Geodesic Deviation
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In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics. In the language of mechanics it measures the rate of relative acceleration of two particles moving forward on neighbouring geodesics.
From the standpoint of counting indices alone a four-index tensor is needed to measure the change, and that Riemann is the right one. There's a 4-vector velocity along one geodesic, which has to be folded into one index. There's an infinitesimal separation vector between the two geodesics, which also eats up an index on Riemann. A third (free) index is needed to exit with the rate of change of the displacement. The fourth (summed) index is less obvious. It is assumed that the two geodesics are neighboring ones that start out parallel. So at first there is no relative velocity. But we have to use the velocity vector (4-velocity) along both neighbors, so it comes in twice and the last index is used. Note that it is not just the distance between the geodesics that can change, there can also be torsion of the bundle and they may twist around each other.
- <math> a^a = - R_{bcd}^{\ \ \ a} X^b T^c T^d</math>
In textbooks, the equation is usually derived in a handwaving manner. However, it can be derived from the second covariant variation of the point particle Lagrangian, or from the first variation of a combined Lagrangian. The Lagrangian approach has two other advantages. First it allows various formal approaches of quantization to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any dynamical system which has a one spacetime indexed momentum appears to have a corresponding generalization of geodesic deviation).
References
General relativity - an introduction to the theory of the gravitation field. Hans Stephani, Cambridge University Press 1982, 1990. ISBN 0-521-37066-3. - ISBN 0-521-37941-5 (pbk.)
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DRAFT
This article largely a condensation of the material in Stephanis text thru § 1.4 and chapter 3 of Will in which I have two goals: 1) to get a better understanding of GR about which I have doubts that I've expressed elsewhere and 2) as a proofing of math ML in wikimedia.
¹ Hans Stephani. General Relativity.
² Clifford M. Will. Theory and Experiment in graviational physics.
