Equivalence of Time Dilation in Different Gravitational and Accelerating Frames

[yuiop]

...
I'll have to brush up on LaTex first.

Hi Chrisc,
to get started click on the equation below. In the window that pops up, you will see an example of the text you need to embed in your post to create a latex image.

w = w_0\frac{(1+v/c)}{\sqrt{1-v^2/c^2}}

 

[Chrisc]

Thanks kev, very handy link. It's come a long way since I last looked at LaTex.

 

[Dale]

Hi Chrisc,

I appreciate the effort. I believe that your eq 2.0 is correct to a first order approximation for low velocities, although your derivation was not very clear. I would have derived it this way:

Consider the momentarily co-moving inertial reference frame where the detector is initially at rest at the origin and accelerates in the positive x direction with an acceleration of g and where the emitter emits a photon while at rest at a position of H on the positive x axis. We can express the worldlines of the photon and detector in this reference frame as:
\begin{array}{l}<br /> x_{\text{detector}}(t)=\frac{g t^2}{2} \\<br /> x_{\text{photon}}(t)=H-c t<br /> \end{array} eq 0.1

We can deterimne the time when the detector meets the photon by setting the two equations in 0.1 equal to each other and solving for t.
\begin{array}{l}<br /> <br /> x_{\text{detector}}\left(t_{\text{detection}}\right)=x_{\text{photon}<br /> }\left(t_{\text{detection}}\right) \\<br /> \frac{g t_{\text{detection}}^2}{2}=H-c t_{\text{detection}} \\<br /> t_{\text{detection}}=\frac{\sqrt{c^2+2 g H}-c}{g}<br /> \end{array} eq 0.2

Multiplying the resulting detection time by g gives us the velocity of the detector relative to the velocity of the emitter.
v=\dot{x}_{\text{detector}}(t_{\text{detection}})=g t_{\text{detection}}=\sqrt{c^2+2 g H}-c eq 0.3

Substituting eq 0.3 into the relativistic Doppler equation (eq 1.0) and simplifying gives us:
\frac{\omega }{\omega _0}=\frac{1}{\sqrt{\frac{2 c}{\sqrt{c^2+2 g<br /> H}}-1}} eq 1.1

And finally, taking a Taylor series expansion about g=0 and simplifying gives.
\frac{\omega }{\omega _0}=1+\frac{H g}{c^2}+O\left(g^2\right) eq 2.0

Please note that your eq 2.0 is constant wrt time since g H and c are all constant wrt time. Note also that eq 1.0, the relativistic Doppler equation, does not depend on the distance between emission and detection, only the relative velocity of the emitter and detector. The fact that this distance becomes smaller as the rocket's velocity increases is irrelevant, although you keep mentioning it.

Finally, note that your eq 3.0 is dimensionally inconsistent and doesn't follow from eq 2.0.

 

[Chrisc]

Thanks DaleSpam, you're equations all make sense and the Taylor series is a handy expression I was not familiar with.
But, you have shown me how to calculate relativistic doppler shift of wavelength between two comoving frames
accelerating with respect to an inertial frame and how to express the infinite function of the same.
This is not the point of this discussion and therefore the reason you disagree with me.
We are dealing with the frequency of detection not the frequency detected.
Which is to say the number of light signals detected at B per unit time (local time at B) not
the number of photons/wavelengths detected per unit time.
The reason I keep mentioning it is because the "frequency of detection" is the evidence B uses
to calculate the rate of the clock at A. The frequency of emission, (again not the frequency of light emitted
but the number of emissions at A per unit time [local time at A]) is not the same as the
frequency of detection at B. This evidence is what B uses to determine the clock at A is
running fast with respect to the clock at B.
I used the doppler shift equations because the "difference" in the frequency of emission and the
frequency of detection as calculated at B, vary "with" the "increase" of the instantaneous velocity
of the ship with respect to an inertial observer.

 

[Dale]

We are dealing with the frequency of detection not the frequency detected.

It is the same thing. Let's say that a transmitter transmits one thousand flashes every second and simultaneously broadcasts a pure sinusoidal 1 kHz radio signal. If the frequency detected of the pure sinusoidal signal is 1.001 kHz then the frequency of detection of the flashes must be 1001 flashes per second.

