- #1
TheMan112
- 43
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Hello everybody, this is my first post here.
I need help with a task in General Relativity.
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Given we have a spherical shell in 2-space of radius [tex]\rho[/tex]. With the line element:
[tex]ds^{2} = g_{ab} dx^{a}dx^{b} = \rho d \theta^{2} + \rho^2 sin^2(\theta) d \phi^2[/tex]
[tex](a,b \in 1,2)[/tex]
Then calculate the following:
[tex]g^{ab}, \Gamma^c_{ab}, R^1_{212}, R^2_{121}, R_{11}, R_{22}, R[/tex]
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To compute the metric, I'm immediately considering the Schwarzschild solution since this is a spherically symmetric problem.
The covariant metric will then be:
[tex]g_{ab} = diag(e^v, -e^\lambda, -\rho^2, -\rho^2 sin^2 \theta)[/tex]
But since:
[tex]e^v dt^2 = 0[/tex] and [tex]e^\lambda d \rho^2 = 0[/tex]
Then I'm not sure what to do. However, since:
[tex](a,b \in 1,2)[/tex]
Can I then set the following?
[tex]g_{ab} = diag(-\rho^2, -\rho^2 sin^2 \theta)[/tex]
If not, then how should I attack the problem?
----
Moreover, I'm not sure how to calculate the Christoffel symbol from the metric, could anyone give me an example how to perform the calculation in my case based on this equation:
[tex]\Gamma^c_{ab} = \frac{1}{2} g^{cl} (g_{la,b} + g_{lb,a} - g_{ab,l})[/tex]
Thanks in advance.
/TheMan112
I need help with a task in General Relativity.
-------
Given we have a spherical shell in 2-space of radius [tex]\rho[/tex]. With the line element:
[tex]ds^{2} = g_{ab} dx^{a}dx^{b} = \rho d \theta^{2} + \rho^2 sin^2(\theta) d \phi^2[/tex]
[tex](a,b \in 1,2)[/tex]
Then calculate the following:
[tex]g^{ab}, \Gamma^c_{ab}, R^1_{212}, R^2_{121}, R_{11}, R_{22}, R[/tex]
-------
To compute the metric, I'm immediately considering the Schwarzschild solution since this is a spherically symmetric problem.
The covariant metric will then be:
[tex]g_{ab} = diag(e^v, -e^\lambda, -\rho^2, -\rho^2 sin^2 \theta)[/tex]
But since:
[tex]e^v dt^2 = 0[/tex] and [tex]e^\lambda d \rho^2 = 0[/tex]
Then I'm not sure what to do. However, since:
[tex](a,b \in 1,2)[/tex]
Can I then set the following?
[tex]g_{ab} = diag(-\rho^2, -\rho^2 sin^2 \theta)[/tex]
If not, then how should I attack the problem?
----
Moreover, I'm not sure how to calculate the Christoffel symbol from the metric, could anyone give me an example how to perform the calculation in my case based on this equation:
[tex]\Gamma^c_{ab} = \frac{1}{2} g^{cl} (g_{la,b} + g_{lb,a} - g_{ab,l})[/tex]
Thanks in advance.
/TheMan112
Last edited: