Required math: algebra
Required physics: special relativity
Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 5; Problems 5.2, 5.3.
We can define a general vector in terms of the basis vectors in a given coordinate system:
This is analogous to the definition of the infinitesimal displacement that we met earlier: . This has a couple of consequences. First, since the basis vectors are not necessarily either unit vectors or orthogonal, this definition may be different from the usual definition of a vector that you’re used to from linear algebra courses.
Second, we require the transformation of a vector’s components between coordinate systems to be the same as the components of , which means that
Finally, the square of a vector follows the same pattern as the square of the increment :
As an example, consider the case of uniform circular motion. From elementary physics, we know that, in polar coordinates, the radial component of the velocity (since the object is always at the same distance from the origin) and the tangential component is . Using the metric for polar coordinates, this means that
To transform this vector to rectangular coordinates, we have
where the primed system is rectangular and the unprimed is polar. So
The square is invariant, since using the rectangular metric
Now let’s look at the inverse problem. This time we have an object moving at a constant speed in the direction, so that , . To convert this to polar coordinates, we need the derivatives
Then we get
If the object starts at at , then and . In polar coordinates we get
At , the motion is entirely in the direction, since the object is moving tangent to the circle at that time. As time increases, the motion gradually transfers over to the radial direction, with .