 

[Chrisc]

Yes DaleSpam, if you set the frequency of emission equal to the frequency of flashes then of course they will be and remain numerically equal as counted by observers in any frame the signal passes through. And although they will remain equal to each other, they will not remain the same over time as A accelerates with respect to any frame.
If A moves with constant velocity with respect to and away from M, the frequency detected at M will be less than the frequency emitted at A, but will remain constant as the motion between A and M is constant. This is "classical" evidence of the relative motion between A and M. In the case where this constant velocity approaches c, there will be an increasingly apparent discrepancy in the classical addition of velocities which is what the principle of relativity reconciles via the Lorentz transformation and the time dilation of A and M become important.

Unlike the "constant" example above, the "continuous" emission of an EM wave from A will, with respect to an inertial observer M, "continue" to shift toward the lower end of EM spectrum with the "continued" acceleration of A with respect to and away from M.
In this case we have the "classical" evidence of acceleration between A and M. This relative motion is also subject to the principle of relativity and will with sufficient velocity between A and M, require the principle of relativity to reconcile the limit of the instantaneous velocity between A and M.

Now we must consider the measurements of B which is "classically" at rest with respect to A and "classically" under the same acceleration as A with respect to M. This is where the difference between "flashes" and "wavelength" becomes apparent and important to the considerations of an observer at B. In the classical limit, the observer can measure a discrepancy in the rate of flashes at a very low number of flashes. If the rate of flashes emitted at A is agreed to be 1 per second, the observer at B only has to mark a single, finite, discrete event - the tick of his clock and the detection of the flash. If they do not agree, he immediately knows there is a reconciliation required to explain the event. If on the other hand he attempts to measure wavelength he must measure the incident wave from trough to trough during which time his position has "potentially" changed. If we allow the wavelength to be very short so as to "nearly" negate the potential motion of B, we approach the classical limits of measurements made by B and the relativistic effects must be considered to significantly affect his measurements.
It is because he does not know if his frame is accelerating or in a gravitational field that he cannot "assume" his motion except with respect to A. Based on the constancy of the speed of light he has no choice but to reconcile the increased rate of flashes is proof of the increased rate of the clock at A. This is true for gravitation and acceleration.

Now consider the rate of flashes and the increasing rate of motion due to acceleration with respect to M. B will as mentioned above, detect each flash as a discrete event. The number of these events will increase per unit time as marked by the clock at B. B has no choice but to reconcile the increased and increasing rate of flashes as proof of the increased and increasing rate of the clock at A. This is NOT true of similar events marked in a gravitational field. B now knows the ship is accelerating in free space, i.e. the force on the ship is inertial.

 

[Dale]

And although they will remain equal to each other, they will not remain the same over time as A accelerates with respect to any frame.

Yes they will remain the same over time. I refer you back to our eq 2.0

This is where the difference between "flashes" and "wavelength" becomes apparent and important to the considerations of an observer at B. In the classical limit, the observer can measure a discrepancy in the rate of flashes at a very low number of flashes. If the rate of flashes emitted at A is agreed to be 1 per second, the observer at B only has to mark a single, finite, discrete event - the tick of his clock and the detection of the flash. If they do not agree, he immediately knows there is a reconciliation required to explain the event. If on the other hand he attempts to measure wavelength he must measure the incident wave from trough to trough during which time his position has "potentially" changed.

I'm sorry, but this is just silly. You cannot measure the frequency of flashes by measuring a single event. You have to measure the time between a minimum of two flashes to get a "frequency of detection" just like you need to measure a minimum of two peaks on a sinusoidal signal to get a "detected frequency". A frequency is, by definition, the inverse of a period. A period is a finite duration of time, not a single event. Your assertions here are absurd.

There is no difference between "frequency of detection" and "detected frequency". If one shifts or dilates so will the other.

Now consider the rate of flashes and the increasing rate of motion due to acceleration with respect to M. B will as mentioned above, detect each flash as a discrete event. The number of these events will increase per unit time as marked by the clock at B.

So derive it already! Despite all of your talk the only thing you have managed to demonstrate so far is that the Doppler shift detected at B is constant over time, which directly contradicts your claims.

