String Theory Demystified

256 String Theory Demystifi ed
A Statement of the Holographic Principle
The holographic principle was fi rst proposed by Gerard t’Hooft in 1993 and has
been worked on extensively by Leonard Susskind. It can be asserted using two
postulates:
• The total information content in a volume of space is equivalent to a theory
that lives only on the surface area that encloses the region.
• The boundary of a region of space-time contains at most a single degree of
freedom per Planck area.
The holographic principle really applies to gravity and we have already seen it in
action when talking about black holes. Information content, which is another way
of saying entropy, is about counting the number of states in a system and so is
proportional to area. We have already seen that in the case of a black hole that
entropy is proportional to the area of the event horizon:
S
A
=
4G
where G is Newton’s gravitational constant. The area A is measured in Planck
units.
This is a surprising result because we would intuitively expect that the number of
states is proportional to the volume of the enclosed region. Following Susskind, we
illustrate that this is in fact the case when gravity is not involved. Imagine that a
volume V contains a set of spins on a lattice. We take the lattice spacing to be a, and
imagine that the lattice fi lls the entire volume. Then the total number of spins
contained in V is
# spins = V
a 3
The total number of states the system can have is
N
V a
= 2 3
/
Using thermodynamics, we arrive at a relationship between the number of states
and entropy S:
N ∝ exp
S