stevendaryl said:
I'm coming into this discussion pretty late, but for the record, it's
not the case that a combination of two Lorentz transformations is another Lorentz transformation. There are 10 independent linear transformations such that any two inertial coordinate systems are related by some combination of the 10:
- A translation in the x-direction (that is, the transformation ##x' = x + \delta x##.
- A translation in the y-direction
- A translation in the z-direction
- A translation in the t-direction (##t' = t + \delta t##)
- A rotation about the x-direction (the transformation ##y' = y cos(\theta) + z sin(\theta), z' = z cos(\theta) - y sin(\theta)##)
- A rotation about the y-direction
- A rotation about the z-direction
- A boost in the x-direction (##x' = \gamma (x - v t), t' = \gamma (t - \frac{v}{c^2} x)##
- A boost in the y-direction
- A boost in the z-direction
If you combine boosts in two different directions, the result is not a boost, but a combination of a boost and a rotation.
Lorentz transformations themselves don't form a group (in more than one spatial dimension), but only the combination of Lorentz transformations + rotations.
Of course the Lorentz group is a group and thus the composition of two Lorentz transformations is again a Lorentz transformation. It's all linear transformations which leave the Minkowski product invariant. For four-vector components it reads
$$V^{\prime \mu} = {\Lambda^{\mu}}_{\nu} V^{\nu},$$
and the matrix must fulfill the pseudo-orthogonality relation
$$\eta_{\mu \nu} {\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma}=\eta_{\rho \sigma}$$
with ##(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)##.
These matrices build the group ##\mathrm{O}(1,3)##. Physically relevant for the space-time description is a priori only the subgroup connected continuously to the identity, and that's the proper orthchronous Lorentz group ##\mathrm{SO}(1,3)^{\uparrow}##, i.e., all Lorentz-trafo matrices with determinent 1 and ##{\Lambda^0}_0 \geq 1##.
While the rotations form a subgroup the rotation-free boosts are not a subgroup; only those in one fixed direction form an Abelian subgroup. The composition of two rotation-free boosts in different directions is of course again a Lorentz transformation but not a rotation-free boost, but a rotation-free boost followed by a rotation (the Wigner rotation).
Minkowski space is the affine pseudo-Euclidean space with a fundamental form of signature (1,3), also called an affine Lorentzian space, and as such the full symmetry group is the Poincare group and is the corresponding semidirect product generated by Lorentz transformations and spatio-temporal translations. Again a priori physically relevant is the proper orthochronous Poincare group, and indeed Nature is described well with a space-time model obeying this symmetry group. The larger group built from the proper orthochronous Poincare group by including time, space, and space-time reflections is not a symmetry group of Nature. The weak interaction violates both time reversal as well as space reflections. Within relativistic local QFT you have, however, necessarily an additional symmetry, which is charge conjugation, i.e., where all particles in a reaction are exchanged by their antiparticles. The weak interaction also breaks the C-symmetry. Today, it has been independently observed that the weak interaction breaks all these discrete symmetries, i.e., T, P, and CP. Relativistic local QFT predicts however that necessarily the "grand reflection" CPT must be a symmetry, and so far this symmetry indeed has passed all tests.
Together with all the other tests of Lorentz symmetry, it is pretty safe to say that Poincare symmetry is obeyed by all phenomena with an amazing accuracy, as is the extension to GR and the corresponding gauge symmetry leading to it from SR and its global Poincare symmetry.