Geodesic deviation equation

In general relativity, geodesic deviation describes the tendency of objects to approach or recede from one another while moving under the influence of a spatially varying gravitational field. Put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towards or away from each other, producing a relative acceleration between the objects.

Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor, and the trajectory of an object solely under the influence of gravity is called a geodesic. The geodesic deviation equation relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In differential geometry, the geodesic deviation equation is more commonly known as the Jacobi equation.

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

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:

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:

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