Geodesic deviation equation
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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 toward on neighbouring geodesics. It is easy to see, from the standpoint of counting indices alone, that one needs a four-index tensor (Riemann is the right one) to handle this problem. 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. So that eats up two indices, and a third (free) index is needed to exit with the rate of change of the displacement. So why is there a fourth (summed) index? Well, it is assumed that the two geodesics are neighboring ones that start out parallel. So there is no relative velocity, at first. 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; they may twist around each other. The details can be found on [1].
In textbooks it is usually derived in a handwaving manner. It can however 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 other advantages: 1) it allows various formal approaches of quantization to be applied to the geodesic deviation system, 2) 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).
[edit] 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.)
