358 Cosmic microwave background anisotropies
It isinvariantunder general coordinate transformations. To prove this, let us go to
another coordinate system
η ̃=η ̃(
η,xi)
,x ̃i=x ̃i(
η,xj)
.
The phase volumed^3 xd ̃^3 ̃pcalculatedat the hypersurfaceη ̃=const in this new
coordinate system is related to (9.1) by
d^3 xd ̃^3 p ̃=Jd^3 xd^3 p, (9.2)where
J=∂(x ̃^1 ,x ̃^2 ,x ̃^3 ,p ̃ 1 , ̃p 2 ,p ̃ 3 )
∂(x^1 ,x^2 ,x^3 ,p 1 ,p 2 ,p 3 ), (9.3)
is the Jacobian of the transformation
xi→x ̃i=x ̃i(xj,η ̃), pi→ ̃pi =(
∂xα/∂x ̃i)
x ̃ pα. (9.4)Note that the new coordinatesx ̃ishould be considered here as functions of the old
coordinatesxjand the new time ̃η.Since(∂x ̃/∂p)= 0 ,we have
J=det(
∂x ̃i
∂xj∣
∣∣
∣
̃η=const)
det(
∂xj
∂x ̃k∣
∣∣
∣
̃η=const)
=det(
δkj)
= 1 , (9.5)
and therefore the phase volume is invariant.
Problem 9.1Verify thatdx^1 dx^2 dx^3 dp^1 dp^2 dp^3 is not invariant under coordinate
transformations.
Liouville’s theorem, which can easily be proved in flat spacetime, states that
the phase volume of the Hamiltonian system isinvariantunder canonical trans-
formations, or in other words, it isconservedalong the trajectory of the particle.
Considering an infinitesimal volume element, one can always go to a local iner-
tial coordinate system (the Einstein elevator) in the vicinity of any point along the
particle trajectory where the same theorem will obviously be valid. Yet, since the
phase volume (9.1) is independent of the particular coordinate system, it must also
be conserved as we move along a trajectory in curved spacetime. Hence, Liouville’s
theorem must continue to hold in General Relativity.
The Boltzmann equationLet us consider an ensemble ofnoninteractingidentical
particles. IfdNis the number of particles per volume elementd^3 xd^3 p, then the
distribution function f,characterizing the number density in one-particle phase
space, is defined by
dN=f(xi,pj,t)d^3 xd^3 p. (9.6)