Equivalence Principle & Gravitational Time Dilation

[nutgeb]

I'm reading in a textbook (Gravity by J. Hartle) that gravitational time dilation is implied by the equivalence principle. The following thought experiment is described. A vertical rocket at rest in a uniform gravitational field (no tidal effects) is compared to a rocket constantly accelerating (at non-relativistic velocities) far from any source of gravitation. Alice in the rocket's nose emits light pulses at equal intervals on a clock at her height. Bob in the tail measures the time interval between receipt of the signals on an identical clock at his location. Bob receives light pulses at shorter intervals than they are emitted because the accelerating tail is always catching up with the signals.

In this thought experiment, it seems clear that Bob's clock does not run at a slower rate than Alice's clock. After all, the differences in velocity and acceleration as between the two clocks are both zero.

Intuitively, Bob can never receive more total pulses than Alice emits, no matter how much (non-relativistically) the rocket's velocity increases with the passage of time and no matter how long the acceleration continues. The only way Bob can experience shorter reception intervals than the emission intervals is by progressively shortening the proper flight path length between Alice and Bob at a given time. For example, let's say that before the rocket starts accelerating, 100 pulses at a time are "in-flight" along the total path between Alice and Bob. Then, when the rocket begins accelerating, no matter over how long a (finite) time period the acceleration continues, Bob can never receive more than 100 pulses in excess of the absolute number emitted by Alice during that time period. If Alice emits 100 billion pulses over the duration of the acceleration, Bob cannot receive more than 100 billion + 100 pulses regardless of the maximum acceleration rate. It seems to me that at first the reception intervals will get shorter with time, but eventually, by the time Bob has received almost all 100 of the extra original "in-flight" pulses, the reception interval must start increasing, asymptotically reverting to the original emission interval.

Moreover, when the period of acceleration ends, Bob and Alice can compare their clocks and see that exactly the same cumulative amount of time has elapsed on both clocks. No cumulative time dilation has occurred.

The same will be true even if the rate of acceleration increases or decreases over time.

This thought experiment does not illustrate actual time dilation. It illustrates only an illusion of time dilation which is caused by the observer receiving subsequent light pulses after progressively shorter and shorter proper flight lengths.

Is this just a case of a weak thought experiment? Is there a more convincing way to use the equivalence principle to demonstrate true gravitational time dilation?

This thought experiment suggests that gravity creates its own illusion of time dilation (perhaps by shortening the flight time, i.e. by progressively increasing the speed of light relative to the emitter). It seems that the same limitation should apply: That the total number of "extra" light pulses received at the gravitating body in a finite period of time due to shortening the flight time can never exceed the absolute number of pulses that are "in-flight" at any instant in time. If the light emissions continue for a duration approaching infinity, eventually the interval between pulses measured by the observer must start increasing again. Causation would be violated if the gravitating body receives more total pulses than the emitter emits during ANY finite period (measured by the emitter's clock) + the number of pulses that were "in-flight" before the counting period started.

In the case of an emitter and observer in concentric orbits close to the event horizon of a black hole, this causation limit presumably could be reached fairly quickly.

 

[JesseM]

In this thought experiment, it seems clear that Bob's clock does not run at a slower rate than Alice's clock. After all, the differences in velocity and acceleration as between the two clocks are both zero.

Not if it is undergoing Born rigid acceleration, in which the distance between the front and back in the instantaneous inertial rest frame of any point on the rocket will remain constant from moment to moment. In Born rigid acceleration, the front and back will have different coordinate acceleration at each moment, and their coordinate acceleration changes over time, although the proper acceleration at any point on the rocket will be constant. See here for some discussion of Born rigidity, as well as this page which discusses it in the context of the Rindler horizon:

We can imagine a flotilla of spaceships, each remaining at a fixed value of s by accelerating at 1/s. In principle, these ships could be physically connected together by ladders, allowing passengers to move between them. Although each ship would have a different proper acceleration, the spacing between them would remain constant as far as each of them was concerned.

