# Atomic clocks in gravitational field

In his paper, Chou et. al., 2010 has demonstrated that an atomic clock at the height of our head ticks faster than an atomic clock at the height of our feet.

I was thinking that one electromagnetic process can be substituted with any other - so that corresponding atoms in our head tick faster than those in our feet so that our head ages faster. If I however simplify the electromagnetic process to a single oszillating field, I concluded I would see the field at my head being blueshifted in comparison to my feet. Faster ticking of (biological)clocks = faster oszillation = more energy/mass (we have to put work in to lift our head in a potential). The scenario for outgoing light would of course be different as light has to fight the potential - it gets redshifted as we all know.

My question now is: Is the head and foot scenario as I understood it in accordance with SR? Does SR state that our head ages faster because it is blueshifted? I am concerned and confused that the effect of time-dilation is attributed to redshift? But I think the experiment by Chou reveals reality? Or is SR not applicaple here?
I was thinking of it in the way that when you sketch a light clock with constant velocity the redshift is an 'apparent' or 'relative' effect. However acceleration (or the equivalence in a gravitational field) will induce real changes. And doesn't energy input cause blueshift? So I conclude that energy input results in 'real' bueshift of everything and with that in faster aging.

I'm not very advanced in my studies yet, but have been thinking about this topic since a year now. So I thought I ask others. Thank you in advance.

## Answers and Replies

Related Special and General Relativity News on Phys.org
tom.stoer
Science Advisor
Redshift in gravitational fields is studied in GR, not in SR (which is applicable to flat spacetime w/o any gravitational fields).

In essence your analysis is correct, your head is aging faster than your feet. In principle it is possible to compare the aging processes, e.g. by making hull bows ;-)

Thx - I was worried about a possible contradiction. So t=to/sqrt(1-(2GM/Rc^2)), implies that when R is larger (at my head) the result of the sqrt is closer to 1 than for my feet so that there is a lesser time-dilation effect? So it agrees that the aging of your feet is slowed down stronger with respect to flat space (where the sqrt would be =1). Does GR say this is because of the redshift of the feet?

Dale
Mentor
In general GR will contradict SR any time tidal gravity (curvature) is significant. It is only locally that they become equivalent.

In this case there the tidal effects are probably negligible, at least for a reasonably short time. This differential aging is a feature of SR also. You should read about Rindler frames.

Thx - I was worried about a possible contradiction. So t=to/sqrt(1-(2GM/Rc^2)), implies that when R is larger (at my head) the result of the sqrt is closer to 1 than for my feet so that there is a lesser time-dilation effect? So it agrees that the aging of your feet is slowed down stronger with respect to flat space (where the sqrt would be =1).
Right.
Does GR say this is because of the redshift of the feet?
Just as SR, GR does not explain deeper causes. However logically it should be rather the other way round, and this is also how it was first deduced:

"Thus the clock goes more slowly if set up in the neighbourhood of ponderable masses. From this it follows that the spectral lines of light reaching us from the surface of large stars must appear displaced towards the red end of the spectrum."
- p.198, Foundation of General Relativity, 1916.

Last edited:
1 person
tom.stoer
Science Advisor
oes GR say this is because of the redshift of the feet?
No. Both redshift and differential aging have a common cause, namely spacetime geometry and its effects to objects moving through.

In his paper, Chou et. al., 2010 has demonstrated that an atomic clock at the height of our head ticks faster than an atomic clock at the height of our feet.

I was thinking that one electromagnetic process can be substituted with any other - so that corresponding atoms in our head tick faster than those in our feet so that our head ages faster. If I however simplify the electromagnetic process to a single oszillating field, I concluded I would see the field at my head being blueshifted in comparison to my feet. Faster ticking of (biological)clocks = faster oszillation = more energy/mass (we have to put work in to lift our head in a potential). The scenario for outgoing light would of course be different as light has to fight the potential - it gets redshifted as we all know.

My question now is: Is the head and foot scenario as I understood it in accordance with SR? Does SR state that our head ages faster because it is blueshifted? I am concerned and confused that the effect of time-dilation is attributed to redshift? But I think the experiment by Chou reveals reality? Or is SR not applicaple here?
I was thinking of it in the way that when you sketch a light clock with constant velocity the redshift is an 'apparent' or 'relative' effect. However acceleration (or the equivalence in a gravitational field) will induce real changes. And doesn't energy input cause blueshift? So I conclude that energy input results in 'real' bueshift of everything and with that in faster aging.

