Has a real clock corrections, with respect to an ideal clock?

[ORF]

Hello

In Special (and General) Relativity, has a physical (real) clock corrections, with respect to an ideal (mathematical) clock? For example, the physical clock has mass, so it will affect its own measure. This is only an example I thought, I don't know if there are more corrections (depending on how the clock works).

If this question is already answered in this forum, just tell me, and I will delete this thread.

Thank you for your time :)

Greetings
PS: My mother language is not English, so I'll be glad if you correct any mistake.

 

[ghwellsjr]

When discussing thought problems in relativity, we assume that the clocks measure time perfectly. For example, when explaining Special Relativity, we talk about two clocks in relative motion approaching the speed of light and accelerating instantly while totally ignoring the influence of gravity. These experiments cannot actually be done for a multitude of reasons. The accuracy of a physical clock is the least of our problems.

But experiments have been done with physical clocks in which their accuracy is good enough to demonstrate that the Lorentz Transformation is the correct understanding of the way physics works and that the Galilean Transformation is not.

 

[harrylin]

Welcome to Physicsforums :)

If I understand you correctly then the answer is yes - but it depends a bit on how you say it, and some details can make a difference.
Suppose that you have two identical clocks in identical conditions except for one thing: one clock has a massive lead bottom. That clock will tick very slightly slower according to general relativity.

Similarly, suppose that you have two identical clocks in identical conditions, except for the design speed with which the balance wheel turns: with a different gear ratio it is made to turn exactly 10 times faster according to classical mathematics. In reality it should then turn very slightly less than 10 times faster according to special relativity.

However, I think that in practice the effects of these examples are too small to measure with mechanical clocks.

 

[ORF]

Hello

Thank you both, for your quickly answer and for the warm welcome.

I suppose that for some problems, the accuracy is truly important; maybe in geoposition, satellite (and spatial waste) movements.

@harrylin: thank you for your graphical examples; I think they clear the question very well. I also think the order of magnitude of the corrections in mechanical clocks will be very small. However, it's said that some atomic clocks are expected to neither gain nor lose a second in more than several hundreds million years. I don't know how they compute the uncertainty, but I thought that it would be possible measure the own clock's effects with this accuracy*.

In the opposite point in the size scale, would the mass have a measurable contribution in the frequency of pulsars as relativistic correction?

Maybe exist more cases where we can find the same idea, but now I can not imagine more. If another example comes to my mind, I will post here. I'll be glad if anyone else thinks in another example.

Greetings.
*or trueness; I don't know the subtle difference between them.

 

[PeterDonis]

I thought that it would be possible measure the own clock's effects with this accuracy*.

No, it isn't. The clocks are very accurate, yes, but they are also very small in terms of their gravitational effect--far too small for that effect to be measurable even with the accuracy of the clocks.

the mass have a measurable contribution in the frequency of pulsars as relativistic correction?

No, because the frequency you are talking about is not frequency of light (or other radiation) emitted by the pulsars (which will indeed be redshifted significantly by the pulsars' gravity). It's the frequency of the pulsars' rotation and orbital motion as we observe them. The only way the mass of the pulsars would affect those observations would be if we were deep inside the pulsars's gravity well (for example, if we were riding on the surface of one of them).

 

[WannabeNewton]

In Special (and General) Relativity, has a physical (real) clock corrections, with respect to an ideal (mathematical) clock?

We commonly make the assumption that given an arbitrarily accelerating observer, at any given event on said observer's worldline the observer's rest frame is equivalent to an instantaneously comoving (local) inertial frame which takes for granted that accelerating clocks are equivalent at any given instant to a momentarily comoving inertial clock. In reality of course this is not true since an accelerating clock undergoes stresses of various kinds causing it to deviate operationally from an ideal (inertial) clock. If the characteristic scales of the system are such that these stresses are negligible then this assumption is valid to great accuracy but the point is there will be corrections due to acceleration/tidal stresses on the accelerating clock causing deviation from the momentarily comoving inertial clock.

