Can Gravity Fields really cause Time-Dilation-Effects inside of them?

[ghostpy]

Hi folks!

I'll start by admiting that I'm a cockroach compared to Einstein or Hawking, and I think that due to Hawking's great mind, maybe he wrote the book in a way he found conspicuous. But I don't have Hawking's IQ, and for me is definitely harder to understand the concept of "Time Dilation" that Hawking was trying to explain with his thought experiment. Could you please help me folks?

Here is my doubt fully explained:

In his book, A Briefer History of Time - Chapter 6 (Curved Space), Hawking explains how gravity affects time through a thought experiment. It starts this way:

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Imagine a rocket ship out in space. For convenience, imagine that the rocket ship is so long that light takes one second to traverse it from top to bottom. Finally, suppose there is an observer at the ceiling of the rocket ship and another at the floor, each with identical clocks that tick once each second.

Suppose the ceiling observer w aits for the clock to tick, and then immediately sends a light signal down to the floor observer. The ceiling observer does this once more the next time the clock ticks. According to this setup, each signal travels for one second and then is received by the floor observer. So just as the ceiling observer sends two light signals a second apart, the floor observer receives two, one second apart.

How would this situation differ if the rocket ship were resting on earth, under the influence of gravity, instead of floating freely out in space? According to Newton’s theory, gravity has no effect on this situation. If the observer on the ceiling sends signals one second apart, the observer will receive them one second apart. But the principle of equivalence does not make the same prediction. We can see what happens, that principle tells us, by considering the effect of uniform acceleration instead of the effect of gravity. This is an example of the way Einstein used the principle of equivalence to create his new theory of gravity.

So let’s now suppose the rocket ship is accelerating. (We will imagine that it is accelerating slowly, so we don’t approach the speed of light!) Since the rocket ship is moving upward, the first signal will have less distance to travel than before and so will arrive sooner than one second later. If the rocket ship were moving at a constant speed, the second signal would arrive exactly the same amount of time sooner, so the time between the two signals would remain one second. But due to the acceleration, the rocket ship will be moving even faster when the second signal is sent than it was when the first signal was sent, so the second signal will have even less distance to traverse than the first and will arrive in even less time. The observer on the floor will therefore measure less than one second between the signals, disagreeing with the ceiling observer, who claims to have sent them exactly one second apart.

This is probably not startling in the case of the accelerating rocket ship—after all, we just explained it! But remember, the principle of equivalence says that it also applies to a rocket ship at rest in a gravitational field. That means that even if the rocket ship is not accelerating but, say, is sitting on a launching pad on the earth’s surface, if the ceiling observer sends signals toward the floor at intervals of one each second (according to his clock), the floor observer will receive the signals at shorter intervals (according to his clock). That is startling!
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To put it simply, please recall that in the accelerating Rocket case, a "Doppler-Effect-like" phenomenon was active: the accelerating Rocket was approaching the light receiver (floor observer) towards the light emitter (ceiling observer).

Thanks to this "Doppler-Effect-like" phenomenon, each time the Rocket accelerated, it shortened the path needed for light to traverse from ceiling to floor, so each subsequent signal would arrive in less time (faster), and give the floor observer "the impression" that the light signal had been emitted "earlier in time".

Since the floor observer would be receiving light signals in a time period shorter than what he would have expected (1 second), he would then have been left no chance but to think that "events were occurring faster than his clock's ticks", and hence experience a "Time Dilation" effect.

But in order to experience a "Time Dilation" effect, the Rocket at rest would need a "Doppler-Effect-like" phenomenon, that is, the light emitter (ceiling observer) would need to be approaching the light receiver (floor observer) each passing second, and since there is not any acceleration to shorten the light-traverse-path, I can't understand how can Time dilate on Earth.

Is it just me, or did someone else extract the same understanding from the experiment's speech?
What part am I missing that could make me understand this experiment?

Thanks in advance!

 

[WannabeNewton]

But in order to experience a "Time Dilation" effect, the Rocket at rest would need a "Doppler-Effect-like" phenomenon, that is, the light emitter (ceiling observer) would need to be approaching the light receiver (floor observer) each passing second, and since there is not any acceleration to shorten the light-traverse-path, I can't understand how can Time dilate on Earth.

