Geodesic deviation

The Deviationsgleichung or geodesic deviation is an equation of the Riemann geometry and general relativity and describes the change in the distance of two neighboring geodesics with the help of the Riemann curvature tensor. By means of this equation can be determined whether and in what way, a space is curved by the relative acceleration between two test pieces is measured in neighboring geodesics. If no relative acceleration between two geodesics measured, so the room is flat. The relative acceleration between the test specimens is due only on the curvature of space, not by their mutual gravitational force that would act additionally in a real experiment.

Formulation of the equation

The mathematical formulation of the Deviationsgleichung is:

And simplified in a torsion space to

The symbols in the equations indicate the following case:

• Denotes the geodesic and whose tangent vector.
• Is the distance vector of two neighboring geodesics, and therefore the linear change in the distance between two infinitesimally nearby geodesics.
• The Torsionstensor the room, in particular, the vector that closes the by and spanned parallelogram. This vector is torsion-free spaces equal to zero.
• Riemann curvature tensor is the.
• In addition, the Einstein summation convention is used, the Greek indices run from and are tensors and 1st stage.
• Denotes the covariant derivative.

In flat space the distance between two intersecting geodesics and increases proportionally to. This is not the case, then this is a symptom for the curvature of the room, and corresponds to the above equation at nonvanishing curvature.