I'm trying to show that [tex]\frac{d}{dt}\; g_{\mu \nu} u^{\mu} v^{\nu} = 0[/tex] in the context of parallel transport (or maybe not zero), and I'm rather insecure about the procedure. This is akin to problem 3.14 in Hobson's et al. book (General Relativity an introduction for physicists).(adsbygoogle = window.adsbygoogle || []).push({});

As a guess, I tried the take the time derivative:

[tex]\frac{d}{dt}\; g_{\mu \nu} u^{\mu} v^{\nu}+g_{\mu \nu} \dot{u^{\mu}} v^{\nu}+g_{\mu \nu} u^{\mu} \dot{v^{\nu}}=0[/tex]

I was assuming a stationary metric, so the first part would be zero, leaving

[tex]g_{\mu \nu} \dot{u^{\mu}} v^{\nu}+g_{\mu \nu} u^{\mu} \dot{v^{\nu}}=0[/tex]

From there I can substitute in for [tex]\dot{u^{\mu}}[/tex] and [tex]\dot{v^{\nu}}[/tex].

Is this the right path to take? It seems there's then some index trickery involved to solve this.

Thanks!

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

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

# Parallel transport invariance

Loading...

Similar Threads - Parallel transport invariance | Date |
---|---|

A Is the Berry connection a Levi-Civita connection? | Jan 1, 2018 |

A Can you give an example of a non-Levi Civita connection? | Oct 30, 2017 |

I Understanding Parallel Transport | May 20, 2017 |

I Conservation of dot product with parallel transport | Jan 15, 2017 |

I Taylor expansion and parallel transport | Jun 15, 2016 |

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