The time-independent Einstein-Vlasov system with the ansatz that every static spherically symmetric solution must have the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f = \Phi(E,L)

[/tex]

is as follows

[tex]

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

[/tex]

[tex]

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

[/tex]

[tex]

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

[/tex]

where

[tex]

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

[/tex]

[tex]

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

[/tex]

I have some very simple questions about this system. I have no background in general relativity.

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

2. What is

[tex]

\mu, \lambda

[/tex]

and

[tex]

\epsilon?

[/tex]

Can you put any restrictions on these variables?

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# Stationary Einstein-Vlasov system

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