# Gravitational Time Dilation . . . Or not?

1. Aug 1, 2006

### Riposte

I recently finished reading Kip Thorne's Black Holes and Time Warps. In it, he proposes a thought experiment (which he attributes to Einstein) which demonstrates gravitational time dilation. I completely disagree with his conclusion and would like to know where I've gone wrong. Here's the problem:

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Take 2 identical clocks. Place one on the floor of a room next to a large hole, and attach the other to the room's ceiling by a short string.

The ceiling clock emits pulses of light at each tick and directs them downwards toward the floor clock. Immediately before the first pulse, cut the string so that the ceiling clock is now falling freely. If it is ticking fast enough, then the duration between the first few ticks will be governed by the 'ceiling' time, as it will not have fallen appreciably yet.

Immediately before the first pulse hits the floor clock, drop the floor clock into the hole. Similarly, this clock will feel 'floor' time for the first few ticks.

Now, because the ceiling clock was dropped before the floor clock, its downward speed is always greater than the floor clock. This implies that the floor clock will see the ceiling clock's pulses Doppler shifted (slightly faster). Since the time between pulses was regulated by the ceiling's time flow, this means that time must flow more slowly near the floor than near the ceiling; in other words, gravity must dilate the flow of time.

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To me, this seems a terrible, incorrect argument. The difference in pulse timing was entirely a result of the difference in speeds, not a difference in heights. Just because this difference in speeds is caused by gravity seems no reason to claim gravity warps time. I could set up the experiment with me pushing things rather than gravity. Does this mean I warp time?

Also, and more importantly, if the experiment was done with the floor clock sending pulses to the ceiling clock (but still drop the ceiling clock first) we would find that time must flow more slowly near the ceiling rather than the floor. This is opposite the result of the first experiment.

Just a guess here, but I'd say Thorne and Einstein are probably correct. Where then am I going wrong in my reasoning?

2. Aug 2, 2006

### MeJennifer

Well all I can say is that despite that fact that the example was made by Kip Thorne who is an expert in relativity, I find it a bad and confusing example!

The explanation for gravitational time dilation has nothing to do with relativistic time dilation. Gravitational time dilation occurs because geodesics in warped space-time are not straight lines while they are straight in flat space-time. The more the lines are warped the longer the wavelengths of light.

Last edited: Aug 2, 2006
3. Aug 2, 2006

### Rach3

Is there anything in all that about the Equivalence principle, by chance? Paragraphs immediately preceding or following? I think the crux of the argument is the EP, that an accelerating frame is locally indistinguishable from a frame in a gravitational field; in particular, an object in free-fall is locally inertial. So you have two objects in free-fall, and locally each one is in an inertial frame; but this expeirment shows that assuming the inertial frames are compatible, that they can be Lorentz-boosted into each other, gives the wrong answer! The conclusion being that global inertial frames do not exist when gravity is involved. Thus there must be a gravitational redshift analogous to the redshift of accelerating sources/observers, otherwise, the effects of gravity and of accelerating frames could be distinguished! Experiments uphold this aspect of the EP, which is ultimately why it's valid. If (hypothetically), there were no gravitational redshift, the EP would be violated, and this thought experiment would give the wrong answer.

http://en.wikipedia.org/wiki/Pound-Rebka_experiment

So the argument in MTW should hinge on some form of the equivalence principle, am I right?

Last edited by a moderator: Aug 2, 2006
4. Aug 2, 2006

### Rach3

It has everything to do with relativistic time dilation, as an observer performing only local experiments cannot distinguish between the two, by the EEP (Einstein equivalence principle).

This is both confusing and wrong, as gravity introduces blue-shift as well as red-shift (the observer at lower "graviational potential" observes blue-shifted light). I'm confused as to why you're trying to explain questions you haven't learned the answers to yourself.

Last edited by a moderator: Aug 2, 2006
5. Aug 3, 2006

### Event_Horizon

I'd like to punch in here and say that jennifer's explanation of gravitational time dilation kind of makes sense to me.

Say an external observer watches a clock approach an event horizon of a BH... the light waves from the clock would slow to a stop and it will never actually seem to enter the EH. Now is there any other explanation to this except that it takes a longer time for light waves to come to the external observer because of extremely curved geodesics at the black hole??

6. Aug 3, 2006

### Event_Horizon

in addition to my post above, I still cannot understand why a clock actually PHYSICALLY slows down in a strong G field in relation to another clock with a G field of earth let's say. I understand how its LIGHT waves would appear slower to a external observer but actually physically affecting the flow of time of it is still beyond my understanding after days and days of reading this crazy relativity theory.

7. Aug 3, 2006

### Garth

Imagine a high tower on a non-rotating massive planet.

Have two identical clocks,A & B, sychronised at the top of the tower and lower B to the bottom and retrieve it later to compare it with A. B will be slow relative to A. This is predicted by GR and verified by the Gravity Probe A experiment see http://cfa-www.harvard.edu/hmc/ [Broken].

The reason clock B recored less time was because the interval along that clock's world-line at the bottom of the tower in a stronger gravitational field was shorter than clock A's.

The interval is given by:

$$\tau = \int d\tau = \int f(x^{\mu})dt$$

Now for a static field and a stationary clock dr = d$\theta$ = d$\phi$ = 0, so the Schwarzschild metric for a sphericaly symmetric gravitational field becomes simply:

$$d\tau^2 = (1 - \frac{2GM}{rc^2})dt^2$$

so d$\tau$, and therefore $\tau$, is less for the clock with the smaller r at the bottom of the tower.

