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.

Mathematically, the tidal force in general relativity is described by the Riemann curvature tensor,

:<math> A^\mu = {R^\mu}_{\nu\rho\sigma} T^\nu T^\rho X^\sigma.</math>

An alternate notation for the directional 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 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).

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, 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 of the Newtonian potential.

See also

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

References

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  • General Relativity and Quantum Cosmology
  • Tensors and Relativity: Geodesic deviation