Page 55 - A Brief History of Time - Stephen Hawking

A Brief History of Time - Stephen Hawking... Chapter 7
hole according to the two definitions would be the same, and hence so would their areas, provided the black hole had
settled down to a state in which it was not changing with time.

The nondecreasing behavior of a black hole’s area was very reminiscent of the behavior of a physical quantity called
entropy, which measures the degree of disorder of a system. It is a matter of common experience that disorder will
tend to increase if things are left to themselves. (One has only to stop making repairs around the house to see that!)
One can create order out of disorder (for example, one can paint the house), but that requires expenditure of effort or
energy and so decreases the amount of ordered energy available.

A precise statement of this idea is known as the second law of thermodynamics. It states that the entropy of an isolated
system always increases, and that when two systems are joined together, the entropy of the combined system is
greater than the sum of the entropies of the individual systems. For example, consider a system of gas molecules in a
box. The molecules can be thought of as little billiard balls continually colliding with each other and bouncing off the
walls of the box. The higher the temperature of the gas, the faster the molecules move, and so the more frequently and
harder they collide with the walls of the box and the greater the outward pressure they exert on the walls. Suppose that
initially the molecules are all confined to the left-hand side of the box by a partition. If the partition is then removed, the
molecules will tend to spread out and occupy both halves of the box. At some later time they could, by chance, all be in
the right half or back in the left half, but it is overwhelmingly more probable that there will be roughly equal numbers in
the two halves. Such a state is less ordered, or more disordered, than the original state in which all the molecules were
in one half. One therefore says that the entropy of the gas has gone up. Similarly, suppose one starts with two boxes,
one containing oxygen molecules and the other containing nitrogen molecules. If one joins the boxes together and
removes the intervening wall, the oxygen and the nitrogen molecules will start to mix. At a later time the most probable
state would be a fairly uniform mixture of oxygen and nitrogen molecules throughout the two boxes. This state would
be less ordered, and hence have more entropy, than the initial state of two separate boxes.

The second law of thermodynamics has a rather different status than that of other laws of science, such as Newton's
law of gravity, for example, because it does not hold always, just in the vast majority of cases. The probability of all the
gas molecules in our first box

found in one half of the box at a later time is many millions of millions to one, but it can happen. However, if one has a
black hole around there seems to be a rather easier way of violating the second law: just throw some matter with a lot
of entropy such as a box of gas, down the black hole. The total entropy of matter outside the black hole would go
down. One could, of course, still say that the total entropy, including the entropy inside the black hole, has not gone
down - but since there is no way to look inside the black hole, we cannot see how much entropy the matter inside it
has. It would be nice, then, if there was some feature of the black hole by which observers outside the black hole could
tell its entropy, and which would increase whenever matter carrying entropy fell into the black hole. Following the
discovery, described above, that the area of the event horizon increased whenever matter fell into a black hole, a
research student at Princeton named Jacob Bekenstein suggested that the area of the event horizon was a measure of
the entropy of the black hole. As matter carrying entropy fell into a black hole, the area of its event horizon would go
up, so that the sum of the entropy of matter outside black holes and the area of the horizons would never go down.

This suggestion seemed to prevent the second law of thermodynamics from being violated in most situations.
However, there was one fatal flaw. If a black hole has entropy, then it ought to also have a temperature. But a body
with a particular temperature must emit radiation at a certain rate. It is a matter of common experience that if one heats
up a poker in a fire it glows red hot and emits radiation, but bodies at lower temperatures emit radiation too; one just
does not normally notice it because the amount is fairly small. This radiation is required in order to prevent violation of
the second law. So black holes ought to emit radiation. But by their very definition, black holes are objects that are not
supposed to emit anything. It therefore seemed that the area of the event horizon of a black hole could not be regarded
as its entropy. In 1972 I wrote a paper with Brandon Carter and an American colleague, Jim Bardeen, in which we
pointed out that although there were many similarities between entropy and the area of the event horizon, there was
this apparently fatal difficulty. I must admit that in writing this paper I was motivated partly by irritation with Bekenstein,
who, I felt, had misused my discovery of the increase of the area of the event horizon. However, it turned out in the end
that he was basically correct, though in a manner he had certainly not expected.

In September 1973, while I was visiting Moscow, I discussed black holes with two leading Soviet experts, Yakov
Zeldovich and Alexander Starobinsky. They convinced me that, according to the quantum mechanical uncertainty
principle, rotating black holes should create and emit particles. I believed their arguments on physical grounds, but I did
not like the mathematical way in which they calculated the emission. I therefore set about devising a better
mathematical treatment, which I described at an informal seminar in Oxford at the end of November 1973. At that time I
had not done the calculations to find out how much would actually be emitted. I was expecting to discover just the
radiation that Zeldovich and Starobinsky had predicted from rotating black holes. However, when I did the calculation, I

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