# Laws of physics and diffeomorphism covariance

Up to my limited understanding, diffeomorphisms on a space-time manifold can be viewed as changes of coordinates. While studying general relativty, I read that the theory has diffemorphism covariance because it expresses its laws on a geometrical form. The same can be said about electromagnetism. However, there is something I don't understand.

General relativty is invariant over any change of coordinates (inertial or not) while special relativity, classical mechanics and every other physical theory I know is invariant over inertial changes of coordinates.

When people say that special relativity or Maxwell's equations have diffeomorphism covariance, are they speaking about restricted (i.e. inertial) diffeomorphisms or every possible diffeomorphism, just like GR?

• "Diffeomorphism invariance" is a misnomer for the symmetry transformations of GR, see this answer of mine. – ACuriousMind Mar 17 '18 at 20:12
• A theory may be expressed in a way that has manifest diffeomorphism invariance or that lacks manifest diff invariance. However, coordinates are just names, so it's guaranteed that any theory can be written in a manifestly diff-invariant form. A more useful question to ask is whether the theory is background-independent. When a theory, such as quantum mechanics, lacks background-independence, the geometry has to be treated as something prescribed in advance, rather than something that can vary dynamically. – user4552 Mar 17 '18 at 21:13

At first it is important to notice, that coordinates don't have any physical reality. They are just a help we use to perform explicit calculations. So it should be our aim to formulate all physical laws in a coordinate independent way.
Historically general relativity was the first theory, where this feature was really implemented on purpose, but today we have got coordinate invariant descriptions of mechanics and electrodynamics as well.

For electrodynamics, a common way inspired from gr is to replace the Minkowski metric by the general metric $g$ and the partial derivatives by covariant derivatives. However there are other ways, such as the use of differential forms or a description in terms of principal bundles, leading to equivalent results. The action for the electromagnetic field then reads: $S=\int F_{\mu \nu}F^{\mu \nu}\sqrt{-g}d^4x$.

The case of classical mechanics is very interesting, because Newtons second axiom $\vec{F}=m\vec{\ddot{x}}$ indeed only holds in inertial system. This leads to weird additional forces such as corioles force if one considers rotating frames. Today we know that the geodesic equation is the correct generalisation of Newtons second axiom to arbitrary frames. If you then calculate the Christoffel symbols this yields exactly to the additional terms we explained as non inertial forces before. So within this generalisation there is no need to introduce additional effects in non inertial systems and the theory is diffeomorphism invariant.

An interesting side remark: in the Lagrangian or Hamiltonian formulation classical mechanics is also explicitly diffeomorphism invariant. I think this is very remarkable, because Lagrange did his work in 1788 way before general relativity entered the game.

As for my understanding the meaning is different.
A diffeomorphism is an invertible and differentiable map between differentiable manifolds.
A diffeomorphism covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates are just artifices of the human mind, thus should have no role in the formulation of physical laws. To that aim the tensorial formalism perfectly applies.
For instance, the Lorentz force or the Maxwell equations written in tensorial form are invariant with respect to both a Lorentz transformation or a GR (general relativity) transformation of coordinates.

• Thanks for your answer! However, newtonian mechanics is in tensorial form but is only invariant under galilean (inertial) changes of coordinates. This is what i dont understand – P. C. Spaniel Mar 14 '18 at 14:57
• Newtonian mechanics is not in tensorial form. The second law of Newton relates a change in time of the classical three-momentum to a three-force. You have to restate the law as a change in proper time of the four-momentum related to a four-force. – Michele Grosso Mar 14 '18 at 17:08
• @MicheleGrosso: You have to restate the law as a change in proper time of the four-momentum related to a four-force. Isn't there a problem because there is no metric (or no nondegenerate metric) for galilean spacetime? – user4552 Mar 17 '18 at 21:08
• @Ben Crowell. Yes, I was hinting to the extension of the law in Minkowski spacetime. – Michele Grosso Mar 18 '18 at 16:38