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
.