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. The road to Special Relativity starts with two fundamental postulates:

  1. Invariant laws of physics
  2. Invariant (and finite) 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. You send some light towards a reflector and measure how long it takes to come back. Split the difference to establish the time of the reflection event and multiply by light speed to establish where that happened. You’ll get a different result if you’re on the move, but we can compensate for that with a bit of geometry. Here’s how it work if light goes at different speeds in different directions:

Here’s how Einstein made it work for invariant light speed: 

The reflector pegs the detector's perception of the reflection event at an earlier time, when they were farther apart. It’s an illusion of sorts, but it's a stubbornly persistent one because it has real world consequences. But that's a story for another day. The problem at hand is to follow the math, which is this:

Fair enough. You can't argue with geometry. That checks the 2nd postulate box, but there's a problem with the 1st one. This is what you get for the reverse transform after a bit of algebra:

That can't be right. It's too lopsided. Postulate #1 demands symmetry. Negating the velocity vector in one transform should yield the other if you also swap the primed and unprimed coordinates. So Einstein takes a page from the Lorentz playbook and comes up with this: 

He says that’s fair game because his postulates are agnostic to a symmetric change of scale. Any fool can do the math to work out the requisite scaling factor for this new constraint (postulate #3) and the rest is history. Here’s how that works out for a collision course at v=0.5c:

Time is the vertical axis in these charts to make it easier to compare the timelines in a side-by-side view. And it’s graduated in units of ct to improve visibility. The problem is, rescaling is tantamount to changing the zoom factor. Why would Mother Nature do that? Doesn’t she have anything better to do? I don’t presume to tell her how to do her job, but there is a better way to balance the books.

Remember that Blue jumped the gun from Red's perspective. If you want to start with Red's perception of events, you have to start with Red's perception of simultaneity. That means starting Red's clock earlier, when Blue was farther away. Otherwise you'd be talking about a different event on Blue's worldline. Here's the math for that:

And here’s that math in action:

The pivot point for the forward transform (xo, to) is where and when Red starts the clock from Blue's point of view. The pivot point for the reverse transform is a bit earlier because Blue jumped the gun from Red's point of view. If that makes you go cross-eyed, you might find the divergent use case easier to digest:

The upshot is, there's no need to meddle with the zoom factor so it’s back to square one unless anyone can think of a better reason for doing that. Algebraic aesthetics is a poor excuse for science. Either way, Mother Nature will have to weigh in again. If I'm right, our recipes for relativistic energy and momentum are going to need adjusting, as will our recipe for gravity. And we’ll need a whole lot of graduate students to grind through the test data archives.

Here's the Excel formulation if you want to play around with it. (I dare say Einstein would have sussed it out if he had access to such a marvellous tool.)

