
1. It should correspond to the case where a particle with its Compton length equal
to its radial size falls into a black hole.
2. The entropy change should be log 2 since the particle is either kept retained or
destroyed.
First, the minimum area change from Assumption 1 is semi-classically calculated
as
δA = 2 ¯
h.
(12.30)
Combining this with Assumption 2 results in
log 2 = δS = δA · f
(A) = 2 ¯
hf
(A) .
(12.31)
In this manner Bekenstein derived
S = f (A) =
log 2
2
A
¯
h
.
(12.32)
This is Bekenstein’s original claim. He mentioned in his paper [6] that “Nevertheless, it should be clear that if the coefficient is not exactly
log 2
2 , then it must be
very close to this, probably within a factor of two.” The precise formula obtained by
Hawking later, in the paper [7] about black hole evaporation with quantized fields,
is
S =
1
4
A
¯
h
.
(12.33)