Geodesic Deviation: Difference between revisions
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[[ | [[:en:Geodesic deviation|English Page]] | ||
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== My Draft == | |||
Body of this § moved to talk page. | |||
From the standpoint of counting indices alone a four-[[Index (mathematics)|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. | == Overview == | ||
In [[:en:differential geometry|differential geometry]], the '''geodesic deviation equation''' is an equation involving the [[:en:Riemann curvature tensor|Riemann curvature tensor]], which measures the change in separation of neighbouring [[:en:geodesic]]s. In the language of mechanics it measures the rate of relative [[:en:acceleration]] of two particles moving forward on neighbouring geodesics. | |||
From the standpoint of counting indices alone a four-[[:en:Index (mathematics)|index]] [[:en:tensor|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 [[:en: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 [[:en:torsion]] of the bundle and they may twist around each other. | |||
: <math> a^a = - R_{bcd}^{\ \ \ a} X^b T^c T^d</math> | : <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). | In textbooks, the equation is usually derived in a [[:en:handwaving]] manner. However, it can be derived from the second [[:en:covariant variation]] of the point particle [[:en:Lagrangian]], or from the first variation of a combined Lagrangian. The Lagrangian approach has two other advantages. First it allows various formal approaches of [[:en:quantization]] to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any [[:en:dynamical system]] which has a one [[:en:spacetime]] indexed momentum appears to have a corresponding generalization of geodesic deviation). | ||
==References== | ==References== | ||
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==See also== | ==See also== | ||
*[[Bernhard Riemann]] | *[[:en:Bernhard Riemann]] | ||
*[[Curvature]] | *[[:en:Curvature]] | ||
*[[Glossary of Riemannian and metric geometry]] | *[[:en:Glossary of Riemannian and metric geometry]] | ||
==External links== | ==External links== | ||
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{{relativity-stub}} | {{relativity-stub}} | ||
Latest revision as of 10:45, 29 December 2023
My Draft
Body of this § moved to talk page.
Overview
In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring en:geodesics. In the language of mechanics it measures the rate of relative en: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 en: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 en: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 en:handwaving manner. However, it can be derived from the second en:covariant variation of the point particle en:Lagrangian, or from the first variation of a combined Lagrangian. The Lagrangian approach has two other advantages. First it allows various formal approaches of en:quantization to be applied to the geodesic deviation system. Second it allows deviation to be formulated for much more general objects than geodesics (any en:dynamical system which has a one en: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.)
See also
External links
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