# Angular momentum of an atom within a rigid body in motion

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• james fairclear
In summary: The hydrogen atom has two nuclear spins, so the angular momentum can be written as:$$\vec{L}=\frac{1}{2}\left( \frac{1}{2m_{\text{p}}}\right)^2+\frac{1}{2}\left( \frac{1}{2m_{\text{e}}}\right)^2.$$The angular momentum of an electron in an atom is usually described relative to the nucleus. If you consider a system where the nucleus is not at rest, then the state of the electron must include an element relating to the state of motion of the nucleus.In a classical system, the angular momentum of an electron is a well-defined quantity
james fairclear said:
To my knowledge there is no evidence to support your claim that "all clocks tick physically at the same rate"?
That's a deficiency in your knowldege!

Can you point me to a link that discusses evidence for this claim?

james fairclear said:
Regardless of the different paths taken how is it possible for 3 previously synchronised clocks to indicate different times without their respective tick rates having changed at some point?
The geometry of spacetime.

Special Relativity is theory of space and time; not a theory about the mechanical workings of clocks.

vanhees71
james fairclear said:
Can you point me to a link that discusses evidence for this claim?
There is a voluminous literature on experimental tests of Special Relativity. Please familiarize yourself with it. The claim in question is called the "clock hypothesis" and is an integral part of Special Relativity.

vanhees71 and PeroK
james fairclear said:
Regardless of the different paths taken how is it possible for 3 previously synchronised clocks to indicate different times without their respective tick rates having changed at some point?
Consider an analogy: you and I both drive from New York to Washington, DC. You take the direct route, straight down the East Coast; I take a roundabout route by way of Pittsburgh, PA. The odometer on my car registers more distance traveled when we meet up again than yours does. Does that mean the "distance rate" of my odometer was different from yours? No; both of our odometers registered one mile per mile. The route I took just had more miles in it than yours: our paths had different lengths.

Similarly, in a "twin paradox" scenario (or a "triplet" scenario such as the H-K experiment), everyone's clock ticks at the same rate: one second per second. But the different paths through spacetime that each observer takes have different numbers of seconds in them: they have different spacetime lengths. That's why the different observers have aged differently when they meet up again: "elapsed age" is a measure of (timelike) distance traveled through spacetime, just as elapsed odometer mileage on a car is a measure of distance traveled through space.

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vanhees71
james fairclear said:
Can you point me to a link that discusses evidence for this claim?
It's the operational definition of a clock. It must measure time accurately. That's what a clock does.

It's a consequence of the principle of relativity, although actually it's not often highlighted. If a clock was physically altered by motion, then we could tell how fast a clock was "really" moving by examining it physically.

The principle of relativity demands that a clock is unaffected by motion.

Let me look for a good reference when I get the chance.

vanhees71
Of course all physics starts with choosing a spacetime model, but of course that spacetime model is only as good as it describes our experience with real-world measurements. Indeed, the great step by Einstein in 1905 was to recognize that the problem with the lack of Galilei invariance of the Maxwell equations is solved when using a different spactime model, i.e., instead of Galilei-Newton spacetime to use the special-relativistic Einstein-Minkowski spacetime model.

Now of course to use this spacetime model one must use it to define the measurement devices to measure times and distances. According to the best of our models concerning matter, relativistic quantum field theory, particularly QED, which describes the electromagnetic field and the interactions of electrically charged particles, the most reliable definitions of a time unit is to use some electromagnetic transition of a specific atomic system, and for practical purposes, i.e., the most reliable practical way to define a unit of time, is to use Cs-133 atoms and a hyperfine transition to define the second as one of the base units of the international system of units, the SI. There are even more precise realizations of the second under investigation, i.e., using transitions in the optical range or even using a nuclear electromagnetic transition to establish a nuclear clock which is even less sensitive to external perturbations than atomic clocks.