Place a ceiling observer and floor observer in a rocket at rest in the uniform gravitational field of the Earth. Then as far as these observers are concerned, they're in a rocket accelerating upwards at a uniform rate. Therefore if they are asked to explain why they observe time dilation between their clocks they will use precisely the "Doppler-Effect-like" phenomenon that you described. In other words, the gravitational field of the Earth is making it so that clocks at rest at different heights along the gravitational potential tick at different rates as if they were clocks placed at the extremities of a rocket ticking at different rates due to the "Doppler-Effect-like" phenomenon. Why? Because of the equivalence principle. GR can't give you a deeper explanation.

However if that doesn't entirely convince you then one can also argue for the existence of gravitational redshift, which is equivalent to gravitational time dilation, using a simple conservation of energy argument. See here: http://books.google.com/books?id=w4...ional redshift conservation of energy&f=false

 

[phinds]

As a total aside, I had to chuckle at the thought of a 1-light-second long spaceship sitting on the surface of the Earth. It would be 186,000 miles tall. Not quite tall enough to bang into the moon, but at more than 20 times the diameter of the Earth, it WOULD stand out a bit

 

[A.T.]

Place a ceiling observer and floor observer in a rocket at rest in the uniform gravitational field of the Earth. Then as far as these observers are concerned, they're in a rocket accelerating upwards at a uniform rate. Therefore if they are asked to explain why they observe time dilation between their clocks they will use precisely the "Doppler-Effect-like" phenomenon that you described.

What if the floor observer is at the center of the Earth, where he doesn't experience any upwards acceleration. Could he still explain the time dilation between their clocks using Doppler?

 

[dauto]

What if the floor observer is at the center of the Earth, where he doesn't experience any upwards acceleration. Could he still explain the time dilation between their clocks using Doppler?

Yes, the gravitational time dilation takes into account the gravitational acceleration along the path between the observers, not just the end points, effectively integrating it. That's why its better stated that time dilation is affected by the gravitational potential, not the local gravitational acceleration

 

[A.T.]

Yes, the gravitational time dilation takes into account the gravitational acceleration along the path between the observers, not just the end points, effectively integrating it. That's why its better stated that time dilation is affected by the gravitational potential, not the local gravitational acceleration

Right, but the OP asks about an explanation in terms of Doppler-Effect, not in terms of potential.

 

[Nugatory]

What if the floor observer is at the center of the Earth, where he doesn't experience any upwards acceleration. Could he still explain the time dilation between their clocks using Doppler?

Not in the easy intuitive way that you get in a uniform gravitational field. The equivalence principle works best when it describes equivalence between locally uniform gravity and constant acceleration.

This is, of course, why we end up integrating the acceleration along a path to find the potential when dealing with non-uniform gravitational fields.

 

[WannabeNewton]

Could he still explain the time dilation between their clocks using Doppler?

I want to say 'no' because it hinges on the use of the equivalence principle which itself relies on a uniform gravitational field. In your case even if we model the Earth as a solid sphere of uniform density the internal gravitational field grows linearly from the origin in a spherically symmetric manner so the field isn't uniform anymore and we can't really apply the equivalence principle when one observer is at the center and the other is at the surface of the sphere. But again I'm not sure if this necessarily throws out any possible Doppler explanation-for sure you could apply the Doppler explanation locally where the field is approximately uniform and then integrate each local Doppler element to get the time dilation between two non-local points.

Sorry if this isn't entirely convincing to you :(

 

[dauto]

Right, but the OP asks about an explanation in terms of Doppler-Effect, not in terms of potential.

My point is that it can be done but you have to integrate the Doppler effect along the way instead of just refer to the behavior of the end points. That happens because the equivalence principle is only valid locally.

 

[ghostpy]

Place a ceiling observer and floor observer in a rocket at rest in the uniform gravitational field of the Earth. Then as far as these observers are concerned, they're in a rocket accelerating upwards at a uniform rate.

I wouldn't say that a rocket at rest is accelerating, after all if it is at rest, it at least has { velocity=0 } and { acceleration=0 } relative to the Earth.

