Atomic clocks in gravitational field

[tom.stoer]

The problem is not the idea, but the " why this idea?" The idea is GR, but there's no deeper reason for GR. GR cannot answer "why GR?" If there would be a more fundamental theory X from which GR emerges in some way, then the same question "why X?" cannot be answered based on X.

So your questions are essentially meta-physically. They are interesting, but not as physical questions.

 

[WannabeNewton]

And if something calculates the right amount, maybe that would also give an idea.

Calculation isn't a problem. Tom gave you one way of calculating the effect. The problem is in answering the much deeper question of "why does this happen?" which as Tom stated is more along the lines of metaphysics than it is physics.

 

[harrylin]

[..] The oszillation velocity is a direct reflection of the clocks energy/mass state methinks. Which suggests the clock at the bottom to have a lower energy/mass content than the one on top.

Concerning mass it's a bit more complex, as also c varies from that non-local perspective.
There was an interesting and informative discussion on that topic in another forum (see in particular the first reply by Jonathan Scott): http://sci.physics.research.narkive.com/tSrjiEsH/mass-of-particles-in-gr-field.

 

[jartsa]

What about the fact that atomic clocks after an experiment can be put next to each other on a table, be examined and leave no doubt, that they have recorded the absolute value of the relative difference between their prior frames? The clock at height of our foot experiences less oszillation periods (phenomenon known as time-dilation) than the clock at our head. But why is that so? Time-dialtion is just how that phenomenon is called. For what reason does one clock oszillate slower than the other? The oszillation velocity is a direct reflection of the clocks energy/mass state methinks. Which suggests the clock at the bottom to have a lower energy/mass content than the one on top.

It seems pretty safe to say that an object hanging at low altitude has less energy than a similar object at higher location.

Because energy is released when two gravitating objects fuse together, and then a mass defect can be measured.

https://en.wikipedia.org/wiki/Binding_energy#Mass_change
https://en.wikipedia.org/wiki/Gravitational_binding_energy

 

[tom.stoer]

nice; but nevertheless time dilation has nothing to do with energy

 

[PeterDonis]

It seems pretty safe to say that an object hanging at low altitude has less energy than a similar object at higher location.

Except that, if the energies of the objects are each measured by observers at the same altitude, they will be the same. So you have to specify relative to what observer the energy of the lower object is less. (For example, the lower object's energy at infinity is less, assuming both are at rest at their respective altitudes.)

Because energy is released when two gravitating objects fuse together, and then a mass defect can be measured.

This is also a change in energy at infinity.

 

[harrylin]

Except that, if the energies of the objects are each measured by observers at the same altitude, they will be the same. [..]

Similarly it seems pretty safe to say that a high energy electron has more kinetic energy because of its high speed, upon which you might comment "Except that, if the energies of the objects are each measured by observers at the same velocity, they will be the same."

 

[harrylin]

nice; but nevertheless time dilation has nothing to do with energy

Time dilation can be written as the combination of the effects of kinetic and potential energy. For example, on the geoid potential energy balances kinetic energy, so that the time dilation effects cancel out. If time dilation has nothing to do with energy, then that must be pure coincidence - which seems highly unlikely to me.

 

[PeterDonis]

Similarly it seems pretty safe to say that a high energy electron has more kinetic energy because of its high speed, upon which you might comment "Except that, if the energies of the objects are each measured by observers at the same velocity, they will be the same."

Yes, exactly, because energy is frame-dependent, which was my point. When you use the term "energy", you have to specify what it's relative to; or else you are implicitly assuming a certain frame of reference. In jartsa's case, he implicitly assumed that "energy" meant "energy at infinity". In your case, you're implicitly assuming that "energy" means "energy relative to the laboratory in which the electron is moving at high speed".

There's nothing wrong with that as long as you realize that that's what you're doing, and don't try to claim that your particular choice of what to measure energy relative to is somehow absolute. (The same applies to other frame-dependent concepts like time dilation and redshift/blueshift.)

 

[PAllen]

Similarly it seems pretty safe to say that a high energy electron has more kinetic energy because of its high speed, upon which you might comment "Except that, if the energies of the objects are each measured by observers at the same velocity, they will be the same."

Yes, of course. There is nothing different about a 'fast' electron or a 'slow' electron except for the observer relative to whom it is fast versus slow. Unless one posits an objective, absolute, frame, there is no conceivable difference.

 

[tom.stoer]

Time dilation can be written as the combination of the effects of kinetic and potential energy. For example, on the geoid potential energy balances kinetic energy, so that the time dilation effects cancel out. If time dilation has nothing to do with energy, then that must be pure coincidence - which seems highly unlikely to me.

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?

 

[tom.stoer]

Let me explain where I see the problem: in my formula we have

$v^\mu = dx^\mu / dt = (1, dx^i/dt)$

expressed in some coordinates.

In order to introduce energy we hae to use the 4-velocity and the 4-momentum

$u^\mu = dx^\mu / d\tau$
$p^\mu = m u^\mu$

For a geodesic along which 4-momentum is conserved we have

$\tau = \int_C \sqrt{g_{\mu\nu}\,dx^\mu\,dx^\nu} = \int_C d\tau \, \sqrt{g_{\mu\nu}\,u^\mu\,u^\nu} = m^{-1} \int_C d\tau \sqrt{g_{\mu\nu} \, p^\mu \, p^\nu}$

But

$g_{\mu\nu} \, p^\mu \, p^\nu = m^2$

and therefore we arive at

$\tau = \int_C d\tau$

therefore 4-momentum trivially drops out.

That's why I doubt that

Time dilation can be written as the combination of the effects of kinetic and potential energy.

does not work

 

[Philosopha]

What seems not possible today might be possible tomorrow. No question should be surrendered to Metaphysics.

 

[tom.stoer]

What seems not possible today might be possible tomorrow. No question should be surrendered to Metaphysics.

This is not metaphysics but math.

In order to say that time dilation for proper times τ is caused by differences in energy E you have to provide a formula like τ = f(E), a function of E.

 

[Philosopha]

Calculation isn't a problem. Tom gave you one way of calculating the effect. The problem is in answering the much deeper question of "why does this happen?" which as Tom stated is more along the lines of metaphysics than it is physics.

Calculating experimental results however from the vantage point of changes in energy/mass content translated into frequency if achieved would allow for an interpretation as of the why. It must be rational and calculated, or it is not an answer.

 

[tom.stoer]

Calculating experimental results however from the vantage point of changes in energy/mass content translated into frequency if achieved would allow for an interpretation as of the why.

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.

 

[WannabeNewton]

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.

 

[tom.stoer]

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.

I think this relation is limited to special cases where you can interpret the 00 coefficient of the metric tensor as gravitational potential. But in general this is not possible - and it's not in the spirit of GR and general covariance.

 

[WannabeNewton]

Well gravitational time dilation really only makes sense for stationary space-times (since we need a notion of clocks being "at rest" in the sense that they follow orbits of the time-like killing vector field) and for stationary space-times we can always define a gravitational potential using the time-like killing vector field.

 

[tom.stoer]

Well gravitational time dilation really only makes sense for stationary space-times and for stationary space-times we can always define a gravitational potential.

The general formula I presented makes sense for arbitrary spacetimes.

 

[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.