A vector field is parallel transported along a curve with tangent
where is the parameter along the curve [ usually taken to be the proper time if the curve is timelike ] if and only if
In an a inertial frame this is
so in a general frame the condition becomes:
i.e. we just replace the partial derivatives (,) with a covariant derivative (;). This is called the `` comma goes to semicolon'' rule, i.e. work things out in a local inertial frame and if it is a tensor equation it will be valid in all frames.
Figure 6.1: Parallel transport of a vector along a timelike curve with tangent .
The curve is a geodesic if it parallel transports its own tangent vector:
This is the closest we can get to defining a straight line in a curved space. In flat space a tangent vector is everywhere tangent only for a straight line. Now
Since and we can write this as
This is the geodesic equation. It is a second order differential equation for , so one gets a unique solution by specifying an initial position and velocity .