So, if the rocket would have been accelerating in a given direction at an uniform rate, it would have been changing its velocity, and that means it couldn't possibly stand still at rest on Earth.

Nevertheless, I agree with you that any object left free to fall (not at rest) would indeed experience an acceleration towards Earth's floor due to Earth's Gravity, and could be applied the Equivalence Principle, because it would resemble the situation of a rocket accelerating at an uniform rate. That would be the case of an apple dropped by the rocket's ceiling observer, for the apple would be accelerating relative to the rocket.

 

[A.T.]

I wouldn't say that a rocket at rest is accelerating, after all if it is at rest

It has proper acceleration, not coordinate acceleration:
http://en.wikipedia.org/wiki/Proper_acceleration

 

[dauto]

I wouldn't say that a rocket at rest is accelerating, after all if it is at rest, it at least has { velocity=0 } and { acceleration=0 } relative to the Earth.

So, if the rocket would have been accelerating in a given direction at an uniform rate, it would have been changing its velocity, and that means it couldn't possibly stand still at rest on Earth.

Nevertheless, I agree with you that any object left free to fall (not at rest) would indeed experience an acceleration towards Earth's floor due to Earth's Gravity, and could be applied the Equivalence Principle, because it would resemble the situation of a rocket accelerating at an uniform rate. That would be the case of an apple dropped by the rocket's ceiling observer, for the apple would be accelerating relative to the rocket.

The whole point of the equivalence principle is the realization that when you look at a falling apple, the apple actually has no proper acceleration and it is you standing on the ground that is accelerating upwards. The rocket blasting through space doesn't just resemble a rocket standing on earth. They are in fact identical physical phenomena. They both have proper acceleration. In General Relativity gravity is a pseudo-force like centripetal force or Coriolis force.

 

[Nugatory]

Nevertheless, I agree with you that any object left free to fall (not at rest) would indeed experience an acceleration towards Earth's floor due to Earth's Gravity, and could be applied the Equivalence Principle, because it would resemble the situation of a rocket accelerating at an uniform rate. That would be the case of an apple dropped by the rocket's ceiling observer, for the apple would be accelerating relative to the rocket.

You have it exactly backwards. The equivalence principle does not apply to the object in free fall even if it is experiencing coordinate acceleration towards the center of the earth. If you were in a compartment inside the free-falling object (think elevator with cable broken) and you dropped something, it wouldn't fall to the floor - it would float in free fall in front lf you.

Conversely, if you're in a rocket accelerating in space a dropped object will fall to the floor, just as it would if the rocket were sitting stationary on the surface of the earth. These are the two cases that are equivalent according to the equivalence principle.

 

[ghostpy]

Hi folks!
Thank you all for your replies.
I ask you for a little patience, since I seem to be the one not getting it yet.

Since this thought experiment may be a little difficult to "imagine", I'll post some graphics to ease up the debate.

The graphics are some Cartesian Planes containing the following:
- Horizontal Axis (t): for Coordinate Time. Represents time passed (in seconds) since the Emission of the 1st Light Signal.
- Vertical Axis (x): for Coordinate Space. Represents altitude (1 means "c") relative to the Bottom of the Rocket.
- An Upper Rising Semi-Horizontal Line or Curve starting at <t=0, x=1>: symbolizes the position of the "Ceiling Observer" (named "Celine" from now on) as time passes.
- A Lower Rising Semi-Horizontal Line or Curve starting at <t=0, x=0>: symbolizes the position of the "Floor Observer" (named "Florence" from now on) as time passes.
- Three Falling Lines starting at { t=0 ; t=1 ; t=2 }: symbolize the Emissions of the three Light Signals traveling from Celine's Line towards Florence's Line as time passes by.
- Circles: symbolize Spacetime coordinates of the Light Signal Emissions departing from Celine and Receipts arriving at Florence.

Using these graphics, I'll examine the conditions of three cases that were described in the 4th Paragraph of the Thought Experiment Speech:
- Rocket at Rest
- Rocket Moving at a Constant Velocity
- Rocket Accelerating at an Uniform Rate

1. Rocket at Rest

- In this case the Rocket is at rest.
- Celine emits 3 Light Signals at { t=0 ; t=1 ; t=2 } .
- Each Light Signal has a flight-time-duration of "1 Second" before arriving at Florence.
- Those Light Signals correctly arrive at Florence at { t=1 ; t=2 ; t=3 } , that is, Florence measures a Δt of "1 Second" between each Light Signal arrival.

