18 An introduction to the physics of cosmology
achievedρc^2 + 3 p<0 would produceantigravity. Although such a possibility
may seem physically nonsensical, it is in fact one of the most important concepts
in contemporary cosmology. The origin of the idea goes back to the time
when Einstein was first thinking about the cosmological consequences of general
relativity. At that time, the universe was believed to be static—although this was
simply a prejudice, rather than being founded on any observational facts. The
problem of how a uniform distribution of matter could remain static was one
that had faced Newton, and Einstein gave a very simple Newtonian solution. He
reasoned that a static homogeneous universe required both the density,ρ,and
the gravitational potential,, to be constants. This does not solve Poisson’s
equation,∇^2 = 4 πGρ, so he suggested that the equation should be changed
to(∇^2 +λ)= 4 πGρ,whereλis a new constant of nature: thecosmological
constant. Almost as an afterthought, Einstein pointed out that this equation has
the natural relativistic generalization of
Gμν+gμν=−8 πG
c^4Tμν.What is the physical meaning of? In the current form, it represents the
curvature of empty space. The modern approach is to move theterm to the
right-hand side of the field equations. It now looks like the energy–momentum
tensor of the vacuum:
Tvacμν=c^4
8 πGgμν.How can a vacuum have a non-zero energy density and pressure? Surely these
are zero by definition in a vacuum? What we can be sure of is that the absence
of a preferred frame means thatTvacμνmust be the same for all observers in special
relativity. Now, apart from zero, there is only one isotropic tensor of rank 2:
ημν. Thus, in order forTvacμνto be unaltered by Lorentz transformations, the
only requirement we can have is that it must be proportional to the metric tensor.
Therefore, it is inevitable that the vacuum (at least in special relativity) will have
a negative-pressure equation of state:
pvac=−ρvacc^2.In this case,ρc^2 + 3 pis indeed negative: a positivewill act to cause a large-
scale repulsion. The vacuum energy density can thus play a crucial part in the
dynamics of the early universe.
It may seem odd to have an energy density that does not change as the
universe expands. What saves us is the peculiar equation of state of the vacuum:
the work done by the pressure is just sufficient to maintain the energy density
constant (see figure 2.1). In effect, the vacuum acts as a reservoir of unlimited
energy, which can supply as much as is required to inflate a given region to any
required size at constant energy density. This supply of energy is what is used
in ‘inflationary’ theories of cosmology to create the whole universe out of almost
nothing.