The Equivalence Principle...

[CraigD]

I think this thread has managed to answer the original question, which I paraphrase

does General Relativity state that it’s impossible to distinguish an isolated room subjected to constant acceleration from one subject to gravitational acceleration?

After trying both possible answers – yes and no – I believe we’ve shown the answer is

no, for any room with non-zero height, it’s possible to distinguish forces due to gravity from forces due to the room’s acceleration.

There are many ways to state this, some involving the terms “equivalence principle” “infinitesimal” and “tidal force”, but all describe, I think, the same underlying principle.

Simple as this is, I’ve long had a problem applying it. In particular, describing what an un-accelerated observer would observe an accurate clock on a body under acceleration to read relative to his own accurate clock, where both have small masses, and are far from large-mass bodies.

Consider two spaceships far from any star or planet. Initially, A has a velocity of 1000 m/s nearly directly toward B. The distance between A and B is 5000 m. A has an acceleration of 100 m/s/s (a bit over 10 gs). After 10 seconds, A and B are near one another, with relative velocity 0.

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

The formula for gravitational time dilation in an accelerated box,

, appears at many reference websites, including this wikipedia page. is acceleration, but is “is the ‘vertical’ distance between the observers”, which I’m unable to understand.

Does anyone know how to calculate the answer to the question above?

[Qfwfq]

Originally Posted by CraigD

Simple as this is, I’ve long had a problem applying it. In particular, describing what an un-accelerated observer would observe an accurate clock on a body under acceleration to read relative to his own accurate clock, where both have small masses, and are far from large-mass bodies.

Consider two spaceships far from any star or planet. Initially, A has a velocity of 1000 m/s nearly directly toward B. The distance between A and B is 5000 m. A has an acceleration of 100 m/s/s (a bit over 10 gs). After 10 seconds, A and B are near one another, with relative velocity 0.

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

Strictly, this is not a problem of GR, it is SR. At that instant, . From the inertial rest frame, the ratio is just according to the usual formula with the instantaneous value of , you can even calculate for a finite interval by integrating the same formula with the time dependance .

Originally Posted by CraigD

The formula for gravitational time dilation in an accelerated box,

, appears at many reference websites, including this wikipedia page. is acceleration, but is “is the ‘vertical’ distance between the observers”, which I’m unable to understand.

Gee that section is rather poorly worded. Here, the observers are mutually at rest so both have the same velocity. The is simply their distance according to the direction of the acceleration and the numerator in the exponent is just the potential difference for a uniform field (and the denominator of course is due if units aren't natural). The exponential simply maps an additive comparison onto a multiplicative one because you want the ratio (no integration is necessary).

[modest]

Originally Posted by CraigD

After 10 seconds, A and B are near one another, with relative velocity 0.

Around that instant, what rate relative to his accurate clock would an observer on B observe of an accurate clock on A?

At that instant neither are accelerating nor have relative velocity,so t = T. No difference in rate. But, I think you mean, what is the rate during the acceleration...? But the rate changes. Because A is accelerating and B is not accelerating there is not a constant difference in rates. The velocity between the two changes so time dilation comes from acceleration and velocity. The equation:

only works if the distance between them (h) is always the same. That is, if both A and B were accelerating at the same rate, in the same direction, then the equation above would give the time dilation between them, where h is the distance between them.

If you want to find the difference in time for your whole thought experiment, from start to finish, between the two you can use the relativistic rock equation. If the trip takes exactly 10 seconds from A's perspective then B's clock would advance:

~modest