Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 5; Problem 5.4.
As an example of a different coordinate system, we can define the semi-log system by introducing coordinates and defined as:
where is a constant. If and have units of length, then must have units of . Note that this means that and have different units, with having the units of length and being dimensionless.
The curves of constant are just the same as the curves of constant , that is, vertical lines. The curves of constant are defined by or . These are horizontal lines, although the spacing for equal steps in will translate into the variable spacing seen on semi-log plots.
If an object has an acceleration in the rectangular system, then we can find its acceleration in the semi-log system in the usual way.
The units of are still those of acceleration, but the units of are .
We can work out the metric of the semi-log system from the rectangular metric by direct calculation:
The length squared of is an invariant, since
The basis vectors in the semi-log system have lengths obtainable from the the metric:
Incidentally, the question part (e) as written in Moore’s book doesn’t make sense; he asks for the length of the basis vector . If we want to use partial derivatives as basis vectors, we need to define a curve along which to take the derivative. Since Moore doesn’t even mention the use of partial derivatives as a basis in the text, I can only assume this is a typo.