- #1

- 918

- 16

Near the middle of the page, the line element is given as

[itex]ds^2 = -dudv + f^2(u)dx^2 + h^2(u)dy^2[/itex]

(I changed g to h so I can talk about the metric tensor) which I think is:

[tex]

\Large

\[

\mathbf{g}_{\alpha\beta} =

\left( \begin{array}{cccc}

0 & -\frac{1}{2} & 0 & 0 \\

-\frac{1}{2} & 0 & 0 & 0 \\

0 & 0 & f^2(u) & 0 \\

0 & 0 & 0 & h^2(u)

\end{array} \right) \]

[/tex]

Is this correct? If so, then:

[tex]

\Large

\[

\mathbf{g}^{\alpha\beta} =

\left( \begin{array}{cccc}

0 & -2 & 0 & 0 \\

-2 & 0 & 0 & 0 \\

0 & 0 & \frac{1}{f^2(u)} & 0 \\

0 & 0 & 0 & \frac{1}{h^2(u)}

\end{array} \right) \]

[/tex]

How am I doing so far?

Then using equation 5.75 on page 143

[itex]\Large \Gamma^{\gamma}_{\beta\mu} = \frac{1}{2}\mathbf{g}^{\alpha\gamma}(g_{\alpha\beta,\mu} + g_{\alpha\mu,\beta} - g_{\beta\mu,\alpha})[/itex]

we have

[itex]\Large \Gamma^{v}_{xx} = \frac{1}{2}\mathbf{g}^{\alpha v}(g_{\alpha x,x} + g_{\alpha x, x} - g_{xx,\alpha})[/itex]

but the only value of [itex]\alpha[/itex] for which [itex]\mathbf{g}^{\alpha v}[/itex] is not zero is u. so

[itex]\Large \Gamma^{v}_{xx} = \frac{1}{2}\mathbf{g}^{uv}(g_{ux,x} + g_{ux, x} - g_{xx,u}) = \frac{1}{2}(-2)(-2f\dot{f}) = 2f\dot{f}[/itex]

but the book has:

[itex]\Large \Gamma^{v}_{xx} = 2\dot{f}/f[/itex]

Is the book wrong or am I?