Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 5; Problem 5.5.
As another example of a different coordinate system, we can define a sinusoidal system by introducing coordinates and defined as:
where and are constants. If and have units of length, then must have units of . If has units of length, then and both have units of length.
The curves of constant are just the same as the curves of constant , that is, vertical lines. The curves of constant are defined by . These are parallel horizontal sine curves displaced vertically by the constant .
The inverted relations are
We can work out the metric of the sinusoidal system from the rectangular metric by direct calculation
Incidentally, we can check this by calculating the reverse transformation metric:
For example (using ):
Similar calculations give the other components of the rectangular metric.
Now suppose we have an object moving with velocity such that and . In the sinusoidal system, we get
where the last line arises because , where is the time.
The square is
Thus the square of the velocity is invariant.
The unit vector is tangent to the constant- curves and points in the direction of increasing . Since the constant- curves are sine curves, this unit vector starts off horizontal (when ), then slopes downward as heads towards at which point it is horizontal again. Then it slopes upwards as heads towards and so on. Its magnitude is given by .
The other unit vector is tangent to the constant- curves, so it always points up, and always has a length of 1. Note that the off-diagonal elements of are zero when is an odd multiple of , which is where is horizontal, and thus perpendicular to , so .
The reason that is not constant, even though the velocity itself is constant is because the unit vector oscillates in both magnitude and direction. Since is a constant, the component multiplying this unit vector (that is, ) must also oscillate to compensate for the non-constant component.
In the rectangular system, the acceleration is zero (since is constant). In the sinusoidal system, , but . Thus the magnitude of would not be invariant under the transformation of coordinates, so this cannot be the correct way of calculating derivatives in a general coordinate system.