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Tensors and Relativity: Chapter 6

# Parallel transport and geodesics

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 .