Time dilation in the field interpretation of GR

[PeterDonis]

that is what I also brought up in my my next sentence.

Here you used the interpretation that the 30°C clock is ticking 1% faster than the 20°C clock so that indeed 1010 s on the 30 C clock is simultaneous with t = 1000 s on the 20 C clock.

I'm confused. What exactly is the argument you are trying to make? You seem to be pulling numbers out of the air. What "three interpretations" are supposed to be in play, and why are the numbers different for different ones?

 

[Sonderval]

@Peter
Thanks for clarifying things and going through the numbers.

@harrylin
As Peter, I feel I'm a bit at a loss about what you are trying to show/do. I am also totally confused by the use of "global clocks" (like artifically slowing down GPS clocks), to me this makes things unnecessarily obscure. I'd rather have a sequence of observers with clocks that were synchronised at teh same point in space+time and are then transported to their destination.
Are you trying to use these clocks to phrase everything in the language of an underlying (unobservable) Minkwoski spacetime where clocks and rods are affected by the field? If so, I still don't see how the argument works out.

 

[harrylin]

@harrylin
As Peter, I feel I'm a bit at a loss about what you are trying to show/do. [/quote]

I proved that your intuition:
Sonderval said: ↑
"but that instead the frequency of radio waves in flight decreases with temperature"
I think my experiment above with the blips shows that this interpretation is problematic - why should the distance between totally unrelated signals be affected in this way?"
was totally correct.

Extending your thought experiment showed that such an interpretation is not just problematic but totally wrong for the chosen example. But of course, as I mentioned earlier, we discussed a truly different interpretation and not just sloppy language for the same interpretation! We could use sloppy language to describe the same interpretation in a poor way, and my request to peterdonis to accompany his phrasings with matching calculations was in part to verify that he really was talking about a different physical interpretation.
If I take pictures of you with a zoom objective, one at 1x and one at 2x, I will interpret the increase of your size on the picture as a change in measurement equipment, and not as a change of your body size. Saying that the difference of your size on the pictures can be interpreted as you having changed in size under influence of my action on the zoom lens is nonsensical and misleading.
Similarly, when I measured that my car was consuming less and less petrol, I quickly found that in fact this was not true: instead it was my measurement that changed and not the fuel consumption of my car (my interpretation that my car was consuming less fuel was disproved by the fact that suddenly I ran out of fuel, with the fuel gauge still at half full!)

If it is possible to be on the same page about a simple classical example, then we can discuss the slightly more difficult topic of relativity. However, if we even cannot be on the same page about the meanings of words for such a simple example then it is useless to discuss more tricky topics.

I am also totally confused by the use of "global clocks" (like artifically slowing down GPS clocks), to me this makes things unnecessarily obscure. I'd rather have a sequence of observers with clocks that were synchronised at teh same point in space+time and are then transported to their destination.

I will gladly do so, as soon as it is clear that we are on the same page with these straightforward classical examples.

Are you trying to use these clocks to phrase everything in the language of an underlying (unobservable) Minkwoski spacetime where clocks and rods are affected by the field? If so, I still don't see how the argument works out.

I phrased everything in accordance with the GR weak field description of Einstein in the passage that I referred to; however you may have problems with that if you have problems with these classical examples.

Meanwhile I found the old discussion on sci.physics.research about what happens to mass in a GR field, here:
http://www.lns.cornell.edu/spr/2007-02/msg0076402.html

And in particular the following reply may be helpful:
http://www.lns.cornell.edu/spr/2007-02/msg0076437.html

Enjoy

 

[Sonderval]

@harrylin
Perhaps I'm a bit dense today, but I still don't really see what exactly you want to agree upon in the classical example - that if we use clocks or other measures that are affected by some field and don't account for that, it will seem as if some physical quantity that is actually constant changes? If so, yes, I agree.
Or do you want to go one step further (as in Feynmans exmaple of the hot plate) that if all our measures are affected in the same way, there is no way to decide whether it is our measure that changes from place to place affected by the field or whether it is the physical quantity that changes (in the opposite way)? Yes, I agree also (and this is exactly what the example in the paper I quoted in my first post is about, IIUC).

Thanks for the links, but I don't understand the second one:
"c varies as Phi^2"
In what reference frame is this statement valid? In a local rest frame at the surface of the earth?

 

[harrylin]

@harrylin
Perhaps I'm a bit dense today, but I still don't really see what exactly you want to agree upon in the classical example - that if we use clocks or other measures that are affected by some field and don't account for that, it will seem as if some physical quantity that is actually constant changes? If so, yes, I agree.

Yes indeed - it's good to see that we are on the same page on the meaning of the terms "constant", "changing" and "physical quantity" .

Or do you want to go one step further (as in Feynmans exmaple of the hot plate) that if all our measures are affected in the same way, there is no way to decide whether it is our measure that changes from place to place affected by the field or whether it is the physical quantity that changes (in the opposite way)? Yes, I agree also (and this is exactly what the example in the paper I quoted in my first post is about, IIUC).

