Calculating the components of the Ricci tensor

[jore1]

Homework Statement

Given the line element $ds^2=a^2dt^2 -a^2dx^2 - \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$
(I) Calculate $\Gamma^{0}_{12}$

Now assume the following values for the connection coefficients: $\Gamma^{0}_{12}=\Gamma^{0}_{21}=\Gamma^{0}_{10}=\Gamma^{0}_{01}=1$, $\Gamma^{1}_{22}=\frac{e^{2x}}{2}, \Gamma^{2}_{10}=-e^{-x}$, $\Gamma^{1}_{02}=\Gamma^{1}_{20}=\frac{e^{x}}{2}$ and all others are zero.

(II) Calculate $R_{22}$


I am currently working through an exercise to calculate the component $R_{22}$ of the Ricci tensor for the line element $ds^2=a^2dt^2 -a^2dx^2 - \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$. The question first asks for the value of $\Gamma^{0}_{12}$, which I calculate to be $\frac{e^{x}}{2}$. I am told to assume the following values for the connection coefficients: $\Gamma^{0}_{12}=\Gamma^{0}_{21}=\Gamma^{0}_{10}=\Gamma^{0}_{01}=1$, $\Gamma^{1}_{22}=\frac{e^{2x}}{2}, \Gamma^{2}_{10}=-e^{-x}$, $\Gamma^{1}_{02}=\Gamma^{1}_{20}=\frac{e^{x}}{2}$ and all others are zero.

Using the relation for the Ricci tensor, I find that the only non-zero components are: $R_{22}=\partial_1(\Gamma^{0}_{12})+\Gamma^{0}_{10}\Gamma^{1}_{22}-\Gamma^{0}_{21}\Gamma^{1}_{02}-\Gamma^{1}_{20}\Gamma^{0}_{12}$. This is where the problem arises: using the assumed values for the connection coefficients (with $\Gamma^{0}_{12}=\Gamma^{0}_{21}=1$) I find that $R_{22}=e^{2x}-e^{x}$, while using the values $\Gamma^{0}_{12}=\Gamma^{0}_{21}=\frac{e^{x}}{2}$ (the rest being those assumed) I find that $R_{22}=e^{2x}$. I am told that the second result is correct. It seems to be the case that the assumed value for $\Gamma^{0}_{12}$ is incorrect.

Could someone provide clarification as to whether there is indeed a mistake in the question? As a beginner in GR, I find myself questioning the basics.

Relevant Equations

$R_{ab}=\Gamma^{d}_{ab,d}-\Gamma^{d}_{da,b}+\Gamma^{d}_{de}\Gamma^{e}_{ab}-\Gamma^{d}_{ae}\Gamma^{e}_{db}$

$\Gamma^{a}_{bc}=\frac{1}{2}g^{ad}(g_{bd,c}+g_{cd,b}-g_{bc,d}$

(I) Using the relevant equation I find this to be $\frac{e^{x}}{2}$.

(II) Using the relation for the Ricci tensor, I find that the only non-zero components are: $R_{22}=\partial_1(\Gamma^{0}_{12})+\Gamma^{0}_{10}\Gamma^{1}_{22}-\Gamma^{0}_{21}\Gamma^{1}_{02}-\Gamma^{1}_{20}\Gamma^{0}_{12}$. This is where the problem arises: using the assumed values for the connection coefficients (with $\Gamma^{0}_{12}=\Gamma^{0}_{21}=1$) I find that $R_{22}=e^{2x}-e^{x}$, while using the values $\Gamma^{0}_{12}=\Gamma^{0}_{21}=\frac{e^{x}}{2}$ (the rest being those assumed) I find that $R_{22}=e^{2x}$.

Could someone provide clarification as to whether there is indeed a mistake in the question? As a beginner in GR, I find myself questioning the basics.

 

[TSny]

I could have slipped up, but I'm getting $\Gamma^0_{1 2} = -\large\frac{e^x} 6$.

Please double-check that there are no typos in your expression for $ds^2$. It will also help if you list the expressions you used for the nonzero $g_{\mu \nu}$ and the nonzero $g^{\mu \nu}$.

 

[jore1]

Yes, I mistyped the metric should be: $ds^2=a^2dt^2 -a^2dx^2 + \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$

This then gives: $g_{ab} = \left[\begin{matrix}a^{2} & 0 & a^{2} e^{x} & 0\\0 & - a^{2} & 0 & 0\\a^{2} e^{x} & 0 & + \frac{a^{2} e^{2 x}}{2} & 0\\0 & 0 & 0 & - a^{2}\end{matrix}\right]$

and $g^{ab} = \frac{1}{a^2}\left[\begin{matrix}-1 & 0 & 2 e^{-x} & 0\\0 & - 1 & 0 & 0\\ 2e^{-x} & 0 & -2e^{-2x} & 0\\0 & 0 & 0 & - 1\end{matrix}\right]$

Apologies for the mistake.

 

[TSny]

the metric should be: $ds^2=a^2dt^2 -a^2dx^2 + \frac{a^2e^{2x}}{2}dy^2 +2a^2e^xdydt -a^2dz^2$

Ok. I agree with you, $\Gamma^0_{12} = \large \frac{e^x} 2$.

 

[jore1]

Thanks for the response. So is my conclusion correct that there is a mistake in the question? I think the idea was that the component $R_{22}$ was supposed to be $e^{2x}$ either way. Though this doesn't seem to be work.

 

[TSny]

So is my conclusion correct that there is a mistake in the question?

Yes, I think there must be a mistake in the question. Here's what I find for the nonzero connection coefficients, $$\Gamma^0_{10} = \Gamma^0_{01} = 1$$ $$\Gamma^0_{12} = \Gamma^0_{21} = \frac {e^x} 2$$ $$\Gamma^2_{10} = \Gamma^2_{01} = -e^{-x}$$ $$\Gamma^1_{22} = -\frac{e^{2x}}{2}$$ $$\Gamma^1_{02} = \Gamma^1_{20} = -\frac{e^{x}}{2}$$ The last two differ in sign from the problem statement.

 

[dx]

R_11 = e^-2x

R_12 = 1 - e^2x

R_21 = 1 + e^2x

R_22 = e^2x