Non-inertial reference frames question

[PeterDonis]

The image compression on approach is a large part of the relavistic beaming effect, increasing brightness. This is a lrger impact than blueshift.

Yep, you're right, I took a look at the formulas. I've deleted my previous post that was in error.

 

[Ibix]

When the rocket stops instantaneously, the last measurement it may make that is unambiguously made when it was moving is the position of Alpha Centauri on the surface of the past light cone of the deceleration event. If it stops at distance $d$ as measured in Centauri's rest frame, this should be $$d\sqrt{\frac{c+v}{c-v}}$$At some point after that, all measurements should agree that the distance is $d$. But how it happens depends on the measurement process.

If you use the angular size of the star, that changes instantaneously at the deceleration event. Since there's a finite speed of light, this means that you interpret the change from one inertial frame to the other as something that happened discontinuously on the surface of the past light cone of your deceleration event. So, the distance changed before you decelerated. The time at which this occurred is either $-d/c$ or $-fd/c$, where $f$ is the Doppler factor (the square root above), depending on which frame you include the surface of the light cone in.

If you use the radar method, the first measurement unambiguously made after stopping is the first radar pulse sent out after the stop. So the last pulse to return before deceleration establishes that Alpha Centauri was at distance $fd$ at ship's time $-fd/c$. The first pulse sent out after the stop establishes that Alpha Centauri was at distance $d$ at ship's time $+d/c$. Dolby and Gull show that pulses sent before the deceleration that return afterwards show a linear change in distance in the intervening period. So this method says that Alpha Centauri slows before the deceleration event and stops after it.

The radar method is actually building one possible non-inertial frame. The angular size method is giving you a physical justification for when to switch inertial frames. The radar method is better, to my way of thinking, because it's assignment of times is never problematic. Stitching together inertial frames leads to the ship asserting that some negative times happened twice in some places, so there's nasty book keeping hidden under a simpler exterior.

 

[PAllen]

Note that radar coordinates are globally well behaved only if, for the defining observer, motion is inertial for all time before some event, and also inertial for all time after some event. Given this constraint, no matter what the world line does in between gives you well behaved global coordinates. However, if this condition is not met, radar coordinates may have no more consistent coverage than Fermi-normal coordinates (which are what you get from stitching MCIF together). For example, for eternal uniform acceleration, the coverage of radar coordinates and Fermi-normal coordinates are identical. Note that Rindler coordinates are just Fermi-normal coordinates with a translation to make x=0 the horizon rather than a given accelerating observer.

 

[Ibix]

Note that radar coordinates are globally well behaved only if, for the defining observer, motion is inertial for all time before some event, and also inertial for all time after some event.

Or just inertial on average, I think. For example I can swing my radar set around my head on a string for all eternity and I should get good coordinates everywhere - if I look on a large enough scale the deviation from the Minkowski frame of my and my radar set's joint centre of mass should be negligible. Or is there something I'm missing?

 

[PAllen]

Or just inertial on average, I think. For example I can swing my radar set around my head on a string for all eternity and I should get good coordinates everywhere - if I look on a large enough scale the deviation from the Minkowski frame of my and my radar set's joint centre of mass should be negligible. Or is there something I'm missing?

That sounds plausible. I did not consider oscillating situations when deriving the rule I stated.