Atomic clocks in gravitational field

[jartsa]

On the other hand, think about what happens to the gravitational redshift at the detection point if instead we replaced the static observer there with a freely falling observer (but kept the static observer at the emission point).

Seems to me that nothing happens to gravitational redshift.

Does the freely falling observer start to fall from infinity? Then the doppler blueshift and the gravitational redshift will cancel out, in which case there is gravitational redshift large enough to cancel the doppler shift, which gravitational redshift is the same gravitational redshift as the static observer's gravitational redshift.

 

[harrylin]

Seems to me that nothing happens to gravitational redshift.

Does the freely falling observer start to fall from infinity? Then the doppler blueshift and the gravitational redshift will cancel out, in which case there is gravitational redshift large enough to cancel the doppler shift, which gravitational redshift is the same gravitational redshift as the static observer's gravitational redshift.

Right. It is of course possible to temporarily mask gravitational redshift on a single object by means of an accelerating reference system. But it is not possible to do that for a plurality of bodies such as satellites orbiting the Earth.

 

[PAllen]

Right. It is of course possible to temporarily mask gravitational redshift on a single object by means of an accelerating reference system. But it is not possible to do that for a plurality of bodies such as satellites orbiting the Earth.

Well, it depends. Two (reasonably close) concentric 2-spheres of free falling bodies will will experience detect no 'gravitational' redshift or blue shift for radial light in either direction.

 

[Philosopha]

Right. It is of course possible to temporarily mask gravitational redshift on a single object by means of an accelerating reference system. But it is not possible to do that for a plurality of bodies such as satellites orbiting the Earth.

You picked something there I believe: "mask". - Very much the essence of what I am wondering about.

 

[harrylin]

Well, it depends. Two (reasonably close) concentric 2-spheres of free falling bodies will will experience detect no 'gravitational' redshift or blue shift for radial light in either direction.

I was thinking about communication satellites around the equator. Some of them are even at opposite sides. As they fall in opposite directions, it is not possible to mask the gravitational redshift of all of them with a single falling reference system.

You picked something there I believe: "mask". - Very much the essence of what I am wondering about.

This is how Einstein put it in 1916 (translated from German):

"It is, for instance, impossible to choose a body of reference such
that, as judged from it, the gravitational field of the Earth (in its entirety) vanishes."
- Relativity: The Special and General Theory
http://www.bartleby.com/173/20.html

 

[Philosopha]

As it stands, yes, because energy is frame-dependent.

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.

 

[WannabeNewton]

If you are asking why as in "why does nature behave this why" then GR doesn't answer that question I'm afraid.

 

[tom.stoer]

For what reason does one clock oszillate slower than the other? The oszillation velocity is a direct reflection of the clocks energy/mass state ...

Here's what GR has to say - and it has nothing to do with energy but is a purely geometrical effect.

Assume we have two clocks located at (t,x) = (0,0) in one specific coordinate system. They will meet again at a later time T but at the same location x=0, i.e. at (T,0).

Assume one clock is traveling along a curve C from point A to point B in spacetime. The second clock is traveling along a different curve C' from point A to point B in spacetime. Of course we could introduce the coordinates for A and B, but that is not necessary for the next steps.

Now you have to believe me that the proper time tau measured by a clock along its curve between A and B is given by the "length" of the curve through spacetime.

\tau = \int_C d\tau

As the two curves C and C' through spacetime are different for the two clocks their proper times will differ.

\Delta\tau_{A\to B} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau

These generic formulas are rather formal. In order to calculate something one introduces a coordinate system (t,x), a spacetime-metric g which fully describes spacetime-geometry and a velocity v=dx/dt along a curve C. Then the above mentioned formula for the proper times can be expressed as

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

Please note that time dilation due to geometry and due to velocity cannot be separated in general. Please note that we have not introduced any expression for energy.

Asking "why this formula explaines time dilation and differential aging" and "why nature behave this why" then GR doesn't answer that question b/c it's like asking "why GR?" which cannot be answered by GR (this is to stress what WannabeNewton said)

 

[Philosopha]

Thank you very much for everything - I will definitely come back to that topic once I studied my way through GR. It is good that you mentioned the "why". Lots of people are thinking about that. I believe there is a why and maybe that observations/ facts will give us an idea.

 

[Philosopha]

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

 

[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.