# How can coordinates be meaningless in General Relativity?

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.

• The problem with your question is that you take a silly statement made by an anonymous user on the web and treat it as a “given truth” asking others to explain it. Perhaps first you should have asked to confirm if the coordinates are meaningless. Both answers you currently have state that they are not thus making your whole question moot. Commented Sep 3 at 6:22
• It's worth remembering that humanity created coordinate systems to better conceptualize the glorious complexity of the universe - and the universe couldn't care less that we did. From the universe's point of view, coordinate systems are truly meaningless. From our point of view, whichever one is most useful is the system that helps you comprehend the concept best. Personally, I think it's fun measuring walking distances in femtoparsecs. :-)
– JBH
Commented Sep 4 at 18:16

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.

• But IIUC, you certainly define a global coordinate system for one specific observer? At least for a specific observer defined as "at rest" at a specific point. Commented Sep 4 at 7:24
• @Peter-ReinstateMonica sure, you can define a global coordinate system that works locally for a particular observer. Such a coordinate system might mean that $dr/dt$ is indeed the velocity measured locally, but $dr/dt$ at some other location would not be the velocity you would find if you went there and measured it. The Schwarzschild/Droste coordinates have this property for an observer at very large $r$ and at rest with respect to the central mass. Commented Sep 4 at 8:29

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).

• So, coordinates as a concept are not meaningless or even irrelevant -- indeed, they are indispensable: But all coordinate systems are all created equal. Commented Sep 4 at 7:27

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.

• Thanks for your reply. Firstly, I wanted to ask more about what is meant by a set of local observers. In SR, for example, the $t$ coordinate would be the time measured by a single observer comoving with the coordinate system. In, say, the Schwarzschild metric, what would the $t$ coordinate represent for a given $(t, r, \theta, \phi)$? Is this what is measured by a stationary observer located at $(r, \theta, \phi)$? Commented Sep 2 at 20:18
• Secondly, after reading the Wikipedia on Schwarzschild coordinates, I can kind of imagine how the $r$ coordinate isn't actually physical distance from the origin, but I can only picture it by embedding the manifold in a higher space, which I imagine shouldn't be necessary. What exactly is the process by which we define the Schwarzschild coordinates? Commented Sep 2 at 20:20
• In Schwarzschild Droste (SD) coordinates the proper time of local stationary observers is the coordinate time divided by the gravitational time dilation factor √|gᵗᵗ|, while in Gullstrand Painlevé (GP) coordinates it is the proper time of local observers infalling with v=-√|gᵗʳgₜᵣ|=-√(rₛ/r). The radial coordinate needs to be integrated with ∫ᵣ₁ʳ²√|gᵣᵣ|dr to give the proper length measured with the stationary rulers in SD, see Flamm's paraboloid. In GP the gravitational depth expansion and kinematic length contraction cancel, so gᵗᵗ=-gᵣᵣ=1 with those free falling clocks and rulers. Commented Sep 2 at 20:53
• @AidanBeecher The Schwarzschild and Droste coordinates are different. What you see in Wiki as the “Schwarzschild” coordinates are in fact the Droste coordinates. The difference is in the definition of radius. Schwarzschild uses the conventional polar coordinates where radius is defined as the distance to the origin. Droste defines radius as a reduced circumference of the sphere at that location. This definition is widely accepted, but is misleading. For example, the radial interval inside the horizon is timelike, but in the Droste definition the radius is spacelike even inside, which is wrong. Commented Sep 3 at 6:37
• @safesphere - see here at equation 15 where you can find the Droste coordinates in Schwarzschild's original work, he just uses R instead of r in that version. But even with his small r the radius is not the proper distance, otherwise the grr would be 1, which it ain't. Commented Sep 3 at 9:36