Relativity
Trace Metric Solution
Solutions with trace metrics where \(g_{\mu\nu} \ne 0\) if and only if \(\mu = \nu\) have fewer non-zero Christoffel symbols. This solution only applies to metrics which only have dependent variables \(r\) and \(\theta\).
Define the metric.
\[g_{tt} = T(r,\theta) \quad g^{tt} = \frac{1}{T}\]
\[g_{rr} = R(r,\theta) \quad g^{rr} = \frac{1}{R}\]
\[g_{\theta\theta} = \Theta(r,\theta) \quad g^{\theta\theta} = \frac{1}{\Theta}\]
\[g_{\phi\phi} = \Phi(r,\theta) \quad g^{\phi\phi} = \frac{1}{\Phi}\]
Calculate Christoffel Symbols
The Christoffel symbols.
\[\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}\left(\frac{\partial g_{\mu\rho}}{\partial\nu} +
\frac{\partial g_{\nu\rho}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial\rho}\right)\]
For the \(t\) symbols, \(\lambda = t\) and for non-zero terms \(\rho = t\).
\[\Gamma^t_{\mu\nu} = \frac{1}{2}g^{tt}\left(\frac{\partial g_{\mu t}}{\partial\nu} +
\frac{\partial g_{\nu t}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial t}\right)\]
The third term is zero as there is no time dependency.
\[\Gamma^t_{\mu\nu} = \frac{1}{2}g^{tt}\left(\frac{\partial g_{\mu t}}{\partial\nu} +
\frac{\partial g_{\nu t}}{\partial\mu}\right)\]
The only non-zero solutions are \((\mu = t, \nu = r)\), \((\mu = r, \nu = t)\), and \((\mu = t, \nu = \theta)\), \((\mu = \theta, \nu = t)\).
\[\Gamma^t_{tr} = \Gamma^t_{rt} = \frac{1}{2}g^{tt}\frac{\partial g_{tt}}{\partial r} =
\frac{1}{2T}\frac{\partial T}{\partial r}\]
\[\Gamma^t_{t\theta} = \Gamma^t_{\theta t} = \frac{1}{2}g^{tt}\frac{\partial g_{tt}}{\partial\theta} =
\frac{1}{2T}\frac{\partial T}{\partial \theta}\]
For the \(r\) symbols, \(\lambda = r\) and for non-zero terms \(\rho = r\).
\[\Gamma^r_{\mu\nu} = \frac{1}{2}g^{rr}\left(\frac{\partial g_{\mu r}}{\partial\nu} +
\frac{\partial g_{\nu r}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial r}\right)\]
The only non-zero solutions are \((\mu = r, \nu = \theta)\), \((\mu = \theta, \nu = r)\), and \((\mu = \nu = t)\), and \((\mu = \nu = r)\), and \((\mu = \nu = \theta)\), \((\mu = \nu = \phi)\).
\[\Gamma^r_{r\theta} = \Gamma^r_{\theta r} = \frac{1}{2}g^{rr}\frac{\partial g_{rr}}{\partial\theta} =
\frac{1}{2R}\frac{\partial R}{\partial\theta}\]
\[\Gamma^r_{tt} = -\frac{1}{2}g^{rr}\frac{\partial g_{tt}}{\partial r} =
-\frac{1}{2R}\frac{\partial T}{\partial r}\]
\[\Gamma^r_{rr} = \frac{1}{2}g^{rr}\frac{\partial g_{rr}}{\partial r} =
\frac{1}{2R}\frac{\partial R}{\partial r}\]
\[\Gamma^r_{\theta\theta} = -\frac{1}{2}g^{rr}\frac{\partial g_{\theta\theta}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Theta}{\partial r}\]
\[\Gamma^r_{\phi\phi} = -\frac{1}{2}g^{rr}\frac{\partial g_{\phi\phi}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Phi}{\partial r}\]
For the \(\theta\) symbols, \(\lambda = \theta\) and for non-zero terms \(\rho = \theta\).
\[\Gamma^\theta_{\mu\nu} = \frac{1}{2}g^{\theta\theta}\left(\frac{\partial g_{\mu\theta}}{\partial\nu} +
\frac{\partial g_{\nu\theta}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial\theta}\right)\]
The only non-zero solutions are \((\mu = r, \nu = \theta)\), \((\mu = \theta, \nu = r)\), and \((\mu = \nu = t)\), and \((\mu = \nu = r)\), and \((\mu = \nu = \theta)\), and \((\mu = \nu = \phi)\).
\[\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = \frac{1}{2}g^{\theta\theta}\frac{\partial g_{\theta\theta}}{\partial r} =
\frac{1}{2\Theta}\frac{\partial \Theta}{\partial r}\]
\[\Gamma^\theta_{tt} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{tt}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial T}{\partial\theta}\]
\[\Gamma^\theta_{rr} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{rr}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial R}{\partial\theta}\]
\[\Gamma^\theta_{\theta\theta} = \frac{1}{2}g^{\theta\theta}\frac{\partial g_{\theta\theta}}{\partial\theta} =
\frac{1}{2\Theta}\frac{\partial \Theta}{\partial\theta}\]
\[\Gamma^\theta_{\phi\phi} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{\phi\phi}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial \Phi}{\partial\theta}\]
For the \(\phi\) symbols, \(\lambda = \phi\) and for non-zero terms \(\rho = \phi\).
\[\Gamma^\phi_{\mu\nu} = \frac{1}{2}g^{\phi\phi}\left(\frac{\partial g_{\mu\phi}}{\partial\nu} +
\frac{\partial g_{\nu\phi}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial\phi}\right)\]
The only non-zero solutions are \((\mu = \phi, \nu = r)\), \((\mu = r, \nu = \phi)\), and \((\mu = \phi, \nu = \theta)\), \((\mu = \theta, \nu = \phi)\).
\[\Gamma^\phi_{\phi r} = \Gamma^\phi_{r\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial r} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial r}\]
\[\Gamma^\phi_{\phi\theta} = \Gamma^\phi_{\theta\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial\theta} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial\theta}\]
Christoffel Symbols
The non-zero Christoffel symbols are:
\[\Gamma^t_{tr} = \Gamma^t_{rt} = \frac{1}{2}g^{tt}\frac{\partial g_{tt}}{\partial r} =
\frac{1}{2T}\frac{\partial T}{\partial r}\]
\[\Gamma^t_{t\theta} = \Gamma^t_{\theta t} = \frac{1}{2}g^{tt}\frac{\partial g_{tt}}{\partial\theta} =
\frac{1}{2T}\frac{\partial T}{\partial\theta}\]
\[\Gamma^r_{r\theta} = \Gamma^r_{\theta r} = \frac{1}{2}g^{rr}\frac{\partial g_{rr}}{\partial\theta} =
\frac{1}{2R}\frac{\partial R}{\partial\theta}\]
\[\Gamma^r_{tt} = -\frac{1}{2}g^{rr}\frac{\partial g_{tt}}{\partial r} =
-\frac{1}{2R}\frac{\partial T}{\partial r}\]
\[\Gamma^r_{rr} = \frac{1}{2}g^{rr}\frac{\partial g_{rr}}{\partial r} =
\frac{1}{2R}\frac{\partial R}{\partial r}\]
\[\Gamma^r_{\theta\theta} = -\frac{1}{2}g^{rr}\frac{\partial g_{\theta\theta}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Theta}{\partial r}\]
\[\Gamma^r_{\phi\phi} = -\frac{1}{2}g^{rr}\frac{\partial g_{\phi\phi}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Phi}{\partial r}\]
\[\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = \frac{1}{2}g^{\theta\theta}\frac{\partial g_{\theta\theta}}{\partial r} =
\frac{1}{2\Theta}\frac{\partial \Theta}{\partial r}\]
\[\Gamma^\theta_{tt} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{tt}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial T}{\partial\theta}\]
\[\Gamma^\theta_{rr} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{rr}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial R}{\partial\theta}\]
\[\Gamma^\theta_{\theta\theta} = \frac{1}{2}g^{\theta\theta}\frac{\partial g_{\theta\theta}}{\partial\theta} =
\frac{1}{2\Theta}\frac{\partial \Theta}{\partial\theta}\]
\[\Gamma^\theta_{\phi\phi} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{\phi\phi}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial \Phi}{\partial\theta}\]
\[\Gamma^\phi_{\phi r} = \Gamma^\phi_{r\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial r} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial r}\]
\[\Gamma^\phi_{\phi\theta} = \Gamma^\phi_{\theta\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial\theta} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial\theta}\]
Geodesics
The geodesic equations are:
\[\frac{d^2x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0\]
The \(\lambda = t\) Geodesic
\[\frac{d^2x^t}{d\tau^2} + \Gamma^t_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0\]
Expand the non-zero \(\Gamma\) terms.
\[\frac{d^2t}{d\tau^2} +
\Gamma^t_{tr}\frac{dt}{d\tau}\frac{dr}{d\tau} +
\Gamma^t_{rt}\frac{dr}{d\tau}\frac{dt}{d\tau} +
\Gamma^t_{t\theta}\frac{dt}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^t_{\theta t}\frac{d\theta}{d\tau}\frac{dt}{d\tau} = 0\]
Substitute Equation (25) and Equation (9) and rearrange.
\[\frac{d^2t}{d\tau^2} +
\frac{1}{T}\frac{dt}{d\tau}\left(\frac{\partial T}{\partial r}\frac{dr}{d\tau} +
\frac{\partial T}{\partial\theta}\frac{d\theta}{d\tau}\right) = 0\]
The \(\lambda = r\) Geodesic
\[\frac{d^2r}{d\tau^2} + \Gamma^r_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0\]
Expand the non-zero terms.
