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 his analysis 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. We can presume the light spends just as much time getting there as it does coming back so split the difference to establish when and where the reflection event occurred.  Travellers will get a different result because they're going to see an event before the rest of us if they're running towards it and we're not for example, but you can do it. It's just geometry. This is the scenario we’re talking about:

You can establish the quantity 2t' by intersecting the red world line with that of 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 can preserve scale for the blue actor (at x=0) by meddling with the zoom factor:

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

That seems more likely if Red is the one who felt the force on the launchpad, but forces shouldn't factor into the equation because that's water under the bridge after everyone has gotten underway. Einstein argued that the roles of bystander and traveller are arbitrary so swapping the primed and unprimed coordinates and reversing the velocity vector should yield the same result. To make that happen, he took a page from the Lorentz playbook and applied a symmetric zoom factor like this:

Any fool can do the math to work out the requisite zoom factor for this new constraint and the rest is history. Painting the red line blue and the blue line red has consequences though because the traveller has a different perception of the distance from here to there

In any case, this is how it all works out for a collision course at  v=0.5c:

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 from Red's point of view in this use case. The question is whether the zoom factor 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 to ask Maxwell. Back to postulate #1...

Digging Deeper

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