- #1
ChrisJ
- 70
- 3
Was not sure weather to post, this here or in differential geometry, but is related to a GR course, so...
I am having some trouble reproducing a result, I think it is mainly down to being very new to tensor notation and operations.
But, given the metric ##ds^2 = -dudv + \frac{(v-u)^2}{4} \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)##
[tex]
g_{\alpha\beta} =
\begin{pmatrix}
0 & -\frac{1}{2} & 0 & 0 \\
-\frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & \frac{(v-u)^2}{4} & 0 \\
0 & 0 & 0 & \frac{(v-u)^2}{4} \sin^2 \theta
\end{pmatrix}
[/tex]
and given this definition of the d'Alambert operator ##\Box := g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}## , reproduce the following given the d'Alambert acting on a function ##f(u,v)##
[tex]
\Box f(u,v) = 4 \left( -\frac{1}{v-u} \frac{\partial f}{\partial u} + \frac{\partial f}{\partial v} - \frac{\partial^2 f}{\partial u \partial v} \right)
[/tex]
And when I try to to reproduce it, I can see from the definition that the only non-zero parts are where the inverse metric components are ##g^{uv} = -2## and ##g^{vu} = -2 ## . The ##g^{\theta \theta} ## and ##g^{\phi \phi}## bits would be zero since the function is just of ##u## and ##v##.
So what I get is this...
[tex]
\Box := g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\\
\Box f(u,v) = g^{uv}\partial_{u}\partial_{v} f+ g^{vu}\partial_{v}\partial_{u}f = -4 \frac{\partial^2 f}{\partial u \partial v}
[/tex]
And I can't seem to see what I am missing here? Any help is really appreciated. Thanks.
I am having some trouble reproducing a result, I think it is mainly down to being very new to tensor notation and operations.
But, given the metric ##ds^2 = -dudv + \frac{(v-u)^2}{4} \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)##
[tex]
g_{\alpha\beta} =
\begin{pmatrix}
0 & -\frac{1}{2} & 0 & 0 \\
-\frac{1}{2} & 0 & 0 & 0 \\
0 & 0 & \frac{(v-u)^2}{4} & 0 \\
0 & 0 & 0 & \frac{(v-u)^2}{4} \sin^2 \theta
\end{pmatrix}
[/tex]
and given this definition of the d'Alambert operator ##\Box := g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}## , reproduce the following given the d'Alambert acting on a function ##f(u,v)##
[tex]
\Box f(u,v) = 4 \left( -\frac{1}{v-u} \frac{\partial f}{\partial u} + \frac{\partial f}{\partial v} - \frac{\partial^2 f}{\partial u \partial v} \right)
[/tex]
And when I try to to reproduce it, I can see from the definition that the only non-zero parts are where the inverse metric components are ##g^{uv} = -2## and ##g^{vu} = -2 ## . The ##g^{\theta \theta} ## and ##g^{\phi \phi}## bits would be zero since the function is just of ##u## and ##v##.
So what I get is this...
[tex]
\Box := g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\\
\Box f(u,v) = g^{uv}\partial_{u}\partial_{v} f+ g^{vu}\partial_{v}\partial_{u}f = -4 \frac{\partial^2 f}{\partial u \partial v}
[/tex]
And I can't seem to see what I am missing here? Any help is really appreciated. Thanks.