The geodesic equation describes a curve whose tangent vector undergoes parallel transport along itself. In local coordinates, for the Levi-Civita connection of a Riemannian metric or pseudo-Riemannian metric, it is
 |
(1)
|
where 
is an affine parameter, 
are Christoffel
symbols of the second kind, and Einstein summation
is used (Carroll 1997, pp. 68-71). A general reparametrization need not preserve
this form of the equation.
For a flat metric tensor in coordinates 
with constant metric
tensor components, the Christoffel symbols
vanish, so the equation reduces to
 |
(2)
|
For a timelike curve in Minkowski space, the proper time 
is an affine parameter along a geodesic.
With the Minkowski metric 
and units in which the speed of light is
1, it satisfies
 |
(3)
|
This last relation defines proper time along a timelike curve; it is not itself the geodesic equation.
