# Stationary Einstein-Vlasov system

1. Jan 24, 2005

### JohanL

The time-independent Einstein-Vlasov system with the ansatz that every static spherically symmetric solution must have the form

$$f = \Phi(E,L)$$

is as follows

$$e^{\mu - \lambda} \frac{v}{\sqrt{1 + \abs{v}^2}}\cdot {\partial_xf}-{\sqrt{1 + \abs{v}^2}}e^{\mu - \lambda}\mu_r\frac{x}{r}\cdot {\partial_rf}=0$$

$$e^{-2 \lambda}(2r \lambda_r -1) + 1 = 8 \pi r^2G_\Phi(r,\mu)$$
$$e^{-2 \lambda}(2r \mu_r +1) - 1 = 8 \pi r^2H_\Phi(r,\mu)$$

where

$$G_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL d\epsilon$$
$$H_\Phi(r,\mu) = \frac{2\pi}{r^2}\int_{1}^{\infty}\int_{0}^{r^2(\epsilon^2-1)} \Phi(e^{\mu(r)\epsilon,L}) \frac{\epsilon}{\sqrt{\epsilon^2-1-L/r^2}}dL d\epsilon$$

1. f is a distrubtion function and describes the distribution of the particles(galaxies or clusters of galaxies), right?

2. What is

$$\mu, \lambda$$
and

$$\epsilon?$$

Can you put any restrictions on these variables?

2. Jan 25, 2005

### hellfire

The non-relativistic equation (Vlasov equation) is used to model a collissionless gas without interactions between the particles (where the phase-space density is conserved). For example: dark matter before recombination, or, on a different scale, stars and galaxies in the current universe, etc. E. Bertschinger gives a nice explanation of this in chapters 3.2 and 3.3 of Cosmological Dynamics. I assume that the relativistic Einstein-Vlasov equation can be used for the same purpose, but, honestly, I have never seen this before.

These parameters seam to be the ones which are used to define a spherical symmetric metric, when written with exponentials as shown e.g. here (but this is only a guess).

3. Jan 25, 2005

### pervect

Staff Emeritus
Google has a number of hits on the Eintstein-Vlassov system, which I'd never head of before either, so I'm fairly sure that it is the relativistic description of a colissionless gas, and that f is indeed a distribution function.

I would also guess that mu and lambda are components of the metric.

I also wanted to give anyone who might know more than I do a chance to answer first.

4. Jan 30, 2005