Geodesic deviation

To quantify geodesic deviation, one begins by setting up a family of closely spaced geodesics indexed by a continuous variable s and parametrized by an affine parameter τ. That is, for each fixed s, the curve swept out by γ_s(τ) as τ varies is a geodesic. When considering the geodesic of a massive object, it is often convenient to choose τ to be the object's proper time. If x^μ(s, τ) are the coordinates of the geodesic γ_s(τ), then the tangent vector of this geodesic is

${\displaystyle T^{\mu }={\frac {\partial x^{\mu }(s,\tau )}{\partial \tau }}.}$

If τ is the proper time, then T^μ is the 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:

${\displaystyle X^{\mu }={\frac {\partial x^{\mu }(s,\tau )}{\partial s}}.}$

The relative acceleration A^μ of the two objects is defined, roughly, as the second derivative of the separation vector X^μ as the objects advance along their respective geodesics. Specifically, A^μ is found by taking the directional covariant derivative of X along T twice:

${\displaystyle A^{\mu }=T^{\alpha }\nabla _{\alpha }\left(T^{\beta }\nabla _{\beta }X^{\mu }\right).}$

The geodesic deviation equation relates A^μ, T^μ, X^μ, and the Riemann tensor R^μ_νρσ:[2][3]

${\displaystyle A^{\mu }={R^{\mu }}_{\nu \rho \sigma }T^{\nu }T^{\rho }X^{\sigma }.}$

An alternate notation for the directional covariant derivative ${\displaystyle T^{\alpha }\nabla _{\alpha }}$ is ${\displaystyle D/d\tau }$, so the geodesic deviation equation may also be written as

${\displaystyle {\frac {D^{2}X^{\mu }}{d\tau ^{2}}}={R^{\mu }}_{\nu \rho \sigma }T^{\nu }T^{\rho }X^{\sigma }.}$

The geodesic deviation equation can be derived from the second variation of the point particle Lagrangian along geodesics, or from the first variation of a combined Lagrangian. The Lagrangian approach has two 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).

The connection between geodesic deviation and tidal acceleration can be seen more explicitly by examining geodesic deviation in the weak-field limit, 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^μ 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

${\displaystyle A^{i}={R^{i}}_{0j0}X^{j},}$

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

${\displaystyle {R^{i}}_{0j0}=-{\frac {\partial ^{2}\Phi }{\partial x^{i}\partial x^{j}}},}$

which is the tidal tensor of the Newtonian potential.

• Bernhard Riemann
• Curvature
• Glossary of Riemannian and metric geometry

• Stephani, Hans (1982), General relativity - an introduction to the theory of the gravitation field, Cambridge University Press, ISBN 0-521-37066-3.
• Wald, Robert M. (1984), General Relativity, ISBN 978-0-226-87033-5.

• General Relativity and Quantum Cosmology
• Tensors and Relativity: Geodesic deviation Archived 2011-11-16 at the Wayback Machine