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 reflector and measure how long it takes to come back. Split the difference to establish when and where the reflection event occurred. You’ll get a different result if you’re on the move. This is the kind of geometry we’re talking about:

The green lines are light rays. The thick red and blue ones are observers and the thin ones are their lines of perceived simultaneity. (i.e. Lines of constant observer time or constant observer space.) The dots are events of interest.

So what’s the big kerfuffle? The age of the reflector at the point of reflection depends on the context in which you start counting, but it ages at the same rate as everyone else. The problem is, there's an ambiguity of scale. Preserving scale for the red observer is an arbitrary choice. You get a completely different result if you start in the red reference frame and preserve scale for the blue observer instead. Things also go off the rails when you look at a tangential light ray. How fast does the light go if it bounces off the ceiling instead of a wall for example? The traveller sees it go straight up and straight down. The bystander sees a zigzag. The zigzag path is longer unless the cabin gets taller in its own reference frame. That seems unlikely because tunnels would have to get taller, too.

Einstein noticed that we can maintain the height of the cabin without changing the speed of light by meddling with the zoom factor instead. Call it his 3rd postulate. Fair enough. You’d be hard pressed to prove him wrong because it’s diabolically hard to get your lab up to ludicrous speed and all other use cases seem to pan out as far as we can tell. But we don’t know for sure that the invariant light speed postulate holds for tangential rays.

The emitter recoils when it gives birth to the light. That reaction establishes the point of origin, 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 its direction of travel. 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 and that could affect the speed of wave propagation.

In any case, here’s how it works out for a collision course at v=0.5c if you preserve scale for the blue observer:

Time is the vertical axis and it’s graduated in units of ct to improve readability. LoS is a line of (perceived) simultaneity. The symmetric transforms are the ones Einstein recommends. You'll notice an extra event at the bottom (the round red dot.) That's when Blue starts counting from Red's point of view. Blue jumps the gun from that perspective, with or without a zoom factor. The question is whether Einstein's zoom factor is indeed the correct one. It's baffling that inertia doesn't factor into the equations for example because that's what it's all about. Observers have inertia and light doesn't. Maybe the zoom factor needs to be established in the centre of mass frame.

Digging Deeper

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