Equations of Motion - Geodesic Equation of Motion

Geodesic Equation of Motion

The above equations are valid in flat spacetime. In curved space spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a geodesic of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor g, the metric provides the notion of arc length (see line element for details), the differential arc length is given by:

and the geodesic equation is a 2nd-order differential equation in the coordinates, the general solution is a family of geodesics:

where Γμ_αβ is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system).

Given the mass-energy distribution provided by the stress–energy tensor Tαβ, the Einstein field equations are a set of non-linear 2nd-order partial differential equations in the metric, and imply the curvature of space time is equivalent to a gravitational field (see principle of equivalence). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation:

where ξα = (x₂)α − (x₁)α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and Rα_βγδ is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.

For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity.