Einstein's 3rd Postulate

by M. R. Gale

I know, I know. The hallways of physics are littered with failed attempts to dethrone Einstein and here we go again apparently. But hear me out. If nothing else, you might learn something new about the road to Special Relativity. It starts with two fundamental postulates:

  1. Invariant laws of physics
  2. Invariant light speed

The 2nd one is a hard pill to swallow because it flies in the face of common sense. How can everyone measure the same speed of light if we’re all mulling about in different directions at different speeds? It’s absurd, but Einstein compels us to keep an open mind and follow the math. The experimentalists will sort us out if we run afoul of Mother Nature.

He starts by clarifying the concept of simultaneity because it’s not so easy if light speed is invariant. Think laser range finder. Send some light towards a wall and measure how long it takes to come back. If we presume the light spends just as much time getting there as it does coming back, we can split the difference to establish when and where the reflection event occurred.  You'll get a different result if you're on the move, but you can do it. It's just geometry. This is the scenario we’re talking about:

The quantity 2t' is established by intersecting the red world line with the reflected light ray. The answer is:

That puts the slope of the thin red line at v/c² and that’s how the red actor perceives simultaneity. All events along that line occurred at the same time as far as Red is concerned so the general transform is:

That preserves scale for the red actor (at x=vt.) You have to meddle with the zoom factor if you want to  preserve scale for the blue actor (at x=0):

Here's how that looks from Red's perspective:

That seems more likely, but Einstein argued that the roles of bystander and traveller are arbitrary. Swapping the primed and unprimed coordinates should yield the same result if you reverse the velocity vector. But it doesn't. Painting the red line blue and the blue line red changes everything because the quantity 2t is then established by intersecting the new blue world line with the reflected light ray:

The new answer is:

That can't be right. The answer can't depend on the direction of travel, let alone your job title or the colour of your shirt. So Einstein took a page from the Lorentz playbook and applied a symmetric zoom factor like this:

Call it his 3rd postulate. Any fool can do the math to work out the requisite zoom factor for this new constraint and the rest is history. This is how it works out for a collision course at  v=0.5c for example:

The "asymmetric" transforms are the ones that preserve scale for the blue actor before swapping roles. Time is the vertical axis and it’s graduated in units of ct to improve readability. LoS is a line of perceived simultaneity. You'll notice an extra event (represented by the round red dot) at the bottom. That's when Blue starts counting from Red's point of view and you can't fix that discrepancy by meddling with the zoom factor. Blue will always jump the gun in this use case as far as Red is concerned. The question is whether the zoom factor really needs to be symmetric. Einstein thought so because a symmetric zoom factor maintains light speed in tangential directions and that's arguably part and parcel to postulate #2. But there's a price to pay.

The emitter recoils when it emits light. That reaction establishes the point of origin in space and time, but also the light ray heading and that’s the direction in which the detector is going to bounce when the light arrives. There’s a distinction between a light ray that goes from my floor to my ceiling and one that goes from your floor to my ceiling, even if they follow the same path from my point of view. Both make my cabin a bit taller for a while, but the latter also imparts some momentum on my vehicle in (or against) its direction of travel so the equations need to account for that. Light speed certainly needs to be conserved in the context where the recoil is parallel to the light ray, but the field is torqued in all other cases. Does that affect the speed of wave propagation? It might. I guess we need Maxwell to weigh in again. Back to postulate #1...

Digging Deeper

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