Atomic clocks in gravitational field

[WannabeNewton]

The general formula I presented makes sense for arbitrary spacetimes.

Oh I was thinking of gravitational time dilation in the sense of "two clocks at rest in the gravitational field ticking at different rates" with the emphasis on the "at rest" part. I don't disagree at all with what you're saying though; even if for stationary space-times we can relate things to gravitational potential I don't see any relation to kinetic energy (which is what I think was originally mentioned).

 

[tom.stoer]

Oh I was thinking of gravitational time dilation in the sense of "two clocks at rest in the gravitational field ticking at different rates" with the emphasis on the "at rest" part.

This case is covered as well. At rest means "at rest w.r.t. some coordinate system" and the gravitational field is the metric.

 

[WannabeNewton]

I was talking about the physical definition of "at rest" meaning following an orbit of a time-like killing vector field in a stationary space-time, which is a physical description of being "at rest". The existence of the time-like killing vector field picks out a preferred family of "static" observers for the space-time (a preferred vector field), including "static" clocks in a gravitational field.

 

[tom.stoer]

Of course you can do this as well. As said in restricted cases you can define a gravitational potential, but in general (no Killing vector field) it will not work

 

[WannabeNewton]

Right, I'm not disagreeing with that at all my friend :) I'm just saying if one wanted to really stretch things and find some special cases of relationships with "energy" then one could do something along the lines of the above.

 

[tom.stoer]

I fully agree ;-)

 

[Philosopha]

Before claiming to know the "why" you should present the calculation. Any (general covariant) result for proper time expressed in terms of energy and momentum is appreciated.

I do not claim to know the why nor that I got any calculations to work. So ?

By the guidelines of Physicsforum one is not ment to discuss any personal unpublished hypothesis I was told. So I was just presenting some definite facts with the atomic clocks. Facts are always good.

My sentence you quoted read "...if achieved would... ". Expressing the consideration of a possibility (by whoever would achieve it, if at all) is not claiming to know the answer. Hope you were not upset at anything :(.

 

[Philosopha]

Merely I'm saying facts are the best providers of answers. One doesn't even need to interpret anything into the clock experiment. The results are as they are. Equating frequency (here of clock) with energy content is just an old basic.

 

[tom.stoer]

Hi Philosopha, no, I am not upset at all. No problem. The only issue I wanted to clarify is that time dilation as a result of energy, gravitational potential etc. is limited to very special conditions (stationary spacetime) and not valid in general (even if sometimes presented that way). I was not asking for any new theory or formula, only for some existing expression unknown to me.

 

[harrylin]

Gravitational time dilation can be related to the gravitational potential of the gravitational field but that's as close a relationship to "energy" as I can think of.

Yes, that's what I meant. One can calculate time dilation for a certain mass as function of potential and kinetic energy. An equal increase of both such as with the equilibrium of the geoid results in zero time dilation, as the two exactly compensate each other.

How do you translate the above mentioned formula

\tau = \int_C d\tau = \int_0^T dt\,\sqrt{g_{\mu\nu}\,v^\mu\,v^\nu}

into an expression containing energy?

I doubt that potential and kinetic energy as commonly defined fit with such GR equations; instead I was referring to the weak field approximations, and surely you know those. Your claim that time dilation has nothing to do with energy implies that the aforementioned correlations are just a coincidence. I find that extremely unlikely.

 

[Philosopha]

Hi Philosopha, no, I am not upset at all. No problem. The only issue I wanted to clarify is that time dilation as a result of energy, gravitational potential etc. is limited to very special conditions (stationary spacetime) and not valid in general (even if sometimes presented that way). I was not asking for any new theory or formula, only for some existing expression unknown to me.

Thanks for that reply Tom.

 

[tom.stoer]

... instead I was referring to the weak field approximations, and surely you know those. Your claim implies that the aforementioned correlations are just a coincidence and I find that extremely unlikely.

For weak fields / stationary spacetimes / appropriate symmetries (Killing vector fields) this is certainly OK. I don't think this is a coincidence, it's a restriction to these special cases.

 

[jartsa]

So, what's wrong with this reasoning:
Always when a gravitational time dilation factor x is observed, then also a gravitational frequency redshift factor x is observed.

Always when a gravitational frequency redshift factor x is observed, then also an energy redshift of gravitational kind is observed, where the factor of energies is x. (This is based on formula E=hf )

Therefore gravitational time dilation factor x is always assosiated with an energy redshift factor x.

