# Einstein Equations of this metric

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1. Apr 12, 2015

### unscientific

1. The problem statement, all variables and given/known data

(a) Find the christoffel symbols
(b) Find the einstein equations
(c) Find A and B
(d) Comment on this metric

2. Relevant equations

$$\Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha \beta} \right)$$

$$R_{v \beta} = \partial_\mu \Gamma_{\beta v}^\mu - \partial_\beta \Gamma_{\mu v}^\mu + \Gamma_{\mu \epsilon}^\mu \Gamma_{v \beta}^\epsilon - \Gamma_{\epsilon \beta}^\mu \Gamma_{v \mu}^\epsilon$$

3. The attempt at a solution

Part(a)

After some math, I found the christoffel symbols to be:
$\Gamma_{11}^0 = \frac{A A^{'}}{c^2}$
$\Gamma_{22}^0 = \frac{B B^{'}}{c^2}$
$\Gamma_{33}^0 = \frac{B B^{'}}{c^2}$
$\Gamma_{01}^1 = \frac{A^{'}}{A}$
$\Gamma_{02}^2 = \frac{B^{'}}{B}$
$\Gamma_{03}^3 = \frac{B^{'}}{B}$

Part (b)
Now brace yourselves for the ricci tensors...
$$R_{00} = -\partial_0 \left( \Gamma_{01}^1 + \Gamma_{02}^2 + \Gamma_{03}^3 \right) - \Gamma_{10}^1 \Gamma_{01}^1 - 2\Gamma_{20}^2 \Gamma_{02}^2$$
$$R_{00} = -\frac{A^{''}}{A} - 2 \frac{B^{''}}{B}$$

By symmetry, $R_{01} = R_{02} = R_{03} = R_{12} = R_{13} = R_{23} = 0$.

Now to find the $11$ component:
$$R_{11} = \partial_0 \Gamma_{11}^0 + \Gamma_{11}^0 \left( \Gamma_{10}^1 + \Gamma_{20}^2 + \Gamma_{30}^3 \right) - \Gamma_{11}^0 \Gamma_{10}^1 - \Gamma_{01}^1 \Gamma_{11}^0$$
$$= \partial_0 \Gamma_{11}^0 + 2 \Gamma_{11}^0 \Gamma_{20}^2 - \Gamma_{11}^0 \Gamma_{10}^1$$
$$R_{11} = \frac{A A^{''}}{c^2} + 2 \left( \frac{A}{B} \right) \frac{A^{'} B^{'}}{c^2}$$

By symmetry, to find $22$ and $33$ components, we swap $A$ with $B$:
$$R_{22} = R_{33} = \frac{B B^{''}}{c^2} + 2 \left( \frac{B}{A} \right) \frac{A^{'} B^{'}}{c^2}$$

The einstein field equations are given by:
$$G^{\alpha \beta} = \frac{8 \pi G}{c^4} T^{\alpha \beta} - \Lambda g^{\alpha \beta}$$

Thus, the simultaneous equations we seek are:
$$G^{00} = \frac{8 \pi G}{c^4} T^{00}$$
For $\mu, v \neq 0$ we have
$$R_{\mu v} = 0$$
So we simply equate $R_11 = 0$, $R_22 = R_{33} = 0$.

However, the equations don't match..

Last edited: Apr 12, 2015
2. Apr 16, 2015

### unscientific

bumpp

3. Apr 17, 2015

### unscientific

bumpp

4. Apr 18, 2015

### unscientific

bumpp

5. Apr 19, 2015

### unscientific

bumpp

6. Apr 20, 2015

### unscientific

bumpp

7. Apr 22, 2015

### unscientific

bumpp

8. Apr 23, 2015

### unscientific

bumpp

9. Apr 24, 2015

### unscientific

bumpp on part (b)/(c)

10. Apr 25, 2015

### unscientific

Would appreciate help with my "ricci-nightmare"

11. Apr 26, 2015

### unscientific

Anyone managed to get a different result for the ricci tensors yet?

12. Apr 30, 2015

### unscientific

anyone else had a go with the ricci tensors?

13. May 4, 2015

### unscientific

tried again, still didn't get the required ricci tensors.

14. May 4, 2015

### thierrykauf

Here is what I find for one term for Ricci

15. May 7, 2015

### unscientific

I think the term is not appearing, do you mind posting it again?

16. May 10, 2015

### unscientific

bumpp ricci

17. May 10, 2015

### thierrykauf

Hold on. Here is what I find

18. May 10, 2015

### thierrykauf

Here is what I find $$R_{00}=\Gamma^x_{x0}\Gamma^x_{x0} + \Gamma^x_{x0}\Gamma^y_{y0} + \Gamma^z_{z0}\Gamma^y_{y0}$$

19. May 14, 2015

### unscientific

Thanks alot for replying. I'll give it a go later today and post my updated work.

20. May 14, 2015

### thierrykauf

Please do. And let's see what you have.