 

[Chrisc]

Yes they will remain the same over time. I refer you back to our eq 2.0

I'm sorry, but this is just silly. You cannot measure the frequency of flashes by measuring a single event. You have to measure the time between a minimum of two flashes to get a "frequency of detection" just like you need to measure a minimum of two peaks on a sinusoidal signal to get a "detected frequency". A frequency is, by definition, the inverse of a period. A period is a finite duration of time, not a single event. Your assertions here are absurd.

I'm sorry, I did not mean to imply one signal was in itself all that was necessary. I meant that the coincidence of detection and the tick of the clock is all that was needed. It does take at least two signals or one signal at a prearranged time according to H/c after the clocks are synchronized. What you're missing or ignoring is that B does not know the nature of his acceleration, so the measuring of an event of A's clock against his clock is the most accurate, "finite" measure of A's clock he can make.
If A and B (the ship) are comoving under inertial acceleration with respect to C, the redshift of an EM wavelength emitted at A with respect to C, will be almost canceled by the (blue shift) of B's motion with respect to M. The difference in the velocity of A with respect to C at time of emission and the velocity of B with respect to C at time of detection will remain constant. This, as far as I can tell is your perception of the "frequency" or "wavelength" proof of constant time dilation. Please let me know if I've misunderstood you.

The difference between "frequency" or "wavelength" and "rate of detection" is that by virtue of the constancy of the speed of light, there is NO cancellation of "rate of emission" because there is no "KNOWN" change in the displacement of A and C. All that is measured by B is an increased rate of detection that predicts an increased rate of the clock at A. Because there is no known displacement of A and C, and because the distance between A and B is constant, in the case of inertial acceleration, the rate of detection at B will continue to increase over time.

So derive it already! Despite all of your talk the only thing you have managed to demonstrate so far is that the Doppler shift detected at B is constant over time, which directly contradicts your claims.

I have derived it already. In fact I offered Feynman's derivation and you not only didn't accept it, you insisted on putting relativistic measures into what an observer at B cannot, unless and until they know the nature of their motion and even then it will not affect the key point of this discussion.

 

[Dale]

I'm sorry, I did not mean to imply one signal was in itself all that was necessary. I meant that the coincidence of detection and the tick of the clock is all that was needed. It does take at least two signals or one signal at a prearranged time according to H/c after the clocks are synchronized.

Same with the sine wave.

If A and B (the ship) are comoving under inertial acceleration with respect to C, the redshift of an EM wavelength emitted at A with respect to C, will be almost canceled by the (blue shift) of B's motion with respect to M. The difference in the velocity of A with respect to C at time of emission and the velocity of B with respect to C at time of detection will remain constant. This, as far as I can tell is your perception of the "frequency" or "wavelength" proof of constant time dilation. Please let me know if I've misunderstood you.

Pretty much. The only slight changes I would make to what you said here are that the blueshift more than cancels the redshift resulting in a net blueshift from A at the front of the rocket to B at the rear. That and the difference in velocity would almost certainly not be constant in C, it would use relativistic velocity addition. Also, "inertial acceleration" is a contradiction in terms.

The difference between "frequency" or "wavelength" and "rate of detection" is that by virtue of the constancy of the speed of light, there is NO cancellation of "rate of emission" because there is no "KNOWN" change in the displacement of A and C. All that is measured by B is an increased rate of detection that predicts an increased rate of the clock at A. Because there is no known displacement of A and C, and because the distance between A and B is constant, in the case of inertial acceleration, the rate of detection at B will continue to increase over time.

You keep saying the same words with no math to back them up.

I have derived it already. In fact I offered Feynman's derivation and you not only didn't accept it, you insisted on putting relativistic measures into what an observer at B cannot, unless and until they know the nature of their motion and even then it will not affect the key point of this discussion.

You have not derived it. I just looked back through the whole thread and the only post where you even came close to deriving anything was https://www.physicsforums.com/showpost.php?p=1748095&postcount=48". Not only was your derivation there extremely sloppy, but it directly contradicted your claim! I don't know how you could possibly think that what you have presented so far is even remotely convincing.

I'm sorry to be so critical, but honestly I am getting a little bored with the discussion. It has become quite repetitive. You really need to learn how to derive things clearly and methodically from first principles. This will give you a much stronger understanding of the physics and allow you to communicate your ideas more clearly. Frankly, the main thing demonstrated by this thread is that English is too vague to accurately convey the real concepts here.

