Mathematics of General Relativity - The Geodesic Equations

The Geodesic Equations

Once the EFE are solved to obtain a metric, it remains to determine the motion of inertial objects in the spacetime. In general relativity, it is assumed that inertial motion occurs along timelike and null geodesics of spacetime as parameterized by proper time. Geodesics are curves that parallel transport their own tangent vector ; i.e., . This condition, the geodesic equation, can be written in terms of a coordinate system with the tangent vector :

where denotes the derivative by proper time, with τ parametrising proper time along the curve and making manifest the presence of the Christoffel symbols.

A principal feature of general relativity is to determine the paths of particles and radiation in gravitational fields. This is accomplished by solving the geodesic equations.

The EFE relate the total matter (energy) distribution to the curvature of spacetime. Their nonlinearity leads to a problem in determining the precise motion of matter in the resultant spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energy-momentum tensor the sum of that for the planet and the star. The gravitational field of the planet affects the total spacetime geometry and hence the motion of objects. It is therefore reasonable to suppose that the field equations can be used to derive the geodesic equations.

When the energy-momentum tensor for a system is that of dust, it may be shown by using the local conservation law for the energy-momentum tensor that the geodesic equations are satisfied exactly.