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Static spherically symmetric perfect fluid solutions

  1. Dec 1, 2006 #1

    Chris Hillman

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    Way back in August 2004, blue_sky asked:

    In many papers and in some books on gtr, including the monograph by Stephani et al., Exact Solutions of Einstein's Field Equations, 2nd ed., Cambridge University Press, 2001.

    This is a huge and fascinating topic which goes right back to the beginnings of gtr, since Schwarzschild's "incompressible" (constant density) static spherically symmetric perfect fluid solution, or Schwarzschild fluid for short (this solution is also sometimes called the "Schwarschild interior solution", meaning the interior of a stellar model), was the second exact solution to the EFE ever discovered (early in 1916). It is just one example of a large, well understood, and important class of exact solutions, the static spherically symmetric perfect fluid solutions. It is pleasant to report that in the past five years, this classic topic has been rejuventated by important new discoveries which are much more elementary than most things involving gtr.

    All of these solutions can be written in various coordinate charts. The most popular are Schwarzschild charts, which have the form
    [itex]ds^2 = -A(r)^2 \, dt^2 + B(r)^2 \, dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right) [/itex]
    and (spatially) isotropic charts, which have the form
    [tex]ds^2 = -\alpha(\rho)^2 dt^2 + \beta(\rho)^2 \; \left( d\rho^2 + \rho^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right) \right) [/itex]
    The difference is this: the Schwarzschild radial coordinate has the property that the surface [tex]t=t_0, r=r_0[/itex] is a round sphere with surface area [tex]A = 4 \pi \, r_0^2[/tex], but the coordinate difference [tex]r_2 -r_1, \, r_2 > r_1 > 0[/tex] does not in general give the length of a radially oriented line segment, and angles in the spatial slices [tex]t=t_0[/tex] are not correctly represented. On the other hand, angles are correctly represented in the spatially isotropic chart (hence the name), but while the surfaces [tex]t=t_0, \, \rho=\rho_0[/tex] still define round spheres, the interpretation of the radial coordinate in terms of the surface area of these spheres breaks down.

    In the Schwarschild chart, the Schwarzschild fluid can be written
    [tex]ds^2 = -1/4 \, \left( 2 B^2 - \sqrt{1-r^2/A^2} \right)^2 \, dt^2 + \frac{dr^2}{1-r^2/A^2} + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), [/itex]
    [itex] -\infty < t < \infty, \; \; 0 < r < B \, \frac{\sqrt{B^2-2A^2}}{\sqrt{A^2-2B^2}},
    \; \; 0 < \theta < \pi, \; \; -\pi < \phi < \pi [/itex]
    In the isotropic chart, it can be written
    [itex] ds^2 = -\left( \frac{1 + r^2/A^2}{1+r^2/B^2} \right)^2 \, dt^2 + \frac{ dr^2 + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)}{\left( 1 + r^2/B^2 \right)^2}, [/itex]
    [itex] -\infty < t < \infty, \; \; 0 < r < A \, \sqrt{1-4 B^4/9},
    \; \; 0 < \theta < \pi, \; \; -\pi < \phi < \pi
    [/itex]
    (In another popular representation, we could use a trigonometric chart in order to take advantage of the fact that geometrically, the constant "time" hyperslices in this model turn out to be three-spherical caps.)

    Depending upon which chart we employ, we will of course obtain different expressions describing how pressure varies with "radius". These will of course agree qualtiatively (the pressure is maximal at the center and decreases to zero at the surface of our fluid ball). The expressions we obtain a somewhat simpler for the isotropic chart!

    Other well known solutions in this class include the Heintzmann fluid (1969), the Martin III fluid (2003), the Buchdahl fluid (1958), the Tolman IV fluid (1939)--- which is still one of the most useful--- the Wyman II fluid (1949), the Kuchowicz fluid (1967), and the Goldman fluid (1978).

    The new ideas I mentioned involve new "solution generating techniques" which are suprisingly easy to use to find new explicit static spherically symmetric perfect fluid solutions, and which are known to generate, in principle, ALL solutions in this class. The most interesting ideas (IMO) appear in a series of papers coauthored by Matt Visser; see http://arxiv.org/find/gr-qc/1/AND+au:+Visser_Matt+ti:+EXACT+perfect_fluid/0/1/0/all/0/1

    I recently mentioned Lie's theory of the symmetry of systems of PDEs; it is interesting that the work of Visser et al. fits into this paradigm, but the most interesting Baecklund automorphism Visser and Martin have concocted does not yet appear to arise from the standard theory in any straightforward way. This is the "pressure change transformation", which maps a given static spherically symmetric perfect fluid solution to another with same density profile but a different central pressure. This is interesting both because it has an immediate physical interpretation and because many other methods often yield solutions in which the central pressure and density agree, which we can then modify to make more realistic using the pressure change transformation.

    By the way, I suspect that blue_sky meant "polytrope", the most important special case of a perfect fluid. Not every perfect fluid admits any equation of state [tex] p = f(\rho)[/tex] at all; only some of the perfect fluid solutions I mentioned above have this property (the Schwarzschild fluid being one of them, rather trivially!).

    Polytropes are required to admit an EOS having a specific form. It is well known to students of astrophysics that even Newtonian polytropes are hard to construct analytically, and things don't get easier in gtr. Nonetheless, we can obtain some interesting expressions relating say the temperature to the three dimensional Riemann tensor of the spatial hyperslices.

    Chris Hillman
     
    Last edited: Dec 1, 2006
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