I'm not very advanced in my studies yet, but have been thinking about this topic since a year now. So I thought I ask others. Thank you in advance.

Out of these alternatives:

A: Gravitationally time dilated foot emits gravitationally time dilated EM-waves
B: Gravitationally time dilated foot emits gravitationally redshifted EM-waves
C: Gravitationally redshifted foot emits gravitationally redshifted EM-waves
D: Gravitationally time dilated foot emits EM-waves that are not gravitationally redshifted, but may become gravitationally redshifted, if climbing upwards

My personal favorite is item A. Because it's very simple. And it seems pretty unlikely that general relativity would disagree with alternative A.

B and C seem to be saying the same thing as A, just not quite as well.

D does not make much sense.

1 person
No. Both redshift and differential aging have a common cause, namely spacetime geometry and its effects to objects moving through.
One must however not neglect that our perception of time depends on the velocity of all processes - that is were the link comes from. Clocks tick faster so we perceive time to run faster.

But I'm still confused if the GR agrees with (my interpretation) of the atomic clock experiment by Chou 2010?

Was I right to say an object (not light for which it is the other way round because of its very nature!!) entering a gravitational well experiences "red"shift because it looses energy - and that is what the clock experiment by Chou 2010 proves with clocks at our feet ticking slower? Slower oszillating fields are "redder". ???

Out of these alternatives:

A: Gravitationally time dilated foot emits gravitationally time dilated EM-waves
B: Gravitationally time dilated foot emits gravitationally redshifted EM-waves
C: Gravitationally redshifted foot emits gravitationally redshifted EM-waves
D: Gravitationally time dilated foot emits EM-waves that are not gravitationally redshifted, but may become gravitationally redshifted, if climbing upwards

My personal favorite is item A. Because it's very simple. And it seems pretty unlikely that general relativity would disagree with alternative A.

B and C seem to be saying the same thing as A, just not quite as well.

D does not make much sense.
Just saw your response then, I pick C then and my question is solved. thank you

Last edited:
tom.stoer
Science Advisor
A system (an emitter) with proper time tau will emit the same el.-mag. waves with the same frequency f regardless which gravitational potential it has. Frequency f is then related to some constant / tau.

An observer located at the same position as the emitter will observe this frequency f. If the emitter and the observer form a co-moving, free-falling pair through spacetime the observed frequency does not change.

But as soon as the emitter and the observer are located at different positions with different gravitational potential then the observed frequency f' will differ from the emitted frequency f.

So all effects regarding time-dilation, redshift etc. are due to different paths through spacetime probing different gravitational fields. The effect is non-local. Thefore it does not make sense to talk about "time-dilated feet", b/c the question is always "time-dilated with respect to what?"

In that sense I think A-D are quite confusing. I would pick none of them.

Thanks Tom - very much as I have learned from SR - all correct of course. But regarding the paper from Chou 2010 it becomes clear that there is a very "real" thing happening - like our head really ages faster - and that has a good "real" reason (I call it space wants energy back - thats why an apple drops - I think!!) - so a respective atom in my feet is "truly" frequencyshifted as oposed to my head (see Chou). I was just wondering if the direction was in accordance or if I did an error in my reasoning. So all is matching - good. To your last sentence I would like to answer, that there is a "real" change in the frequency of matter when it enters a gravitational field as compared to when it would be in flat space. I'm an ex-Biologist who after her PhD arrived on this topic because of my head and feet - so still sort of Biology. But since a while I study basic University Physics and Math on top because of my topic. Frequencyshift of all matter to do with something else and the involvement of human (internal observer) perception. This forum has helped me much. Thank you all.

WannabeNewton
Science Advisor
...that there is a "real" change in the frequency of matter when it enters a gravitational field as compared to when it would be in flat space...
What "real" change are you talking about?

tom.stoer
Science Advisor
... there is a very "real" thing happening ...
I never said something else.

All what I am saying is that all these effects are entirely non-local and relative.

It does not make sense - neither physically nor semantically - to say that "something is redshifted" or "something is slower". This something can only be redshifted or slower "relative to something else". And in the case of gravitational effects this means "relative to something else located somewhere else where the gravitational potential is different".

And btw., this what you said in your first post ;-)

In his paper, Chou et. al., 2010 has demonstrated that an atomic clock at the height of our head ticks faster than an atomic clock at the height of our feet.