\[\frac{d^2r}{d\tau^2} + \Gamma^r_{tt}\frac{dt}{d\tau}\frac{dt}{d\tau} +
\Gamma^r_{rr}\frac{dr}{d\tau}\frac{dr}{d\tau} +
\Gamma^r_{\theta\theta}\frac{d\theta}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^r_{\phi\phi}\frac{d\phi}{d\tau}\frac{d\phi}{d\tau} = 0\]
Substitute Equation (12), Equation (29), Equation (30) and Equation (31) and rearrange.
\[\frac{d^2r}{d\tau^2} + \frac{1}{2R}\left[\frac{\partial T}{\partial r}\left(\frac{dt}{d\tau}\right)^2 +
\frac{\partial R}{\partial r}\left(\frac{dr}{d\tau}\right)^2 +
\frac{\partial \Theta}{\partial r}\left(\frac{d\theta}{d\tau}\right)^2 +
\frac{\partial \Phi}{\partial r}\left(\frac{d\phi}{d\tau}\right)^2\right] = 0\]
The \(\lambda = \theta\) Geodesic
\[\frac{d^2\theta}{d\tau^2} + \Gamma^\theta_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0\]
Expand the non-zero terms.
\[\frac{d^2\theta}{d\tau^2} +
\Gamma^\theta_{r\theta}\frac{dr}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^\theta_{\theta r}\frac{d\theta}{d\tau}\frac{dr}{d\tau} +
\Gamma^\theta_{tt}\frac{dt}{d\tau}\frac{dt}{d\tau} +
\Gamma^\theta_{rr}\frac{dr}{d\tau}\frac{dr}{d\tau} +
\Gamma^\theta_{\theta\theta}\frac{d\theta}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^\theta_{\phi\phi}\frac{d\phi}{d\tau}\frac{d\phi}{d\tau} = 0\]
Substitute the \(\Gamma\) terms and rearrange.
\[\frac{d^2\theta}{d\tau^2} + \frac{1}{2\Theta}\left[
\frac{\partial\Theta}{\partial r}\left(\frac{dr}{d\tau}\frac{d\theta}{d\tau} +
\frac{d\theta}{d\tau}\frac{dr}{d\tau}\right) -
\frac{\partial T}{\partial\theta}\left(\frac{dt}{d\tau}\right)^2 -
\frac{\partial R}{\partial\theta}\left(\frac{dr}{d\tau}\right)^2 +
\frac{\partial \Theta}{\partial\theta}\left(\frac{d\theta}{d\tau}\right)^2 -
\frac{\partial \Phi}{\partial\theta}\left(\frac{d\phi}{d\tau}\right)^2
\right] = 0\]
The \(\lambda = \phi\) Geodesic
\[\frac{d^2\phi}{d\tau^2} + \Gamma^\phi_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0\]
Expand the non-zero terms.
\[\frac{d^2\phi}{d\tau^2} +
\Gamma^\phi_{\theta r}\frac{d\theta}{d\tau}\frac{dr}{d\tau} +
\Gamma^\phi_{r\theta}\frac{dr}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^\phi_{\phi\theta}\frac{d\phi}{d\tau}\frac{d\theta}{d\tau} +
\Gamma^\phi_{\theta\phi}\frac{d\theta}{d\tau}\frac{d\phi}{d\tau} = 0\]
Substitute the \(\Gamma\) terms and rearrange.
\[\frac{d^2\phi}{d\tau^2} + \frac{1}{\Phi}\left[
\frac{\partial \Phi}{\partial r}\frac{d\phi}{d\tau}\frac{dr}{d\tau} +
\frac{\partial \Phi}{\partial\theta}\frac{d\theta}{d\tau}\frac{d\phi}{d\tau}
\right] = 0\]
Geodesic Equations
\[\frac{d^2t}{d\tau^2} +
\frac{1}{T}\frac{dt}{d\tau}\left(\frac{\partial T}{\partial r}\frac{dr}{d\tau} +
\frac{\partial T}{\partial\theta}\frac{d\theta}{d\tau}\right) = 0\]
\[\frac{d^2r}{d\tau^2} + \frac{1}{2R}\left[\frac{\partial T}{\partial r}\left(\frac{dt}{d\tau}\right)^2 +
\frac{\partial R}{\partial r}\left(\frac{dr}{d\tau}\right)^2 +
\frac{\partial \Theta}{\partial r}\left(\frac{d\theta}{d\tau}\right)^2 +
\frac{\partial \Phi}{\partial r}\left(\frac{d\phi}{d\tau}\right)^2\right] = 0\]
\[\frac{d^2x^\theta}{d\tau^2} + \frac{1}{2\Theta}\left[
\frac{\partial\Theta}{\partial r}\left(\frac{dr}{d\tau}\frac{d\theta}{d\tau} +
\frac{d\theta}{d\tau}\frac{dr}{d\tau}\right) -
\frac{\partial T}{\partial\theta}\left(\frac{dt}{d\tau}\right)^2 -
\frac{\partial R}{\partial\theta}\left(\frac{dr}{d\tau}\right)^2 +
\frac{\partial \Theta}{\partial\theta}\left(\frac{d\theta}{d\tau}\right)^2 -
\frac{\partial \Phi}{\partial\theta}\left(\frac{d\phi}{d\tau}\right)^2
\right] = 0\]
\[\frac{d^2\phi}{d\tau^2} + \frac{1}{\Phi}\left[
\frac{\partial \Phi}{\partial r}\frac{d\phi}{d\tau}\frac{dr}{d\tau} +
\frac{\partial \Phi}{\partial\theta}\frac{d\theta}{d\tau}\frac{d\phi}{d\tau}
\right] = 0\]
Ricci Tensor
The Ricci tensor is quite complex, but symmetrical. It makes extensive use of the Einstein summation convention. Given that the metric tensor is symmetric, it is easy to miss a summation term and lose a factor of two.
