How can coordinates be meaningless in General Relativity?

Question

I am fairly new to the subject of General Relativity. While looking for answers to some questions I had about it, I came across this post: Whose coordinates are the Schwarzschild coordinates?

One of the answers to this person's question stated that "coordinates are meaningless". While I can imagine that maybe a particular choice of coordinates don't correspond to a specific observer, I do not understand two things.

1. Why isn't it possible to find any choice of coordinates that correspond to a particular observer, like in Special Relativity, where it is possible to Lorentz transform into any inertial frame?
2. How can it be that coordinates are "meaningless"? If we wanted to state the Pythagorean theorem, for example, we would need to specify that $a$, $b$, and $c$ were particular side lengths of a right triangle to give meaning to the statement that $a^2+b^2=c^2$. Likewise, how does it make sense to assert the form of, say, the Schwarzschild metric in $(t, r, \theta, \phi)$ if these coordinates mean nothing? Wouldn't that mean anything defined in terms of them (i.e. the metric) would also be meaningless? If I'm not mistaken, $r$, for example, isn't exactly the usual radial distance from the origin, since we're in a curved spacetime, but it seems like it should mean something in order to give meaning to the form of the metric.

For whatever reason, the answers in the post linked above didn't really clarify much to me, so I hope someone might be able to do that here.

Answer 1

Coordinates are not meaningless. But perhaps a better word would be unimportant - in the sense that the physics does not care what coordinate system you use and all measurements that you could make are also not dependent on your system of coordinates.

In GR you can certainly define a local coordinate system that is closely equivalent to the coordinate system of an observer in SR. However, unlike SR, this will not be a coordinate system that applies globally because of spacetime curvature.

Or, you can define/design a global coordinate system (like the Schwarzschild/Droste coordinates), but their increments will not measure increments of proper distance or time for all (or perhaps any) observers.

Answer 2

The quote that prompted this question is my fault, so perhaps I should answer this.

In context I wrote

[C]oordinates are meaningless. You can calculate any physically meaningful quantity using any coordinate system and you'll get the same result, because they all describe the same world. As such, there's no such thing as the coordinate system "of" any object or person.

I was contrasting coordinate systems with "physically meaningful quantities". Defining the latter may be philosophically tricky, but you can start with Philip K. Dick's definition: "Reality is that which, when you stop believing in it, doesn't go away." You can change the number that represents the temperature of the sun's core by deciding in your head to use a different temperature scale, but you can't change the temperature of the sun's core that way. The temperature scale is "meaningless" in that sense. I suppose it's a poor choice of words, since there is a meaning to Celsius and Fahrenheit temperatures. It's just not a meaning that bears on the temperature of the sun.

In other answers I've said "the universe doesn't care about your choice of coordinates", which is probably a better way of putting it.

I wasn't contrasting general relativity with special relativity. In my experience, the root of most confusion about both special and general relativity is a failure to understand the irrelevance of coordinates to the physical reality of whatever you're studying—especially the idea that there's a particular coordinate system that you must use to get correct answers to questions about an object (namely its instantaneous inertial rest frame).

Answer 3

Aidan Beecher asked: "Why isn't it possible to find any choice of coordinates that correspond to a particular observer, like in special relativity, where it is possible to Lorentz transform into any inertial frame?"?"

You can have coordinates that correspond to a set of local observers. In classic Schwarzschild Droste coordinates for example this is a set of stationary observers, while in Gullstrand Painlevé coordinates the local observers are infalling from infinity and have an r-dependend velocity.

Aidan Beecher asked: "How can it be that coordinates are "meaningless"?"

The coordinates are not meaningless, you just have to know what they mean. They are in general not a direct 1:1 representation of the proper times and distances, but you can get them with the corresponding metric tensor, so the coordinates in combination with that tensor do of course mean something, otherwise this would be fooling around instead of physics.

Answer 4

The phrase "coordinates are meaningless" is a shorthand for "coordinates have no physical meaning". In spite of this, they allow the defining of the physical (measurable) quantities. For example, if one writes the infinitesimally length element as $$ds^2=g_{\mu\nu}~dx^{\mu}dx^{\nu}$$ then $dx^{\mu}$ represents the grid lines (coordinate components) and $dX^{\mu}\equiv \sqrt{g_{\mu\mu}}~dx^{\mu}$ (no summation) represents the physically measured distances (physical components).

A more comprehensive analysis of this topic can be found in the article by Robert D. Klauber Physical Components, Coordinate Components, and the Speed of Light.