 

[Chrisc]

...Also, "inertial acceleration" is a contradiction in terms.

It's a common term used to distinguish between "inertial" and "gravitational" acceleration.

You keep saying the same words with no math to back them up.

Please look at the chart posted below and let me know if you agree or disagree.
I think this will put it to rest.

 

[stillwonder]

Two clocks in a gravitational field separated by an altitude of x, exhibit constant time dilation.
Two clocks accelerating in free space separated by the length x of their ship, exhibit non-constant time dilation.
How is the principle of equivalence sustained when this fundamental measure of acceleration differs?

gravitational field is unfortunately not uniform. that's the reason of time difference of two clocks at different altitudes. IF you assume the g. filed is uniform, then both redshifts should match and stay constant (gravitational and accelerated).

Edit: there will be a redshift, but no time dilation

I was wondering about similar scenario since the consideration of uniform g. field goes against how gravitational redshift is explained.

 

[Dale]

Please look at the chart posted below and let me know if you agree or disagree.
I think this will put it to rest.

I agree with the drawing, it is exactly what you have been saying over and over and over. It doesn't support your claim.

It supports the claim that the time between A0 and B0 as measured by M is longer than the time between A1 and B1 as measured by M. That is not your claim. Your claim is that the time between B0 and B1 as measured by B is longer than the time between B1 and B2 as measured by B. This drawing doesn't demonstrate that, nor does anything you have written anywhere in this thread in either English or math.

 

[Mentz114]

Chrisc,

from your diagram I would say that the relative velocity (v) between the emitter and detector ( separated by length L) at some time t is -

v = at - a\left(t + \frac{L}{c}\right) = -\frac{aL}{c} ( assuming at t=0, a=0 )

which is constant if acceleration a is constant. So the red-shift would not change over time.

I did this during my lunch break - could be wrong, or not relevant even.

M

 

[Dale]

Edit: there will be a redshift, but no time dilation

That is a logical contradiction. If the distance is not changing then the only way to get a redshift is to have time dilation. I refer you back to https://www.physicsforums.com/showpost.php?p=1749296&postcount=55".

 

[Chrisc]

Thanks DaleSpam and Mentz114, it is now quite clear why you're disagreeing, and thanks for putting the event IDs on my image DaleSpam, it helps make the problem much more obvious. The issue is not math it is the justification of B's measurement.
My claim has always been that the time between A0 and B0 as measured by B, is less than B's calculation of H/c and larger than the time between A1 and B1 measured by B. Where H is the distance between B and A at rest with respect to each other. Before you take issue with the claim that B is even capable of measuring A0 with respect to his own clock, remember all the events of emission are pre-designated by A and B to occur at regular intervals according to the clocks they synchronized before moving to the front and back of the ship.
Since you agree the time between A0 and B0 as measured by M is longer than the time between A1 and B1 as measured by M, then you must agree that the time between A0 and B0 as measured by B is less than H/c as measured by B and longer than the time between A1 and B1. If you do not agree with this you must explain how B measures the speed of light between A and B to decrease over time.
Once again, before you begin to calculate the relativistic addition of velocities, the point is not how "you" will explain away the speed of light, with the knowledge you have of the situation, it is how B explains away the speed of light, as we are discussing the principle of equivalence not the principle of a God's Eye frame.
The only options B has to reconcile the time of detection with the agreed rate of emissions are:
1) reject the constancy of the speed of light.
2) consider A's clock to be running faster than his
He will likely pick 2. and conclude:
the force he measures on the ship is a) gravitational or b) inertial acceleration
The testing of light signals as described above will, over time, tell him:
the ship is in inertial acceleration in free space.

 

[Mentz114]

Chris,

If you do not agree with this you must explain how B measures the speed of light between A and B to decrease over time.

I don't think this is what B will experience. At constant acceleration only external quantities like relative velocity and distance will change. The observer in the ship will experience constant ( unchanging with time) effects. Like a red-shift between the front and back of the ship.

M

 

[Chrisc]

Chris,

I don't think this is what B will experience.

Hi Mentz144
I am not suggesting it is.
I said if you do not agree with the previous sentence, you will be making the claim that the speed of light changes as measured by B.

 

[yuiop]

Could someone bring me up to speed here as this thread has got rather big to read through it all.