Garth

Last edited by a moderator: May 2, 2017
8. Aug 3, 2006

### Riposte

I thought of a better argument against this experiment. If this experiment were performed in a uniform gravitational field, the result would still be the same. The floor clock would still see blue-shifted pulses (the ceiling clock got a slight headstart, and with a uniform gravitational field, it will maintain a slightly higher velocity than the floor clock). The floor clock would then apparently be forced to assume that time flows faster at the ceiling, even though both objects feel the same gravity. This is in conflict with GR and therefore the experiment is wrong.

9. Aug 3, 2006

### Rach3

The situation is not symmetric - lower clock is at lower graviational potential, even though the fields are the same to first order. (actually these terms are not defined in GR, I'm cheating a bit - you can translate this into the different r coordinates of the Schwarzschild solution. Note how r appears explicitly in the Schwarzschild metric.) GR does work.

Last edited by a moderator: Aug 3, 2006
10. Aug 3, 2006

### Riposte

First off, the thought experiment made no mention of the equivalence principle.

Now, while I agree with your logic, that if an acceleration produces a redshift, then gravity must produce one also, I don't think that has been shown here.

The redshift in this experiment does not come from gravity or from the acceleration directly, but comes from the different speeds of the falling clocks. Doppler shifts take place all the time without gravity or acceleration and this seems to be one of them. A gravity induced doppler shift happens because the flow of time is actually different. This has not been shown to be the cause of the Doppler shift mentioned in the experiment.

11. Aug 3, 2006

### pervect

Staff Emeritus
I have to agree that the example is confusing.

There shouldn't be any controversy that gravitational time dilation is a prediction of GR, or even of SR + equivalence principle however, in spite of the fact that this particular example may be confusing.

12. Aug 3, 2006

### Janus

Staff Emeritus
No, this is not in conflict with GR. GR predicts that clocks at different heights in a uniform gravitational field will run at a different rate.

13. Aug 3, 2006

### Riposte

I agree. I have no problem with the concept of gravitational time dilation. My complaint is that this experiment does not demonstrate/prove what it claims to.

I am surprised to hear this. I admit, my knowledge of GR has come only from physics books for the layman. (I will be taking a GR course next semester, though, so hopefully in a couple months I'll be able to answer my own questions!) I had always assumed that time dilation was related to the strength/curvature of the gravity rather than gravitational potential energy. (i.e stronger gravity = stronger time dilation)

However, if a uniform gravitational field does produce different times, shouldn't a uniform acceleration do the same?

If two astronauts are in a rocket accelerating somewhere, you're telling me that even though they are at rest with respect to each other, they are feeling different times? This bothers me because a) They're in the same frame. Even though it's non-inertial, they're still in the same reference frame. And b) How do we tell whose clocks are running slower? I don't see a parallel to gravitational potential here. (Although it seems like the guy furthest along the direction of acceleration would have a higher potential, what has he done to deserve this? For gravity, work must be done to move to a higher gravitational potential. When accelerating, however, it takes the same amount of work to accelerate someone close to the engine as it does for someone further away.)

Since it looks like this uniform gravity isn't a good argument against this experiment, I'd like to reiterate my previous question:

If the experiment was done with the floor clock sending pulses to the ceiling clock (but still drop the ceiling clock first) we would find that time must flow more slowly near the ceiling rather than the floor. This is opposite the result of the first experiment.

14. Aug 3, 2006

### pervect

Staff Emeritus
The metric of an accelerating observer can be written something like this, in units where c=1 (geometrized units) for someone accelerating in the 'z' direction.

ds^2 = -(1+gz)^2 dt^2 + dx^2 + dy^2 + dz^2

Thus an accelerating observer does experience "gravitational" time dilation. The ratio between coordinate time t and proper time dtau will be

dtau = -ds

thus

dtau / dt = (1+gz)

"proper time" is the time that one measures on a clock, and is also the invariant lorentz interval integrated along a path.

Thus at z=0, dtau = dt. But as z increased, dtau/dt increases. So the ratio of coordinate time to proper time depends on height for an accelerated observer.

Note that this happens solely because of the choice of coordinate systems - it's an illustration of how time dilation is a coordinate dependent notion (and not a coordinate independent notion, like the Lorentz interval).

15. Aug 3, 2006

### MeJennifer

Why do you need a coordinate system? Looks like this can be solved coordinate independently:

Take an elevator in flat space-time in an inertial frame. Exactly in the middle are two synchronized clocks, bring one to the ceiling and one to the bottom. Now accelerate the elevator constantly at 1g for say 10 hours (measured from the clock at the bottom). Now stop accelerating, and go inertial. Now bring both clocks back to the middle. Are they showing a different time?

Personally I do not care much for an SR type explanation for gravitational time dilation, I think GR can explain this much better.

16. Aug 4, 2006

### pervect

Staff Emeritus
The idea that the "top of the box" is a constant distance away from the "bottom of the box" needs to be codified. Coordinates are one way of doing this. Another possible way of doing this is to insist that the "radar distance" between the top and bottom of the box is constant. This issue is a litltle more subtle than it looks, because the distance one measures with a ruler is a non-linear function of radar distance. But as long as one can agree that a constant radar distance implies a constant ruler distance (even if it's a non-linear function for large enough distance), everything is OK.

Note that the particles at the 'top of the box' and the 'bottom of the box' will have different proper accelerations - i.e. if you mount an accelerometer on each particle, you'll get a different reading on the 'top' of the box than on the 'bottom'.

On a space-time diagram, the "top of the box" and the "bottom of the box" will both be hyperbolas on a Minkowski space-time diagram.

See for instance

http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)

The two hyperbolas (top & bottom) will both have the same asymptotes.