\[R_{\mu\nu} =
\frac{\partial}{\partial\nu}\Gamma^\lambda_{\mu\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{\mu\nu} +
\Gamma^\lambda_{\mu\eta}\Gamma^\eta_{\nu\lambda} -
\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\mu\nu}\]
As it is trace then \(\nu = \mu\).
\[R_{\mu\mu} =
\frac{\partial}{\partial\mu}\Gamma^\lambda_{\mu\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{\mu\mu} +
\Gamma^\lambda_{\mu\eta}\Gamma^\eta_{\mu\lambda} -
\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\mu\mu}\]
The Ricci \(R_{tt}\) Term
\[R_{tt} =
\frac{\partial}{\partial t}\Gamma^\lambda_{t\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{tt} +
\Gamma^\lambda_{t\eta}\Gamma^\eta_{t\lambda} -
\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{tt}\]
The first term is zero as there are no time dependencies.
Expand the second term given that \(\Gamma^t_{tt} = 0\) and there is no dependency on \(\phi\).
\[-\frac{\partial}{\partial\lambda}\Gamma^\lambda_{tt} =
-\frac{\partial}{\partial r}\Gamma^r_{tt}
-\frac{\partial}{\partial\theta}\Gamma^\theta_{tt} = \\
\frac{\partial}{\partial r}\left(\frac{1}{2R}\frac{\partial T}{\partial r}\right)
+ \frac{\partial}{\partial\theta}\left(\frac{1}{2\Theta}\frac{\partial T}{\partial\theta}\right)
= \\
\frac{1}{2R}\frac{\partial^2 T}{\partial r^2}
+ \frac{1}{2\Theta}\frac{\partial^2 T}{\partial\theta^2}
- \frac{1}{2R^2}\frac{\partial R}{\partial r}\frac{\partial T}{\partial r}
- \frac{1}{2\Theta^2}\frac{\partial \Theta}{\partial\theta}\frac{\partial T}{\partial\theta}\]
Expand the third term \(\lambda\) terms.
\[\Gamma^\lambda_{t\eta}\Gamma^\eta_{t\lambda} =
\Gamma^t_{t\eta}\Gamma^\eta_{tt} +
\Gamma^r_{t\eta}\Gamma^\eta_{tr} +
\Gamma^\theta_{t\eta}\Gamma^\eta_{t\theta} +
\Gamma^\phi_{t\eta}\Gamma^\eta_{t\phi} = \\
\Gamma^t_{tr}\Gamma^r_{tt} + \Gamma^t_{t\theta}\Gamma^\theta_{tt} +
\Gamma^r_{tt}\Gamma^t_{tr} +
\Gamma^\theta_{tt}\Gamma^t_{t\theta} = \\
2\Gamma^t_{tr}\Gamma^r_{tt} + 2\Gamma^t_{t\theta}\Gamma^\theta_{tt} =
-\frac{1}{2TR}\left(\frac{\partial T}{\partial r}\right)^2
-\frac{1}{2T\Theta}\left(\frac{\partial T}{\partial \theta}\right)^2\]
Expand the fourth term \(\eta\) terms.
\[-\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{tt} =
-\Gamma^\lambda_{\lambda r}\Gamma^r_{tt} - \Gamma^\lambda_{\lambda\theta}\Gamma^\theta_{tt} = \\
-(\Gamma^t_{tr} + \Gamma^r_{rr} + \Gamma^\theta_{\theta r} + \Gamma^\phi_{\phi r})\Gamma^r_{tt} -
(\Gamma^t_{t\theta} + \Gamma^r_{r\theta} + \Gamma^\theta_{\theta\theta} + \Gamma^\phi_{\phi\theta})\Gamma^\theta_{tt} =
\frac{1}{4R}\frac{\partial T}{\partial r}\left(
\frac{1}{T}\frac{\partial T}{\partial r} +
\frac{1}{R}\frac{\partial R}{\partial r} +
\frac{1}{\Theta}\frac{\partial \Theta}{\partial r} +
\frac{1}{\Phi}\frac{\partial\Phi}{\partial r}
\right) +
\frac{1}{4\Theta}\frac{\partial T}{\partial \theta}\left(
\frac{1}{T}\frac{\partial T}{\partial \theta} +
\frac{1}{R}\frac{\partial R}{\partial \theta} +
\frac{1}{\Theta}\frac{\partial \Theta}{\partial \theta} +
\frac{1}{\Phi}\frac{\partial\Phi}{\partial \theta}
\right)\]
Combine the terms.
