# Is there a natural coordinate system in general relativity?

After reviewing some different coordinates systems that describe Schwarzschild spacetime (such as Gullstrand–Painlevé coordinates) it seems like we can always make a coordinates transformation very far from us such that locally we are in Minkowski coordinates.

So the question if whether one of those coordinates will be "more natural" to work in (so are there useful general condition that give rise to a unique coordinates).

Of course that locally all the laws of physics will not prefer any system, but in this condition I allow also extrinsic experiments (similarly to the uniqueness (up to translations and rotations) of the coordinates used in the Friedman's metric that follows when we require Homogenic and Isotropic coordinates)

Edit:

If I measure different velocities to the same object that is significant difference for me. The fact that you can deduce the same conclusion from experiment only mean that that every measurements correspond to some invariant property. Saying that there is no general conditions that we can define in order to get always some defined properties doesn't follow from that. Note that this is not a formal mathematical question but rather a question on what we can define if we want to and there is always a possibility that after we define something it will turn out to be natural to most of us

It seems to me like there are classes of equivalence that we can define that have some meaning. In this case I mean that we can define all the coordinates transformation that does not involve time to be in the same class and then for example cartesian and spherical be in the same class. And I argue that this class is unique in some sense and I am trying to formulate specific (but more general) condition for other systems.

I want to create the uniqueness in some sense that is meaningful to me using those terms and I wonder what such condition will look like

• I think the lack of "natural coordinates" is one of the various things that makes GR so difficult to think through. For example, what does it mean to talk about simultaneity for two distance observers? Can you even do that? I've had a very painstaking argument with someone here one time (neither of us changed our minds) and the trouble largely came from this very issue. Feb 27 at 15:22
• Also, you make a good point (in the comments of one of the answers) that if one describes the desideratum for what they would want out of coordinates they deem natural, then you could identify the natural coordinates (because you've defined what you mean by physical/natural). Unfortunately, I've not ever come across such a desideratum. If you (or anyone else) do find anyone stating the condition clearly, I would be interested. Feb 27 at 15:30
• One way to induce "natural coordinates" is to use asymptotic flatness. Asymptotic flatness means that far away from matter the curvature will go to zero, which means the metric will look like the Minkowski metric. We can then define our coordinates such that a far away, static observer locally sees a Minkowski metric. The Schwarzschild metric is an example of this. This is not always possible. Feb 27 at 15:47
• @MaximalIdeal thanks for your comment, I will let you know if I will found something. By the way, I looked at your profile, you have nice questions
– ziv
Feb 27 at 15:51
• @AccidentalTaylorExpansion thanks, finally someone give me a nice answer that I can study
– ziv
Feb 27 at 18:28

Coordinates literally make no difference in any physical sense. Is there any sense in which polar coordinates in the plane are a more ‘true’ description of (an open subset of) the plane? Absolute not. Thus, there is no natural/ inherently preferred coordinate system.

A completely separate question is whether there are situations where we might favour one set of coordinates over another. For that the answer is yes. The choice of coordinates depends entirely on the problem at hand. Some coordinates may make some features of geometry more apparent, for instance:

• in the usual Schwarzschild coordinates $$(t,r,\theta,\phi)$$, the metric components do no depend on $$t$$ or $$\phi$$ so $$\frac{\partial}{\partial t}$$ and $$\frac{\partial}{\partial \phi}$$ are Killing fields, whereas in Kruskal coordinates $$(T,R,\theta,\phi)$$, the metric components do depend on $$T,R$$, so $$\frac{\partial}{\partial T}$$ is no longer a Killing field.
• if you’re interested in studying things like propagation of waves, or just null curves, then it is a good idea to not use the usual $$t,r$$ coordinates, but replace them with some form of $$u,v$$ null-coordinates which will more transparently reflect the geometry (e.g in Minkowksi, we use the two $$45^{\circ}$$ rotated lightlike coordinates instead of the usual $$t,x$$).

Anyway, this is not at all a feature specific to GR. Even in classical mechanics, the coordinate system we use is the one which is most conveniently adapted to the problem at hand (rotation: use polar coordinates, translations: use cartesian, for a mix of both, make a choice as to which aspect you wish to simplify etc).

• Of course coordinate make real physical difference. This just correspond to what you mean by "physical"
– ziv
Feb 25 at 13:06
• @ziv Coordinates do not make any physical difference, they are just a representation of the physical configuration. A solution in one set of coordinates can always be mapped to an equivalent solution in a different set of coordinates. Feb 25 at 16:29
• I still think this is overly repeated statement that depend on your definition of "physical meaning". For example, don't you think that locally inertial coordinates are have physical meaning?
– ziv
Feb 25 at 20:48
• @ziv Now you seem to be introducing a colloquial definition of the word physical. To be concrete, a diffeomorphism $\phi$ (coordinate transformation) from the set $\{M,g,\Psi\}$ to $\{M, \phi(g), \phi(\Psi) \}$ represents the same physical configuration (same solutions, dynamics, etc). Here $M$ is the spacetime manifold, $g$ the metric and $\Psi$ all the matter field content. Any measurement or prediction you make within the theory remains unchanged under this transformation, which most people will say as the 'physics remaining the same'. Feb 25 at 22:28
• Of course coordinates make a difference in a physical sense. Any physical measurement occurs in a coordinate system defined by the measuring device. Now, of course, a good theory can relate those measurements to an invariant abstraction, but that's mathematical modeling, a step removed from the physical. Feb 27 at 15:58

The natural coordinates don't depend so much on the theory as on the experiments you perform. The phenomena, not the models, are fundamental to physics.