("energy redshift" means that phenomenom that is sometimes explained as light losing energy while climbing upwards)

 

[WannabeNewton]

"Associated with" is meaningless in the way you have presented it. Unless there is a mathematical relation that you can provide, this is moot.

 

[tom.stoer]

Looking at the general expressions the two entities "proper time of massive bodies" and "redshift of massless phoons" are nearly unrelated. The close relationship between these two effects seems to be an artifact of simplified models.

 

[harrylin]

For weak fields / stationary spacetimes / appropriate symmetries (Killing vector fields) this is certainly OK. I don't think this is a coincidence, it's a restriction to these special cases.

For the relationship between time dilation and energies that I indicated I know no exception. Could you give an example where an equal increase in potential and kinetic energy results in time dilation?

 

[harrylin]

So, what's wrong with this reasoning:

[..]
Always when a gravitational frequency redshift factor x is observed, then also an energy redshift of gravitational kind is observed, where the factor of energies is x. (This is based on formula E=hf )

[..]
("energy redshift" means that phenomenom that is sometimes explained as light losing energy while climbing upwards)

The basis of formula E=hf is misleading in that context: although it's no doubt connected through conservation of energy, it's the wrong explanation. See https://www.physicsforums.com/showthread.php?p=3162897

 

[tom.stoer]

For the relationship between time dilation and energies that I indicated I know no exception. Could you give an example where an equal increase in potential and kinetic energy results in time dilation?

In order to give you an example where this relationship fails you have to present it here explicitly. But I still miss the formula which demonstrates this relationship. I indicated above why I think that such a formula cannot be given in general.

 

[jartsa]

"Associated with" is meaningless in the way you have presented it. Unless there is a mathematical relation that you can provide, this is moot.

I'll declare some restrictions: The gravity field must be static. And static measuring instruments measure the redshift of energy and the time dilation. Those restriction should suffice.

Now we can say the time dilation factor and energy redshift factor are the same number.

Well, now there's a mathematical relation.

 

[WannabeNewton]

Ok if you are using a static gravitational field then I completely agree with you. See also here: http://physics.stackexchange.com/qu...redshift-imply-gravitation-time-dilation?rq=1

 

[tom.stoer]

Yes, we never had doubts that in special cases as described above (in several posts) there is this direct relation between proper times and energy or frequency.

The only thing we wanted to stress is that you do have redshift even if these restrictions do not apply, i.e. where a "gravitational potential" does no longer make sense. And b/c this is the case we should not say that differences in the gravitational potential energies cause redshift, simply b/c this does not apply in general.

 

[jartsa]

The only thing we wanted to stress is that you do have redshift even if these restrictions do not apply, i.e. where a "gravitational potential" does no longer make sense.

Like scattering of light from a fast moving black hole?

 

[harrylin]

Yes, we never had doubts that in special cases as described above (in several posts) there is this direct relation between proper times and energy or frequency.

The only thing we wanted to stress is that you do have redshift even if these restrictions do not apply, i.e. where a "gravitational potential" does no longer make sense. And b/c this is the case we should not say that differences in the gravitational potential energies cause redshift, simply b/c this does not apply in general.

Inversely, what I (we) wanted to stress is that as long as kinetic and potential energy are unambiguously defined, there is a clear relationship with time dilation (for undefined or ambiguously defined quantities we can't say of course!). Therefore we should not say that time dilation has nothing to do with energy, as, I think, for all the cases that those concepts make sense, that relationship does apply.

 

[tom.stoer]

Like scattering of light from a fast moving black hole?

No, in this case the conditions do apply ;-) Instead of a fast moving black hole you study a black hole at rest. Outside the Schwarzschild radius this is exactly the same as near a normal star or a planet. As a last step you apply a transformation from the black hole rest frame to a different, fast-moving frame. That's all.

 

[tom.stoer]

... we should not say that time dilation has nothing to do with energy, as, I think, for all the cases that those concepts make sense, that relationship does apply.

Yes, I think so, too.

The statement "that relationship does apply" is OK, of course. But there were statements that redshift is caused by differences in potential energies, and this is not OK b/c then we would have to answer the question "what caused redshift when no potential energy can be defined?"

There is another problem, namely the idea that the difference in kinetic energy (difference in frequency) is due to a difference in potential energy. This cannot be true in general, either. Kinetic energy and frequency of a photon is always defined locally w.r.t. to two independent, local reference frames. So the two kinetic energies are not related at all, a priori. The difference in potential energy relies on a unique definition of "gravitational potential", which is a global entity and which cannot be given in general.

I only wanted o make clear that redshift is a much more general phenomenon and applies to every spacetime geometry, whereas for the definition of potential energy rather special conditions must apply.

So yes, if these conditions do apply, the relation is there, and this is not a coincidence. But no, there are situations where these conditions to not apply, but there's still redshift (the best known example is cosmological redshift in an expanding, anisotropic universe we live in)

 

[TrickyDicky]

...anisotropic universe we live in)

You meant isotropic, right?

 

[The17YearOld]

It seems that the most recent question in this forum is if time dilation and energy are related in someway. It is said in the beginning of the forum that an atomic clock at someones feet would tick slower than an atomic clock at someones head. I wish I could quote this person but I am new to the forum and still have not figured out how to quote people so if anyone could assist me it would be appreciated. Now is is known that the closer something gets to a massive object like Earth, the gravitational pull that effects it increases. This is due to the space surrounding the object being more curved closer to the surface of a planet. General Relativity states that gravity is causes by the curvature of space due to matter displacing it.

What I am trying to relate this to is the spinning of a wheel. Say a bicycle wheel is spinning. The wheel is one object therefore all parts of the wheel have the same energy. However, the outer part of the wheel spins faster than the inner rim of the wheel. If the inner part of the wheel was taken out and made to be its own separate wheel, and the wheels took the same amount of time to make a full rotation, the larger wheel would have covered more distance. Now to relate it back to the clock, a person walking with an atomic clock at their head and feet would experience the clock at their head ticking faster. Their head is technically moving faster than their feet. If the world is thought of as a giant bicycle wheel, the head being the outermost part of the rubber and the feet being the innermost part of the rubber where it touches the metal rim, the persons whole body is moving as one but at different speeds.

Now the simple formula for time is s=d/t. All numbers are made up and are not necessarily accurate but say a person is walking along the Earth and the Earth's circumference is 15 ft. Now the persons head travels along a circle with the feet but the circumference of the path the head travels is 30 ft. The person moves at 5 feet per second. It should take the feet 3 seconds to makes the journey when plugged into the equation s=d/t. It should take the head 6 seconds to make the journey, but it moves with the feet so it really takes 3 seconds. I'm not really sure what all of this means but to me there must be some relation in this to time dilation and energy. Speed and energy are somewhat interchangable I believe as speed is the measurement of how much distance is covered in an amount of time. The more energy as object has the more distance it will cover in less time. The head traveling along the outer wheel must have more energy than the feet because it travels faster than the feet. This is only the case in curved space like that of a planet. In perfectly flat space, both the head and the feet would travel at the same speed covering the same distance in the same amount of time, but when traveling through curved space, the head somehow travels faster and possibly with more energy due to its increased speed.

Just some things to think about and possibly contribute to the discussion
Dan

 

[tom.stoer]

You meant isotropic, right?

no, I mean anisotropic; anisotropy is tiny, but it's there, as can be seen in the CMB (Planck data)

 

[harrylin]

Yes, I think so, too.

The statement "that relationship does apply" is OK, of course. But there were statements that redshift is caused by differences in potential energies, and this is not OK b/c then we would have to answer the question "what caused redshift when no potential energy can be defined?" [..]
[rearranged] The difference in potential energy relies on a unique definition of "gravitational potential", which is a global entity and which cannot be given in general.

I can't imagine a situation where no definition is possible.

There is another problem, namely the idea that the difference in kinetic energy (difference in frequency) is due to a difference in potential energy. This cannot be true in general, either. Kinetic energy and frequency of a photon is always defined locally w.r.t. to two independent, local reference frames. [..]

That sounds like another definition issue, and I think even a wrong one. When comparing energies one must use the same reference system (compare my post #17).

[..]So yes, if these conditions do apply, the relation is there, and this is not a coincidence. But no, there are situations where these conditions to not apply, but there's still redshift (the best known example is cosmological redshift in an expanding, anisotropic universe we live in)

Of course Doppler redshift must be taken in account too; while that complicates things, I don't see it as a problem of principle.

 

[Dale]

I can't imagine a situation where no definition is possible.

FLRW. Or any spacetime that isn't asymptotically flat.