Tolman-Oppenheimer-Volkoff equation

[K.J.Healey]

Lets see. The SC metric is correct, as it is the presumptive metric for a spherically symmetric body about a mass "M", PRIOR to solution. The actual solution for the metric would be to replace the exponents with (1-2M/r) and (1-2M/r)^-1 respectively. (but that's in the limit of a point M)

Continuing, as you can see from the line element in the Kerr metric, there exists an off-diagonal component that mixes time and Phi.

Don't mix up the Einstein tensor and the metric. I assume you just made a typo. But from the metric you can get to the Einstein tensor by using that code I showed you.

The .m file is just a package you run (basically just a function), then look at the demo. It defines a metric "metric" (aka "g") and from it and the normal coordinate vector calculates the Ricci tensor & scalar (thru christoffel) automatically and spits out the Einstein tensor ("G").As for whether or not the code can handle the Kerr metric, I don't see why not. But it may not be entirely useful yet. Once you have the "G" you set up all of your equations using the GR field equation, equating it to T. You should then have a set of equations that are solvable, with the connection you listed (P vs density).

Actually, now that I'm looking at your line element. Wouldn't it make sense that your matrix form be:
$$g_{\mu\nu} = \left(<br /> \begin{array}{llll}<br /> e^{\nu (r)} & 0 & 0 & \frac{r_s r \alpha sin^2 \theta}{\rho^2} \\<br /> 0 & -e^{\lambda (r)} & 0 & 0 \\<br /> 0 & 0 & - \rho^2 & 0 \\<br /> \frac{r_s r \alpha sin^2 \theta}{\rho^2} & 0 & 0 & - \left(r^2 + \alpha^2 + \frac{r_s r \alpha^2}{\rho^2} \sin ^2 \theta \right) <br /> \end{array}<br /> \right)$$

The of diagonals give that extra term. THIS ABOVE IS NOT CORRECT AT ALL. I am just showing you where to start, there needs to be off diagonals.

http://www.astro.ku.dk/~milvang/RelViz/000_node12.html

has a way. Take a look.

 

[K.J.Healey]

remember your line element must be made from

$$ds^2=g_{\mu\nu} dx^\mu dx^\nu$$

where $$dx^\mu$$ is like $$(dt,dr,r d\theta, r sin\theta d\phi)$$ I believe.

 

[Orion1]

Kerr metric...


The Kerr metric:
$$c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} <br /> - \frac{\rho^{2}}{\Lambda^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi$$
$$\alpha = \frac{J}{Mc} \; \; \; \; \; \; \rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta \; \; \; \; \; \; \Lambda^{2} = r^{2} - r_{s} r + \alpha^{2}$$


I reduced each matrix element into the most trigonometric form.

$$g_{00} = 1 - \frac{r_{s} r}{\rho^{2}} = 1 - \frac{r_{s} r}{r^{2} + \alpha^{2} \cos^{2} \theta}$$
$$\boxed{g_{00} = 1 - \frac{r_{s} r}{r^{2} + \alpha^{2} \cos^{2} \theta}}$$
$$g_{03} = g_{30} = \frac{r_s r \alpha sin^2 \theta}{\rho^2} = \frac{r_s r \alpha sin^2 \theta}{r^{2} + \alpha^{2} \cos^{2} \theta}$$
$$\boxed{g_{03} = g_{30} = \frac{r_s r \alpha sin^2 \theta}{r^{2} + \alpha^{2} \cos^{2} \theta}}$$
$$g_{11} = - \frac{\rho^{2}}{\Lambda^{2}} = - \frac{r^{2} + \alpha^{2} \cos^{2} \theta}{r^{2} - r_{s} r + \alpha^{2}}$$
$$\boxed{g_{11} = - \frac{r^{2} + \alpha^{2} \cos^{2} \theta}{r^{2} - r_{s} r + \alpha^{2}}}$$
$$g_{22} = - \rho^2 = - (r^{2} + \alpha^{2} \cos^{2} \theta)$$
$$\boxed{g_{22} = - (r^{2} + \alpha^{2} \cos^{2} \theta)}$$

$$g_{\mu \nu} = \left( \begin{array}{llll} 1-\frac{r r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} \\ 0 & - \frac{r^2+\alpha ^2 \cos ^2(\theta )}{r^2-r_s r+\alpha ^2} & 0 & 0 \\ 0 & 0 & -r^2-\alpha ^2 \cos ^2(\theta ) & 0 \\ \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \sin ^2(\theta ) \left(-r^2-\alpha ^2 \sin ^2(\theta ) r_s r-\alpha ^2\right) \end{array} \right)$$