Having now a very precise definition of the unit of time, the Einstein-Minkowski spacetime model tells us that you can use the universal limiting speed, which to the best of our knowledge is identical with the speed of light in vacuum, to define a unit of length, and indeed that's how it's done in the SI, the meter is defined via the second by defining a specific value for the speed of light in vacuo, ##c##.

Having such definitions of time and length units, one can make consistency tests. There are very accurate tests of the "clock hypothesis", i.e., if you have something defining a specific time, measurements of this time should follow the predictions of the underlying spacetime model. One example is to use instable particles, which define (in a statistical way) a time by their lifetime. The lifetime is defined as the expectation value of their survival time when kept at rest wrt. an inertial reference frame. Now, the clock hypothesis says that the only changes of time, when the particle is moving wrt. the inertial frame (the "lab frame") should be do to the kinematical effect of time dilation, and that should hold even when the particles are accelerated. This indeed has been tested to be true in many experiments, e.g., in storage rings where radioactive nuclei are moving in a circle or also with elementary particles like muons (in fact with atmospheric muons this was one of the first tests of this kind).

Other tests use the relativistic Doppler effect for light (particularly the transverse Doppler effect).

https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation

Famously, then also the special-relativistic spacetime model has to be further refined as soon as the gravitational interaction becomes relevant. Then one has to use the general-relativistic spacetime model, and also the predictions of this spacetime model concerning time measurements, particularly the gravitational time dilation, Shapiro delay, etc. have also been tested at high accuracy, particularly also for strong gravitational interactions as with pulsar timing or via gravitational lensing. At very high precision these predictions of general relativity have been confirmed by all observations. The standard review article for this is (open access)

https://doi.org/10.12942/lrr-2014-4

PeroK said:
The geometry of spacetime.

Special Relativity is theory of space and time; not a theory about the mechanical workings of clocks.
As you say SR is a theory of space and time. The definition of time is "that which is measured by clocks" so the workings of clocks are inextricably bound up with time.

Regardless of whether or not time is a separate entity something must be physically affecting the tick rate of clocks in motion as otherwise their indicated times would not vary.

weirdoguy
PeroK said:
The principle of relativity demands that a clock is unaffected by motion
If so then the results of the Hafele Keating experiment disprove the principle.

Motore
james fairclear said:
If so then the results of the Hafele Keating experiment disprove the principle.
I doubt that Hafele and Keating would have agreed with that!

Doc Al and berkeman
james fairclear said:
Regardless of whether or not time is a separate entity something must be physically affecting the tick rate of clocks in motion as otherwise their indicated times would not vary.
That's simply not the case. Motion is relative, not physical. Absolute motion cannot be detected. So, there is nothing physical about relative motion for a clock to be affected by. A good reference for this is Einstein's original 1905 paper:

It is known that Maxwell’s electrodynamics—as usually understood at the
present time—when applied to moving bodies, leads to asymmetries which do
not appear to be inherent in the phenomena ...

Examples of this sort, together with the unsuccessful attempts to discover
any motion of the Earth relatively to the “light medium,” suggest that the
phenomena of electrodynamics as well as of mechanics possesses no properties
corresponding to the idea of absolute rest
.

They suggest rather that the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport
of which will hereafter be called the “Principle of Relativity”) to the status of a postulate.

In summary, you are missing the first postulate of relativity and thereby misinterpreting all of the theory of relativity.

vanhees71
PeroK said:
That's simply not the case. Motion is relative, not physical. Absolute motion cannot be detected. So, there is nothing physical about relative motion for a clock to be affected by. A good reference for this is Einstein's original 1905 paper:

It is known that Maxwell’s electrodynamics—as usually understood at the
present time—when applied to moving bodies, leads to asymmetries which do
not appear to be inherent in the phenomena ...

Examples of this sort, together with the unsuccessful attempts to discover
any motion of the Earth relatively to the “light medium,” suggest that the
phenomena of electrodynamics as well as of mechanics possesses no properties
corresponding to the idea of absolute rest
.

