Mathematical Expression
A timelike geodesic is a worldline which parallel transports its own tangent vector and maintains the magnitude of its tangent as a constant. If a curve has tangent then this can be expressed as
which says that the covariant derivative of the tangent in the direction of the tangent is zero. The above equation can be restated in terms of components of :
where
and
The full geodesic equation is therefore:
where s is the proper time or distance and is the Levi-Civita connection.
Proof- ,
- ,
- ,
- ,
- ,
The parameter s typically represents proper time for a timelike curve, or distance for a spacelike curve. This parameter cannot be chosen arbitrarily. Rather, it must be chosen so that the tangent vector has constant magnitude. This is referred to as an affine parametrization. Any two affine parameters are linearly related. That is, if r and s are affine parameters, then there exist constants a and b such that .
Read more about this topic: Geodesic (general Relativity)
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