Required math: algebra, calculus
Required physics: none
Reference: d’Inverno, Ray, Introducing Einstein’s Relativity (1992), Oxford Uni Press. – Section 6.4; Problem 6.8.
Parallel transport of a vector along a curve parametrized by a variable occurs if we can satisfy the condition
where is the tangent to the curve.
Parallel transport is usually defined as moving the vector (or tensor) along the curve without changing it, either in magnitude or direction (or in any of the dimensions, if we’re dealing with a higher-rank tensor). This condition is expressed by requiring the total derivative of the tangent vector to be zero, as the above equation specifies.
If we relax this condition a bit, and require only that the vector has the same direction, but not necessarily the same magnitude, as it propagates along the curve, then the condition becomes instead
where is some scalar function of . This is easier to see if we consider a 3-d example in flat space. If we have a 3-d vector field where are constants (so the vector field is constant over all space), then if we propagate this vector along any curve, it remains the same, so it satisfies the condition for parallel transport along any curve.
Now suppose we multiply this field by some scalar function , where is the position vector. We then get
If we consider a particular curve then we can write as a function of along this curve, so that along the curve we have
Since is merely multiplied by a scalar as we move along the curve, all instances of are parallel everywhere on the curve. If we take the derivative with respect to , we get
or, if we set we get
(OK, we could have just required that the vector field at each point on the curve is itself is a scalar multiple of the vector at a given point, but the condition here is a bit more general in that it doesn’t make reference to any particular point on the curve.)
Now we can look at a very special case. If the vector being transported along the curve is the tangent vector itself, then that tangent vector will be propagated parallel to itself if it satisfies the above condition, that is
It’s worth clearing up a bit of confusion that arises (at least for me) in d’Inverno’s section 6.4. He initially says that a tensor is parallely propagated along a curve if , which is fine. However, he then says the tangent vector is parallely propagated if it satisfies 7, which is not the same thing. Equation 7 means only that the tangent vector remains parallel to itself, but does not guarantee that it retains the same magnitude; that happens only if everywhere on the curve. The terminology is confusing, since a vector is actually propagated parallel to itself in both cases, but he defines (in his equation 6.32) the particular case where as parallel propagation (that is, the tensor is propagated unchanged in direction or magnitude), and then goes on to use the more general definition 7 (where only the direction is unchanged) in discussing tangent vectors. He does return to the case in equation 6.36.
If a curve on which the tangent vector satisfies 7 can be found, such a curve is called an affine geodesic. Further, if the curve can be parametrized in such a way then the parameter is called an affine parameter. An affine parameter is often given the symbol instead of .
It’s important to note that is not a parameter that defines a curve. It serves only to give the proportionality between the total derivative of a vector and the vector itself as you move along the curve.
Starting from 7, we have
using the chain rule to condense the first term. Thus we get the relation
In the case of a curve parametrized by an affine parameter, we get