You could also take a look at the spaceship paradox[/url], which deals with the fact that if two ships have identical coordinate acceleration from the perspective of some inertial observer, the distance between the ships is actually increasing from their own perspective (in terms of the distance in each ship's instantaneous inertial rest frame from one moment to the next), with the consequence that a taut string between the ships will experience increasing stress until it snaps if they accelerate this way.

 

[atyy]

Is this equivalent? Alice and Bob start off in the tail of an unaccelerated rocket, and compare their clocks side by side to make sure they tick at the same rate. Alice agrees to send Bob a light pulse every ten ticks after the rocket accelerates. She moves to the tail nose of the, while Bob stays at the tail. The rocket undergoes constant acceleration (Rindler frame), and Alice starts to send out the light pulses every ten ticks of her clock. Does Bob receive them at less than, equal to, or greater than ten ticks of his clock?

 

[Al68]

After all, the differences in velocity and acceleration as between the two clocks are both zero.

That statement isn't generally true. Neither will be zero with respect to any inertial reference frame.

Moreover, when the period of acceleration ends, Bob and Alice can compare their clocks and see that exactly the same cumulative amount of time has elapsed on both clocks.

They will see that their clocks show a difference in elapsed time.

This thought experiment does not illustrate actual time dilation. It illustrates only an illusion of time dilation which is caused by the observer receiving subsequent light pulses after progressively shorter and shorter proper flight lengths.

That sounds like time dilation to me. The time elapsed between the pulses is different for Bob than for Alice. The fact that the number of pulses sent by Alice equals the number of pulses received by Bob is the reason that the time elapsed is different for Bob than Alice, since total time elapsed equals the number of pulses times the elapsed time between pulses.

An easy way to see this is to have Alice start at the rear of the ship, start sending pulses, go to the front of the ship for a while, then return to the rear with Bob, sending pulses the whole time. If the elapsed times between the pulses are shorter for Bob than for Alice, then Bob's clock must have less elapsed time than Alice's clock when she returns precisely because the number of pulses sent equals the number received.

 

[nutgeb]

Not if it is undergoing Born rigid acceleration, in which the distance between the front and back in the instantaneous inertial rest frame of any point on the rocket will remain constant from moment to moment.

Born acceleration is an interesting and complex topic, but I don't think it's directly relevant to my post because I specified non-relativistic velocities at which the Born effect is insignificant. I also specified a uniform gravitational field, and the mathpages article says:

"The only truly stressless "acceleration" would be of objects in a perfectly uniform gravitational field, in which case the intrinsic curvature of spacetime conforms identically to the skewed spatio-temporal relations usually associated with acceleration, so that in a local sense the object is actually moving inertially."

Bell's paradox effects also should be insignificant at non-relativistic velocities.

 

[nutgeb]

Is this equivalent? Alice and Bob start off in the tail of an unaccelerated rocket, and compare their clocks side by side to make sure they tick at the same rate. Alice agrees to send Bob a light pulse every ten ticks after the rocket accelerates. She moves to the tail nose of the, while Bob stays at the tail. The rocket undergoes constant acceleration (Rindler frame), and Alice starts to send out the light pulses every ten ticks of her clock. Does Bob receive them at less than, equal to, or greater than ten ticks of his clock?

The scenario you describe is the kind of thing I had in mind. As the textbook says, Bob will receive the light pulses after shortened intervals.

My point is that although Bob indeed sees shortened intervals between pulses, his own clock is always running at (vanishingly close to) the same rate as Alice's clock throughout the exercise, and will show about the same time when they are compared after the experiment ends. And if the experiment continues long enough, the causality limitation I explained may become relevant.

 

[ZikZak]

I'm sorry, but you can't ask a Relativity question and then reject the answers as being too Relativistic. If the spacecraft is relativistic enough for the two observers to disagree on rates of their respective clocks, then it is relativistic enough for Born rigidity to be the reason.

 

[nutgeb]

Neither will be zero with respect to any inertial reference frame.They will see that their clocks show a difference in elapsed time. That sounds like time dilation to me. The time elapsed between the pulses is different for Bob than for Alice.

