Singularity in coordinate transformations

Question

Ok. I would like to apologise if the question is somewhat 'stupid'. So, initially I didn't pay much attention to it, but we know that when the Jacobian vanishes for a particular point, we say that the point is a singularity of the transformation, but if we are using the transformation to describe a Physical phenomena, then shouldn't we have a complete description of the system at every time? (I am really not sure if this should be the case always)

Taking the simplest example itself to illustrate my point, a motion in the Cartesian coordinaate system, if is well defined, it necessarily won't be, while considering it in let's say, Polar coordinate system, since singularities are present at $r,\theta = 0$

If not then the OP raises a very fair argument related to the topic here, but my question from this is :

1. If let's say we solve this issue by taking the limits on both the sides (as suggested by one of the answers there). then what is the guarantee that the limits will match and give a unique answer?

2. What if I don't wish to change the coordinate system (in the question there's an answer suggesting that we can transition from Cartesian to polar and vice-versa to solve the problem of singularity) and what if choose to remain only in 1 coordinate system?

Thanks.

Answer 1

What you’re seeing is the difficulty surrounding manifolds.

Manifolds are geometric objects that, by definition, cannot be completely mapped in one coordinate system with dimensions less than it’s natural form. Spheres are manifolds, as seen by the reason you pointed out; you can’t flatten a sphere, you will always lose information. And it makes sense if you think about it; there’s not a single map of the earth in two dimensions that totally captures all of the information of the earth without adding another map with a different coordinate system.

Another example is spacetime, which is a four dimensional manifolds. In a standard treatment of general relativity, coordinate singularities come up all the time (for example, the event horizon) and so we change our coordinate system, losing information about one part of the manifold to gain insight on another, and between the two mappings better understanding the whole.

In short, you can’t only have one coordinate system for a mapping of a manifold of lesser dimensions; it’s geometrically impossible to not lose information doing so. But for some purposes, that’s ok, because the information lost is not particularly relevant to what you’re doing.

In regards to taking limits, what you’ll sometimes find in coordinate singularities is an infinite discontinuity; this is the case for the event horizon pre-coordinate change.