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## Main Question or Discussion Point

Hello, I've enjoyed reading these forums for a while now as they have lots of great insight! Today I decided to register so I can ask a question that's been bothering me for a few days now. (By the way, I'm moving onto my senior year as a physics major)

So yea, to the question. I have been trying to derive the Schwarzschild metric (a static, spherically symmetric, vacuum solution) in a more rigorous fashion than I have seen documented in a few books and I'm having a little trouble. I have been able to make all [itex]g_{\mu\nu} = 0[/itex] for all [itex]\mu \neq \nu[/itex] to give me a metric of the form [tex]g_{\mu \nu} = A(r) dr^2 + C(r, \theta, \phi) d\theta^2 + D(r,\theta,\phi) d\phi^2 + B(r) dt^2[/tex].

I then defined my r coordinate such that when keeping [itex]\theta[/itex] and [itex]\phi[/itex] constant, the metric is just [tex]g_{\mu \nu} = r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2[/tex] (i.e. Setting r such that it is not physical radius but so that the area of the 2-sphere at r is [itex]4 /pi r^2[/itex])

Then I have:

[tex]g_{\mu \nu} = A(r) dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 + B(r) dt^2[/tex]

I know that I must next use the vacuum field constraint so that the Ricci tensor, [itex]R_{\mu \nu} = 0[/itex], so I computed all of the Christoffel symbols and have them to my disposal (if you wish to see them just let me know and I'll post them up... luckily there are only 13 nonzero ones). Now here is the problem: I'm not very comfortable yet with tensor algebra, so I don't know exactly how to use my Christoffel symbols to put them into my vacuum field constraint. I know the equation of the Riemann curvature tensor (posted at the end). I tried taking the Riemann curvature tensor [itex]R^\omega{}_{\beta \gamma \lambda}[/itex] such that [itex]\lambda = \gamma[/itex] and we have [itex]R^\gamma{}_{\beta \gamma \lambda}=R_{\beta \lambda}[/itex]. So I know I only need two equations to solve for A(r) and B(r) so I tried solving for [itex]R_{1 1}[/itex] and [itex]R_{4 4}[/itex] (where the [itex]r, \theta, \phi, t \rightarrow 1, 2, 3, 4[/itex]) the long way and actually plugging in all the Christoffel symbols for all [itex]R^\gamma{}_{1 \gamma 1}[/itex](

[tex]\frac{6}{r^2} + \frac{7}{4} \frac{(\frac{dB(r)}{dr})^2}{B(r)} - 2\frac{\frac{d^2B(r)}{dr^2}}{B(r)} + \frac{1}{r} \frac{\frac{dA(r)}{dr}}{A(r)} + \frac{1}{4} \frac{\frac{dA(r)}{dr}}{A(r)} \frac{\frac{dB(r)}{dr}}{B(r)} = 0[/tex]

and

[tex]2\frac{\frac{dA(r)}{dr} \frac{dB(r)}{dr}}{A^2(r)} - 2\frac{\frac{d^2A(r)}{dr^2}}{A(r)} - \frac{\frac{dA(r)}{dr} \frac{dB(r)}{dr}}{4 A^2(r)} - \frac{\frac{dB(r)}{dr}}{r A(r)} - \frac{(\frac{dB(r)}{dr})^2}{4 A(r) B(r)} = 0[/tex]

Apart from taking out a factor of [itex]\frac{1}{A(r)}[/itex] from the second equation, I really don't know how to solve for A(r) and B(r). Am I doing something wrong? I'm concerned that my lack of familiarity with tensors is leading me to incorrect equations. So basically, am I using the [itex]R_{\mu \nu} = 0[/itex] requirement correctly?

Sorry for the long post, but any help would be greatly appreciated. Thanks!