Al, I don't follow your logic. At non-relativistic speeds, both Alice and Bob are effectively stationary within the same inertial rest frame at each individual instant in time. What effect would cause a difference between the rates of their own clocks? Sure Bob observes a shortened interval between the pulses he receives from Alice, but as I said that is an illusion of time dilation caused by the progressive shortening of the proper flight path of each pulse compared to the prior pulse. As its velocity steadily increases, the tail of the ship intercepts each succeeding pulse earlier in that pulse's flight path than was the case with the immediately prior pulse.

Applying the label "time dilation" to this scenario would be analogous to applying that label to classical Doppler shift.

The fact that the number of pulses sent by Alice equals the number of pulses received by Bob is the reason that the time elapsed is different for Bob than Alice, since total time elapsed equals the number of pulses times the elapsed time between pulses.

An easy way to see this is to have Alice start at the rear of the ship, start sending pulses, go to the front of the ship for a while, then return to the rear with Bob, sending pulses the whole time. If the elapsed times between the pulses are shorter for Bob than for Alice, then Bob's clock must have less elapsed time than Alice's clock when she returns precisely because the number of pulses sent equals the number received.

In your scenario, if Alice starts sending pulses after she returns to the nose of the ship, then Bob will experience an "initial waiting period" (which we'll call A) in which he receives no pulses after Alice begins emissions, while he waits for the first one to arrive. Then the pulses arrive with shortened intervals between them. Let's call the time period Alice measures on her own clock between the emission of the first and last pulses as period "B". Let's call the shorter time period Bob measures on his own clock between his reception of the first and last pulses as period "C".

B - C can never exceed A. In other words, the total amount of time saved by the shortened intervals between pulses can never exceed the initial waiting period A during which no pulses arrive. The ship's ever-increasing velocity causes the flight length (and therefore the flight duration) of each succeeding pulse to decrease, until in the extreme limit, the flight length approaches zero. But the flight length can never be equal to or less than zero. So as the flight length for pulses approaches zero, the interval between pulses must begin to increase again and eventually to asymptotically approach the original interval at which Alice emits them. Because of that turnaround in the pulse interval, B - C will never exceed A. (Again, all of this is at non-relativistic velocities).

In the scenario as you describe it, when Alice and Bob meet up at the tail again at the end of the experiment, any difference in synchronization between their own clocks will be vanishingly close to zero.

 

[nutgeb]

I'm sorry, but you can't ask a Relativity question and then reject the answers as being too Relativistic. If the spacecraft is relativistic enough for the two observers to disagree on rates of their respective clocks, then it is relativistic enough for Born rigidity to be the reason.

My goodness. I didn't invent the scenario, it's straight out of a widely used textbook on GR. The author specified that all velocities remain non-relativistic. The author used the scenario to make a very basic point about the equivalence principle, not about more advanced specialty topics like Born rigidity.

If you like, let's revise the scenario to assume that the Born rigidity applies even though it is insignificant at non-relativistic velocities, and then we will intentionally factor out the Born rigidity effects (if any) before analyzing the results of the scenario.

Also as I said, Alice and Bob will find that their clocks ARE synchronized when they meet up again at the tail.

Let me be clear that I believe gravitational time dilation reflects a "real" difference in clock rates between the emitter and receiver. That's the only way that shortened intervals between received pulses could continue in perpetuity. My point in this thread is that the textbook scenario confuses the illusion of time dilation with the real thing. In doing so, the scenario fails to convince that the equivalence principle can explain what time dilation is.

 

[atyy]

The scenario you describe is the kind of thing I had in mind. As the textbook says, Bob will receive the light pulses after shortened intervals.

My point is that although Bob indeed sees shortened intervals between pulses, his own clock is always running at (vanishingly close to) the same rate as Alice's clock throughout the exercise, and will show about the same time when they are compared after the experiment ends. And if the experiment continues long enough, the causality limitation I explained may become relevant.

If Bob sees shortened time intervals between the pulses, I would understand that as gravitational time dilation (gravitational redshift). Looking at the Rindler metric, it seems intuitive that this should happen. It's not intuitive to me that this should happen at slow speeds - does it work out just considering the non-relativistic normal doppler effect?

 

[nutgeb]

If Bob sees shortened time intervals between the pulses, I would understand that as gravitational time dilation (gravitational redshift). Looking at the Rindler metric, it seems intuitive that this should happen. It's not intuitive to me that this should happen at slow speeds - does it work out just considering the non-relativistic normal doppler effect?

Yes that's a good thought, I think the slow speed scenario can be described simply as a twist on the classical Doppler shift. Although Alice and Bob have the same velocity both at the time of emission and reception, the receiver's velocity at the time of reception is different from the emitter's velocity at the time of emission. That net velocity differential gives rise to a classical Doppler shift, in this case blueshift.

 

[JesseM]

Born acceleration is an interesting and complex topic, but I don't think it's directly relevant to my post because I specified non-relativistic velocities at which the Born effect is insignificant.

But I think clocks undergoing Born rigid acceleration will have the same time dilation factor between them (as observed by someone moving along with them) at all times, both initially when velocity is low and later when velocity is high (relative to some external inertial observer). I know they both experience a constant proper acceleration, so by the equivalence principle you should be able to look at this in terms of two clocks at different fixed heights in a uniform gravitational field (also see post #75 here from a while ago by 'pervect', a GR expert who used to be a frequent poster).

This is not to say the author of the textbook was actually talking about Born rigid acceleration, necessarily...there may be another way to conceptualize gravitational time dilation in terms of the equivalence principle, I don't know.

 

[Al68]

Al, I don't follow your logic. At non-relativistic speeds, both Alice and Bob are effectively stationary within the same inertial rest frame at each individual instant in time. What effect would cause a difference between the rates of their own clocks? Sure Bob observes a shortened interval between the pulses he receives from Alice, but as I said that is an illusion of time dilation caused by the progressive shortening of the proper flight path of each pulse compared to the prior pulse. As its velocity steadily increases, the tail of the ship intercepts each succeeding pulse earlier in that pulse's flight path than was the case with the immediately prior pulse.

If they were at rest in an inertial frame, there would be no difference in the time elapsed between pulses between Bob and Alice. At rest in an accelerating ship is not at rest in an inertial frame.

In your scenario, if Alice starts sending pulses after she returns to the nose of the ship, then Bob will experience an "initial waiting period" (which we'll call A) in which he receives no pulses after Alice begins emissions, while he waits for the first one to arrive. Then the pulses arrive with shortened intervals between them. Let's call the time period Alice measures on her own clock between the emission of the first and last pulses as period "B". Let's call the shorter time period Bob measures on his own clock between his reception of the first and last pulses as period "C".

B - C can never exceed A. In other words, the total amount of time saved by the shortened intervals between pulses can never exceed the initial waiting period A during which no pulses arrive. The ship's ever-increasing velocity causes the flight length (and therefore the flight duration) of each succeeding pulse to decrease, until in the extreme limit, the flight length approaches zero. But the flight length can never be equal to or less than zero. So as the flight length for pulses approaches zero, the interval between pulses must begin to increase again and eventually to asymptotically approach the original interval at which Alice emits them. Because of that turnaround in the pulse interval, B - C will never exceed A. (Again, all of this is at non-relativistic velocities).

I don't follow this at all. The proper distance between the front and the back of the ship wouldn't change. Why is it relevant that B-C doesn't exceed A? A is itself larger for Alice than for Bob, and B exceeds C. That's what time dilation is.

In my scenario, pulses are sent the whole time to make it obvious that there must be a difference in total elapsed times between the clocks between the time Bob and Alice separate and the time they reunite. Since the total number of pulses sent/received are equal and the elapsed time between them is shorter for Bob.

In the scenario as you describe it, when Alice and Bob meet up at the tail again at the end of the experiment, any difference in synchronization between their own clocks will be vanishingly close to zero.

