# Whether nonlinear coordinate transformations are symmetries of flat spacetime

I am editing this question after the answers are posted just to present my question a little clearly (without changing the main theme of the question). Moreover, this question is solely about flat spacetime and special relativity!

According to the principle of special relativity, "The laws of physics must have the same form under transformations from one inertial frame to another inertial frame".

Lorentz/Poincare transformations are linear transformations from one set of Cartesian coordinates to another set of Cartesian coordinates related by rotations, boosts and/or space-time translations. If the first set is inertial, the second set related to the first by Poincare transformation is also inertial. Under this transformation laws of physics indeed remain the same.

Now, nonlinear transformations from Cartesian to curvilinear coordinate systems (say, Cartesian to Spherical), are still transformations from inertial to inertial, I think. But under Cartesian to curvilinear transformation, the form of the equation does not maintain form invariance as postulated by SR! See comment by Nikos M.

So my question is (the main question), does the above postulate of SR exclude nonlinear transformations from Cartesian to curvilinear coordinate systems? If you say 'yes', then we are using the word 'inertial' in a very restricted sense.

On the other hand, if you say 'no', laws of physics in SR are also form-invariant under nonlinear coordinate transformations (though I need to understand, in what sense), why is the group of Poincare transformations called the only symmetries of physical laws in flat spacetime?

• SR does not state that the equations muist have the same form under all inertial coordinate systems. Rather it states that the values of certain quantities are the same for frames of reference related by relativity principle (ie principle of relativity). Commented Apr 22 at 16:44
• One of the postulates of special relativity is that the laws of physics must have the same form in all inertial reference frames. Commented Apr 22 at 16:49
• Simply express the same physical system in SR in cartesian and curvelinear coordinates. Does it have the same form? Commented Apr 22 at 16:50
• Commented Apr 22 at 17:01

Poincaire invariance in special relativity is a global symmetry that relates one physical state to another, different physical state. By Noether's theorem, the existence of this symmetry implies the existence of conserved charges (such as momenta and angular momenta).

Coordinate transformations (or diffeomorphisms) are a local symmetry or gauge symmetry that amount to a redundancy of our mathematical description. The same physical state can be represented in multiple coordinate systems. As a result, this kind of symmetry has trivial physical content (although it can be extremely convenient, mathematically, to carry it around). Noether's (first) theorem does not apply in this situation. (There is another theorem, Noether's second theorem, that does apply, but it doesn't give you a conserved charge as in the global symmetry case).

There is a lot written about the difference between global and local symmetries. One place (not necessarily the best or most comprehensive, just one that I know off the top of my head) you can read a little bit about this is Section 6.1.1 of Tong's QFT notes: https://www.damtp.cam.ac.uk/user/tong/qft.html

• I am not asking about the difference between local and global symmetries. Let me phrase it a little better perhaps. In SR, when we say that laws of physics are the same in all inertial reference frames, what do we mean? Do we allow transformation only from Cartesian to Cartesian system related by rotation, boosts and translation? Does it exclude transforming from Cartesian to curvilinear coordinates? If yes, why? Does transforming from Cartesian to curvilinear amount to going from an inertial to a noninertial frame and therefore, excluded? This is the point I wanted to be clarified. Commented Apr 23 at 6:07
• @Solidification The underlying point of special relativity is that there is a global Poincaire symmetry. The symmetry is manifest if you use inertial reference frames with Cartesian coordinates. However, the symmetry is also present if you use other coordinates. So the physics is the same in any coordinate system, but is most clear in inertial/Cartesian coordinates. Different authors have different opinions about whether "special relativity" includes other coordinates, etc, but to me the invariant description is that it is physics on a fixed spacetime with a global Poincaire symmetry. Commented Apr 23 at 14:04

"The laws of physics must have the same form under transformations from one inertial frame to another inertial frame". …

nonlinear transformations from Cartesian to curvilinear coordinate systems (say, Cartesian to Spherical), are still transformations from inertial to inertial, I think. But under Cartesian to curvilinear transformation, the form of the equation does not maintain form invariance as postulated by SR

These statements are contradictory. Such transformations do not maintain form invariance of the laws. The transformations between inertial frames do maintain form invariance. So such transformations cannot be transformations between inertial frames. That is a clear and direct consequence of the above definition.

If you say 'yes', then we are using the word 'inertial' in a very restricted sense

That is correct. This definition of inertial is restricted. Different authors may define things differently. So it is important to read and understand the implications of the specific definitions used by a particular author.

why is the group of Poincare transformations called the only symmetries of physical laws in flat spacetime?

Usually symmetries are given by Killing vector fields. The Rindler transformations also define some Killing flows. So you would need to check with the specific text that makes that claim to understand in what sense the claim is made.

It is important to understand that different authors may make different formal statements about such basic issues. However, they usually have no practical scientific consequence. Physics is an experimental science.

As long as author $$A$$ and $$B$$ share the same mapping between the math and experiment then they are using the same scientific theory. It doesn’t matter if $$A$$ and $$B$$ use the same English words. $$A$$ may call non-accelerating polar coordinates “inertial” and $$B$$ may call them “non inertial”. If they agree on what a polar grid of clocks and rulers and accelerometers will read, then they are doing the same science.

My one cent.

A rotating frame or accelerating frame or other non-inertial frame is not inertial frame of reference. On the other hand, merely a different choice of coordinates description (eg curvelinear instead of cartesian) does not make an inertial frame become non-inertial (these coordinate transformations that do not change the character nor content of the physical system go by various names).

Form invariance of equations is used in General Relativity only (ie Principle of General Covariance). In other theories (eg Special Relativity) what matters is their Principle of Relativity which describes equivalent inertial frames of reference (regardless of form of equations in these frames)(*). In this sense, the Poincare Group of transformations (of Special Relativity) indeed expresses the relativity principle of the theory.

(*) If a system in an inertial frame of reference is expressed in the simplest possible form or a form of coordinate system is agreed upon and fixed (eg cartesian instead of curvelinear), then indeed the form of the equations can be the same between equivalent/inertial frames.

• If I understand you correctly, then you're saying that if in one inertial frame, if the form of an equation is written in some $\xi$-coordinate system, then in another inertial frame connected to the first by rotation, boost and/or translation, the form of the equation will remain the same in the $\xi$-ccordinate system. Did I get you right? Then, if we say that "laws of physics are the same in all inertial frames" in SR (which is a very common postulate in the SR books), we have to exclude nonlinear transformations (say, Cartesian to Spherical). Commented Apr 23 at 6:23