In general relativity, 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.
[1] Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor,[1] and the trajectory of an object solely under the influence of gravity is called a geodesic.
Specifically, Aμ is found by taking the directional covariant derivative of X along T twice: The geodesic deviation equation relates Aμ, Tμ, Xμ, and the Riemann tensor Rμνρσ:[2][3] An alternate notation for the directional covariant derivative
First it allows various formal approaches of quantization to be applied to the geodesic deviation system.
[citation needed] 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 Tμ is approximately (1, 0, 0, 0); i.e., only the timelike component is nonzero.