In principle these deviations can affect standard SR formulas such as time dilation. See e.g. https://www.physicsforums.com/threads/radar-distance-rindler-observer.730009/#post-4619764 and https://www.physicsforums.com/threads/spring-with-2-masses-free-fall.731181/#post-4620023

 

[harrylin]

No, it isn't. The clocks are very accurate, yes, but they are also very small in terms of their gravitational effect--far too small for that effect to be measurable even with the accuracy of the clocks.

I fully agree. Moreover atomic clocks don't have a balance wheel, so that my example is not applicable.

No, because the frequency you are talking about is not frequency of light (or other radiation) emitted by the pulsars (which will indeed be redshifted significantly by the pulsars' gravity). It's the frequency of the pulsars' rotation and orbital motion as we observe them. The only way the mass of the pulsars would affect those observations would be if we were deep inside the pulsars's gravity well (for example, if we were riding on the surface of one of them).

I also concur with that answer, but for a different reason. I'm pretty sure that astronomers can't calculate how fast a particular pulsar is expected to rotate according to classical mechanics vs. relativity (or, surely, not with enough accuracy; compare https://en.wikipedia.org/wiki/Pulsar#Formation).

 

[PeterDonis]

I'm pretty sure that astronomers can't calculate how fast a particular pulsar is expected to rotate according to classical mechanics vs. relativity

I think it's even more basic than that: we can't observe directly the parameters that would determine the rotation rate (some of them are aspects of how the pulsar was formed, so they aren't even observable at all today). So when we observe a given rotation rate, we could construct both a classical (Newtonian) model consistent with that rate, and a relativistic model consistent with that rate; the two models would have somewhat different parameter values, but we have no way of checking the parameter values to see which model fits the data.

However, the OP was not asking how pulsars confirm a relativistic (as opposed to Newtonian) model; he was assuming that the relativistic model is correct (which is what virtually all physicists in the field do), and asking about its consequences. My answer was based on the same assumption.

 

[harrylin]

I think it's even more basic than that: we can't observe directly the parameters that would determine the rotation rate (some of them are aspects of how the pulsar was formed, so they aren't even observable at all today). So when we observe a given rotation rate, we could construct both a classical (Newtonian) model consistent with that rate, and a relativistic model consistent with that rate; the two models would have somewhat different parameter values, but we have no way of checking the parameter values to see which model fits the data.

However, the OP was not asking how pulsars confirm a relativistic (as opposed to Newtonian) model; he was assuming that the relativistic model is correct (which is what virtually all physicists in the field do), and asking about its consequences. My answer was based on the same assumption.

Indeed, my answer was based on that same assumption, that he wants to know the relativistic corrections. :)

 

[ghwellsjr]

Indeed, my answer was based on that same assumption, that he wants to know the relativistic corrections. :)

What is the difference between the relativistic corrections for a physical (real) clock and an ideal (mathematical) clock?

I got the impression that he is thinking that Time Dilation is the deviation of a physical clock from the "trueness" (as he mentioned in post #4) of an ideal clock and therefore needs a correction. A mass in the vicinity of a real clock (whether part of the clock or not) causes it to tick at a different rate than if the mass were not there, but this would be the same for an ideal clock because time is progressing differently in the vicinity of a mass.

 

[PeterDonis]

my answer was based on that same assumption, that he wants to know the relativistic corrections.

Relativistic corrections to "rate of time flow" without any consideration of the particular dynamics of a system (such as a pulsar). Not relativistic corrections to a classical model of the internal dynamics of pulsars. At least, that's how I'm interpreting the OP--he's welcome to clarify if I'm wrong.

 

[harrylin]

Relativistic corrections to "rate of time flow" without any consideration of the particular dynamics of a system (such as a pulsar). Not relativistic corrections to a classical model of the internal dynamics of pulsars. At least, that's how I'm interpreting the OP--he's welcome to clarify if I'm wrong.

