Notes on Relativity and Cosmology - Physics Department, UCSB

10.2. DYNAMICS! (A.K.A. TIME EVOLUTION) 293
principle why the pressure P must be positive. Let’s think about what a negative
pressure would mean. A positive pressure is an effect that resists something
being squeezed. So, a negative pressure is an effect that resists something being
stretched. This is also known as a ‘tension.’ Imagine, for example, a rubber
band that has been stretched. We say that it is under tension, meaning that it
tries to pull itself back together. A sophisticated relativistic physicist calls such
an effect a ‘negative pressure.’
Let’s look closely at equation (10.4). We see that the universe can in fact ‘sit
still’ and remain static if ρ + 3P = 0. If ρ + 3P is negative, then gravity will
in fact be repulsive (as opposed to attractive) the various bits of matter will
accelerate apart. Now, because ρ is typically very large (since it is the density
of energy and E = mc 2 ) this requires a truly huge negative pressure. The kinds
of matter that we are most familiar with will never have such a large negative
pressure. However, physicists can imagine that their might possibly be such a
kind of matter.
The favorite idea along these lines is called “vacuum energy.” The idea is that
empty space itself might somehow have energy. At first, this is a rather shocking
notion. I mean, if it is empty, how can it have energy? But, some reflection will
tell us that this may simply be a matter of semantics: given the space that we
think is empty (because we have cleared it of everything that we know how to
remove), how empty is it really? In the end, like everything else in physics, this
question must be answered experimentally. We need to find a way to go out
and to measure the energy of empty space.
Now, what is clear is that the energy of empty space must be rather small.
Otherwise, it’s gravitational effects would screw up our predictions of, for example,
the orbits of the planets. However, there is an awful lot of ‘empty’ space
out there. So, taken together it might still have some nontrivial effect on the
universe as a whole.
OK, so why should vacuum energy (the energy density of empty space) have
negative pressure? Well, an important fact here is that energy density and
pressure are not completely independent. Pressure, after all is related to the
force required to change the size of a system: to smash it or to stretch it out.
On the other hand, force is related to energy: for example, we must add energy
to a rubber band in order fight the tension forces and stretch it out. The fact
that we must add energy to a spring in order to stretch it is what causes the
spring to want to contract; i.e., to have a negative pressure when stretched.
Now, if the vacuum itself has some energy density ρ and we stretch the space
(which is just what we will do when the universe expands) then the new (stretched)
space has more vacuum and therefore more energy. So, we again have to add
energy to stretch the space, so there is a negative pressure. It turns out that pressure
is (minus) the derivative of energy with respect to volume P = −dE/dV .
Here, E = ρV , so P = −ρ. Clearly then for pure vacuum energy we have
ρ + 3P < 0 and gravity is repulsive. On the other hand, combining this with
the appropriate amount of normal matter could make the two effects cancel out
and could result in a static universe.