disclaimer: this is my first time asking a question.
In general relativity, I have been told that everything stems from the coordinates we have and that from these, we can derive the metric and, thus, everything else about the system such as the geodesic equations. I was also told that this means that there are proper coordinates to use in general relativity, we cannot use any coordinate system we like. What I want to know is how we know what coordinates to use for a given situation? For example (oversimplified for sure), if we have polar coordinates, we can derive a metric and geodesic equations of motion for these, but how do we know that the space we are describing is correctly described by polar coordinates?
update, I have asked my professor for clarification on what he meant, and this is what he said:
"1. You can always pick any four vectors, let's call them t, a, b, c to be the basis of your coordinate system.
2. Once you've made this choice, the dot product between these two vectors is NOT arbitrary, but is the corresponding value in the metric g_uv which, as we say, is related to the stress-energy tensor through Einstein's Equations.
3. The way I think of it, that means that you practically can't pick any coordinate system. For example, if you pick "spherical coordinates" outside a mass M (the Schwarzschild Metric), it is NOT normal spherical coordinates: r dot r is NOT equal to 1 as you would think from the equations defining this coordinate system, but has to be g_rr in that metric. Therefore, coordinates are not really as arbitrary as the poster implied they are — yes, you can pick whatever basis you want, but the relationship between those basis vectors is given entirely by the metric g_uv derived for the stress-energy tensor T_uv."
my confusion is, I think, as follows: take where it says the dot product is not necessarily what we would expect, so these aren't spherical coordinates in the traditional sense. so it seems there is something imposed that I'm not sure where it comes from, and I think this imposition is at the heart of my confusion.
Best Answer
It seems as if the source was saying exactly what General Relativity is not. As pointed out in the comments, General Relativity a is covariant (coordinate independent) theory. In other words, it is formulated geometrically in terms of invariant geometric objects (tensors), and coordinates are essentially just a tool in order to compute calculations. They say nothing about the physical system itself.
That being said, there is often a 'canonical' choice of coordinates: a chosen set of coordinates that mirror the symmetry of the problem you're trying to solve. So if you're dealing with spherical symmetry, it may be more natural to use polar coordinates (and this may make the equations take a simpler form). Certain coordinates may also make the symmetries, known as isometries in GR, more obvious. But this does not mean there are 'proper coordinates' as any choice is equally valid. This freedom to choose your coordinate system is sometimes seen as intrinsically linked with GR$^1$. Studying how to find exact solutions in GR is a huge area of research, and a lot of effort goes into adapting useful coordinate systems for this purpose, but this is purely for utility and says nothing about which coordinate description is more correct$^2$.
$^1$I should also clarify that this coordinate independence is true for all physical theories (if they're formulated correctly), not just General Relativity. When we do physics, make measurements, etc, the choice of coordinates we choose to use do not at all impact what's actually happening. What is perhaps less trivial about GR, that sets it apart from other theories like Special Relativity, is that there's no preferred forumulation that lends itself to a particular set of coordinates. This is really a consequence of the more important property of being background independent.
$^2$ There's also the other caveat that different coordinate charts may be defined in different neighbourhoods/regions, which means some may be be better adapted to certain spacetimes, given what you're trying calculate. Again, this is a practical issue.
On covariance: https://en.m.wikipedia.org/wiki/General_covariance https://en.m.wikipedia.org/wiki/Principle_of_covariance