I understood relativistic corrections to the frequency of clocks, starting from a calculation without those corrections - like the GR correction to the perihelion of Mercury. Indeed, it will be good if he clarifies what he is looking for exactly. :)

 

[harrylin]

What is the difference between the relativistic corrections for a physical (real) clock and an ideal (mathematical) clock?

I got the impression that he is thinking that Time Dilation is the deviation of a physical clock from the "trueness" (as he mentioned in post #4) of an ideal clock and therefore needs a correction. A mass in the vicinity of a real clock (whether part of the clock or not) causes it to tick at a different rate than if the mass were not there, but this would be the same for an ideal clock because time is progressing differently in the vicinity of a mass.

"trueness" was there used as synonym for "accuracy"; that's a different issue. also, I don't linger on "ideal" as that's personal as well as topic related. What he asked about concerns "a measurable contribution in the frequency of pulsars as relativistic correction", in the sense of how (in the first post) "it will affect its own measure". For me that's a clear question; let's see if I understood him correctly. :)

 

[ORF]

Hello

Sorry for the delay, I broke my little toe yesterday and I spent the day in the hospital.

*-Thank you for answering; I didn't expect so many people here :)

*-Yes, I assume the GR is tested enough.

*-I used the word [relativistic] "corrections", but maybe it's an error of translation; other expressions, as "mass factors" or "mass contributions" (with respect to SR, or even to classical mechanics), might sound better.

if we were deep inside the pulsars's gravity well

You're right, my fault.

Relativistic corrections to "rate of time flow" without any consideration of the particular dynamics of a system. Not relativistic corrections to a classical model of the internal dynamics of pulsars.

Yes; I thought the question should be about the simplest case.

@ghwellsjr, @PeterDonis: maybe the first question wasn't enough clear (sorry for that). I will try to remake it:

-has a real clock a "rate of time flow" different from a mathematical/ideal clock?*
-If the answer is yes, what is the order of magnitude of this difference? (and, what can be the origin of the difference? I supposed the GR)

we can't observe directly the parameters that would determine the rotation rate [...] we have no way of checking the parameter values to see which model fits the data.

I read that some pulsars are more precise than atomic clocks, and because of that, I used the idea in the example. I'm not an astronomer (and I've never met any), so if the next sentence sounds stupid, please just overlook it: I've heard the pulsars lose their kinetic energy by radiation, so, is it possible to compare the rate of radiation and the rate of frequency, and use them as a way to check the theories? (classical mechanics, SR and GR)

@harrylin: mmm... I had not thought about the trueness (accuracy) of the astronomers' measurements, good remark :)

On the other hand, I think GR affects atomic clocks too, although they works at quantum levels (Wu et al made an experiment with quantum interference, testing successfully the GR). My doubt is the order of magnitude of the difference between an ideal clock (the clock which measure the "time" in the mathematical expressions of SR and GR), and the atomic clock of 2 lb which is in a lab :)

let's see if I understood him correctly.

Yes, yes, thank you very much :) :)

Sorry for such a long statement; I tried to sum it up. And again, please, forgive my grammar errors, and I'll be glad if you correct any mistake.

Greetings.
*Maybe I should make a clarification: I refer to an ideal clock as a clock which measure the parameter "time" that appears in the equations; and to a real clock as a clock which is used in the real life.
PS: I thought the words accuracy, trueness and precision have the meaning which is explained here:
http://en.wikipedia.org/wiki/Accuracy_and_precision#Terminology_of_ISO_5725

 

[ghwellsjr]

@ghwellsjr, @PeterDonis: maybe the first question wasn't enough clear (sorry for that). I will try to remake it:

-has a real clock a "rate of time flow" different from a mathematical/ideal clock?*
...
*Maybe I should make a clarification: I refer to an ideal clock as a clock which measure the parameter "time" that appears in the equations; and to a real clock as a clock which is used in the real life.