Are we specifically analysing:

1) Born rigid acceleration
The case of an accelerating rocket that has greater acceleration at the rear than at the tail at any given instant according to a non accelerating observer. That same obeserver would measure the rocket to be length contracting progressively more over time. An observer on the ship would measure the proper length of the ship to remain constant over time. Observers on the rocket would notice that an accelerometer indicates greater proper acceleration at the tail than at the nose.

2) Bell type acceleration.
The case of accelerating rocket that has equal acceleration at the nose and at the tail at any given instant, according to a non accelerating observer. That same observer would say the length of the rocket remains constant over time while an observer onboard the rocket would consider the rocket to be getting longer over time. Observers on the rocket would notice that an accelerometer indicates the same proper acceleration at the tail than at the nose.

 

[Dale]

Could someone bring me up to speed here as this thread has got rather big to read through it all.

Are we specifically analysing:

1) Born rigid acceleration
The case of an accelerating rocket that has greater acceleration at the rear than at the tail at any given instant according to a non accelerating observer. That same obeserver would measure the rocket to be length contracting progressively more over time. An observer on the ship would measure the proper length of the ship to remain constant over time. Observers on the rocket would notice that an accelerometer indicates greater proper acceleration at the tail than at the nose.

2) Bell type acceleration.
The case of accelerating rocket that has equal acceleration at the nose and at the tail at any given instant, according to a non accelerating observer. That same observer would say the length of the rocket remains constant over time while an observer onboard the rocket would consider the rocket to be getting longer over time. Observers on the rocket would notice that an accelerometer indicates the same proper acceleration at the tail than at the nose.

The proper distance (in the accelerating frame) was stipulated to be constant in https://www.physicsforums.com/showpost.php?p=1743187&postcount=7". So it is Born rigid acceleration.

 

[Mentz114]

Hi Mentz144
I am not suggesting it is.
I said if you do not agree with the previous sentence, you will be making the claim that the speed of light changes as measured by B.

Your logic is too tortuous for me.

It's a strange way to communicate, asserting that if I disgree with proposition A then I'll be claiming that proposition B is true etc. How do you know what I'll be claiming ? Maybe your logic is faulty and I'll claim something completely different.

Its all hand waving if you can't state whatever your problem is in equations.

 

[stillwonder]

-----------------------------------------------------------------------------------
That is a logical contradiction. If the distance is not changing then the only way to get a redshift is to have time dilation. I refer you back to post 55.
------------------------------------------------------------------------------------


assume segment AB move into a *uniform* gravitational field which is perpendicular to the uniform motion of segment AB.

-------------------UGF-------------->
^^^^ A--------B ^^^^

why would there be a time dilation between A and B?
A redshift is needed since there is still energy gained or lost moving
from A/B to B/A.

Of course, UGF and field with a boundary are just hypothetical, not encounterred in practice.

 

[Chrisc]

kev, check DaleSpam's revision of my diagram in #62. I think it will bring you up to speed on my point.

Your logic is too tortuous for me.

It's a strange way to communicate, asserting that if I disgree with proposition A then I'll be claiming that proposition B is true etc. How do you know what I'll be claiming ? Maybe your logic is faulty and I'll claim something completely different.

Mentz114, it shouldn't difficult or tortuous. I simply meant if you do not agree with the constancy of the speed of light then you are claiming the speed of light is not constant. Which in my opinion requires explanation.

Its all hand waving if you can't state whatever your problem is in equations.

Equations are not the problem, as previously thought. The problem is stating the observations from B's perspective without including external information that leads to the temptation of irrelevant equations.
The external information needed to confirm acceleration with respect to an inertial frame and set up the thought experiment in context of the principle of equivalence is "not" information that B (the test observer) possesses when considering the time of light signals.

stillwonder, I don't know if you're speaking to me. If so you seem to be missing a few points made earlier,
we are not talking about spectral shifts.

 

[Dale]

Equations are not the problem, as previously thought.

Equations are the problem: specifically that you still can't derive any to prove your point.

 

[Dale]

why would there be a time dilation between A and B?
A redshift is needed since there is still energy gained or lost moving
from A/B to B/A.

Since the disance is constant the redshift is due to time dilation.

 

[Dale]

My claim has always been that the time between A0 and B0 as measured by B, is less than B's calculation of H/c and larger than the time between A1 and B1 measured by B.

