Tensors and Relativity: Chapter 6

Parallel transport and geodesics

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A vector field tex2html_wrap_inline3398 is parallel transported along a curve with tangent


where tex2html_wrap_inline3612 is the parameter along the curve [ usually taken to be the proper time tex2html_wrap_inline3614 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 tex2html_wrap_inline3398 along a timelike curve with tangent tex2html_wrap_inline3622 .

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 tex2html_wrap_inline3624 and tex2html_wrap_inline3626 we can write this as


This is the geodesic equation. It is a second order differential equation for tex2html_wrap_inline3628 , so one gets a unique solution by specifying an initial position tex2html_wrap_inline3418 and velocity tex2html_wrap_inline3632 .