I can't answer your question any better than I already have in post #2:

When discussing thought problems in relativity, we assume that the clocks measure time perfectly. For example, when explaining Special Relativity, we talk about two clocks in relative motion approaching the speed of light and accelerating instantly while totally ignoring the influence of gravity. These experiments cannot actually be done for a multitude of reasons. The accuracy of a physical clock is the least of our problems.

But experiments have been done with physical clocks in which their accuracy is good enough to demonstrate that the Lorentz Transformation is the correct understanding of the way physics works and that the Galilean Transformation is not.

If that answer doesn't suit you, then I would like to ask you what your concern is and how will it help you understand relativity any better?

 

[harrylin]

[..] My doubt is the order of magnitude of the difference between an ideal clock (the clock which measure the "time" in the mathematical expressions of SR and GR), and the atomic clock of 2 lb which is in a lab :)
[..]
*Maybe I should make a clarification: I refer to an ideal clock as a clock which measure the parameter "time" that appears in the equations; and to a real clock as a clock which is used in the real life. [..]

Thanks for your clarification of "ideal"! The mass of a clock is included in those mathematical expressions, and to my knowledge no deviation from theory has been observed. However, real atomic clock frequency can also depend on temperature for example. You may like the discussion by clock experts here: http://www.nist.gov/pml/div688/2013_1_17_newera_atomicclocks.cfm

 

[ghwellsjr]

Thanks for your clarification of "ideal"! The mass of a clock is included in those mathematical expressions, and to my knowledge no deviation from theory has been observed. However, real atomic clock frequency can also depend on temperature for example. You may like the discussion by clock experts here: http://www.nist.gov/pml/div688/2013_1_17_newera_atomicclocks.cfm

Let's not confuse the issue. At the atomic level, temperature is motion. If the atoms are moving with respect to the lab frame, then they are experiencing Time Dilation and so they are properly displaying a different time than that the lab rest frame. That is the reason it is important to get their temperature as low as possible which simply means that you want to reduce the relative motions between all the atoms in the cloud.

 

[ORF]

Hello

@ghwellsjr: I insisted because it seemed to me that your answer wasn't complete enough. The question is just out of curiosity. :)

@harrylin: thank you for your interest. Yes, the NIST discussion is a good starting point.

Best regards.
PS: I thought that (in statistical mechanics) temperature is not just "atomic motion" :)

 

[harrylin]

Let's not confuse the issue. At the atomic level, temperature is motion. If the atoms are moving with respect to the lab frame, then they are experiencing Time Dilation and so they are properly displaying a different time than that the lab rest frame. That is the reason it is important to get their temperature as low as possible which simply means that you want to reduce the relative motions between all the atoms in the cloud.

The NIST article that I linked to mentions a number of influences on time keeping, including temperature and magnetic fields. Anyway, you suggest that the main effect of temperature on real cesium clocks is due to time dilation - good thinking!
However, NIST appears to disagree with you: "cooling dramatically lowers the background radiation and thus reduces some of the very small measurement errors" - http://www.nist.gov/pml/div688/nist-f2-atomic-clock-040314.cfm

 

[ghwellsjr]

Hello

@ghwellsjr: I insisted because it seemed to me that your answer wasn't complete enough. The question is just out of curiosity. :)

Is my last answer complete enough?

@harrylin: thank you for your interest. Yes, the NIST discussion is a good starting point.

Best regards.
PS: I thought that (in statistical mechanics) temperature is not just "atomic motion" :)

We're talking about a cloud of many thousand cesium atoms in a "gaseous" state in an otherwise evacuated chamber except that unlike normal gases which would randomly fill the chamber and bounce off the walls and each other due to thermal excitations (which is what their random motions are), these atoms are constrained by laser beams shining on them from different directions to hopefully bring them to a stop within the overlap of the beams which is what forms the cloud.

