robwilson said:
I wanted to see if I could get some good physical reason for assuming the Lorentz group is the correct symmetry group for spacetime, if indeed spacetime is a meaningful concept at all, and on my terms the answer is no, I couldn't.
I arrive late to this thread, and maybe I shouldn't get involved, but it's a Sunday afternoon here and I'm a bit bored. So,...
Starting from the relativity principle, we ask "
what is the largest group of coordinate transformations that maps inertial (non-accelerating) worldlines among themselves". I.e., coordinate transformations ##(t,x^i) \to (t',x'^i)## such that $$\frac{d^2x^i}{dt^2} ~=~ 0 ~~~ \leftrightarrow~~~ \frac{d^2x'^i}{dt'^2} ~=~ 0 ~.$$ Hidden in this is already an assumption of spatial isotropy and a technical assumption about continuity and differentiability of the admissible coordinate systems (which I gloss over for now).
The largest such group is the 24-parameter fractional linear transformations ("FLT" hereafter). Although it's possible to work with this group, I'll sketch the part of the argument relevant to ordinary Lorentz transformations by restricting the FLT group down to the affine group. It's obvious that ordinary translations in space and in time form subgroups, so by using this freedom I'll restrict to the case where coordinate origin is preserved by the transformation. I.e., by an ordinary translation, the primed origin can be made to coincide with the unprimed origin. Then, it's also obvious that spatial rotations are a subgroup here which preserves coordinate origins.
Now we can investigate velocity-changing transformations. Pick an arbitrary direction in space (anchored at the origin), and rename the spatial coordinates so that one spatial coordinate lies along this arbitrary direction -- I'll call it ##x##. By a spatial rotation of the primed coordinates we can make its spatial axes coincide with the unprimed coordinates.
(This also assumes we've set up our spatial axes to be Euclidean-orthogonal.)
Now we ask for the most general coordinate transformation (using the few degrees of freedom remaining) that maps $$\frac{dx}{dt} ~=~ 0 ~~~ \to ~~~ \frac{dx'^i}{dt'} ~=~ v ~,$$where ##v \ne 0##. We further restrict such transformations by requiring that they form a 1-parameter Lie group (with parameter ##v##), where ##v=0## corresponds to the identity, and is well-defined on a open set in ##v##-space containing ##v=0##.
Introducing such a Lie group requirement here is physically motivated by using (again) the relativity principle, i.e., that no inertial reference frame is in any way physically distinguished.
There's also an implicit assumption that if observer A is moving with velocity ##v_1## relative to observer B, who is moving with velocity ##v_2## relative to observer C (all reference frames having been set up as described above, i.e., all origins and all spatial axes coincide),
then there should exist a well-defined velocity ##v_3## with which C is moving relative to A.
Cranking the mathematical handle on the above yields the Lorentz transformations along spatial direction ##x##. We can repeat the whole procedure along directions ##y## and ##z## separately, then determine the full Lie algebra among the generators of those transformations (and discover that we much include the spatial rotations for the algebra to close). Then we can add the translation generators and discover the Poincare algebra.
You asked for a "physical" reason for assuming the Lorentz group is the correct symmetry group for spacetime. The answer is that we do NOT "assume" it. Rather, it is derived via analysis of the most general symmetry group that maps non-accelerating worldlines amongst themselves.
All my explanation above was in terms of abstract coordinates. Although most people then "promote" these coordinates to a "physically real" spacetime entity, but there is (imho) no need to do so. The basic (intrinsic) physical properties (mass, spin, etc) of elementary particles can be found from the unitary irreducible representations of the Poincare group, and the possible interactions between particles (and exchange of these properties) is also strongly constrained by the Poincare group.
[...]if indeed spacetime is a meaningful concept at all,[...]
It is meaningful only in the sense of being isomorphic to a homogeneous space for the Poincare group. The
symmetry group governing inertial reference frames is what really matters.
<Phew!> ... I sure do hope this has
actually helped...