Nothing very thrilling happens here.

2. Rocket Moving at a Constant Velocity

- In this case the Rocket is moving at the constant velocity of "0.1c".
- Celine emits 3 Light Signals at { t=0 ; t=1 ; t=2 } .
- Each Light Signal has a flight-time-duration of "10/11 Seconds" (~0.909090... Seconds) before arriving at Florence.
- Those Light Signals correctly arrive at Florence at { t~0.909090... ; t~1.909090... ; t~2.909090... } , that is, Florence measures a Δt of "1 Second" between each Light Signal arrival.

Note that even though the flight-time-duration is now smaller ("0.909090... Seconds" instead of "1 Second"), Florence doesn't notice this effect. As far as Florence is concerned, she has measured exactly "1 Second" of difference between each Light Signal arrival.

3. Rocket Accelerating at an Uniform Rate

- In this case the Rocket is constantly accelerating at "0.1c/second".
- Celine emits 3 Light Signals at { t=0 ; t=1 ; t=2 } .

- The 1st Light Signal emitted at { t=0 } :

a) has a flight-time-duration of "~0,95445 Seconds" before arriving at Florence.
b) correctly arrives at Florence at { t~0,95445 Seconds } .
​

- The 2nd Light Signal emitted at { t=1 } :

a) has a flight-time-duration of "~0,87434 Seconds" before arriving at Florence.
b) correctly arrives at Florence at { t~1,87434 Seconds } .
c) Florence measures a Δt of "~0,91989 Seconds" between this signal and the previous.
​

- The 3rd Light Signal emitted at { t=2 } :

a) has a flight-time-duration of "~0,80625 Seconds" before arriving at Florence.
b) correctly arrives at Florence at { t~2,80625 Seconds } .
c) Florence measures a Δt of "~0,93191 Seconds" between this signal and the previous.
​

Here can be noted that Florence is now measuring a smaller time difference (Δt) between each two Light Signal Arrivals. Hence, this is one case where Florence is experiencing a "Time Dilation Effect".

In that graphic we have a table were:
- The Orange Column is the flight-time-duration of each Light Signal from Celine to Florence
- The Yellow Column is the time difference (Δt) between each Light Signal Arrival and the previous's arrival.

Note that as time passes by, the flight-time-duration decreases because Florence's Velocity is approaching to "c", and that causes Florence almost to be instantenously in the spacetime coordinate where the Events are happening (i.e., the Light Signals are being emitted), and hence there is not much space/time needed for a Light Signal to "travel through". This is the "Doppler-Effect-like" phenomenon I described earlier in this post.

Also note that as time passes by, the time difference (Δt) increases because since Florence is approaching the spacetime coordinate where the Events are happening, her "awareness of events" is resembling that of Celine, and hence Florence is experiencing the Light Signals separated by a time difference (Δt) exactly as Celine herself would experience.

Lastly, note that the difference in Velocity (Δv) measured at the 11th Light Signal Arrival is an invalid one, because there is no Light Signal that could have a velocity greater than "1c" relative to the previous signal.

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Until now I think that the Rocket at Rest would be the First exposed case, where there is no reason to believe that the Floor Observer (Florence) will experience a time difference (Δt) of less than "1 Second".

This is my comprehension of the Experiment, at top detail.
Nevertheless, I may be wrong.
If you think you have a good and detailed explanation of why this would be wrong, or even better, you know where is the error in my judgments, please do reply.

 

[dauto]

One glaring error is the assumption that the rocket can accelerate all the way to c. A second mistake is to forget about time dilation due to large speeds (that's why it's better to assume small accelerations and speeds, that way you can understand gravitational time dilation without having to worry about relative speed time dilation. One time dilation mechanism at a time. A third mistake is to assume that a rocket with uniform proper acceleration would describe a uniformly accelerated motion from the point of view of external observers. Again, that fails to take time dilation into account. You really need to have Lorentz transformations down before attempting to understand accelerated motion.