 

[ghwellsjr]

The NIST article that I linked to mentions a number of influences on time keeping, including temperature and magnetic fields. Anyway, you suggest that the main effect of temperature on real cesium clocks is due to time dilation - good thinking!
However, NIST appears to disagree with you: "cooling dramatically lowers the background radiation and thus reduces some of the very small measurement errors" - http://www.nist.gov/pml/div688/nist-f2-atomic-clock-040314.cfm

There's no disagreement. You can't get the atoms in the cloud to stop moving when they are bombarded by random thermal radiation from the walls of the chamber. They don't have to touch the walls to be excited by the radiation emitted by the walls even if it is at a very low temperature.

 

[harrylin]

There's no disagreement. You can't get the atoms in the cloud to stop moving when they are bombarded by random thermal radiation from the walls of the chamber. They don't have to touch the walls to be excited by the radiation emitted by the walls even if it is at a very low temperature.

Maybe yes; in any case, you identified one more possible correction of a real clock due to a relativistic effect "on itself". :)

I imagined that peaks in the microwave background can directly affect the observed apparent resonance frequency. That as well as relativistic frequency reduction could be part of the suggested causes for corrections due to the microwave background radiation.

Out of curiosity I dug a little deeper in the literature about cesium clocks to see what is said about this. To my surprise, several papers mention inaccuracy due to first order Doppler shift! Thus, depending on the configuration, even ordinary Doppler can significantly affect the measured frequency which serves as time reference in a real atomic clock.

And of course NIST mentions still other causes for corrections in the link I gave. Moreover, a real clock has issues with instabilities ("noise") so that regular calibrations remain required.

 

[ghwellsjr]

Maybe yes; in any case, you identified one more possible correction of a real clock due to a relativistic effect "on itself". :)

The specific transition frequency of an individual cesium atom is exact, trying to exploit that frequency in a real atomic clock is a challenge.

I imagined that peaks in the microwave background can directly affect the observed apparent resonance frequency.

There's no resonant frequency in a cesium atom or in an atomic cesium clock. It's based on the transition frequency of the atom. When you have a cloud of cesium atoms, it's the combined effect of all of them being excited at different times and if they are not all at rest with respect to each other, there can be a spread in the observed transition frequency.

That as well as relativistic frequency reduction could be part of the suggested causes for corrections due to the microwave background radiation.

What is "relativistic frequency reduction"? If you are thinking that it is only a one-way shift caused by Time Dilation and therefore can be calculated out, keep in mind that there will be a Doppler shift in both directions depending on the relative speeds of each atom and the detector. The thermal noise increases the spread of the observed frequency. You have to reduce the temperature of the whole apparatus to eliminate the problem. That's the whole point of the second-generation of atomic cesium clocks.

Out of curiosity I dug a little deeper in the literature about cesium clocks to see what is said about this. To my surprise, several papers mention inaccuracy due to first order Doppler shift! Thus, depending on the configuration, even ordinary Doppler can significantly affect the measured frequency which serves as time reference in a real atomic clock.

After a stable cloud of cesium atoms is formed in an atomic clock on earth, the cloud is launched in an upward direction in an evacuated tube to allow the cloud to be in free fall while it is going up and back down the tube. This is why it is called a fountain clock. It's during this time that the transition frequency is being measured and of course, in the frame of the lab, it is in motion so there will be a first order Doppler shift but this can be calculated out since the speed profile of the cloud is known.

And of course NIST mentions still other causes for corrections in the link I gave. Moreover, a real clock has issues with instabilities ("noise") so that regular calibrations remain required.

Is the real clock you are referring to here the atomic cesium clock or the other clocks that are calibrated against the atomic cesium clock? Keep in mind, that for purposes of establishing UTC, the second that we get from the atomic cesium clock is not the same duration as the second of UTC.

 

[PeterDonis]

Keep in mind, that for purposes of establishing UTC, the second that we get from the atomic cesium clock is not the same duration as the second of UTC.

It's supposed to be, isn't it? The cesium transition defines the SI second, and the SI second is the UTC second. Are you just referring to the measurement inaccuracies, or something else?