Last edited:
[..] how it was first deduced:

"Thus the clock goes more slowly if set up in the neighbourhood of ponderable masses. [..]."
- p.198, Foundation of General Relativity, [Einstein ]1916.
[..] It does not make sense - neither physically nor semantically - to say that "something is redshifted" or "something is slower". This something can only be redshifted or slower "relative to something else". And in the case of gravitational effects this means "relative to something else located somewhere else where the gravitational potential is different".
[..]
It can make sense if the context makes it clear - which certainly is the case in the text that I cited. The standard reference (the "more slowly") is taken at infinite distance - not in the neighbourhood of masses.

I suppose that with "real", Philosopha refers to our rejection of interpretation D.
See also an earlier, similar thread and a paper in post #14 there that discusses that issue: www.physicsforums.com/showthread.php?p=4067937

Last edited:
PAllen
Science Advisor
2019 Award
It can make sense if the context makes it clear - which certainly is the case in the text that I cited. The standard reference (the "more slowly") is taken at infinite distance - not in the neighbourhood of masses.

I suppose that with "real", Philosopha refers to our rejection of interpretation D.
See also an earlier, similar thread and a paper in post #14 there that discusses that issue: www.physicsforums.com/showthread.php?p=4067937
My preferred mental model (the math does not specify a unique mental model) is rather close to D: that light is emitted the same everywhere by the same physical process, and does not change while it travels. How it is received by some other detector (or family of detectors) depends on the world line(s) of the emitter(s) and detector(s) and the spacetime geometry traversed [the emitter world line and the intervening geometry and light path determine a specific function: of shift as a function of receiver world line tangent]. Gravitational time dilation or redshift, in this view, is simply an extremely convenient computational model possible due the existence of a well defined family of static observers (which, for example, does not exist in the vicinity of a binary pulsar; despite gravitational shift being undefinable in that case, the shift between a specific emission and absorption is perfectly well defined; you just can't factor it into gravitational and kinematic).

Last edited:
tom.stoer
Science Advisor
It can make sense if the context makes it clear ...
A comparison of an observation does not make sense of there is no second observation to compare with. If the context specifies the second one it's fine, but there is always a relation between two observers, no matter whether it's specified explictly or implicitly.

Alternative D is not too bad, but I don't agree with the "time dilated foot" b/c it seems that this is a property of the foot only. It isn't.

Last edited:
WannabeNewton
Science Advisor
I like D too in a sense if one interprets the statement that the EM waves "are not gravitationally redshifted" to mean that gravitational redshift has no absolute meaning, that the EM waves are not gravitationally redshifted in and of themselves in any absolute sense, and that it only has meaning through comparison of measurement by two different observers.

PeterDonis
Mentor
2019 Award
Out of these alternatives:
You left out one:

E: Your foot emits EM waves which, according to clocks at your foot, are emitted at a certain frequency. The EM waves don't change at all during their travel upward, but when they are observed by your head, they are observed to have a different, lower, frequency, because the 4-velocity of your head is different from the 4-velocity of your feet.

E is the alternative that most closely matches the actual math: the photon's 4-momentum is fixed by the frequency of emission; that 4-momentum is parallel transported along its worldline, which is the standard GR definition of "doesn't change"; the change in observed frequency is entirely due to the change in 4-velocity between your feet and your head, since the observed frequency is the contraction of the photon's 4-momentum with your head's 4-velocity.

Also, E generalizes to *all* cases of observed "time dilation" or "redshift", and even covers cases where the standard versions of those concepts don't apply or have fuzzy boundaries, such as cosmological redshift. You can always parallel transport a photon's 4-momentum along its worldline and contract it with the 4-velocity of the detector, regardless of the spacetime you're in.

PAllen
Science Advisor
2019 Award
You left out one:

E: Your foot emits EM waves which, according to clocks at your foot, are emitted at a certain frequency. The EM waves don't change at all during their travel upward, but when they are observed by your head, they are observed to have a different, lower, frequency, because the 4-velocity of your head is different from the 4-velocity of your feet.

E is the alternative that most closely matches the actual math: the photon's 4-momentum is fixed by the frequency of emission; that 4-momentum is parallel transported along its worldline, which is the standard GR definition of "doesn't change"; the change in observed frequency is entirely due to the change in 4-velocity between your feet and your head, since the observed frequency is the contraction of the photon's 4-momentum with your head's 4-velocity.

Also, E generalizes to *all* cases of observed "time dilation" or "redshift", and even covers cases where the standard versions of those concepts don't apply or have fuzzy boundaries, such as cosmological redshift. You can always parallel transport a photon's 4-momentum along its worldline and contract it with the 4-velocity of the detector, regardless of the spacetime you're in.
This, of course, is my preference. Slight correction: it is the emitter's 4-velocity you need to parallel transport along the null path, not the photon's. That's obvious from the rest of your description - comparing 4-velocity of feet versus head via parallel transport.

