The Missing Ingredient

People have been pushing back on my assertion that the rest frame for a Lorentz transformation cannot be dealer’s choice. It boggled my mind for the longest time because it seemed perfectly obvious to me. Why did Hafele and Keating start with a non-rotating Earth for example? Because that context has no angular momentum. The cosmic rest frame has no net linear momentum so the same logic should apply.

The universe has no net momentum so the cosmic rest frame is the best context for distinguishing between hitting the gas and hitting the brakes. But momentum doesn’t factor into the equations. Time dilation is agnostic to the direction of motion and proper time intervals are invariant so who cares about the cosmic rest frame?

We can transform events into the cosmic rest frame before calculating proper time intervals:

  • Elapsed cosmic time from launch to reunion is Tb = t / sqrt(1-Vb^2/c^2)
  • Elapsed traveller time from launch to waypoint is To = (Tb/2) * sqrt(1-Vo^2/c^2)
  • Elapsed traveller time from waypoint to reunion is Ti = (Tb/2) * sqrt(1 - Vi^2/c^2)
The elapsed traveller time from launch to reunion is then:

To + Ti = (t/2) * (sqrt(1-Vo^2/c^2) + sqrt(1 - Vi^2/c^2)) / sqrt(1-Vb^2/c^2)

Where:
  • t = elapsed bystander time from launch to reunion
  • Vb = bystander velocity in cosmic rest frame
  • Vo = traveller velocity on outbound leg in cosmic rest frame
  • Vi = traveller velocity on inbound leg in cosmic rest frame
It all boils down to the context in which the events in question are simultaneous. You have to think in terms of observables. Your age on arrival over there is entirely objective because the evidence is right there in your vacation footage. Same goes for my age when the image of your arrival arrives. I can correct for the propagation delay in order to work out how old I was when that image started its journey towards my eyeballs. That establishes simultaneity and SR can take it from there. It's not entirely obvious at first glance, but this analysis gives the same result as the conventional one. You just have to be careful to use the velocity addition formula when calculating relative velocities in a different contexts. The cosmic rest frame is not special in this regard. It's just the best place to establish simultaneity because it's a context everyone can agree on.

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