Sasha [They/Them]

Cool yeah, dug through the maths and took the time to understand the situation you were describing and I understand now.

I thought you were describing a something else, and then slightly confused myself by only considering the metric and not the global picture. I was trying to abuse some old heuristic techniques and they don’t quite work for this case, though it was fine for what I was picturing, where Jane and Alice are symmetric about Bob.

Thanks for taking the time to convince me.

Actually reading back over this is hilariously dumb, forgive my bad reading comprehension, that is how redshift works.

Would you agree that, at the moment they pass each other, Alice and Jane would see different points in Bob’s history as being simultaneous in their respective reference frames?

I wouldn’t say that’s necessarily true, no. It’s only true if their speeds are different, the direction they’re travelling doesn’t factor into it. In either case, their lines of constant time are the same because space is rotationally symmetric. I’m not thinking of them adjusting their calculations to some “true” sense of simultaneous, if that’s what you meant in your edit, because there isn’t one by definition.

And it’s related to distance because the further they are from Bob, the greater the discrepancy in their calculations of Bob’s relative time will be.

That really depends on what you mean by time shift in your original comment. If you mean the shift in what they perceive as simultaneous, then it’s not, but it seemed to me that’s what you meant. If you mean the difference in their age then I honestly can’t remember how it factors in for the accelerating case, I haven’t had to think about SR problems in a while.

I don’t think this is true, you’re right that what’s considered simultaneous changes*, but it’s not related to distance and that’s not how redshift works.

At best Alice could use the redshift to work out how fast she’s moving relative to Bob, if she’s moving towards and away from him at the same speed, she’ll always get the same result. She’d actually think the same moments are simultaneous regardless of direction.

The time difference is only accumulated during her acceleration so it can only be measured during it.

*but only when her speed is different

If you want to get really technical, it’s because the symmetries of the Minkowski metric are the Poincaré group. Which includes only rotations, translations and boosts, none of which correspond to acceleration. Meaning it’s inherently impossible to make acceleration look like being stationary because of the geometry of spacetime.

If Alice flies by Bob at some relativistic speed, then there’s a very simple coordinate transform (a Lorentz boost) that flips our perspective to Alice’s pov; she’s stationary and Bob is moving.

If Alice were to accelerate and we did the same thing, we’d end up with a “momentarily comoving reference frame,” in which Alice is only “stationary” for an instant and Bob is moving at a constant speed as before. Or we could create a non-inertial reference frame which would look nothing like Bob’s perspective, but Alice would be stationary.

Physics in non-inertial frames behaves differently, as a simple example: if stationary (or constant speed) Bob dropped an object while floating in space, it would remain there. If accelerating Alice tried the same thing, it would accelerate away from her. You can test this out in an accelerating car or train or whatever and see that it’s fundamentally asymmetrical even before considering SR.

In terms of things like length contraction and time dilation, these are a little more complicated mathematically, but it’s just an extension of the above asymmetry when spacetime is Minkowski rather than Euclidean. The difference in observed time is clear when looking at each person’s worldline, Alice’s isn’t straight like Bob’s and so she unambiguously experiences a different proper time and proper length.

Ultimately this means that even if Alice accelerates then passes Bob at a constantly speed, they’ll both see one another’s clocks running slow by the same amount, when Alice decelerates and returns to compare her stopwatch with Bob’s they’ll have very different totals which corresponds to how much time Alice lost during her acceleration.

It’s satire, but it’s not fake, it does exactly what it says it does.