They suggest rather that the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport
of which will hereafter be called the “Principle of Relativity”) to the status of a postulate.

In summary, you are missing the first postulate of relativity and thereby misinterpreting all of the theory of relativity.

There is nothing stated or implied in the first postulate that motion is not something physical.

What exactly do you mean by your statement "motion is not physical"?

My original question is "Considering an atom within a rigid body, does the angular momentum of an electron within the atom vary when the body is put in motion?"

Some of the responses provided on this forum suggest that the answer is yes which implies that motion will have a physical effect. If there is no physical effect on a body when put into motion then there should be no need to take into account relativistic effects.

james fairclear said:
There is nothing stated or implied in the first postulate that motion is not something physical.

What exactly do you mean by your statement "motion is not physical"?
That a state of motion cannot be experimentally distinguished from a state of rest. For example, you might claim to be at rest sitting at your computer. But, the Earth is orbitting the Sun at about ##30,000 m/s##. You can, of course, measure that relative motion. The question is: what is physically different about you to the tune of ##30,000m/s##?

Then, the Milky Way is moving towards the Andromeda galaxy at ##300,000 m/s##. How does that physically affect you?

What all physics since Galileo, through Newton and Einstein answers is that these motions are not physically measurable. This is what Einstein was referring to by the unsuccessful attempts to discover
any motion of the Earth relatively to the “light medium,”
.

In other words, there is no experiment that can tell you the Earth's absolute physical state of motion. Only motion relative to other objects. This is the principle of relativity and the first postulate of special relativity.

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vanhees71
You have just restated the principles of relative motion.

As to whether there any physical changes to an object in motion is a separate issue. Observationally there are physical changes to synchronised clocks when put in motion as they cease to be synchronised with the relatively stationary clock.

A clock ticking at its highest possible tick rate could be considered to be at rest relative to any other clock ticking at a slower rate.

james fairclear said:
As to whether there any physical changes to an object in motion is a separate issue. Observationally there are physical changes to synchronised clocks when put in motion as they cease to be synchronised with the relatively stationary clock.
The "stationary" clock is in motion relative to the "moving" clock. It, likewise, ought to be physically altered by the relative motion. Neither clock can claim to be "really" at rest or "really" moving. That is what Einstein was saying above.
james fairclear said:
A clock ticking at its highest possible tick rate could be considered to be at rest relative to any other clock ticking at a slower rate.
All clocks tick at their highest tick rate in a reference frame where they are at rest; and all other clocks are ticking relatively slowly as measured by that clock. Every clock can claim to be ticking normally. As indeed it is.

vanhees71
james fairclear said:
If so then the results of the Hafele Keating experiment disprove the principle.
The discrepancy between the clock on the ground and the clock on the aircraft is not explained by the motion of the aircraft. Consider that the ground clock was moving with the Earth's rotation, with the orbit of the Earth around the sun, with the sun's orbit around the galactic core, and the galaxy's drift through intergalactic space... What makes the trajectory of the Earth clock so special?

james fairclear said:
something must be physically affecting the tick rate of clocks in motion as otherwise their indicated times would not vary.
This argument is wrong. You have already been told why it is wrong in this thread. Consider the odometer analogy in post #40. Do the different odometer readings for drivers who take different routes between two points mean that "something must be physically affecting the tick rate" of the odometers?

james fairclear said:
If so then the results of the Hafele Keating experiment disprove the principle.
No, they don't. You have already been told why in this thread. See above.

james fairclear said:
There is nothing stated or implied in the first postulate that motion is not something physical.
"Not physical" in this connection means "not invariant" (because "motion" is frame-dependent) or "not directly observable" (because there is no experiment you can do that will tell you that you are "moving" in any absolute sense). So your statement is wrong; the first postulate does say that motion is "not physical" in this sense.

james fairclear said:
My original question is "Considering an atom within a rigid body, does the angular momentum of an electron within the atom vary when the body is put in motion?"
And the question as you originally phrased it is not well-defined, which is why it does not have a single well-defined answer.

james fairclear said:
Some of the responses provided on this forum suggest that the answer is yes
Because if "angular momentum" includes "orbital angular momentum around some chosen external point", then an electron in an atom at rest relative to that point does have different "angular momentum" than an electron in an atom that is moving relative to that point. But this definition of "angular momentum" is frame-dependent; it's not something you directly measure, it's something you calculate after you've chosen a particular point, which implies choosing a particular frame in which that point is at rest.

