- #1

latentcorpse

- 1,444

- 0

I have that [itex]g=L^2 \left( e^{-2U} \left( e^{2A} \left( -dt^2 + d \theta^2 \right) + R^2 dy^2 \right) + e^{2U} dx^2 \right)[/itex] is the metric on my spacetime.

taking [itex]\{ t, \theta, x , y \}[/itex] as a coordinate system for the manifol M, i can write this in matrix form as

[itex]g_{ab}=L^2 \left( \begin {array}{cccc} -{e}^{2 \left( A-U \right)}&0&0&0

\\ \noalign{\medskip}0&{e}^{2 \left(A-U \right)}&0&0\\ \noalign{\medskip}0&0&{e}

^{2U}&0\\ \noalign{\medskip}0&0&0&{R}^{2}{e}^{-2U}\end {array}

\right)[/itex]

now i need to show the vacuum einstein equations for g are

[itex]\partial_t^2 R - \partial_{\theta}^2 R =0[/itex]

[itex]\partial_t (R \partial_t U ) - \partial_{\theta} ( R \partial_{\theta} U ) =0[/itex]

[itex]\partial_t^2 A - \partial_{\theta}^2 A = ( \partial_{\theta} U )^2 - ( \partial_t U)^2[/itex]

and

[itex]\partial_{\theta} \partial_+ R = ( \partial_+ A)(\partial_+ R) - R ( \partial_+ U)^2[/itex]

[itex]\partial_{\theta} \partial_- R = ( \partial_- A)( \partial_- R) - R ( \partial_- U )^2[/itex]

where [itex]\partial_{\pm} = \partial_{\theta} \pm \partial_{t}[/itex]

so i want to start by computing the christoffel symbols andyway.

this is done using [itex]\Gamma^{\sigma}_{\mu \nu} = \frac{1}{2} \displaystyle \sum_{\rho} g^{\sigma \rho} \left( \frac{ \partial g_{\nu \rho}}{\partial x^{\mu}} + \frac{ \partial g_{\mu \rho}}{\partial x^\nu} - \frac{ \partial g_{\mu \nu}}{ \partial x^\rho} \right)[/itex]

however in previous examples I've worked with, [itex]\sigma, \mu, \nu, \rho \in \{ 1,2,3 \}[/itex] but now i have a problem because of this fourth index due to the presence of time in my metric and i don't know how to deal with it. any advice?

taking [itex]\{ t, \theta, x , y \}[/itex] as a coordinate system for the manifol M, i can write this in matrix form as

[itex]g_{ab}=L^2 \left( \begin {array}{cccc} -{e}^{2 \left( A-U \right)}&0&0&0

\\ \noalign{\medskip}0&{e}^{2 \left(A-U \right)}&0&0\\ \noalign{\medskip}0&0&{e}

^{2U}&0\\ \noalign{\medskip}0&0&0&{R}^{2}{e}^{-2U}\end {array}

\right)[/itex]

now i need to show the vacuum einstein equations for g are

[itex]\partial_t^2 R - \partial_{\theta}^2 R =0[/itex]

[itex]\partial_t (R \partial_t U ) - \partial_{\theta} ( R \partial_{\theta} U ) =0[/itex]

[itex]\partial_t^2 A - \partial_{\theta}^2 A = ( \partial_{\theta} U )^2 - ( \partial_t U)^2[/itex]

and

[itex]\partial_{\theta} \partial_+ R = ( \partial_+ A)(\partial_+ R) - R ( \partial_+ U)^2[/itex]

[itex]\partial_{\theta} \partial_- R = ( \partial_- A)( \partial_- R) - R ( \partial_- U )^2[/itex]

where [itex]\partial_{\pm} = \partial_{\theta} \pm \partial_{t}[/itex]

so i want to start by computing the christoffel symbols andyway.

this is done using [itex]\Gamma^{\sigma}_{\mu \nu} = \frac{1}{2} \displaystyle \sum_{\rho} g^{\sigma \rho} \left( \frac{ \partial g_{\nu \rho}}{\partial x^{\mu}} + \frac{ \partial g_{\mu \rho}}{\partial x^\nu} - \frac{ \partial g_{\mu \nu}}{ \partial x^\rho} \right)[/itex]

however in previous examples I've worked with, [itex]\sigma, \mu, \nu, \rho \in \{ 1,2,3 \}[/itex] but now i have a problem because of this fourth index due to the presence of time in my metric and i don't know how to deal with it. any advice?

Last edited: