Physical Foundations of Cosmology

6.4 Beyond linear approximation 281

Problem 6.7Prove that

tr

(

J ̇·J−^1

)

=(lnJ)

• , (6.87)

whereJ(q,t)≡detJ.

After substitution of (6.87) into (6.85), the resulting equation can easily be
integrated to give

ε(q,t)=

! 0 (q)

J(q,t)

, (6.88)

where! 0 (q)is an arbitrary time-independent function of the Lagrangian coordi-
nates. With (6.87) and (6.88), (6.86) simplifies to

(lnJ)

••

+tr

[(

J ̇·J−^1

) 2 ]

+ 4 πG! 0 J−^1 = 0. (6.89)

This resulting equation forJcan be solved exactly for a few interesting cases.

6.4.1 Tolman solution

Let us consider a spherically symmetric inhomogeneity. In this case one can always
find a coordinate system where the strain tensor is proportional to the unit tensor:

Jki=a(R,t)δki, (6.90)

whereR≡|q|is the radial Lagrangian coordinate. Substituting

J=a^3 , tr

[(

J ̇·J−^1

) 2 ]

= 3

(

a ̇

a

) 2

(6.91)

into (6.89), we obtain

a ̈(R,t)=−

4 πG! 0 (R)

3 a^2 (R,t)

. (6.92)

Multiplying this equation bya ̇, we easily derive its first integral

a ̇^2 (R,t)−

8 πG! 0 (R)

3 a(R,t)

=F(R), (6.93)

whereF(R)is a constant of integration. Note that for a homogeneous matter dis-
tribution! 0 ,aandFdo not depend onRand (6.93) coincides with the Friedmann
equation for a matter-dominated universe.