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Kerr metric hydrostatic equilibrium

  1. Nov 13, 2008 #1

    I basically understand how the Tolman-Oppenheimer-Volkoff equation for hydrostatic equilibrium was derived from the Schwarzschild metric in General Relativity and from the equation derivatives listed. However, when I attempt to derive the equation derivatives for the Kerr metric, I obtain these extremely large solutions which do not contain any derivatives.

    Has the relative analogous equivalent to the Tolman-Oppenheimer-Volkoff equation for hydrostatic equilibrium for the Kerr metric ever been derived?

    I am not specifically interested in the proof, only the solution. Although it would be very interesting to examine the proof.

    Schwarzschild metric line element:
    [tex]ds^2 = e^{\nu (r)} c^2 dt^2 - e^{\lambda (r)} dr^2 - r^2 ( \sin^2 \theta d \phi^2 + d \theta^2)[/tex]

    General Relativity:
    [tex]G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}[/tex]

    Equation derived from Einstein tensor matrix element [tex]T_{11}[/tex]:
    [tex]\frac{8 \pi G P(r)}{c^4} = e^{- \lambda} \left( \frac{1}{r} \frac{d \nu}{dr} + \frac{1}{r^2} \right) - \frac{1}{r^2}[/tex]

    [itex]e^{-\lambda} = r^{-1}(r - r_s)[/itex]

    Hydrostatic equilibrium equation derived from Einstein tensor matrix elements [tex]T_{11}[/tex] and [tex]T_{22}[/tex]:
    [tex]\frac{d \nu}{dr} = \frac{2}{P(r) + \rho(r) c^2} \left( \frac{dP(r)}{dr} \right)[/tex]

    Hydrostatic equilibrium:
    [tex]\frac{dP(r)}{dr} = - \left( P(r) + \rho(r)c^2 \right) \frac{d \phi}{dr}[/tex]

    Tolman-Oppenheimer-Volkoff equation:
    [itex]\boxed{\frac{dP(r)}{dr} = -(P(r) + \rho(r) c^2) \left( \frac{4 \pi G r^3 P(r)}{c^4} + \frac{r_s}{2} \right) \left[ r \left( r - r_s \right) \right]^{-1}} \; \; \; (r > r_p) \; \; \; r \neq r_p[/itex]

    The Kerr metric line element:
    [tex]c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Lambda^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi[/tex]

    General Relativity Kerr metric tensor matrix elements:
    [tex]g_{\mu \nu} = \left( \begin{array}{llll} 1 - \frac{r r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} \\ 0 & -\frac{r^2+\alpha ^2 \cos ^2(\theta )}{r^2-r_s r+\alpha ^2} & 0 & 0 \\ 0 & 0 & -r^2-\alpha ^2 \cos ^2(\theta ) & 0 \\ \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \sin ^2(\theta ) \left(-r^2-\frac{\alpha ^2 \sin ^2(\theta ) r_s r}{r^2+\alpha ^2 \cos ^2(\theta )}-\alpha ^2\right) \end{array} \right)[/tex]

    Mathematica source code used by application in reference 1:
    Code (Text):

    ToFileName[{$TopDirectory, "AddOns", "Applications"}]
    << einsteintensor.m
    x = {t, r, \[Theta], \[Phi]}
    (metric = {{1 - (Subscript[r, s]*r)/(r^2 + \[Alpha]^2*Cos[\[Theta]]^2), 0, 0, (Subscript[r, s]*r*\[Alpha]*Sin[\[Theta]]^2)/(r^2 + \[Alpha]^2*Cos[\[Theta]]^2)}, {0, -((r^2 + \[Alpha]^2*Cos[\[Theta]]^2)/(r^2 - Subscript[r, s]*r + \[Alpha]^2)), 0, 0}, {0, 0, -(r^2 + \[Alpha]^2*Cos[\[Theta]]^2), 0}, {(Subscript[r, s]*r*\[Alpha]*Sin[\[Theta]]^2)/(r^2 + \[Alpha]^2*Cos[\[Theta]]^2), 0, 0, -(r^2 + \[Alpha]^2 + (Subscript[r, s]*r*\[Alpha]^2*Sin[\[Theta]]^2)/(r^2 + \[Alpha]^2*Cos[\[Theta]]^2))*Sin[\[Theta]]^2}}) // MatrixForm
    (tensor = {{\[Rho][r]*c^2, 0, 0, Subscript[\[CapitalPhi], \[Epsilon]][r]}, {0, -P[r], 0, 0},{0, 0, -P[r], 0}, {Subscript[\[Rho], p][r], 0, 0, -P[r]}}) // MatrixForm
    (Einstein = Inverse[metric].Simplify[EinsteinTensor[metric, x], TimeConstraint -> 3600]) // MatrixForm
    MaxMemoryUsed[]
     

    Reference:
    Calculating the Einstein Tensor - Wolfram
    Kerr metric - Wikipedia
    Schwarzschild_metric - Wikipedia
    Tolman-Oppenheimer-Volkoff equation - Wikipedia
    Tolman-Oppenheimer-Volkoff equation #59 - K.J. Healey
    General Relativity Einstein tensor chart
     
    Last edited: Nov 14, 2008
  2. jcsd
  3. Nov 14, 2008 #2

    atyy

    User Avatar
    Science Advisor

    I thought the TOV equation was derived by considering how matter in a spherical star might behave, then joining it to the Schwarzschild vacuum solution outside because of Birkhoff's theorem? Or are you referring to the Schwarzschild constant density solution? I think the equation you listed for the Schwarzschild line element is for a rotationally symmetric spacetime, which would be more general than either the Schwarzschild vacuum or constant density solutions.
     
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