# Geodesic deviation

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{{Short description|Bending of trajectories in general relativity by a tidal force}}
In [general relativity](/source/general_relativity), if two objects are set in motion along two initially parallel trajectories, the presence of a [tidal gravitational force](/source/tidal_force) will cause the trajectories to bend towards or away from each other, producing a relative [acceleration](/source/acceleration) between the objects.<ref name="ohanian">{{cite book|last1=Ohanian|first1=Hans|title=Gravitation and Spacetime|edition=1st|year=1976|pages=271&ndash;6}}</ref>

Mathematically, the tidal force in general relativity is described by the [Riemann curvature tensor](/source/Riemann_curvature_tensor),<ref name="ohanian" /> and the trajectory of an object solely under the influence of gravity is called a ''[geodesic](/source/geodesic)''. The '''geodesic deviation equation''' relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In [differential geometry](/source/differential_geometry), the geodesic deviation equation is more commonly known as the [Jacobi equation](/source/Jacobi_field).

==Mathematical definition==
To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable {{mvar|s}} and parametrized by an [affine parameter](/source/affine_parameter) {{mvar|τ}}. That is, for each fixed ''s'', the curve swept out by {{math|''γ''<sub>''s''</sub>(''τ'')}} as {{mvar|τ}} varies is a geodesic. When considering the geodesic of a massive object, it is often convenient to choose {{mvar|τ}} to be the object's [proper time](/source/proper_time). If {{math|''x''<sup>''μ''</sup>(''s'',''τ'')}} are the coordinates of the geodesic {{math| γ<sub>''s''</sub>(''τ'')}}, then the [tangent vector](/source/tangent_vector) of this geodesic is

:<math>T^\mu = \frac{\partial x^\mu(s, \tau)}{\partial \tau}.</math>

If {{mvar|τ}} is the proper time, then {{math|''T''<sup>''μ''</sup>}} is the [four-velocity](/source/four-velocity) of the object traveling along the geodesic.

One can also define a ''deviation vector'', which is the displacement of two objects travelling along two infinitesimally separated geodesics:

:<math>X^\mu = \frac{\partial x^\mu(s, \tau)}{\partial s}.</math>

The ''relative acceleration'' ''A''<sup>μ</sup> of the two objects is defined, roughly, as the second derivative of the separation vector ''X''<sup>μ</sup> as the objects advance along their respective geodesics. Specifically, ''A''<sup>μ</sup> is found by taking the directional [covariant derivative](/source/covariant_derivative) of ''X'' along ''T'' twice:
:<math> A^\mu = T^\alpha \nabla_\alpha \left(T^\beta \nabla_\beta X^\mu\right).</math>

The geodesic deviation equation relates ''A''<sup>μ</sup>, ''T''<sup>μ</sup>, ''X''<sup>μ</sup>, and the [Riemann tensor](/source/Riemann_curvature_tensor) ''R''<sup>μ</sup><sub>νρσ</sub>:<ref name="carroll">{{cite book|last=Carroll|first=Sean|title=Spacetime and Geometry|year=2004|pages=144&ndash;6}}</ref><ref>{{Cite book |last=Wald |first=Robert |title=General Relativity |year=1984 |pages=46–47}}</ref>
:<math> A^\mu = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>

An alternate notation for the directional [covariant derivative](/source/covariant_derivative) <math>T^\alpha \nabla_\alpha</math> is <math>D/d\tau</math>, so the geodesic deviation equation may also be written as
:<math>\frac{D^2 X^\mu}{d\tau^2} = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>

The geodesic deviation equation can be derived from the [second variation](/source/second_variation) of the point particle [Lagrangian](/source/Lagrangian_mechanics) along geodesics, or from the first variation of a combined Lagrangian.{{Clarify|date=September 2009}} The Lagrangian approach has two advantages. First it allows various formal approaches of [quantization](/source/Quantization_(physics)) 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](/source/dynamical_system) which has a one [spacetime](/source/spacetime) indexed momentum appears to have a corresponding generalization of geodesic deviation).{{Citation needed|date=September 2009}}

==Weak-field limit==
The connection between geodesic deviation and tidal acceleration can be seen more explicitly by examining geodesic deviation in the [weak-field limit](/source/Linearized_gravity), where the metric is approximately Minkowski, and the velocities of test particles are assumed to be much less than ''c''. Then the tangent vector ''T''<sup>μ</sup> is approximately (1, 0, 0, 0); i.e., only the timelike component is nonzero.

The spatial components of the relative acceleration are then given by
:<math> A^i = {R^i}_{0j0} X^j,</math>
where ''i'' and ''j'' run only over the spatial indices 1, 2, and 3.

In the particular case of a metric corresponding to the Newtonian potential Φ(''x'', ''y'', ''z'') of a massive object at ''x'' = ''y'' = ''z'' = 0, we have
:<math> {R^i}_{0j0} = -\frac{\partial^2\Phi}{\partial x^i \partial x^j},</math>
which is the [tidal tensor](/source/tidal_tensor) of the Newtonian potential.

==See also==
*[Bernhard Riemann](/source/Bernhard_Riemann)
*[Curvature](/source/Curvature)
*[Glossary of Riemannian and metric geometry](/source/Glossary_of_Riemannian_and_metric_geometry)

==References==
{{reflist}}

*{{Citation|title=General relativity - an introduction to the theory of the gravitation field|first=Hans|last=Stephani|publisher=Cambridge University Press|year=1982|isbn=0-521-37066-3}}.
*{{Citation | last1=Wald | first1=Robert M. | author1-link=Robert Wald | title=[General Relativity](/source/General_Relativity_(book)) | isbn=978-0-226-87033-5 | year=1984| publisher=University of Chicago Press }}.

==External links==
*[http://www.arXiv.org/abs/gr-qc/0404094 General Relativity and Quantum Cosmology]
*[http://www.mth.uct.ac.za/omei/gr/chap6/node11.html Tensors and Relativity: Geodesic deviation] {{Webarchive|url=https://web.archive.org/web/20111116174208/http://www.mth.uct.ac.za/omei/gr/chap6/node11.html |date=2011-11-16 }}

{{DEFAULTSORT:Geodesic Deviation Equation}}
Category:Geodesic (mathematics)
Category:Riemannian geometry
Category:Equations

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Adapted from the Wikipedia article [Geodesic deviation](https://en.wikipedia.org/wiki/Geodesic_deviation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Geodesic_deviation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.