Well, the difference in the elapsed time between pulses between Bob and Alice will be "vanishingly close to zero" also at non-relativistic accelerations. I assumed from the title of this forum we were talking about relativistic effects. The amount of gravitational time dilation will be as significant as the difference in elapsed time between pulses.

 

[atyy]

Yes that's a good thought, I think the slow speed scenario can be described simply as a twist on the classical Doppler shift. Although Alice and Bob have the same velocity both at the time of emission and reception, the receiver's velocity at the time of reception is different from the emitter's velocity at the time of emission. That net velocity differential gives rise to a classical Doppler shift, in this case blueshift.

That's good, I'll buy your intuitive explanation until I work out the maths for myself (after I retire ). The Doppler shift has never been very intuitive to me - I never hear the ambulance change pitch as it passes me - and I think I'm not "tone deaf"!

 

[nutgeb]

The proper distance between the front and the back of the ship wouldn't change.

That's right, but because the ship is accelerating, by the time the pulse has flown to the point where it meets the receiver at the ship's tail, the ship's tail has a higher velocity than the emitter had at the time when the pulse was emitted. So in fact there is a "net" velocity differential between the emitter velocity and the receiver velocity. That velocity differential results in plain old classical Doppler shift, making the interval between the pulses (as received by the receiver) shorter and blushifting their spectrum. Plain old classical Doppler shift at non-relativistic velocities involves no SR time dilation or gravitational time dilation.

Why is it relevant that B-C doesn't exceed A? A is itself larger for Alice than for Bob, and B exceeds C. That's what time dilation is.

Yes that's what time dilation is, but there is no time dilation in this scenario. So Alice's clock runs the same as Bob's, and if they compare notes on the duration of A, they'll both calculate it be the same when the include the effect of classical Doppler shift.

Well, the difference in the elapsed time between pulses between Bob and Alice will be "vanishingly close to zero" also at non-relativistic accelerations.

It should be clear now why this statement is incorrect. Classical Doppler effect can be significant at non-relativistic velocities, at which it can far exceed any effect of SR time dilation.

 

[yuiop]

... the receiver's velocity at the time of reception is different from the emitter's velocity at the time of emission. That net velocity differential gives rise to a classical Doppler shift, in this case blueshift.

This part is correct. If we temporarily suspend the laws of relativity there would still be a purely Newtonian doppler shift making the signals look blue shifted from Bob's point of view.

When the Relativistic effects are re-introduced then length contraction of the ship would exagerate the effect making the interval between each signal from Alice look even shorter from Bob's point of view. There is the additional effect that Bob's clock at the time of reception is running slower than Alice's clock at the time of emmision due to real SR velocity time dilation and this causes Bob to measure an even shorter time interval between succesive signals from Alice. When all 3 effects are added together, the situation is indistinguishable )in the vertical plane) from the measurements made in a rocket sitting on the surface of a planet in a gravitational field.

MyMy point in this thread is that the textbook scenario confuses the illusion of time dilation with the real thing. In doing so, the scenario fails to convince that the equivalence principle can explain what time dilation is.

The EP does not claim to explain what time dilation is or offer a "mechanism" for time dilution. It simply states that no measurement (in the vertical plane) made inside an accelerating rocket can be distinguished from measurements made inside a stationary rocket in a gravitational field. The two situations are equivalent.

In the scenario as you describe it, when Alice and Bob meet up at the tail again at the end of the experiment, any difference in synchronization between their own clocks will be vanishingly close to zero.

In the purely Newtonian universe, you are right in thinking that when Bob and Alice get together again then there would be no difference in the elapsed times of their clocks because in the Newtonian universe there is no real relativistic time dilation and any apparent difference in clocks rates is just a simple doppler effect (assuming ideal clocks).

 

[nutgeb]

The EP does not claim to explain what time dilation is or offer a "mechanism" for time dilution.

I agree that the EP will never be able to explain relativistic time dilation or provide a mechanism for it. To the extent that anyone says or implies that it can, confusion will result.