PeterDonis
Mentor
2019 Award
Slight correction: it is the emitter's 4-velocity you need to parallel transport along the null path, not the photon's.
Well, the two are equivalent, since the frequency of emission fixes the contraction of the emitter's 4-velocity with the photon's 4-momentum. It really depends on whether you want an answer in energy units or time units (as well as on which description you prefer; I don't really have much of a preference either way--one could even combine the two, so to speak, by saying that the photon "transports" information about the emitter's 4-velocity to the detector).

PAllen
Science Advisor
2019 Award
Well, the two are equivalent, since the frequency of emission fixes the contraction of the emitter's 4-velocity with the photon's 4-momentum. It really depends on whether you want an answer in energy units or time units (as well as on which description you prefer; I don't really have much of a preference either way--one could even combine the two, so to speak, by saying that the photon "transports" information about the emitter's 4-velocity to the detector).
Ok, yes I see they would be equivalent. The way I described is closer to traditional Doppler descriptions, but either is universal to all GR and SR situations.

PAllen
Science Advisor
2019 Award
This, of course, is my preference. Slight correction: it is the emitter's 4-velocity you need to parallel transport along the null path, not the photon's. That's obvious from the rest of your description - comparing 4-velocity of feet versus head via parallel transport.
Well, the two are equivalent, since the frequency of emission fixes the contraction of the emitter's 4-velocity with the photon's 4-momentum. It really depends on whether you want an answer in energy units or time units (as well as on which description you prefer; I don't really have much of a preference either way--one could even combine the two, so to speak, by saying that the photon "transports" information about the emitter's 4-velocity to the detector).
Ok, yes I see they would be equivalent. The way I described is closer to traditional Doppler descriptions, but either is universal to all GR and SR situations.
One further thought on this:

- Viewing it as parallel transport of emitter 4-velocity to target world line along light path, and then applying SR Doppler, emphasizes the influence of relative motion of the world lines (in the only way this can be sensible in GR).

- Viewing it as parallel transport of light 4-momentum along light path and contracting this per the target world line local basis emphasizes how nothing changes about the light (in the only way this can be sensible in GR) along the way.

tom.stoer
Science Advisor
We should have a look at the math which we discussed a couple of weeks ago.

We have a null-geodesic C connecting two spacetime points P and Q. This null geodesic represents the k-4-vector of the el.-mag. wave. Then have two observers OP and OQ located at P and Q, with 4-velocities uP and uQ. Then we have an infinitesimally neighbored null-geodesic C' connecting two points P' and Q' on the worldlines of the observers defined via their 4-velocities in P and Q. We identify the two geodesics C,C' starting at P,P' and ending at Q,Q' with two light signals. The frequency is replaced by the (inverse) proper time intervals defined via the 4-velocities on the observer worldlines connecting P with P' and Q with Q', respectively.

Therefore the redshift can be defined via the proper times

$$1+z = \frac{d\tau_Q}{d\tau_P}$$

The null-geodesic is $x^\mu(\lambda)$ with affine parameter $\lambda$

Then we have for the redshift z

$$1+z = \frac{\langle\dot{x},u\rangle_P}{\langle\dot{x},u\rangle_Q}$$
$$\langle x,y \rangle = g_{\mu\nu}\,x^\mu\,y^\nu$$

Ref.: eq. (37) in http://relativity.livingreviews.org/open?pubNo=lrr-2004-9&amp;page=articlesu4.html [Broken]

Last edited by a moderator:
My preferred mental model (the math does not specify a unique mental model) is rather close to D: that light is emitted the same everywhere by the same physical process, and does not change while it travels. [..]
I understood Jartsa's D as meaning that light waves change as they go upwards - as he said, that doesn't make much sense. And as was discussed earlier in the other thread, that's simply wrong (elaborated there by grav-universe in post #12 and by me in post #14).

PeterDonis
Mentor
2019 Award
I understood Jartsa's D as meaning that light waves change as they go upwards - as he said, that doesn't make much sense.
Well, if that's the particular thing he thought didn't make much sense (light waves changing as they go upwards), then his A, B, and C don't make much sense either, since they all talk about redshifted light waves. That's why I added E to the list: to emphasize that it's not the light that changes, it's the 4-velocity of the detector as compared to that of the emitter.