\[R_{tt} =
\frac{1}{2R}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 T}{\partial\theta^2} -
\frac{1}{4TR}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{4R^2}\frac{\partial T}{\partial r}\frac{\partial R}{\partial r} -
\frac{1}{4T\Theta}\left(\frac{\partial T}{\partial \theta}\right)^2 -
\frac{1}{4\Theta^2}\frac{\partial T}{\partial\theta}\frac{\partial\Theta}{\partial\theta} +
\frac{1}{4R\Theta}\frac{\partial T}{\partial r}\frac{\partial\Theta}{\partial r} +
\frac{1}{4R\Phi}\frac{\partial T}{\partial r}\frac{\partial\Phi}{\partial r} +
\frac{1}{4R\Theta}\frac{\partial T}{\partial\theta}\frac{\partial R}{\partial\theta} +
\frac{1}{4\Theta\Phi}\frac{\partial T}{\partial\theta}\frac{\partial\Phi}{\partial\theta}\]
The Ricci \(R_{rr}\) Term
\[R_{rr} =
\frac{\partial}{\partial r}\Gamma^\lambda_{r\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{rr} +
\Gamma^\lambda_{r\eta}\Gamma^\eta_{r\lambda} -
\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{rr}\]
Expand the first term.
\[\frac{\partial}{\partial r}\Gamma^\lambda_{r\lambda} =
\frac{\partial}{\partial r}(\Gamma^t_{rt} + \Gamma^r_{rr} +
\Gamma^\theta_{r\theta} + \Gamma^\phi_{r\phi}) =
\frac{1}{2}\frac{\partial}{\partial r}\left(
\frac{1}{T}\frac{\partial T}{\partial r} +
\frac{1}{R}\frac{\partial R}{\partial r} +
\frac{1}{\Theta}\frac{\partial \Theta}{\partial r} +
\frac{1}{\Phi}\frac{\partial \Phi}{\partial r}
\right) =
\frac{1}{2}\left(
\frac{1}{T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{R}\frac{\partial^2 R}{\partial r^2} +
\frac{1}{\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{\Phi}\frac{\partial^2 \Phi}{\partial r^2} -
\frac{1}{T^2}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{R^2}\left(\frac{\partial R}{\partial r}\right)^2 -
\frac{1}{\Theta^2}\left(\frac{\partial \Theta}{\partial r}\right)^2 -
\frac{1}{\Phi^2}\left(\frac{\partial \Phi}{\partial r}\right)^2
\right)\]
Expand the second term.
\[-\frac{\partial}{\partial\lambda}\Gamma^\lambda_{rr} = -\frac{\partial}{\partial r}\Gamma^r_{rr} -
\frac{\partial}{\partial\theta}\Gamma^\theta_{rr} =
-\frac{\partial}{\partial r}\left(\frac{1}{2R}\frac{\partial R}{\partial r}\right) -
\frac{\partial}{\partial\theta}\left(\frac{1}{2\Theta}\frac{\partial R}{\partial\theta}\right) =
\frac{1}{2}\left(
-\frac{1}{R}\frac{\partial^2 R}{\partial r^2} -
\frac{1}{\Theta}\frac{\partial^2 R}{\partial\theta^2} +
\frac{1}{R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{\Theta^2}\left(\frac{\partial R}{\partial\theta}\right)^2
\right)\]
Combine the first two terms.
\[\frac{\partial}{\partial r}\Gamma^\lambda_{r\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{rr} = \\
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2R}\frac{\partial^2 R}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{2\Phi}\frac{\partial^2 \Phi}{\partial r^2} -
\frac{1}{2T^2}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{2R^2}\left(\frac{\partial R}{\partial r}\right)^2 -
\frac{1}{2\Theta^2}\left(\frac{\partial \Theta}{\partial r}\right)^2 -
\frac{1}{2\Phi^2}\left(\frac{\partial \Phi}{\partial r}\right)^2 \\
-\frac{1}{2R}\frac{\partial^2 R}{\partial r^2} -
\frac{1}{2\Theta}\frac{\partial^2 R}{\partial\theta^2} +
\frac{1}{2R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{2\Theta^2}\left(\frac{\partial R}{\partial\theta}\right)^2 = \\
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{2\Phi}\frac{\partial^2 \Phi}{\partial r^2} -
\frac{1}{2T^2}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{2\Theta^2}\left(\frac{\partial \Theta}{\partial r}\right)^2 -
\frac{1}{2\Phi^2}\left(\frac{\partial \Phi}{\partial r}\right)^2 -
\frac{1}{2\Theta}\frac{\partial^2 R}{\partial\theta^2} +
\frac{1}{2\Theta^2}\left(\frac{\partial R}{\partial\theta}\right)^2\]
Expand the third term.
