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].(adsbygoogle = window.adsbygoogle || []).push({});

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.

Laura

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# Lie derivative and Riemann tensor

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