However, when physicists talk about the "angular momentum" of an electron in an atom, they normally don't mean the frame-dependent thing above. They mean the angular momentum (orbital plus spin) of the electron relative to the nucleus of the atom, which is not frame-dependent and which can be directly observed, and which is the same regardless of the atom's state of motion relative to some external point. As has already been explained in this thread.

james fairclear said:
which implies that motion will have a physical effect.
No, it implies that some definitions of "angular momentum" are frame-dependent. See above.

james fairclear said:
If there is no physical effect on a body when put into motion
"Put into motion" is ambiguous. If it just means choosing a different frame of reference without doing anything to the body itself, obviously this has no physical effect. But if it means exerting an actual force on the body, which will change its motion relative to other bodies, obviously this does have a physical effect. But if you use the same term to refer to both of these things, obviously you're just going to confuse yourself.

james fairclear said:
there should be no need to take into account relativistic effects.
This is fallacious reasoning, both because of the issue just stated above and because "relativistic effects" include things that are directly observable.

james fairclear said:
A clock ticking at its highest possible tick rate could be considered to be at rest relative to any other clock ticking at a slower rate.
Wrong, for two reasons.

First, as has already been explained (see my reference to the odometer analogy above), "tick rate" is the wrong way to think about this issue. The correct way to think about it is "distance through spacetime, i.e., elapsed time, along different paths".

Second, there is no such thing as "highest distance through spacetime" in any general sense. To give that any meaning, you first have to pick two particular events in spacetime--for example, "clocks all start out together in a run of the H-K experiment", and "clocks all come back together after completing a run of the H-K experiment". Once you have picked two particular events, then yes, there will be some particular path through spacetime (worldline) that has the longest possible length (elapsed time) between those events. Or, more generally, you can always compare elapsed times along different paths through spacetime between those two chosen events, and those comparisons will be invariant--all observers will agree on them and they are independent of any choice of reference frame. (Note, btw, that in the actual H-K experiment, none of the paths taken in the experiment were the absolute longest possible paths between the two relevant events, but, as just noted, the comparisons between the elapsed times on the different clocks are still invariant.)

PeroK
@james fairclear you have gotten multiple responses saying the same things in different ways in this thread. There is no point in continuing to go around in circles. If all you can do is to continue to restate your (wrong) claims based on your (wrong) understanding of what relativity says, this thread can simply be closed as it will go nowhere.

weirdoguy
PeroK said:
The "stationary" clock is in motion relative to the "moving" clock. It, likewise, ought to be physically altered by the relative motion. Neither clock can claim to be "really" at rest or "really" moving. That is what Einstein was saying above.

All clocks tick at their highest tick rate in a reference frame where they are at rest; and all other clocks are ticking relatively slowly as measured by that clock. Every clock can claim to be ticking normally. As indeed it is.
"All clocks tick at their highest tick rate in a reference frame where they are at rest".

By your definition a clock is always in a reference frame where it is at rest! From this definition it follows that the tick rate of a clock will never vary which is contradicted by the evidence of the Hafele Keating experiment.

'Neither clock can claim to be "really" at rest or "really" moving'.

A clock can claim to be at rest if its tick rate is the highest possible tick rate. If 2 previously synchronised clocks are set in relative motion and then brought back together, the clock indicating an earlier time can be considered to be the one that was moving faster than the other clock (ignoring gravitational effects).

weirdoguy
james fairclear said:
A clock can claim to be at rest if its tick rate is the highest possible tick rate.
Can you give an example of an object that you believe is at rest? That definitely is not moving?

PeterDonis said:
Consider an analogy: you and I both drive from New York to Washington, DC. You take the direct route, straight down the East Coast; I take a roundabout route by way of Pittsburgh, PA. The odometer on my car registers more distance traveled when we meet up again than yours does. Does that mean the "distance rate" of my odometer was different from yours? No; both of our odometers registered one mile per mile. The route I took just had more miles in it than yours: our paths had different lengths.

Similarly, in a "twin paradox" scenario (or a "triplet" scenario such as the H-K experiment), everyone's clock ticks at the same rate: one second per second. But the different paths through spacetime that each observer takes have different numbers of seconds in them: they have different spacetime lengths. That's why the different observers have aged differently when they meet up again: "elapsed age" is a measure of (timelike) distance traveled through spacetime, just as elapsed odometer mileage on a car is a measure of distance traveled through space.
Your driving analogy is only applicable to time dilation if the odometers measure the same distance despite the fact that we have traveled different distances.

The implication then is that 1 odometer was measuring distance at a different rate to the other. We could then conclude for example that the the average speed on the roundabout route was much faster and that therefore speed affects the measuring rate of an odometer.

weirdoguy
Nugatory said:
The discrepancy between the clock on the ground and the clock on the aircraft is not explained by the motion of the aircraft. Consider that the ground clock was moving with the Earth's rotation, with the orbit of the Earth around the sun, with the sun's orbit around the galactic core, and the galaxy's drift through intergalactic space... What makes the trajectory of the Earth clock so special?
The discrepancy between the clock on the ground and the clock on the aircraft can only be explained by the relative motion between the clocks as that was the only material difference between the 2 clocks in the experiment.

weirdoguy
james fairclear said:
From this definition it follows that the tick rate of a clock will never vary which is contradicted by the evidence of the Hafele Keating experiment.
No, from that definition it follows that what you are calling the tick rate is a frame-dependent quantity.

And - although I'm not sure that saying this again is going to do you any good - the different elapsed time measurements in the HK experiment and the twin paradox are not explained by different tick rates. If the airborne experimenters in the HK experiment were to be in continuous communication with the ground station, they would find that the effect of their relative speed is that the ground clock was ticking more slowly than their own at all times. So if what you're calling the "tick rate" were what mattered, the ground-based clock would be the one that ticked less often.

(This is using the simultaneity convention of the momentarily comoving inertial frame, the only sensible way of comparing tick rates. The alternative, ticks per second of proper time, leads to the tautological conclusion that both clocks are ticking at a rate of one second per second).

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james fairclear said:
Your driving analogy is only applicable to time dilation if the odometers measure the same distance despite the fact that we have traveled different distances.
Wrong. Odometers measure distance traveled through space. Clocks measure distance traveled through spacetime. The analogy is valid.

What is invalid is your belief that any path through spacetime between two given events must have the same "length", which is just as nonsensical as the belief that any two paths in space between two locations must have the same length. The fact that there is a path of shortest length in space between two locations does not mean all paths through space between those two locations have that length. Similarly, the fact that there is a path of longest timelike length between two given events in spacetime does not mean all paths through spacetime between those two events have that timelike length.

@james fairclear There is no point in you continuing to post into a thread when you are refusing to listen.

You will no longer be able to post into this thread.

PeroK
james fairclear said:
The discrepancy between the clock on the ground and the clock on the aircraft can only be explained by the relative motion between the clocks as that was the only material difference between the 2 clocks in the experiment.
No, the relative motion was not the only material difference. The different clocks took different paths through spacetime; that is the material difference. The relative motion is one side effect of that, but not the only one.

Nugatory said:
You will no longer be able to post into this thread.
Which, since he was the OP of the thread, means the thread is now closed.

weirdoguy