- #1

addaF

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## Homework Statement

I'd like to calculate the form of Liouville operator in a Robertson Walker metric.

## Homework Equations

The general form is

$$ \mathbb{L} = \dfrac{\text{d} x^\mu}{\text{d} \lambda} \dfrac{\partial}{\partial x^\mu} - \Gamma^{\mu}_{\nu \rho} p^{\nu} p^{\rho} \dfrac{\partial}{\partial p^\mu} $$

where ##x^\mu## are the coordinates, ##\lambda## is an affine parameter and ##p^\mu = \frac{\text{d} x^\mu}{\text{d} \lambda}## is the particle four momentum, such as ##g_{\mu \nu} p^\mu p^\nu = m^2##.

Given the Robertson-Walker metric

$$

\text{ds}^2 = \text{d}t^2 - a^2(t) \left( \dfrac{\text{d}r^2}{1-kr^2} + r^2 \text{d} \theta^2 + r^2 \sin^2 \theta \text{d}\varphi^2 \right)

$$

I know that I should get the following expression, considering a mean distribution of the particles ##f(E,t)##

$$ \mathbb{L}[f] = E \dfrac{\partial f}{\partial t} - \dfrac{\dot{a}}{a} \left| \vec{p} \right|^2 \dfrac{\partial f}{\partial E} $$

where ##\frac{\dot{a}}{a} = H## is the Hubble parameter and ##\left| \vec{p} \right| ## is the modulus of the three momentum.

## The Attempt at a Solution

I'm actually struggling a bit on this problem, but I do not really know where I'm making the mistake. Since the metric is diagonal, and the distribution ## f ## do not depend on spatial components of ##x^\mu## and ##p^\mu##, the only needed Christoffel connection components should be:

$$

\Gamma^{0}_{\nu \rho} = \dfrac{1}{2} g^{0 \sigma} \left( \partial_{\nu} g_{\rho \sigma} +\partial_{\rho} g_{\sigma \nu} - \partial_{\sigma} g_{\nu \rho} \right)

$$

again because the metric is diagonal, ##\sigma = 0## and the first two terms in the brackets are vanishing. Then I calculate

$$

\Gamma^{0}_{11} = - \dfrac{1}{2} g^{0 0} \partial_{0} g_{11} = \dfrac{a \dot{a}}{1 - k r^2}

$$

$$

\Gamma^{0}_{22} = - \dfrac{1}{2} g^{0 0} \partial_{0} g_{22} = a \dot{a} r^2

$$

$$

\Gamma^{0}_{33} = - \dfrac{1}{2} g^{0 0} \partial_{0} g_{33} = a \dot{a} r^2 \sin^2 \theta

$$

Now there are the problems: if i suppose that ##p^\mu = \left(E,\vec{p} \right)##, the sum over ##\nu, \rho## gives

$$

\Gamma^{0}_{\nu \rho} p^\nu p^\rho= \dfrac{a \dot{a}}{1 - k r^2} p^1 p^1 + a \dot{a} r^2 p^2 p^2 + a \dot{a} r^2 \sin^2 \theta p^3 p^3 \neq H \left| \vec{p} \right|^2

$$

Since i cannot find the error, can anyone help me? Thanks in advance.