# What is “a general covariant formulation of newtonian mechanics”?

I am a little confused: I read that there are general covariant formulations of Newtonian mechanics (e.g. here). I always thought:

1) A theory is covariant with respect to a group of transformations if the form of those equations is conserved.

2) general covariance means not only transformations defined by arbitrary velocities between different systems, but also transformations defined by arbitrary accelerations conserve the form of such equations.

But in that case the principle of general relativity (the form of all physical laws must be conserved under arbitrary coordinate transformations) would not be unique to general relativity. Where is my error in reasoning, or stated differently which term do I misunderstand?

• Nothing stops you from making Newtonian mechanics covariant. We do it in crystallography all the time when we deal with non-orthogonal bases and their duals. Does it buy much for understanding? If it does, I am not aware of it. General coordinate transformations are handled by the Lagrange formalism, if needed, but an arbitrary coordinate transformation is of as little use in Newtonian mechanics as it is in GR. The important ones are problem specific and simplify the solution or highlight its symmetries/properties, which requires more than the ability to formulate a general transformation. – CuriousOne Jun 21 '16 at 22:49
• Indeed, the Lagrange formalism is the result of a general covariant formulation of Newtonian mechanics. – Lewis Miller Jun 21 '16 at 23:54
• Although not quite the same thing, there is a lovely discussion of a geometric formulation of Newtonian mechanics by Élie Cartan in Chapter 12 of Misner Thorne Wheeler "Gravitation". You'll be quite horrified at how awkward it is and how natural the Einstein field equations seem in comparison – WetSavannaAnimal Jun 30 '16 at 4:16

The development of general relativity has led to a lot of misconceptions about the significance of general covariance. It turns out that general covariance is a manifestation of a choice to represent a theory in terms of an underlying differentiable manifold.

Basically, if you define a theory in terms of the geometric structures native to a differentiable manifold (i.e. tangent spaces, tensor fields, connections, Lie derivatives, and all that jazz), the resulting theory will automatically be generally-covariant when expressed in coordinates (guaranteed by the manifold's atlas).

It turns out that most physical theories can be expressed in this language (e.g. symplectic manifolds in the case of Hamiltonian Mechanics) and can therefore be presented in a generally covariant form.

What turns out to be special(?) about the general theory of relativity is that space and time combine to form a (particular type of) Lorentzian manifold and that the metric tensor field on the manifold is correlated with the stuff occupying the manifold.

In other words, general covariance was not the central message of general relativity; it just seemed like it was because it was a novelty at the time, and a poorly understood one at that.

• Thanks a lot for you clear answer. But I have two remaining questions: 1) Is the difference between covariance and general-covariance that for general-covariance the group of transformations conserving the form of the physical laws must include accelerations also? 2) Is the general principle of relativity ("All systems of reference are equivalent with respect to the formulation of the fundamental laws of physics.") still unique to gr, since a general-covariant formulation of Newtonian mechanics would e.g. not give constant speed of light in all frames? Kind regards – DonkeyKong Jun 25 '16 at 12:07
• @DonkeyKong special-covariance is a necessary condition for a law to be invariant with respect to a group of transformations, whereas general-covariance is a necessary condition for a law to have an invariant form with respect to a (much larger) group of transformations. To have the same form does not mean the frames are equivalent. There could, for example, be a centrifugal force term in the general form of a law which happens to vanish in inertial coordinate systems but not in others. There is no "general principle of relativity". Einstein was wrong about that. – Physics Footnotes Jun 26 '16 at 12:45

This is not what I, and I would posit most physicists, understand as a physical treatment of what general covariance is in physics. General covariance is that the equations look the same in any coordinate frame - any meaning that the transformations can be any function. The only limitation is that the functions be differentiable, maybe n or infinite times (diffeomorphisms).

Newtonian mechanics has the Galilean group as its definition of covariance. It means transformations to inertial frames only. That is not general covariance in the way physics either works or is understood.

Possibly you can define mathematical contraptions that allows something more general, but it would be a mathematical technique, not a deep property of the physics. The Lagrange or Hamilton equations (which yes are more than contraptions but still no newer physics than Newtonian mechanics) may look the same if you change the p and q's to another coordinate system, but the equations of motion are different in non-inertial frames. The centrifugal and Coriolis forces are not real forces but appear in rotating coordinate frames. And yes, you do have relativity covariant (or Lorentz covariant in special relativity) but it is not space, it is 4D spacetime and a +/- 2 signature, and not 3D.

I saw the Wikipedia article, and for physicists also Landau and Lifshitz and Goldstein/etc, on Newtonian covariance and this is consistent. Wikipedia simply calls the Galilean group the covariance group. In physics we call it the symmetry group. In that article that the OP referred to it clearly says covariant for special relativity, but they say'Lorentz covariant'. Newtonian mechanics is similarly 'I Galilean covariant'. BTW, the article referred to by the OP is about philosophy of science, not about science.

Did some new physics that makes Newtonian mechanics generally covariant get discovered more than 200'years after Newton?

Or am I not understanding something you are trying to say? And if so, does it make any difference in physics and is it represented in accepted refereed physics journals?

By the way, in general relativity having the general covariance makes a difference. In radial normal coordinates the Schwarzchild metric has a singularity at the horizon. In Penrose or Kruskal coordinates it is seen that it is not, and in fact you can use them to understand the causal structure of black holes. Many other reasons. It is not for naught.

• +1; I'm sure you know this, but it is worth pointing out here because it re-emphasizes the point of your last paragraph: that we can switch between observers freely and and get nonsingular horizon behavior from one observer's standpoint and not from another has a physical meaning: that $\mathrm{d} t$ blows up as $\mathrm{d} \tau$ remains finite means that an object crossing the horizon takes forever as reckoned $\mathrm{d} t$ by a distant observer's clock, whereas its a perfectly straighforward, finite time process for the hapless soul plunging through. – WetSavannaAnimal Jun 30 '16 at 4:10
• I think we probably always need to call the Schwarzschild "singularity" at the horizon a "co-ordinate singularity" in this kind of discussion. Also a good tl;dr for your first paragraph: covariance with respect to an obviously highly restricted class of transformations is not "general" covariance, by definition; unless the class of transformation is very broad (e.g. diffeomorphism), it's downright misleading and a poor use of words to call something "generally covariant". – WetSavannaAnimal Jun 30 '16 at 4:13
• It generally tends to be used to mean that. It's a fine point, but anyway, yes, certainly diffeomorphism. – Bob Bee Jun 30 '16 at 4:16

As it is said in the article referred to in my post

we can take any theory and reformulate it so that it is covariant under any group of transformations we pick; the problem is not physical, it is merely a challenge to our mathematical ingenuity.

As @Lewis Miller pointed out the Lagrangian formulation of Newtonian mechanics is general-covariant, since the equations do not change their form under arbitrary coordinate transformations (also including accelerations).

As Bob Bee pointed out: general-covariance means that the transformations may be any function, whereas in (normal) covariance the group of transformations can be one of small cardinality with many restrictions (e.g. only permitting inertial frames).