 

[ghwellsjr]

The second is different at different elevations. An atomic clock at Greenwich, England near sea level ticks at a different rate than one at Boulder, Colorado at about a mile elevation and neither one ticks at the UTC rate.

 

[PeterDonis]

The second is different at different elevations.

Ah, ok. Yes, the UTC second, as I understand it, is defined based on the cesium transition on the geoid (mean sea level, more or less). The UTC time signals that get sent out by, for example, ntp servers and GPS satellites are corrected for that.

 

[harrylin]

There's no resonant frequency in a cesium atom or in an atomic cesium clock. It's based on the transition frequency of the atom.

I cited NIST who call that the "natural resonance frequency of the cesium atom".
- http://www.nist.gov/pml/div688/how-nist-f2-works.cfm
I don't have the opinion that that is wrong, but it's not relevant for this topic.

[..] Is the real clock you are referring to here the atomic cesium clock [..].

Yes, I merely paraphrased NIST's referral to cesium clocks in http://www.nist.gov/pml/div688/2013_1_17_newera_atomicclocks.cfm :

"NIST-F1 and NIST-F2 are microwave clocks, based on a particular vibration in cesium atoms [..]
the aluminum ion logic clock’s world-record timekeeping performance was due in part to this clock’s insensitivity to changes in magnetic fields, electrical fields, and temperature. This insensitivity is highly desirable for the best timekeeping results. But it also means the aluminum ion clock is not a good candidate for measuring magnetic and electrical fields or temperature, whereas other NIST atomic clocks have greater sensitivity to those quantities.
[..]
NIST-F1 must be averaged for about 400,000 seconds (about five days) to achieve its best performance of about 1 second in 100 million years. In contrast, the ytterbium and strontium lattice clocks reach that level of performance in a few seconds of averaging, and after a few hours of averaging are about 100 times more stable than NIST-F1."

Concerning stability, I also recalled Hafele&Keating's experience with real clocks (cesium clocks).
From their 1972 paper:

"Because the stability of a single
cesium beam clock would probably be
inadequate to permit an absolutely unambiguous
detection of the expected
relativistic effects, we used four flying
clocks [..]
In retrospect, it is
clear that the use of only one or
two flying clocks would have substantially
decreased the feasibility of our
experiments."

Indeed, if you have their paper (hint: you can find it on Internet), just take a look at their fig.1 :)

 

[ghwellsjr]

The second is different at different elevations.

Ah, ok. Yes, the UTC second, as I understand it, is defined based on the cesium transition on the geoid (mean sea level, more or less).

That's true now but it wasn't the way the UTC second was originally defined. Originally, for the SI second which is the basis for the UTC second, they chose the number of cycles of the cesium transition to match the same duration as the standard second previously in effect based on astronomical considerations without regard to what elevation that comparison was made at. I guess they didn't realize it would matter. After a few years of comparing atomic clocks from around the world at different elevations, they realized that there was a systematic deviation due to elevation so they adopted the definition that you stated. If they had realized it from the beginning, the SI second would probably contain a different number of cycles than what it does now.

 

[PeterDonis]

I guess they didn't realize it would matter.

I'd be surprised if they didn't realize it at all. The SI second was defined in terms of the cesium transition in 1967; certainly gravitational time dilation was known well before then, and the Pound-Rebka experiment which measured it (technically it measured gravitational redshift but that is sufficient to demonstrate gravitational time dilation) was done in 1959, so there was experimental support for GR's prediction.

I suspect they just didn't stop to consider how quickly atomic clocks might become accurate enough to be affected by elevation differences.

 

[russ_watters]

I was going to say I'm confused by that, but really all they did is add an additional constraint that essentially declares the UTC second to actually be a Unviersal time, which would seem to me to violate the spirit of Relativity. I wonder if that would annoy Einstein?

It kind of annoys me: it simplifies life for GPS satellites, but if we ever learn to travel fast enough for time dilation to matter for chemical or biological processes it will mess up our time measurement for such processes.