Three Roads To Quantum Gravity

AREA AND INFORMATION
103
Both loop quantum gravity and string theory assert that
there is an atomic structure to space. In the next two chapters
we shall see that loop quantum gravity in fact gives a rather
detailed picture of that atomic structure. The picture of the
atomic structure one gets from string theory is presently
incomplete but, as we shall see in Chapter 11, it is still
impossible in string theory to avoid the conclusion that there
must be an atomic structure to space and time. In Chapter 13
we shall discover that both pictures of the atomic structure of
space can be used to explain the entropy and temperature of
black holes.
But even without these detailed pictures there is a very
general argument, based simply on what we have learned in
the last few chapters, that leads to the conclusion that space
must have an atomic structure. This argument rests on the
simple fact that horizons have entropy. In previous chapters
we have seen that this is common to both the horizons of
black holes and to the horizon experienced by an accelerated
observer. In each case there is a hidden region in which
information can be trapped, outside the reach of external
observers. Since entropy is a measure of missing information,
it is reasonable that in these cases there is an entropy
associated with the horizon, which is the boundary of the
hidden region. But what was most remarkable is that the
amount of missing information measured by the entropy had
a very simple form. It was simply equal to one-quarter of the
area of the horizon, in Planck units.
The fact that the amount of missing information depends on
the area of the boundary of the trapped region is a very
important clue. It becomes even more signi®cant if we put
this dependence together with the fact that spacetime can be
understood to be structured by processes which transmit
information from the past to the future, as we saw in Chapter
4. If a surface can be seen as a kind of channel through which
information ¯ows from one region of space to another, then
the area of the surface is a measure of its capacity to transmit
information. This is very suggestive.
It is also strange that the amount of trapped information is
proportional to the area of the boundary. It would seem more