Einsteintensor source code:
In[n]=

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)*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[]

The resulting evaluation function required a lot of memory:
MaxMemoryUsed[] = 1745544760 bytes

Unfortunately, my computer does have enough memory to complete all the transformations and the result:[/Color]
Simplify::time: Time spent on a transformation exceeded 300 seconds, and the transformation was aborted.
General::stop: Further output of Simplify::time will be suppressed during this calculation.
[/Color]
Reference:
http://en.wikipedia.org/wiki/Kerr_metric"
http://library.wolfram.com/infocenter/MathSource/162/"

 

[Orion1]

Correction, the matrix and source code from previous post should be:
$$\boxed{g_{33} = \sin ^2(\theta ) \left(-r^2-\frac{\alpha ^2 \sin ^2(\theta ) r_s r}{r^2+\alpha ^2 \cos ^2(\theta )}-\alpha ^2\right)}$$

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$$g_{\mu \nu} = \left( \begin{array}{llll} 1 - \frac{r r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} \\ 0 & -\frac{r^2+\alpha ^2 \cos ^2(\theta )}{r^2-r_s r+\alpha ^2} & 0 & 0 \\ 0 & 0 & -r^2-\alpha ^2 \cos ^2(\theta ) & 0 \\ \frac{r \alpha \sin ^2(\theta ) r_s}{r^2+\alpha ^2 \cos ^2(\theta )} & 0 & 0 & \sin ^2(\theta ) \left(-r^2-\frac{\alpha ^2 \sin ^2(\theta ) r_s r}{r^2+\alpha ^2 \cos ^2(\theta )}-\alpha ^2\right) \end{array} \right)$$

Einsteintensor source code:
In[n]=

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[]

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[Orion1]

neutron star mass-radius relation...


The neutron star mass-radius relation is dependent on a particular neutron star model, however the mass-radius relation for my model based upon the Proton charge radius and Tolman mass equation solution VII:

$$m_n = 1.6749272928 \cdot 10^{-27} \; \text{kg}$$ - Neutron mass
$$r_p = 0.8757 \cdot 10^{-15} \; \text{m}$$ - Proton charge radius

Proton charge radius neutron density:
$$\rho_n = \frac{3 m_n}{4 \pi r_p^3}$$

Neutron star core density equivalent to Proton charge radius neutron density:
$$\rho_c = \rho_n$$

Total Tolman mass equation solution VII:
$$M_0(R) = \frac{8 \pi \rho_c R^3}{15} = \frac{8 \pi R^3}{15} \left( \frac{3 m_n}{4 \pi r_p^3} \right) = \frac{2 m_n R^3}{5 r_p^3}$$

Total mass-radius relation equation for Tolman solution VII:
$$\boxed{M_0(R) = \frac{2 m_n R^3}{5 r_p^3}}$$

Mass of a 10 km radius Tolman VII neutron star:
$$\boxed{M_0(10 \; \text{km}) = 9.976 \cdot 10^{29} \; \text{kg}}$$

$$\boxed{M_0(10 \; \text{km}) = 0.501 \cdot M_{\odot}}$$

Note that the lower limit for total radius R, is equivalent to the Schwarzschild radius and the upper limit for total mass M(R), is equivalent to the Tolman-Oppenheimer-Volkov mass limit.
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Reference:
Neutron - Wikipedia
TOV #39 - Orion1
TOV #47 - Orion1
Schwarzschild radius - Wikipedia
Tolman-Oppenheimer-Volkoff mass limit - Wikipedia