Riemann curvature tensor:

[tex]R^\omega{}_{\beta \gamma \lambda} = \Gamma^\omega{}_{\beta \lambda, \gamma} - \Gamma^\omega{}_{\beta \gamma, \lambda} + \Gamma^\omega{}_{\gamma \sigma} \Gamma^\sigma{}_{\beta \lambda} - \Gamma^\omega{}_{\lambda \sigma} \Gamma^\sigma{}_{\beta \gamma}[/tex]

(where the ',' in the subscript denotes the usual notation for derivative)

So yea, to the question. I have been trying to derive the Schwarzschild metric (a static, spherically symmetric, vacuum solution) in a more rigorous fashion than I have seen documented in a few books and I'm having a little trouble. I have been able to make all [itex]g_{\mu\nu} = 0[/itex] for all [itex]\mu \neq \nu[/itex] to give me a metric of the form [tex]g_{\mu \nu} = A(r) dr^2 + C(r, \theta, \phi) d\theta^2 + D(r,\theta,\phi) d\phi^2 + B(r) dt^2[/tex].

I then defined my r coordinate such that when keeping [itex]\theta[/itex] and [itex]\phi[/itex] constant, the metric is just [tex]g_{\mu \nu} = r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2[/tex] (i.e. Setting r such that it is not physical radius but so that the area of the 2-sphere at r is [itex]4 /pi r^2[/itex])

Then I have:

[tex]g_{\mu \nu} = A(r) dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2 + B(r) dt^2[/tex]

I know that I must next use the vacuum field constraint so that the Ricci tensor, [itex]R_{\mu \nu} = 0[/itex], so I computed all of the Christoffel symbols and have them to my disposal (if you wish to see them just let me know and I'll post them up... luckily there are only 13 nonzero ones). Now here is the problem: I'm not very comfortable yet with tensor algebra, so I don't know exactly how to use my Christoffel symbols to put them into my vacuum field constraint. I know the equation of the Riemann curvature tensor (posted at the end). I tried taking the Riemann curvature tensor [itex]R^\omega{}_{\beta \gamma \lambda}[/itex] such that [itex]\lambda = \gamma[/itex] and we have [itex]R^\gamma{}_{\beta \gamma \lambda}=R_{\beta \lambda}[/itex]. So I know I only need two equations to solve for A(r) and B(r) so I tried solving for [itex]R_{1 1}[/itex] and [itex]R_{4 4}[/itex] (where the [itex]r, \theta, \phi, t \rightarrow 1, 2, 3, 4[/itex]) the long way and actually plugging in all the Christoffel symbols for all [itex]R^\gamma{}_{1 \gamma 1}[/itex](

*quite*long... 2 free indeces to sum over which gives 16 sets of sums with 4 terms each). I did the same for [itex]g_{4 4}[/itex], and got the following out of the two:[tex]\frac{6}{r^2} + \frac{7}{4} \frac{(\frac{dB(r)}{dr})^2}{B(r)} - 2\frac{\frac{d^2B(r)}{dr^2}}{B(r)} + \frac{1}{r} \frac{\frac{dA(r)}{dr}}{A(r)} + \frac{1}{4} \frac{\frac{dA(r)}{dr}}{A(r)} \frac{\frac{dB(r)}{dr}}{B(r)} = 0[/tex]

and

[tex]2\frac{\frac{dA(r)}{dr} \frac{dB(r)}{dr}}{A^2(r)} - 2\frac{\frac{d^2A(r)}{dr^2}}{A(r)} - \frac{\frac{dA(r)}{dr} \frac{dB(r)}{dr}}{4 A^2(r)} - \frac{\frac{dB(r)}{dr}}{r A(r)} - \frac{(\frac{dB(r)}{dr})^2}{4 A(r) B(r)} = 0[/tex]

Apart from taking out a factor of [itex]\frac{1}{A(r)}[/itex] from the second equation, I really don't know how to solve for A(r) and B(r). Am I doing something wrong? I'm concerned that my lack of familiarity with tensors is leading me to incorrect equations. So basically, am I using the [itex]R_{\mu \nu} = 0[/itex] requirement correctly?

Sorry for the long post, but any help would be greatly appreciated. Thanks!

Riemann curvature tensor:

[tex]R^\omega{}_{\beta \gamma \lambda} = \Gamma^\omega{}_{\beta \lambda, \gamma} - \Gamma^\omega{}_{\beta \gamma, \lambda} + \Gamma^\omega{}_{\gamma \sigma} \Gamma^\sigma{}_{\beta \lambda} - \Gamma^\omega{}_{\lambda \sigma} \Gamma^\sigma{}_{\beta \gamma}[/tex]

(where the ',' in the subscript denotes the usual notation for derivative)