Compute Christoffel Symbols for Metric w/ Time Variable

In summary: It might be worth trying to find the equation for g in terms of the Ricci tensor and then solving it.
  • #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?
 
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  • #2
just run rho from 0 to 3

[itex]
g^{\sigma 0} ... + g^{\sigma 1} ... + g^{\sigma 2} ... + g^{\sigma 3}

[/itex]
 
  • #3
so do i keep [itex]\sigma, \mu, \nu[/itex] running from 1 to 3 only?
 
  • #4
so does this look ok so far...

[itex]\Gamma^1{}_{11}=\frac{1}{2} g^{11} ( \partial_{\theta} g^{11} ) = \frac{1}{2} L^2 e^{2(A-U)} \partial_{\theta} (L^2 e^{2(A-U)})[/itex]
[itex]=\frac{1}{2}L^4 e^{2(A-U)} e^{2(A-U)} ( \partial_{\theta} ( 2(A-U)))[/itex]
[itex]=L^4 e^{4(A-U)} ( \partial_{\theta} A- \partial_{\theta} U)[/itex]

[itex]\Gamma^1{}_{12}=\frac{1}{2} g^{11} ( \partial_x (L^2 e^{2(A-U)}))=0[/itex]
as [itex]A,U[/itex] are functions of [itex]\theta,t[/itex] only.

to be honest i don't see how there's ever going to be a situation where we use [itex]\rho=0[/itex] as if [itex]\sigma \in \{ 1,2,3 \}[/itex], then the [itex]g^{\sigma \rho}[/itex] term in front of the brackets in the formula will always be zero when [itex]\rho=0[/itex] will it not?
 
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  • #5
bump.
 
  • #6
latentcorpse said:
so do i keep [itex]\sigma, \mu, \nu[/itex] running from 1 to 3 only?
No, all indices here run from 0 to 3.

(I can't speak for general usage but in Eddington's "The Mathematical Theory of Relativity", he specifically uses Greek letters for indices from 0 to 3, Latin letters for indices from 1 to 3.)
 
  • #7
ok. thanks.
so now that i have the non-zero Christoffel symbols, the next step in getting to the Einstein equations there would be to copmute the Ricci tensor using the eqn

[itex]R_{\mu \rho}= \displaystyle \sum_{\nu} \frac{\partial}{\partial x^{\nu}} \Gamma^{\nu}{}_{\mu \rho} - \frac{\partial}{\partial x^{\mu}} \left( \displaystyle \sum_\nu \Gamma^{\nu}{}_{\nu \rho} \right) + \displaystyle \sum_{\sigma, \nu} \left( \Gamma^{\alpha}{}_{\mu \rho} \Gamma^{\nu}{}_{\alpha \nu} - \Gamma^{\alpha}{}_{\nu \rho} \Gamma^{\nu}{}_{\alpha \mu} \right)[/itex]


then i would write the Ricci tensor as a matrix

then i calculate [itex]R=R_{a}{}^{a}[/itex]

and then i put these into [itex]R_{ab}-\frac{1}{2}Rg_{ab}=8 \pi T_{ab}[/itex]
yeah?

and if these are the vacuum Einstein eqns i can set [itex]T_{ab}=0[/itex], yeah?

thanks.
 
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  • #8
latentcorpse said:
then i would write the Ricci tensor as a matrix

then i calculate [itex]R=R_{a}{}^{a}[/itex]

and then i put these into [itex]R_{ab}-\frac{1}{2}Rg_{ab}=8 \pi T_{ab}[/itex]
yeah?

and if these are the vacuum Einstein eqns i can set [itex]T_{ab}=0[/itex], yeah?

Yeah, that's the plan. But why do you keep the big sigma?

summation convection should get rid of it.
 
  • #9
ye, i think it was just left in in the book i was working from (Wald's General Relativity).

anyway i found that copmuting, even [itex]R_{00}[/itex] was about 4 pages of wrok and my final answer was about 10 lines long...has something gone wrong or is this typical?
 
  • #10
latentcorpse said:
anyway i found that copmuting, even [itex]R_{00}[/itex] was about 4 pages of work

yup that's normal if you write small

and my final answer was about 10 lines long...has something gone wrong or is this typical?

I'm not sure what the answer is to this metric, but the components of the Ricci tensor usually simplify to simple second order differential equations.

Hobson has a nice step by step calculation on this.
 

1. What are Christoffel symbols?

Christoffel symbols are mathematical objects used to describe the curvature of a space. They are a set of numbers that represent the connection between the coordinates of a space and the metric of that space.

2. Why do we need to compute Christoffel symbols for a metric with time variable?

In physics, we often encounter situations where both space and time are involved, such as in the theory of relativity. In order to accurately describe the dynamics of these systems, we need to take into account the curvature of spacetime. Computing Christoffel symbols for a metric with a time variable allows us to do this.

3. How do you compute Christoffel symbols for a metric with time variable?

The process for computing Christoffel symbols for a metric with a time variable involves first defining the metric in terms of both space and time coordinates, and then using the appropriate equations to calculate the Christoffel symbols. This can be a complex process and may require advanced mathematical techniques.

4. What is the significance of Christoffel symbols for a metric with time variable?

Christoffel symbols are important in understanding the curvature of spacetime and its effects on physical systems. They can be used to calculate quantities such as geodesic equations, which describe the paths that objects follow in curved space.

5. Are there any applications of computing Christoffel symbols for a metric with time variable?

Yes, there are many applications of computing Christoffel symbols for a metric with a time variable. Some examples include general relativity, cosmology, and black hole physics. Understanding the curvature of spacetime is essential in these fields and computing Christoffel symbols is a crucial step in this process.

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