B Gravity vs Acceleration: The Impact on Clocks in Reference Frame R

[PeroK]

If "we" are inertial observers, you are right, but I suspect what cianfa72 meant to say was "each observer claims the other is at rest relative to themself", and, in that case, that is true.

That's what we all meant! Starting with:

The precise way of putting it is this: if we have two observers inside a rocket accelerating in flat spacetime ("accelerating" meaning "experiencing proper acceleration"), at rest relative to each other (as measured by them exchanging round-trip light signals and seeing that the round-trip travel time by each of their clocks remains constant), the observer at the bottom of the rocket will be time dilated (clock running slower) relative to the observer at the top (as measured by noting the elapsed time on both of their clocks between two successive round-trip light signals).

 

[jartsa]

EDIT: Just to clarify possibly ambiguous language:

• Observer A claims that Observer B is at rest relative to Observer A
• Observer B claims that Observer A is at rest relative to Observer B

Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer" means.

Last sentence is ambiguous. It should be said like this: Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer, according to the observer himself" means.

Relative to what, and according to who.

 

[DrGreg]

Last sentence is ambiguous. It should be said like this: Both these claims are true; they can be taken as a definition of what "at rest relative to a non-inertial observer, according to the observer himself" means.

Relative to what, and according to who.

I think in relativity, when you say "A is at rest relative to B" it is automatically assumed it's "...according to B". Or in other words, it means "A is at rest in B's coordinate system". I think it's a bad idea to say "A is at rest relative to B according to C". I'm not even sure it has a well-defined meaning. It would be better to say "A and B have the same velocity relative to C", if that's what you mean, or "A and B have a constant separation as measured by C". The details depend on exactly how C chooses to do the measurement. It may also depend on whether spacetime is curved or flat. ("Same velocity" doesn't always make sense.)

 

[PeterDonis]

I think in relativity, when you say "A is at rest relative to B" it is automatically assumed it's "...according to B".

Actually, I think the proper meaning of "at rest relative to" should be that it is an invariant: the best way to say it is "A and B are at rest relative to each other", and this is to be taken as meaning that, for example, round-trip light signals between them have a constant round-trip travel time, according to either of their clocks, which is an invariant, independent of any choice of coordinates. That is the meaning @cianfa72 was using, and I think it's correct.

 

[PeterDonis]

All inertial observers say that there is some relative motion.

In coordinate terms, yes. But what invariant indicates "relative motion" in this case? Note that, per my previous post just now, there is an invariant (timing of round-trip light signals between A and B) that says there is no "relative motion".

 

[jartsa]

In coordinate terms, yes. But what invariant indicates "relative motion" in this case? Note that, per my previous post just now, there is an invariant (timing of round-trip light signals between A and B) that says there is no "relative motion".

Velocity reading on an inertial navigation system bolted on the rocket floor minus velocity reading on an inertial navigation system bolted on the rocket ceiling. That number is the same in all frames.

 

[Ibix]

Velocity reading on an inertial navigation system bolted on the rocket floor minus velocity reading on an inertial navigation system bolted on the rocket ceiling. That number is the same in all frames.

You need to compare readings taken at the same time, though. What simultaneity condition are you using? You can get growing, shrinking, or steady distances depending on your choice.

 

[cianfa72]

Velocity reading on an inertial navigation system bolted on the rocket floor

Btw, what does a such inertial navigation system actually measure ?

 

[Ibix]

Btw, what does a such inertial navigation system actually measure ?

The proper acceleration and possibly rotation rates with respect to the Fermi-normal axes, all as functions of the instrument's proper time, and integrals thereof.