# Lie derivative and Riemann tensor

1. Apr 12, 2008

### lark

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 $$\tau x^b {R_{0b0}}^d$$, where $$\tau$$ is the proper time as measured by the observer, $$x^b$$ is the position vector on the surface at t = 0, and $${R_{0b0}}^d$$ is the Riemann tensor at the origin. So the 4-velocity of points starting from rest at t = 0 is $$\tau x^b {R_{0b0}}^d + (1,0,0,0)$$.
So can you get the Riemann tensor, at least the $${R_{0b0}}^d$$ 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.
Laura

Last edited: Apr 12, 2008