\[\Gamma^\lambda_{r\eta}\Gamma^\eta_{r\lambda} =
\left(\Gamma^t_{rt}\right)^2 +
\left(\Gamma^r_{rr}\right)^2 +
2\Gamma^r_{r\theta}\Gamma^\theta_{rr} +
\left(\Gamma^\theta_{r\theta}\right)^2 +
\left(\Gamma^\phi_{r\phi}\right)^2 =
\frac{1}{4}\left(
\frac{1}{T^2}\left(\frac{\partial T}{\partial r}\right)^2 +
\frac{1}{R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{\Theta^2}\left(\frac{\partial\Theta}{\partial r}\right)^2 +
\frac{1}{\Phi^2}\left(\frac{\partial\Phi}{\partial r}\right)^2 -
\frac{2}{R\Theta}\frac{\partial\Theta}{\partial r}\frac{\partial R}{\partial\theta}
\right)\]
Combine the first three terms.
\[\frac{\partial}{\partial r}\Gamma^\lambda_{r\lambda} -
\frac{\partial}{\partial\lambda}\Gamma^\lambda_{rr} +
\Gamma^\lambda_{r\eta}\Gamma^\eta_{r\lambda} = \\
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{2\Phi}\frac{\partial^2 \Phi}{\partial r^2} -
\frac{1}{2\Theta}\frac{\partial^2 R}{\partial\theta^2} -
\frac{1}{4T^2}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{2\Theta^2}\left(\frac{\partial \Theta}{\partial r}\right)^2 -
\frac{1}{4\Phi^2}\left(\frac{\partial \Phi}{\partial r}\right)^2 +
\frac{1}{2\Theta^2}\left(\frac{\partial R}{\partial\theta}\right)^2 + \\
\frac{1}{4R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{4\Theta^2}\left(\frac{\partial\Theta}{\partial r}\right)^2 -
\frac{1}{2R\Theta}\frac{\partial\Theta}{\partial r}\frac{\partial R}{\partial\theta}\]
Expand the fourth term \(\eta\) terms.
\[-\Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{rr} =
-\Gamma^\lambda_{\lambda r}\Gamma^r_{rr} - \Gamma^\lambda_{\lambda\theta}\Gamma^\theta_{rr} =
-(\Gamma^t_{tr} + \Gamma^r_{rr} + \Gamma^\theta_{\theta r} + \Gamma^\phi_{\phi r})\Gamma^r_{rr} -
(\Gamma^t_{t\theta} + \Gamma^r_{r\theta} + \Gamma^\theta_{\theta\theta} + \Gamma^\phi_{\phi\theta})\Gamma^\theta_{rr} = \\
\frac{1}{4R}\frac{\partial R}{\partial r}\left(
\frac{1}{T}\frac{\partial T}{\partial r} +
\frac{1}{R}\frac{\partial R}{\partial r} +
\frac{1}{\Theta}\frac{\partial \Theta}{\partial r} +
\frac{1}{\Phi}\frac{\partial\Phi}{\partial r}
\right) -
\frac{1}{4\Theta}\frac{\partial R}{\partial \theta}\left(
\frac{1}{T}\frac{\partial T}{\partial \theta} +
\frac{1}{R}\frac{\partial R}{\partial \theta} -
\frac{1}{\Theta}\frac{\partial \Theta}{\partial \theta} +
\frac{1}{\Phi}\frac{\partial\Phi}{\partial \theta}
\right) = \\
\frac{1}{4TR}\frac{\partial R}{\partial r}
\frac{\partial T}{\partial r} +
\frac{1}{4R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{4R\Theta}\frac{\partial R}{\partial r}
\frac{\partial \Theta}{\partial r} +
\frac{1}{4R\Phi}\frac{\partial R}{\partial r}
\frac{\partial\Phi}{\partial r} -
\frac{1}{4T\Theta}\frac{\partial R}{\partial \theta}
\frac{\partial T}{\partial \theta} -
\frac{1}{4R\Theta}\left(\frac{\partial R}{\partial \theta}\right)^2 +
\frac{1}{4\Theta^2}\frac{\partial R}{\partial \theta}
\frac{\partial \Theta}{\partial \theta} -
\frac{1}{4\Theta\Phi}\frac{\partial R}{\partial \theta}
\frac{\partial\Phi}{\partial \theta}\]
Combine the terms.
\[R_{rr} =
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{2\Phi}\frac{\partial^2 \Phi}{\partial r^2} -
\frac{1}{2\Theta}\frac{\partial^2 R}{\partial\theta^2} -
\frac{1}{4T^2}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{2\Theta^2}\left(\frac{\partial \Theta}{\partial r}\right)^2 -
\frac{1}{4\Phi^2}\left(\frac{\partial \Phi}{\partial r}\right)^2 +
\frac{1}{2\Theta^2}\left(\frac{\partial R}{\partial\theta}\right)^2 + \\
\frac{1}{4R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{4\Theta^2}\left(\frac{\partial\Theta}{\partial r}\right)^2 -
\frac{1}{2R\Theta}\frac{\partial\Theta}{\partial r}\frac{\partial R}{\partial\theta} + \\
\frac{1}{4TR}\frac{\partial R}{\partial r}
\frac{\partial T}{\partial r} +
\frac{1}{4R^2}\left(\frac{\partial R}{\partial r}\right)^2 +
\frac{1}{4R\Theta}\frac{\partial R}{\partial r}
\frac{\partial \Theta}{\partial r} +
\frac{1}{4R\Phi}\frac{\partial R}{\partial r}
\frac{\partial\Phi}{\partial r} -
\frac{1}{4T\Theta}\frac{\partial R}{\partial \theta}
\frac{\partial T}{\partial \theta} -
\frac{1}{4R\Theta}\left(\frac{\partial R}{\partial \theta}\right)^2 +
\frac{1}{4\Theta^2}\frac{\partial R}{\partial \theta}
\frac{\partial \Theta}{\partial \theta} -
\frac{1}{4\Theta\Phi}\frac{\partial R}{\partial \theta}
\frac{\partial\Phi}{\partial \theta}\]
Schwarzschild Outer Solution Revisited
We will revisit the Schwarzschild outer solution using the trace solution described. We have \(T_{\mu\nu} = 0 \implies R_{\mu\nu} = 0\). We have the unknown functions \(T(r)\) and \(R(r)\).
\[\Theta = -r^2 \\
\frac{\partial \Theta}{\partial r} = -2r \\
\frac{\partial^2 \Theta}{\partial r2} = -2\]
\[\Phi = -r^2\sin^2\theta \\
\frac{\partial \Phi}{\partial r} = -2r\sin^2\theta \\
\frac{\partial^2 \Phi}{\partial r2} = -2\sin^2\theta\]
Christoffel Symbols
The non-zero Christoffel symbols are:
\[\Gamma^t_{tr} = \Gamma^t_{rt} = \frac{1}{2}g^{tt}\frac{\partial g_{tt}}{\partial r} =
\frac{1}{2T}\frac{\partial T}{\partial r}\]
\[\Gamma^r_{tt} = -\frac{1}{2}g^{rr}\frac{\partial g_{tt}}{\partial r} =
-\frac{1}{2R}\frac{\partial T}{\partial r}\]
\[\Gamma^r_{rr} = \frac{1}{2}g^{rr}\frac{\partial g_{rr}}{\partial r} =
\frac{1}{2R}\frac{\partial R}{\partial r}\]
\[\Gamma^r_{\theta\theta} = -\frac{1}{2}g^{rr}\frac{\partial g_{\theta\theta}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Theta}{\partial r} = \frac{r}{R}\]
\[\Gamma^r_{\phi\phi} = -\frac{1}{2}g^{rr}\frac{\partial g_{\phi\phi}}{\partial r} =
-\frac{1}{2R}\frac{\partial \Phi}{\partial r} = \frac{r}{R}\sin^2\theta\]
\[\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = \frac{1}{2}g^{\theta\theta}\frac{\partial g_{\theta\theta}}{\partial r} =
\frac{1}{2\Theta}\frac{\partial \Theta}{\partial r} = \frac{1}{r}\]
\[\Gamma^\theta_{\phi\phi} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{\phi\phi}}{\partial\theta} =
-\frac{1}{2\Theta}\frac{\partial \Phi}{\partial\theta} = -\sin\theta\cos\theta\]
\[\Gamma^\phi_{\phi r} = \Gamma^\phi_{r\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial r} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial r} = \frac{1}{r}\]
\[\Gamma^\phi_{\phi\theta} = \Gamma^\phi_{\theta\phi} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi\phi}}{\partial\theta} =
\frac{1}{2\Phi}\frac{\partial \Phi}{\partial\theta} = \cot\theta\]
Construct the Ricci tensor.
\[R_{tt} =
\frac{1}{2R}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{4TR}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{4R^2}\frac{\partial T}{\partial r}\frac{\partial R}{\partial r} -
\frac{1}{4R\Theta}\frac{\partial T}{\partial r}\frac{\partial\Theta}{\partial r} +
\frac{1}{4R\Phi}\frac{\partial T}{\partial r}\frac{\partial\Phi}{\partial r}\]
\[R_{rr} =
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{2\Theta}\frac{\partial^2 \Theta}{\partial r^2} +
\frac{1}{2\Phi}\frac{\partial^2 \Phi}{\partial r^2} +
\frac{1}{4T^2}\left(\frac{\partial T}{\partial r}\right)^2 +
\frac{1}{2R^2}\left(\frac{\partial R}{\partial r}\right)^2 -
\frac{1}{4\Theta^2}\left(\frac{\partial\Theta}{\partial r}\right)^2 -
\frac{1}{4\Phi^2}\left(\frac{\partial\Phi}{\partial r}\right)^2\]
Substitute the values for \(\Theta\) and \(\Phi\) and their derivatives and evaluate terms as \(T\) and \(R\) are independent of \(\theta\).