It simply states that no measurement (in the vertical plane) made inside an accelerating rocket can be distinguished from measurements made inside a stationary rocket in a gravitational field. The two situations are equivalent.

I know that's what the EP is supposed to mean. I think it's probably true that classical Doppler shift (including an insignificant element of relativistic time dilation) cannot be distinguished from true gravitational time dilation inside a closed ship. But in no way does that convert the classical Doppler effect itself into a relativistic effect.

In the purely Newtonian universe, you are right in thinking that when Bob and Alice get together again then there would be no difference in the elapsed times of their clocks because in the Newtonian universe there is no real relativistic time dilation and any apparent difference in clocks rates is just a simple Doppler effect (assuming ideal clocks).

I'm not engaging in purely Newtonian thinking; as the textbook says: "When (V / c)² is negligible, Newtonian mechanics can be used and Lorentz contraction and time dilation neglected. Also, since we will just be comparing time intervals, issues of simultaneity can be neglected."

*****

If you are interested, please think about the other point I made: Causation prevents any observer from receiving more total pulses than the emitter has emitted during a finite period, regardless of which party's clock is used to measure that period (as long as the correct Doppler and relativistic adjustments are made in the calculation).

Consider an emitter orbiting somewhat farther away from a black hole than an observer in concentric orbit relatively close to the event horizon. Let's say that gravitational time dilation causes the observer's clock to run 1000x faster than the emitter's clock. The emitter emits a single pulse per day, and the observer eventually begins receiving these pulses at the rate of approximately 1000 per day (plus or minus some small corrections for the difference in their orbital velocities.) In this scenario perhaps the causation limit I described could be reached fairly quickly. If that occurs, I see no alternative but for the interval between pulses as received by the observer to increase asymptotically back toward the original interval (as if the observer were then measuring the pulses with the emitter's faster (undilated) clock instead of his own). Does that mean gravitational time dilation no longer occurs, or is something else going on here?

 

[atyy]

Consider an emitter orbiting somewhat farther away from a black hole than an observer in concentric orbit relatively close to the event horizon. Let's say that gravitational time dilation causes the observer's clock to run 1000x faster than the emitter's clock. The emitter emits a single pulse per day, and the observer eventually begins receiving these pulses at the rate of approximately 1000 per day (plus or minus some small corrections for the difference in their orbital velocities.) In this scenario perhaps the causation limit I described could be reached fairly quickly. If that occurs, I see no alternative but for the interval between pulses as received by the observer to increase asymptotically back toward the original interval (as if the observer were then measuring the pulses with the emitter's faster (undilated) clock instead of his own). Does that mean gravitational time dilation no longer occurs, or is something else going on here?

Is this an objection or another question? Assuming your intuitive argument that there will be a Doppler shift for constantly accelerated observers, then by the EP, wouldn't there be a red shift/time dilation for constantly stationary observers in a gravitational field?

 

[nutgeb]

Yes, and I just figured out that causation can never become a problem for a scenario involving either time dilation or Doppler shift. Fuzzy thinking on my part.

Thanks for the helpful answers.

 

[yuiop]

Yes, and I just figured out that causation can never become a problem for a scenario involving either time dilation or Doppler shift. Fuzzy thinking on my part.

Just to clarify...

In your scenario, if Alice starts sending pulses after she returns to the nose of the ship, then Bob will experience an "initial waiting period" (which we'll call A) in which he receives no pulses after Alice begins emissions, while he waits for the first one to arrive. Then the pulses arrive with shortened intervals between them. Let's call the time period Alice measures on her own clock between the emission of the first and last pulses as period "B". Let's call the shorter time period Bob measures on his own clock between his reception of the first and last pulses as period "C".

B - C can never exceed A. In other words, the total amount of time saved by the shortened intervals between pulses can never exceed the initial waiting period A during which no pulses arrive. The ship's ever-increasing velocity causes the flight length (and therefore the flight duration) of each succeeding pulse to decrease, until in the extreme limit, the flight length approaches zero. But the flight length can never be equal to or less than zero. So as the flight length for pulses approaches zero, the interval between pulses must begin to increase again and eventually to asymptotically approach the original interval at which Alice emits them. Because of that turnaround in the pulse interval, B - C will never exceed A. (Again, all of this is at non-relativistic velocities).

