- #1

- 918

- 16

## Main Question or Discussion Point

Can someone help me understand something on page 220 of the book 'A First Course in General Relativity' by Bernard Schutz?

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?

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?