Does Galilean relativity constitute a dynamical symmetry or an isometry?

Question

There are many papers which derive the form of the Lorentz transform from elementary symmetry principles (usually homogeneity of spacetime, isotropy of space, and the fact that boosts form a group), e.g. here or here (warning: PDF). These derivations result in an expression for the Lorentz factor of the form $$\gamma(v) = \frac{1}{\sqrt{1 - K v^2}},$$ where $K$ is some non-negative constant. When $K$ is strictly positive, the Lorentz transform is recovered, whereas when $K$ is zero, the Galilean transform is recovered. This style of derivation implies that the Galilean transform and the Lorentz transform occupy similar roles in non-relativistic and relativistic physics, respectively.

However, in relativity, boosts between reference frames (which are described by the Lorentz transform) are part of the isometry group of the Minkowski metric. Galilean boosts, on the other hand, are never (to my knowledge) presented as isometries of the Euclidean metric; in fact, I'm having difficulty showing that a Galilean boost is an isometry.

The other option is that Galilean relativity (i.e. invariance under Galilean boosts) is actually a dynamical symmetry (i.e. a symmetry of the equations of motion, specifically the Euler-Lagrange equations), and not an isometry. Is this the case, and if so, can invariance under Lorentz boosts be similarly conceived as a dynamical symmetry (of the E-L equations for a relativistic field, perhaps)?

Answer 1

When you derive coordinate transformations based on spacetime symmetries, you get 3 results, not just 2. In addition to Lorentz and Galilean, you also get the Euclidean transformation for K<0. This transformation is ruled out, because it makes no physical sense since time is non reversible. Nevertheless it is 1 of 3 mathematical results. Accordingly, the Galilean transformation does NOT correspond to the Eucledian spacetime and therefore cannot be viewed as a result of the symmetries in this spacetime. For example, time is reversible in the Euclidean spacetime, but not reversible in the Galilean transformation. The Galilean transformation corresponds to a degenerative spacetime where the ratio between space and time coordinates is infinitely small (same as the speed of light being infinite). Therefore the Galilean transformation does not reflect a symmetry between space and time, as there isn't any in the Galilean spacetime.

Answer 2

In physics it is straightforward and Galilean and Lorentz transformation are similar, and do correspond to the isometries.

First, since it seems less confusing to you but historically it was much more confusing, the Poincare group (Lorentz boosts plus rotations and translations) transformation all came because of the isometry of the 4D spacetime on which special relativity (SR) resides. The isometries are translation (in all 4 dimensions, i.e., space and time), rotations in space, and boosts which are the uniform motion transformations (to a constant velocity wrt first frame). The conserved quantities are 4 momentum (energy and momentum), angular momentum and for the boosts it's a pretty useless one (so not usually mentioned, it's simply the position of the center of mass at t=0). For the boosts it's not common knowledge, you can see the answers at What is the invariant associated with the symmetry of boosts?, as well as the note there that it is a duplicate for another one that preceded it. The words in the last answer explains it, the math is in the first answer.

For the Galilean transformations it's also energy, 3 momentum, and angular momentum. The uniform motion transformation just gives you the Galilean uniform motion transformation to a prime frame moving with uniform velocity wrt the original inertial frame.

$x\prime$ = x-vt, with K in your equation 0 as you also noted. The conservation law is the same useless one for the conservation of the position of the center of mass at t=0. The Galilean Group is a Group Contraction of the Poincare Group for c= infinity. You can see the Galilean Group and Lie Algebra at https://en.m.wikipedia.org/wiki/Galilean_transformation

BTW, both classical physics and relativistic physics also allow the discrete symmetries for reflection and time inversion. We already know that reflection symmetry is only approximate, as parity is not conserved in the weak interactions. For time reversal we still are not sure, but it seems to be in the microscopic domain, except usually one describes the 4D spacetime as time oriented, i.e., t only increases, which allows us to insure causality.

The Galilean group is not more than the limit of the Poincare group for c going to infinity. No reason to get confused. Neither depends on the equations of motion, both give you Lie algebra generators (eg the Galilean wiki reference noted above has them for the Galilean group) as part of the group (symmetry) structure.