Can someone help me understand something on page 220 of the book 'A First Course in General Relativity' by Bernard Schutz?(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question from Schutz, A First Course in GR

Loading...

Similar Threads - Question Schutz Course | Date |
---|---|

I Some geometry questions re Swartzchild metric | Friday at 9:27 AM |

I Schutz: question regarding geodesic deviation | Mar 21, 2017 |

Question from Schutz A First Course in GR | Sep 26, 2015 |

Schutz - A First Course in GR - Simple Summation Question | Feb 18, 2010 |

Another question from Schutz | Oct 4, 2005 |

**Physics Forums - The Fusion of Science and Community**