Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Problem P17.5.
The invariant interval is
so the metric is
with the expression for the Christoffel symbols:
The Christoffel symbols are thus
Now consider a vector field (where is a constant) in rectangular coordinates. In these coordinates, its covariant derivative is just the normal derivative, so
In the semi-log system, we have
To use this formula we need in the system, which we can get by the usual transformation, using:
With these transformations, we get
The difference in gradients is because of the different scales used in the vertical direction. Consider first the vertical component of . In rectangular coordinates this has a constant value for a given value of namely . However, if we use to measure vertical distance, a unit change in results in a larger and larger change in the higher up the vertical axis we go, so must increase as increases, even if is held constant. This is reflected by 17, where is proportional to as well as .
If we used ordinary derivatives to calculate the gradient of in the system, we would get . However, the ‘true’ value of the vertical component of doesn’t change as we move up or down a vertical line, and this is reflected by the correction term that is present in the covariant derivative 15, with the result that .
The value of again reflects the fact that a vertical component that is constant in rectangular coordinates must get numerically larger with increasing height in the system.
As a final test that all is well, we can use the standard tensor transformation to transform 19 back to rectangular coordinates, where we denote rectangular coordinates by and the semi-log system by :
The only non-zero component of is , so the RHS has only one non-zero term, which occurs when , , and . For this term we have
The overall transformation thus gives 14 back again.