280 Gravitational instability in Newtonian theory
These new coordinates can be used until the trajectories of the matter elements
start to cross one other. The velocity of a matter element with the given Lagrangian
coordinatesqis equal to
Vi≡dxi
dt=
∂xi(q,t)
∂t∣
∣∣
∣
q. (6.78)
The derivatives of the velocity field with respect to the Eulerian coordinates can
then be written as
∇jVi=∂qk
∂xj∂
∂qk(
∂xi(q,t)
∂t)
=
∂qk
∂xj∂Jki
∂t, (6.79)
where we have introduced the strain tensor
Jki(q,t)≡∂xi(q,t)
∂qk. (6.80)
Taking into account that
(
∂
∂t
∣
∣∣
∣
x+Vi∇i)
=
∂
∂t∣
∣∣
∣
x+
∂xi(q,t)
∂t∣
∣∣
∣
q∂
∂xi=
∂
∂t∣
∣∣
∣
q, (6.81)
and substituting (6.79) into (6.75) and (6.76), we obtain
∂ε
∂t
+ε∂qk
∂xi∂Jki
∂t= 0 , (6.82)
∂
∂t(
∂qk
∂xi∂Jki
∂t)
+
(
∂qk
∂xi∂Jkj
∂t)(
∂ql
∂xj∂Jli
∂t)
+ 4 πGε= 0 , (6.83)where the time derivatives are taken at constantq.The elements of the strain tensor
(6.80) form a 3×3 matrixJ≡
∥
∥Jki∥
∥, and since∂qk
∂xj·
∂xi
∂qk=
∂xi
∂xj=δij, (6.84)the derivatives∂qk/∂xiare the elements of the inverse matrixJ−^1 .Consequently,
we can rewrite (6.82) and (6.83) in matrix notation as
ε ̇+εtr(
J ̇·J−^1
)
= 0 , (6.85)
[
tr(
J ̇·J−^1
)]•
+tr[(
J ̇·J−^1
) 2 ]
+ 4 πGε= 0 , (6.86)where a dot denotes the partial time derivative.