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In Relativity, both the old Galilean theory or Einstein's Special Relativity, one of the most important things is the discussion of whether or not physical laws are invariant. Einstein's theory then states that they are invariant in all inertial frames of reference.

The books usually states that invariant means that the equations take the same form. So, for example, if in one frame $\mathbf{F} = m\mathbf{a}$ holds one expect this same equation to hold on other inertial frame.

If one studies first relativity and then differential geometry, there seems to be really important to postulate that: there's no guarantee whastoever that the equations will be equal. I, however, studied differential geometry first and this led me to this doubt.

On differential geometry everything is defined so that things don't depend on coordinates. So for example: vectors are defined as certain differential operators, or as equivalence classes of curves. Both definitions makes a vector $V$ be one geometrical object, that although has representations on each coordinate system, is independent of them.

Because of that, any tensor is also defined without coordinates and so equalities between vectors and tensors automatically are coordinate-independent. This of course, is valid for vector and tensor fields.

Scalar functions follow the same logic: a function $f : M\to \mathbb{R}$ has one coordinate representation $\tilde{f} : \mathbb{R}^n\to \mathbb{R}$ which is just $\tilde{f} = f\circ x^{-1}$ but still, $f$ is independent of the coordinates. So if $f = g$ this doesn't refer to a coordinate system, but to the functions themselves.

So, it seems that math guarantees that objects are coordinate-independent by nature. So in that case, what are examples where a Physical law is not invariant and why my reasoning fails for those examples?

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  • $\begingroup$ centrifugal force is not real. In an inertial frame, one sees centripetal force keeping an object in nice circular motion. In the frame of that object (non-inertial), you observe no net acceleration even though there is a net force (the source of centripetal force). $F\ne ma$. Thus, we invent the idea of centrifugal force to explain why we see no net acceleration in the non-inertial frame. To make us feel like $F=ma$ still holds even though there is only one force source and there's no acceleration in that frame. The law doesn't hold, we just invent things to make it look like it holds $\endgroup$ – Jim Mar 11 '15 at 21:18
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    $\begingroup$ You shouldn't assume that the generality of differential geometry carries over into physical laws that might be expressed in geometrical language. There definitely are coordinate systems when you are discussing physical laws. (They may only be local, though.) I would dispute the claim that using differential geometry creates the "guarantee" that you infer. $\endgroup$ – DWin Mar 11 '15 at 21:26
  • $\begingroup$ user1620696: "On differential geometry everything is defined so that things don't depend on coordinates." -- I wonder ... "So for example: vectors are defined as certain differential operators" -- So what constitutes a "differential operator" wrt. some given set $M$? Required is apparently (at least) a certain "topology" assigned to set $M$; and the debate (in which I'm interested) is whether and how it can be derived from the elements of $M$ itself, and not only by its (possible) representation as "real n-tuple topology" of certain equivalence classes of coordinates. $\endgroup$ – user12262 Mar 11 '15 at 22:27
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    $\begingroup$ Consider he Christoffel symbols associated with a connection. These are not tensors and so physical `laws' one might associate with them are then not invariant $\endgroup$ – Arthur Suvorov Mar 12 '15 at 1:14
  • $\begingroup$ C.f. the difference between general invariance and general covariance: pitt.edu/~jdnorton/papers/decades.pdf $\endgroup$ – FenderLesPaul Mar 18 '15 at 14:44
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An example of a law that is not invariant: $F = -\mu mv$. That is, some kind of universal friction slows down all moving objects. This requires a point of reference they are slowing down compared to, so it is not invariant.

Any law that can be written in the form of a tensor is invariant, but this law cannot be written in that form. Not unless you have some kind of vector specifying what zero velocity is.

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  • $\begingroup$ Nice explanation. $\endgroup$ – Ján Lalinský Mar 12 '15 at 8:20
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Because of that, any tensor is also defined without coordinates and so equalities between vectors and tensors automatically are coordinate-independent. This of course, is valid for vector and tensor fields.

Although this is true, it is a purely mathematical statement and doesn't have any implications in physics. A law of physics as a set of equalities between vectors and tensors only is a law of physics if you can actually describe these quantities. Typically you would do that by choosing coordinates and describe the quantities in terms of these coordinates. You could then postulate that in any coordinatization the laws of physics can be stated in terms of equalities between vectors and tensors. When this is assumed, the description of the laws of nature would be coordinate independent.

Of course, no such description can exist in such generality, and restrictions should be imposed on the allowed coordinatizations.

To be explicit, consider special relativity. Here it is assumed or postulated that there is a class of coordinatizations of spacetime, called inertial frames, in which the laws of physics can be written down in such a way that they are the same in all these frames, namely as equalities between Lorentz tensors. In other classes of coordinatizations the tensor equations would still hold, only they wouldn't have the same description in terms of the coordinates.

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any tensor is also defined without coordinates and so equalities between vectors and tensors automatically are coordinate-independent ... it seems that math guarantees that objects are coordinate-independent by nature. So in that case, what are examples where a Physical law is not invariant and why my reasoning fails for those examples?

What is obvious to us today was not so in Einstein's day! He had to make us believe in this point of view, which seems so obvious now from a differential geometry viewpoint.

We still need to make sure we do not mix random quantities as components of something we define as a "tensor". And we cannot, for instance, simply state that the stress-energy tensor equals the anti-symmetric electromagnetic field tensor. We have to be sure that this works in at least one frame of reference. The beauty of differential geometry can then extend this to all frames.

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