MODERN COSMOLOGY

The energy–momentum tensor 15

The idea here is that the changes experienced by an observer moving with the
fluid are inevitably a mixture of temporal and spatial changes. This two-part
derivative arises automatically in the relativistic formulation through the 4-vector
dot productUμ∂μ, which arises from the 4-divergence of an energy–momentum
tensor containing a term∝UμUν.
The equations that result from unpackingTμν,ν =0inthiswayhavea
familiar physical interpretation. Theμ= 1 , 2 ,3 components ofT,νμν=0give
the relativistic generalization ofEuler’s equationfor momentum conservation in
fluid mechanics (not to be confused with Euler’s equation in variational calculus):

d

dt

v=−

1

γ^2 (ρ+p/c^2 )

(∇p+ ̇pv/c^2 ),

and theμ=0 component gives a generalization of conservation of energy:

d

dt

[γ^2 (ρ+p/c^2 )]= ̇p/c^2 −γ^2 (ρ+p/c^2 )∇·v,

where p ̇ ≡ ∂p/∂t. The meaning of this equation may be made clearer by
introducing one further conservation law: particle number. This is governed by a
4-current having zero 4-divergence:

d

dxμ

Jμ= 0 , Jμ≡nUμ=γn(c,v).

If we now introduce therelativistic enthalpyw = ρ+p/c^2 , then energy
conservation becomes
d
dt

(γw

n

)

=

p ̇

γnc^2

.

Thus,insteadyflow,γ×(enthalpy per particle)is constant.
A very useful general procedure can be illustrated bylinearizingthe fluid
equations. Consider a small perturbation about each quantity (ρ→ρ+δρetc)
and subtract the unperturbed equations to yield equations for the perturbations
valid to first order. This means that any higher-order term such asδv·∇δρis set
equal to zero. If we take the initial state to have constant density and pressure and
zero velocity, then the resulting equations are simple:

∂

∂t

δv=−

1

ρ+p/c^2

∇δp

∂

∂t

δρ=−(ρ+p/c^2 )∇·δv.

Now eliminate the perturbed velocity (via the divergence of the first of these
equations minus the time derivative of the second) to yield the wave equation:

∇^2 δρ−

(

∂ρ

∂p

)

∂^2 δρ

∂t^2

= 0.