This is a huge problem. If you honestly believe that this is the same claim you have been making throughout this thread then it is plain that you don't even understand the terminology. Your claim has always been that the time dilation of A relative to B is time varying. In this scenario time dilation is the ratio of A1-A0 as measured by A to B1-B0 as measured by B. It is not "the time between A0 and B0 as measured by B".

Since you are so fond of the "if you don't agree with X then you will be making the claim that Y" form of argument, then I will post my own:

If you do not clearly and rigorously derive your claim from first principles then you will be making the claim that your assertions are illogical.

 

[yuiop]

...

Are we specifically analysing:

1) Born rigid acceleration
The case of an accelerating rocket that has greater acceleration at the rear than at the tail at any given instant according to a non accelerating observer. That same obeserver would measure the rocket to be length contracting progressively more over time. An observer on the ship would measure the proper length of the ship to remain constant over time. Observers on the rocket would notice that an accelerometer indicates greater proper acceleration at the tail than at the nose.

2) Bell type acceleration.
The case of accelerating rocket that has equal acceleration at the nose and at the tail at any given instant, according to a non accelerating observer. That same observer would say the length of the rocket remains constant over time while an observer onboard the rocket would consider the rocket to be getting longer over time. Observers on the rocket would notice that an accelerometer indicates the same proper acceleration at the tail than at the nose.

The proper distance (in the accelerating frame) was stipulated to be constant in https://www.physicsforums.com/showpost.php?p=1743187&postcount=7". So it is Born rigid acceleration.

OK, I have uploaded a spacetime diagram for Born-rigid type acceleration. In the digaram it can be seen that signals sent at regular intervals of 15.88 seconds from the rear of the rocket (as timed by a clock at the rear of the rocket) will arrive at the nose of the rocket (in this example) at regular intervals of 21.02 seconds as timed by the observer at the nose of the rocket. In other words the time dilation and red shift of internal signals observed inside the rocket undergoing Born-rigid acceleration remains constant over time. I used the equation of acceleration described here http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html which take length contraction, time dilation, syncronicity and changing acceleration over time (as observered by the non accelerating observer into account. (Almost none of these things are taken into account by the diagram posted by Chrisc.


On the other hand it can be shown that an observer at the nose of a rocket undergoing Bell type acceleration WILL see the clock at the rear of the rocket get progressively slower over time, while the the observer riding at the rear of the rocket will see the clock at the nose progressively speeding up over time. Therefore redshift of internal signals is not constant over time in this model, which can be described as the equivalent of a uniform gravitational field. The second diagram shows the proper time of the arrival of the signals at the nose of the rocket when signals are sent at regular intervals from the tail of the rocket undergoing Bell type acceleration.

 

[Dale]

OK, I have uploaded a spacetime diagram for Born-rigid type acceleration. In the digaram it can be seen that signals sent at regular intervals of 15.88 seconds from the rear of the rocket (as timed by a clock at the rear of the rocket) will arrive at the nose of the rocket (in this example) at regular intervals of 21.02 seconds as timed by the observer at the nose of the rocket. In other words the time dilation and red shift of internal signals observed inside the rocket undergoing Born-rigid acceleration remains constant over time. I used the equation of acceleration described here http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html which take length contraction, time dilation, syncronicity and changing acceleration over time (as observered by the non accelerating observer into account. (Almost none of these things are taken into account by the diagram posted by Chrisc.

Thanks kev, these are excellent diagrams. The most important thing for Chrisc to note here is that the one effect that he repeatedly mentioned, the progressive reduction in time between emission at the front and reception at the back, is included in this diagram. That effect, by itself, does not imply a time varying time dilation.

On the other hand it can be shown that an observer at the nose of a rocket undergoing Bell type acceleration WILL see the clock at the rear of the rocket get progressively slower over time, while the the observer riding at the rear of the rocket will see the clock at the nose progressively speeding up over time. Therefore redshift of internal signals is not constant over time in this model, which can be described as the equivalent of a uniform gravitational field. The second diagram shows the proper time of the arrival of the signals at the nose of the rocket when signals are sent at regular intervals from the tail of the rocket undergoing Bell type acceleration.