... I guess by now you have figured out that there is no "turnaround" in the pulse interval. The pulse interval asymptotically aproaches zero as the velocity of the rocket asymptotically approaches infinite. (Here I am assuming infinite velocity is the maximum speed limit in the purely Newtonian universe where the speed of light is not a maximum velocity limit for physical objects).

If you are interested, please think about the other point I made: Causation prevents any observer from receiving more total pulses than the emitter has emitted during a finite period, regardless of which party's clock is used to measure that period (as long as the correct Doppler and relativistic adjustments are made in the calculation).

Consider an emitter orbiting somewhat farther away from a black hole than an observer in concentric orbit relatively close to the event horizon. Let's say that gravitational time dilation causes the observer's clock to run 1000x faster than the emitter's clock. The emitter emits a single pulse per day, and the observer eventually begins receiving these pulses at the rate of approximately 1000 per day (plus or minus some small corrections for the difference in their orbital velocities.) In this scenario perhaps the causation limit I described could be reached fairly quickly. If that occurs, I see no alternative but for the interval between pulses as received by the observer to increase asymptotically back toward the original interval (as if the observer were then measuring the pulses with the emitter's faster (undilated) clock instead of his own). Does that mean gravitational time dilation no longer occurs, or is something else going on here?

Again there is no "turnaround" in the pulse interval in this scenario. The observer would see the number of pulses received per day going towards infinity the closer his hovering position gets to the event horizon. If the observer stayed at his original orbit position, he would continue to see 1000 signals per day indefinitely. After 10 days he would have received 10,000 signals. It would have taken the emitter 10,000 days to emit those those signals but the observer near the event horizon would only perceive that 10 days had passed by his own clock. Just to nitpick a little, a time dilation ratio of 1,000 would require being very near the the event horizon at 2M and way below the photon orbit radius of 3M. It is not possible to orbit below r=3M as that requires an orbital speed that is greater than the speed of light. (Hovering is better )

If you wish to progress to Born rigid motion and accelerated clocks there is a nice thread with diagrams here: https://www.physicsforums.com/showthread.php?t=216113 and there is another thread here on Bell's spaceship paradox here https://www.physicsforums.com/showthread.php?t=236681 which is a closely related subject.

 

[Al68]

Plain old classical Doppler shift at non-relativistic velocities involves no SR time dilation or gravitational time dilation.

Well, of course that's right, but the title of the thread implies relativistic acceleration. Obviously the effect is negligible at low acceleration/small distance between observers.

Yes that's what time dilation is, but there is no time dilation in this scenario. So Alice's clock runs the same as Bob's, and if they compare notes on the duration of A, they'll both calculate it be the same when the include the effect of classical Doppler shift.

Again, that's true at low acceleration, where the time lapse between the emission of each pulse is the same for both observers.

Well, the difference in the elapsed time between pulses between Bob and Alice will be "vanishingly close to zero" also at non-relativistic accelerations.

It should be clear now why this statement is incorrect. Classical Doppler effect can be significant at non-relativistic velocities, at which it can far exceed any effect of SR time dilation.

I was referring to the difference in elapsed time (between the emission of each pulse) between Bob and Alice after they adjust for the light transit time. Which obviously would only be significant at relativistic accelerations. But note that gravitational time dilation between the front and rear of the ship doesn't depend on the ship's velocity relative to any inertial frame. It only depends on the ship's proper acceleration and the distance between observers.

How about this: We just have Alice send pulses long enough so that the time it takes her to travel from the rear to the front of the ship (and back) is insignificant. After all, we can make the number of total pulses arbitrarily large, lasting for years if we want. Then it's just simple math to show that for both Bob and Alice, the total time elapsed is equal to the time elapsed between each pulse times the total number of pulses.