JesseM said:
How are you defining "proper velocity"? I haven't seen that term.
There is a couple of definitions for the velocity of an object in relativity: 1- coordinate velocity which is measured by an observer at rest and thus equals v=dx^{\alpha}/dt. 2- proper velocity relative to an observer divides observer-measured distance by the time elapsed on the clocks of the traveling object and thus equals v=dx^{\alpha}/d\tau. When an instantaneously at rest observer is assumed, then the proper velocity is zero. This automatically suggests from dx^{\alpha}/d\tau \gamma^{-1} that the coordinate velocity vanishes as well.
You're right, they do say that, but I'm confused about how that fits with what they said about time dilation...after all if you have a family of observers with the same proper acceleration, then in an inertial frame where they all begin to accelerate simultaneously, they will all have the same velocity as a function of time and thus their clocks won't get out of sync. So if they're talking about this type of acceleration rather than Born rigid acceleration,
This is correct! Since in an inertial frame all observers at rest move with a uniform velocity, then their proper accelerations are the same, thus the clocks will remain in sync!
Is it because although they don't get out of sync in the inertial frame, we're considering a non-inertial frame whose definition of simultaneity at each moment matches up with the definition of simultaneity in one of the observer's instantaneous inertial rest frame at that moment, so in the non-inertial frame the clocks are getting progressively further out-of-sync?
Two scenarios can be imagined:
1-It is just because they want to call, for example, Rindler coordinates fallaciously 'uniform' (or similarly inertial) and justify why such position-dependent field would make the time dilation find its meaning through the height difference!
2- They define some sort of non-inertial frame wherein they map all information at each moment into an observer's instantaneous inertial rest frame to only have the definition of simultaneity met in a possible way. But since the non-inertia of the first frame does not keep the traveling clock in sync with the observer's clock, then the time dilation occurs!
Both scenarios are flawed. The First one has this problem that a uniform field makes the proper acceleration stay at a constant everywhere so all clocks will experience the same time dilation at any point! This only works if the spacetime is Minkowski! The second one is contradictory: if the clocks are simultaneously set with an inertial observer's clock, then how does the property of "being in sync" from frame to frame stay in agreement with the fact that the non-inertial clocks are getting out of sync?
Also, if they all had the same proper acceleration, then in each observer's instantaneous inertial rest frame, the instantaneous velocity of the other observers would be nonzero, so if there were measuring-rods connecting the observers they would be physically stretching (experiencing changing internal stresses, as in the spaceship paradox[/url])...only Born rigid acceleration guarantees that each observer sees the other observers' instantaneous velocity as zero in their own instantaneous inertial rest frame (and thus the distance between neighboring observers doesn't change from one moment to the next if you consider their instantaneous inertial rest frame at each moment), correct?
Correct! But as bcrowell said, we don't have a spacetime that admits "uniform field" globally in GR. This can only be discussed along with a consideration borrowed from the equivalence principle that in a small region of spacetime the field can be uniform thus the local inertia and Born rigidity get resurrected again!
So if you want to draw an equivalence between a small accelerating lab and a small laboratory in a gravitational field where you can put up a ruler to measure the distance between the ceiling and floor and the ruler can have stable structure (its internal stresses aren't increasing until it breaks), don't you need to assume the accelerating lab is experiencing Born rigid acceleration rather than uniform proper acceleration at both the ceiling and floor? Or does it not really matter because you're only considering what would be measured in each lab during an infinitesimally brief span of time?
Of course it does not matter because as I earlier said, both the implications of "uniform acceleration" and "Born rigid acceleration" are the same in a small region of spacetime. But I recall that if you have a spacetime admitting a uniform gravitational field, then it is necessary for proper accelerations to be position-independent and thus both the above accelerations will be the same! The only difference between the two pictures taken into account by Born and by the uniform gravitational field is that considering an instantaneously co-moving inertial rest frame in Born's picture is mandatory, but in the latter we already have the constant distances between particles everywhere.
But you'd agree that if we transform from Rindler coordinates into an inertial frame, the grounds are all moving inertially, right?
Of course! But remember that such a coordinate transformation changes the nature of spacetime, if considered globally true, and thus is completely artificial. In the equivalence principle, we do a metric transformation to reach the Minkowski metric in a small region, but this does not destroy the nature of spacetime because we have already intuited this in the real world that in small regions the flatness is approximately guaranteed.
AB