The important thing for Chrisc to note here is that the proper distance is increasing. So the equivalent gravitational situation would be that H is increasing also. Since the gravitational time dilation depends on H then in such a case the time dilation in the gravitational field would also be non-constant.

There is no reason to expect that the equivalence principle would not hold in either case.

 

[Chrisc]

Thanks kev, your diagrams are very helpful and address the crux of my question.
They also show my question should have been qualified in context of a "uniform" gravitational field.
DaleSpam, I take it you're saying I have compared the equivalence of a "uniform" field
with Bell type acceleration, instead of Born type. In other words an accurate comparison
of equivalence should be either "uniform field - Born type acceleration" or "non-uniform field - Bell type"
That does make sense, but I'm not sure it resolves the problem.

My question is a matter of principle. I am very stubborn when it comes
to principles (and I am grateful for your patience) which is why this thought
experiment has always caused me angst.

kev, in your description of Bell type acceleration you mention the observer
onboard the rocket would "consider" the rocket to be getting longer
over time. This is what DaleSpam has been telling me.
Can you explain or point me to where I can find an explanation
of the line of reasoning that justifies this conclusion of length contraction?
I was under the impression the length of the rocket could only be determined as a
measure of time, which would require some principle (or least metaphysical
modeling) to make the leap from decreased rate of time to increased length.

 

[yuiop]

Thanks kev, your diagrams are very helpful and address the crux of my question.
They also show my question should have been qualified in context of a "uniform" gravitational field.
DaleSpam, I take it you're saying I have compared the equivalence of a "uniform" field
with Bell type acceleration, instead of Born type. In other words an accurate comparison
of equivalence should be either "uniform field - Born type acceleration" or "non-uniform field - Bell type" ...

I think it should be the other way round. A Uniform field is the same as Bell type acceleration because the proper acceleration anywhere in the rocket (or the field) is measured to be the same. In Bell type acceleration the proper acceleration at the rear is greater than the at the front so that represents a non uniform gravitational field. I think also there is some confusion as to just what is meant by a uniform gravitational field. Some people seem to use it in the context of uniform horizontally (so that objects without initial horizontal motion all fall parallel to each other) and are not too concerned with changes in acceleration with height. I prefer to think a uniform gravitational field is unvarying with height and horizontal position. I would love to know what the official definition is.

kev, in your description of Bell type acceleration you mention the observer
onboard the rocket would "consider" the rocket to be getting longer
over time. This is what DaleSpam has been telling me.
Can you explain or point me to where I can find an explanation
of the line of reasoning that justifies this conclusion of length contraction?
I was under the impression the length of the rocket could only be determined as a
measure of time, which would require some principle (or least metaphysical
modeling) to make the leap from decreased rate of time to increased length.

In this thread https://www.physicsforums.com/showthread.php?t=236681 I posted another spacetime diagram that attempts to make the Bell's spaceship paradox easy to understand, along with links to more complete articles on the subject. Feel free to ask more questions in that thread if still puzzled. It is one of the more tricky subjects in relativity, more so because it less commonly addressed than paradoxes like the twins for example. The implications of applying the Equivalence principle to the Bell's type situation leads to some pretty bizarre conclusions that makes it difficult to be absolutely confident about what should happen in the gravitational equivalent. The implication seems to be that everything physical in such a field would be ripped apart including the gravitational body that is responsible for the uniform field. I am still puzzling over that one ;)

 

[stillwonder]

kev,

stillwonder, I don't know if you're speaking to me. If so you seem to be missing a few points made earlier,
we are not talking about spectral shifts.

actually yes, since i stumbled into similar contradiction thru a separate path
i noticed you were equating non uniform Gfiled with uniform acceleration, hence
the comment.

but as it turned out, the formulae have been cleverly retrofitted to take care of
redshift/time dilation wrt uniform / non-uniform G fields

 

[Chrisc]

kev, thanks, I will look into both and follow up with the links your mentioned.
I've been too busy to get back to it and I will be for a few days. But I wanted to take a minute to thank you for your help.
DaleSpam, I also want to thank you again for your help and patience. I know it must be frustrating to go through so many post of this nature. I am grateful you did not give up. If you hadn't kept pushing me to derive my point, I wouldn't have seen your point (Bell vs Born).
stillwonder, sorry for the confusion. I will post more about this later when I have time, as I am not convinced of the "cleverness" argument.