Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lie derivative and Riemann tensor

  1. Apr 12, 2008 #1
    Suppose you have a spacetime with an observer at rest at the origin, and the surface at t = 0 going through the origin, and passing through the surface there are geodesics along increasing time. Then as you get a small ways away from the surface, the geodesics start to deviate from each other. Within a small region around the origin, the geodesic deviation is [tex]\tau x^b {R_{0b0}}^d[/tex], where [tex]\tau[/tex] is the proper time as measured by the observer, [tex]x^b[/tex] is the position vector on the surface at t = 0, and [tex]{R_{0b0}}^d[/tex] is the Riemann tensor at the origin. So the 4-velocity of points starting from rest at t = 0 is [tex]\tau x^b {R_{0b0}}^d + (1,0,0,0)[/tex].
    So can you get the Riemann tensor, at least the [tex]{R_{0b0}}^d[/tex] components, as some kind of Lie derivative - what happens to the position vector of a point as it's carried along by the time flow vector field? I think it'd be a second Lie derivative, the points start out at rest so the first Lie derivative would be 0.
    Last edited: Apr 12, 2008
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted