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Stationary Einstein-Vlasov system

  1. Jan 24, 2005 #1
    The time-independent Einstein-Vlasov system with the ansatz that every static spherically symmetric solution must have the form

    [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?
     
  2. jcsd
  3. Jan 25, 2005 #2

    hellfire

    User Avatar
    Science Advisor

    Since your questions still remain unanswered, I will try to answer some, but you should know that my knowledge about this is very limited.

    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).
     
  4. Jan 25, 2005 #3

    pervect

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
  5. Jan 30, 2005 #4
    Thank you for your answers! :smile:

    the links were helpfull.
    I think i need to get some book which explains gen. relativity and this system in a simple way.
    If anybody know about such a book plz tell me.
     
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