v/c,0.5,,,,,,,,,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,
,Blue POV,,,,Forward Lorentz Transform,,,,"=SUBSTITUTE(F3,""Lorentz"",""Lattice"")",,,,"=SUBSTITUTE(F3,""Forward"",""Reverse"")",,,,"=SUBSTITUTE(J3,""Forward"",""Reverse"")",,,
Event,x,ct,=B4,=C4,"=IF(RIGHT(B4,1)=""'"",LEFT(B4,LEN(B4)-1),B4&""'"")","=IF(RIGHT(C4,1)=""'"",LEFT(C4,LEN(C4)-1),C4&""'"")",=F4,=G4,=F4,=G4,=H4,=I4,"=IF(RIGHT(J4,1)=""'"",LEFT(J4,LEN(J4)-1),J4&""'"")","=IF(RIGHT(K4,1)=""'"",LEFT(K4,LEN(K4)-1),K4&""'"")",=N4,=O4,=N4,=O4,=P4,=Q4
Prestart,,,=D$6+D9/E9*(E5-E$6),=E6-ABS($C$1*(B6-D6))/(1-$C$1^2),,,=(D5-$E$6-$C$1*(E5-$F$6))/SQRT(1-$C$1^2)+D$6,=(E5-$F$6-$C$1*(D5-$E$6))/SQRT(1-$C$1^2)+$F$6,,,=D5-$C$1*($F5-$F$6),=E5-$C$1*($E5-$E$6),,,=(H5-$G$6+$C$1*(I5-$H$6))/SQRT(1-$C$1^2),=(I5-I$6+$C$1*(H5-H$6))/SQRT(1-$C$1^2)+$F$6,,,=(L5+$C$1*(M5-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(M5+$C$1*(L5-($E$6+$C$1*$F$6)))/(1-$C$1^2)
Start,0,=E6,-1,0,=(B6-$E$6-$C$1*(C6-$D$6))/SQRT(1-$C$1^2)+$E$6,=(C6-$F$6-$C$1*(B6-$E$6))/SQRT(1-$C$1^2)+E$6,=(D6-$E$6-$C$1*(E6-$F$6))/SQRT(1-$C$1^2)+D$6,=(E6-$F$6-$C$1*(D6-$E$6))/SQRT(1-$C$1^2)+$F$6,=B6-$C$1*($D6-$D$6),=C6-$C$1*($C6-$E$6),=D6-$C$1*($F6-$F$6),=E6-$C$1*($E6-$E$6),=(F6-$G$6+$C$1*(G6-$H$6))/SQRT(1-$C$1^2),=(G6-G$6+$C$1*(F6-F$6))/SQRT(1-$C$1^2)+$F$6,=(H6-$G$6+$C$1*(I6-$H$6))/SQRT(1-$C$1^2),=(I6-I$6+$C$1*(H6-H$6))/SQRT(1-$C$1^2)+$F$6,=(J6+$C$1*(K6-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(K6+$C$1*(J6-($E$6+$C$1*$F$6)))/(1-$C$1^2),=(L6+$C$1*(M6-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(M6+$C$1*(L6-($E$6+$C$1*$F$6)))/(1-$C$1^2)
Split,=B6,=E7,=D$6+D9/E9*(E7-E$6),"=E6+IF(B6=D6,0,ABS($C$1/(B6-D6)))",=(B7-$E$6-$C$1*(C7-$D$6))/SQRT(1-$C$1^2)+$E$6,=(C7-$F$6-$C$1*(B7-$E$6))/SQRT(1-$C$1^2)+$F$6,=(D7-$E$6-$C$1*(E7-$F$6))/SQRT(1-$C$1^2)+D$6,=(E7-$F$6-$C$1*(D7-$E$6))/SQRT(1-$C$1^2)+$F$6,=B7-$C$1*($D7-$D$6),=C7-$C$1*($C7-$E$6),=D7-$C$1*($F7-$F$6),=E7-$C$1*($E7-$E$6),=(F7-$G$6+$C$1*(G7-$H$6))/SQRT(1-$C$1^2),=(G7-G$6+$C$1*(F7-F$6))/SQRT(1-$C$1^2)+$F$6,=(H7-$G$6+$C$1*(I7-$H$6))/SQRT(1-$C$1^2),=(I7-I$6+$C$1*(H7-H$6))/SQRT(1-$C$1^2)+$F$6,=(J7+$C$1*(K7-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(K7+$C$1*(J7-($E$6 + $C$1*$F$6)))/(1-$C$1^2),=(L7+$C$1*(M7-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(M7+$C$1*(L7-($E$6+$C$1*$F$6)))/(1-$C$1^2)
Stop,=B7,=E8,0,2,=(B8-$E$6-$C$1*(C8-$D$6))/SQRT(1-$C$1^2)+$E$6,=(C8-$F$6-$C$1*(B8-$E$6))/SQRT(1-$C$1^2)+$F$6,=(D8-$E$6-$C$1*(E8-$F$6))/SQRT(1-$C$1^2)+D$6,=(E8-$F$6-$C$1*(D8-$E$6))/SQRT(1-$C$1^2)+$F$6,=B8-$C$1*($D8-$D$6),=C8-$C$1*($C8-$E$6),=D8-$C$1*($F8-$F$6),=E8-$C$1*($E8-$E$6),=(F8-$G$6+$C$1*(G8-$H$6))/SQRT(1-$C$1^2),=(G8-G$6+$C$1*(F8-F$6))/SQRT(1-$C$1^2)+$F$6,=(H8-$G$6+$C$1*(I8-$H$6))/SQRT(1-$C$1^2),=(I8-I$6+$C$1*(H8-H$6))/SQRT(1-$C$1^2)+$F$6,=(J8+$C$1*(K8-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(K8+$C$1*(J8-($E$6 + $C$1*$F$6)))/(1-$C$1^2),=(L8+$C$1*(M8-($F$6+$C$1*$E$6)))/(1-$C$1^2),=(M8+$C$1*(L8-($E$6+$C$1*$F$6)))/(1-$C$1^2)
Displacement,=B8-B6,=C8-C6,=D8-D6,=E8-E6,=F8-F6,=G8-G6,=H8-H6,=I8-I6,=J8-J6,=K8-K6,=L8-L6,=M8-M6,=N8-N6,=O8-O6,=P8-P6,=Q8-Q6,=R8-R6,=S8-S6,=T8-T6,=U8-U6
Velocity,=B9/C9,,=D9/E9,,=F9/G9,,=H9/I9,,=J9/K9,,=L9/M9,,=N9/O9,,=P9/Q9,,=R9/S9,,=T9/U9, 

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