Historical vs modern presentation of special relativity

Question

I have noticed that historical or brief introductions of special relativity will discuss it in terms of inertial frames and postulates:

• Principle of Relativity - (from Einstein's 1905 paper) "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good"
• Constant speed of light - "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body"

While modern descriptions state it instead as the symmetries of the physical laws and space-time. For example, Poincare symmetry of the action.

While these are obviously compatible, does their content differ slightly in precision and reach? Or are they entirely equivalent, differing only in pedagogy?

For example, some thoughts

• Can we derive angular momentum conservation from the first, or must we just take that as a consequence of what the physical laws happened to be?
• I could see how the first could be claimed to predict parity symmetry, but not the later.

Answer 1

The Lorentz group and Poincaré group symmetries are a more general starting point. If you take the historical postulates as a starting point (which most intro special relativity course still do I think), you can, for example, derive time dilation using the light clock argument and the constancy of $c$ (which derives directly from the principle of relativity):

$$t' = \gamma t $$

You can extend this argument to derive length contraction by considering a rod of length $l$ in its rest frame moving relative to clock at speed $v$.

$$ l' = \frac{l}{\gamma} $$

From these you can derive the Lorentz boosts along a single axis by considering the relative position of an event in two different reference frames, and you get the usual:

$$ t' = \gamma\left(t - \beta x\right) \\ x' = \gamma\left(x - \beta ct\right) $$

In matrix form it's clear that this is a hyperbolic rotation:

$$ \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} = \begin{pmatrix} ct' \\ x' \\ y' \\ z' \end{pmatrix} $$

with $\gamma = \cosh\phi$. Given that this holds true for all time-space coordinate pairs, it suggests that each time-space pair forms a 2D hyperbolic subspace. Hence the $s^2 = (ct)^2 - x^2 - y^2 - z^2$ metric. However, implicit here is the assumption that the 3D spatial subspace is Euclidean and has standard $\mathrm{SO}(3)$ rotational symmetry. So no, you can't derive conservation of angular momentum from the postulates, it has to be taken a priori.

The main problem with this approach is that it limits you to considering only spacetime coordinates and 'real' spacetime vectors and tensors (i.e., things that live in the tangent spaces to the Lorentzian manifold). There's nothing here about spinors, for example. It's useful pedagogy since it motivates the Poincaré group as the correct symmetry group, but once you've reached that point you can use the group theory as the starting point.

By taking the Poincaré group as the fundamental object in physics, you can not only derive the form that spacetime must take, but you also get the unitary representations for quantum mechanics, and the spinor representation for spin $\frac{1}{2}$ fermion fields. Also parity symmetry is included in the Poincaré group. It's also easier to generalise physics to different symmetry groups for higher dimensional spaces if you use Poincaré as a starting point, since your maths is already constructed around building up from group theory.

Answer 2

While the historical postulate approach is entirely compatible with the modern symmetry approach, issues such as the one of parity symmetry you bring up, make it clear that they are not entirely equivalent in their precision.

If you postulate the physical laws are the same in all inertial frames and the speed of light is the same in all inertial frames, to be precise this begs the question of what the precise definition of an inertial frame is. This is one of those concepts we intuitively understand, but which would be very difficult to make precise. Starting from these principles, and mainly drawing from experience from electrodynamics, it would be very tempting to claim special relativity predicts time reversal invariance and parity symmetry.

Then when parity violation is later observed, how would one adjust their understanding to this? Abandon SR, or just absorb it into the definition of inertial frame, thus preserving SR? Again, to avoid these issues, if using the historical postulates approach an inertial frame would need to be defined precisely. This would likely involve a procedural definition, thus depend on the physics, and so one would have to be very careful not to reason circularly. And parts of it would feel very artificial, like demanding an inertial coordinate system was right-handed instead of left-handed.

Postulating a symmetry is much more precise. Does Poincare symmetry include parity transformations? No. Therefore SR doesn't predict one way or the other regarding parity symmetry. Simple, done.

Continuing along the idea of trying to absorb broken symmetries into the definition of an inertial frame, what if one found there was a preferred direction, but Lorentz boosts perpendicular to this axis was still a symmetry? Sure I guess if part of physics involved this "conserved vector", then we could still define an infinite set of frames which are "inertial" frames, move relative to each other, and physics looked the same. In the parity case, it would be, do you want to demand inertial frames using a right-handed coordinate system? Here it would be, do you want to demand that space looks the same in all directions? As you can see, answering such questions in trying to precisely define an inertial frame, is really just inching our way towards the symmetry case.

So while it would be straining credulity: it depends on how you define an inertial frame, whether we could obtain full rotational symmetry of physics from the postulates. Did you include rotational symmetry in the definition or not? And in that sense, it would not even be a derivation, it just has to be taken a priori.