\[R_{tt} =
\frac{1}{2R}\frac{\partial^2 T}{\partial r^2} -
\frac{1}{4TR}\left(\frac{\partial T}{\partial r}\right)^2 -
\frac{1}{4R^2}\frac{\partial T}{\partial r}\frac{\partial R}{\partial r} +
\frac{1}{4Rr^2}\frac{\partial T}{\partial r}2r +
\frac{1}{4Rr^2\sin^2\theta}\frac{\partial T}{\partial r}2r\sin^2\theta =
\frac{1}{2R}\frac{\partial^2 T}{\partial r^2} -
\frac{1}{4R}\frac{\partial T}{\partial R}\left(\frac{1}{T}\frac{\partial T}{\partial r} + \frac{1}{R}\frac{\partial R}{\partial r}\right) +
\frac{1}{rR}\frac{\partial T}{\partial r}\]
\[R_{rr} =
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{r^2} +
\frac{1}{r^2} +
\frac{1}{4T^2}\left(\frac{\partial T}{\partial r}\right)^2 +
\frac{1}{2R^2}\left(\frac{\partial R}{\partial r}\right)^2 -
\frac{1}{r^2} -
\frac{1}{r^2} =
\frac{1}{2T}\frac{\partial^2 T}{\partial r^2} +
\frac{1}{4T^2}\left(\frac{\partial T}{\partial r}\right)^2 +
\frac{1}{2R^2}\left(\frac{\partial R}{\partial r}\right)^2\]
Ricci Tensor
As the functions are only dependent on \(r\) the partial derivatives become regular derivatives.
\[R_{tt} =
\frac{1}{2R}\frac{d^2T}{dr^2} -
\frac{1}{4R}\frac{dT}{dR}\left(\frac{1}{T}\frac{dT}{dr} + \frac{1}{R}\frac{dR}{dr}\right) +
\frac{1}{rR}\frac{dT}{dr} = 0\]
\[R_{rr} =
\frac{1}{2T}\frac{d^2T}{dr^2} +
\frac{2}{r^2} +\]
Christoffel Symbols Trace Metric With Cross Term
Rotating solutions have a trace metric with a cross term \(g_{t\phi} = g_{\phi t} \ne 0\). This term will be a function of \(r\) and \(\theta\).
This complicates the inverse \(g^{\mu\nu}\)!
This adds Christoffel symbols in addition to the trace symbols.
For the \(t\) symbols, \(\lambda = \rho = t\).
\[\Gamma^t_{\mu\nu} = \frac{1}{2}g^{tt}\left(\frac{\partial g_{\mu t}}{\partial\nu} +
\frac{\partial g_{\nu t}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial t}\right)\]
This has additional symbols for \((\mu = \phi, \nu = r)\), \((\mu = r, \nu = \phi)\), \((\mu = \phi, \nu = \theta)\), and \((\mu = \theta, \nu = \phi)\).
\[\Gamma^t_{\phi r} = \Gamma^t_{r\phi} = \frac{1}{2}g^{tt}\frac{\partial g_{\phi t}}{\partial r}\]
\[\Gamma^t_{\phi\theta} = \Gamma^t_{\theta\phi} = \frac{1}{2}g^{tt}\frac{\partial g_{\phi t}}{\partial\theta}\]
For the \(r\) symbols, \(\lambda = \rho = r\).
\[\Gamma^r_{\mu\nu} = \frac{1}{2}g^{rr}\left(\frac{\partial g_{\mu r}}{\partial\nu} +
\frac{\partial g_{\nu r}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial r}\right)\]
This has additional symbols for \((\mu = \phi, \nu = t)\), and \((\mu = t, \nu = \phi)\).
\[\Gamma^r_{\phi t} = \Gamma^r_{t\phi} = -\frac{1}{2}g^{rr}\frac{\partial g_{\phi t}}{\partial r}\]
For the \(\theta\) symbols, \(\lambda = \rho = \theta\).
\[\Gamma^\theta_{\mu\nu} = \frac{1}{2}g^{\theta\theta}\left(\frac{\partial g_{\mu\theta}}{\partial\nu} +
\frac{\partial g_{\nu\theta}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial\theta}\right)\]
This has additional symbols for \((\mu = \phi, \nu = t)\), and \((\mu = t, \nu = \phi)\).
\[\Gamma^\theta_{\phi t} = \Gamma^\theta_{t\phi} = -\frac{1}{2}g^{\theta\theta}\frac{\partial g_{\phi t}}{\partial\theta}\]
For the \(\phi\) symbols, \(\lambda = \rho = \phi\).
\[\Gamma^\phi_{\mu\nu} = \frac{1}{2}g^{\phi\phi}\left(\frac{\partial g_{\mu\phi}}{\partial\nu} +
\frac{\partial g_{\nu\phi}}{\partial\mu} - \frac{\partial g_{\mu\nu}}{\partial\phi}\right)\]
This has additional symbols for \((\mu = t, \nu = r)\), \((\mu = r, \nu = t)\), \((\mu = t, \nu = \theta)\), and \((\mu = \theta, \nu = t)\).
\[\Gamma^\phi_{tr} = \Gamma^\phi_{rt} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi t}}{\partial r}\]
\[\Gamma^\phi_{t\theta} = \Gamma^\phi_{\theta t} = \frac{1}{2}g^{\phi\phi}\frac{\partial g_{\phi t}}{\partial\theta}\]