Kerr metric hydrostatic equilibrium

[Orion1]

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:
$$ds^2 = e^{\nu (r)} c^2 dt^2 - e^{\lambda (r)} dr^2 - r^2 ( \sin^2 \theta d \phi^2 + d \theta^2)$$

General Relativity:
$$G_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$$

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

$e^{-\lambda} = r^{-1}(r - r_s)$

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

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

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

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

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

Mathematica source code used by application in reference 1:

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:
http://library.wolfram.com/infocenter/MathSource/162/"
http://en.wikipedia.org/wiki/Kerr_metric"
http://en.wikipedia.org/wiki/Schwarzschild%20metric"
http://en.wikipedia.org/wiki/Tolman-Oppenheimer-Volkoff_equation"
https://www.physicsforums.com/showpost.php?p=1801307&postcount=59"
https://www.physicsforums.com/attachment.php?attachmentid=14705&d=1216037026"

 

[atyy]

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.

 

[Entropia]

Thank you for your question. The derivation of the Tolman-Oppenheimer-Volkoff equation from the Schwarzschild metric is a well-established result in General Relativity. However, the derivation for the Kerr metric is not as straightforward and has been a subject of ongoing research and debate.

Currently, there is no universally agreed upon equivalent to the Tolman-Oppenheimer-Volkoff equation for the Kerr metric. Some researchers have attempted to derive an analogous equation using different approaches, but there is no consensus on the validity of these equations.

One possible reason for the difficulty in deriving an equivalent equation for the Kerr metric is that the Kerr metric is more complex and has additional parameters (such as the spin parameter) compared to the Schwarzschild metric. This makes the derivation more challenging and prone to errors.

In addition, the Kerr metric also has a more complicated singularity structure, which further complicates the derivation of an equivalent equation for hydrostatic equilibrium. Therefore, it is not surprising that when you attempt to derive the equation derivatives for the Kerr metric, you obtain large and complicated solutions.

In conclusion, while there have been attempts to derive an equivalent to the Tolman-Oppenheimer-Volkoff equation for the Kerr metric, there is currently no universally accepted result. Further research is still needed in this area to fully understand the hydrostatic equilibrium in the presence of the Kerr metric.