I don't know that example by Feynman, but I discussed (and will clarify more) how in certain cases we can decide which interpretation is the only known correct one, by disproving all the others. For example, once in another discussion group, someone proposed that accelerating electrons and protons have a limit speed of c because the electric / magnetic field "pushes" them less and less the faster they go; and that such an alternative interpretation can produce the same equations. However for experiments with kinetic energy such an interpretation messes up so that it must be discarded.

Thanks for the links, but I don't understand the second one:
"c varies as Phi^2"
In what reference frame is this statement valid? In a local rest frame at the surface of the earth?

I guess (I'm busy now and should read it again) that it's valid for a non-local frame in which the Earth is in rest. See "again" §23 of https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity (sorry only now I notice that I had not linked to the chapter of interest). A local measurement of a vertical rod must give the same result in the field as outside at infinity at zero speed (but with acceleration g).

 

[Sonderval]

"I discussed (and will clarify more) how in certain cases we can decide which interpretation is the only known correct one, by disproving all the others."
Unfortunately I still don't get your discussion...

 

[harrylin]

[..] What "three interpretations" are supposed to be in play, and why are the numbers different for different ones?

The ones I put forward in my post #25
I'll put them here once more:
1. the one of the proposed theory: clock frequency increases with 0.1% / °C.
2. alternative interpretation of the observed frequency difference: clocks are unaffected by temperature, instead it is the radio waves in transit that are affected by temperature such that the waves arrive with a 1% lower frequency on the hot side.
3. Both of the foregoing: clock frequency increases with temperature and temperature affects radio waves in space, modifying the frequency.

"I discussed (and will clarify more) how in certain cases we can decide which interpretation is the only known correct one, by disproving all the others."
Unfortunately I still don't get your discussion...

In my classical example only interpretation 1. remains as possibly valid explanation.

 

[Sonderval]

"In my classical example only interpretation 1. remains as possibly valid explanation."
Yes, unless we find a way to make explanation 2 valid by having everything in transit being affected in exactly the same way (don't see how this would work for the blips, though).
So let's agree on this point and see where it leads us.
BTW, here is the link to the Feynman lecture:
http://www.feynmanlectures.caltech.edu/II_42.html

 

[PeterDonis]

In my classical example only interpretation 1. remains as possibly valid explanation.

These aren't three different "interpretations" of the same theory; they are three different theories, because they make three different predictions for the values of direct observables. (They all make the same prediction for one observable, the 1% difference in observed frequency at the 30 C clock; but they make different predictions for other observables, which is how you rule out #2 and #3 as not possibly valid.) As I have said several times now, different "interpretations" don't change the math or the physical predictions; they only change what ordinary language words we use to describe the same math and predictions.

 

[harrylin]

These aren't three different "interpretations" of the same theory; they are three different theories, because they make three different predictions for the values of direct observables. (They all make the same prediction for one observable, the 1% difference in observed frequency at the 30 C clock; but they make different predictions for other observables, which is how you rule out #2 and #3 as not possibly valid.) [..].

Quite so. More precisely, of the three suggested possible interpretations of that example theory, two were shown to be wrong as they lead to other predictions for observables that were not considered at first. I'm glad that we are now all on the same page.

I'll try to come back to this thread soon.

 

[PeterDonis]

of the three suggested possible interpretations of that example theory, two were shown to be wrong as they lead to other predictions for observables that were not considered at first.

It isn't the "possible interpretations" that lead to different predictions; it's the different theories that you arrive at by adding additional assumptions to the "possible interpretations". For example, in "possible interpretation" #2, you add the assumption that there is absolute time and space. If you had added different assumptions, you might possibly have arrived at a theory that made the same predictions as theory #1. Or, instead of adding any assumptions, you could have just specified a way of mapping the words in interpretation #2 to the same math that you used in theory #1, so that interpretation #2 is not a different theory, it's just a different way of describing theory #1 in ordinary language.

Your basic position in this thread has been that those alternate descriptions of the same theory in ordinary language don't make sense to you. But that doesn't invalidate the alternate descriptions, because they're not being proposed as alternate theories; they're just alternate descriptions of the same theory. If you don't want to use them, fine, don't. Different descriptions can be useful for different people and different purposes.

 

[harrylin]

[..] in "possible interpretation" #2, you add the assumption that there is absolute time and space.

The example theory was specified as belonging to classical physics in order to keep it simple and no assumptions needed to be added.

If you had added different assumptions, you might possibly have arrived at a theory that made the same predictions as theory #1. Or, instead of adding any assumptions, you could have just specified a way of mapping the words in interpretation #2 to the same math that you used in theory #1, so that interpretation #2 is not a different theory, it's just a different way of describing theory #1 in ordinary language. [..]

Restating a physical interpretation in what you call "ordinary language" is not a different interpretation - it's a fake difference. Anyway, I came to this thread (in post no.3) with the request to map words to calculations for clarification of meaning. And it now looks like we're going to get there after all.

 

[PeterDonis]

The example theory was specified as belonging to classical physics in order to keep it simple

Yes, and that amounts to an additional assumption (or rather an assumption that must be satisfied by all of the alternative theories you consider). If you assumed that all theories must be relativistic, that would make a difference.

Restating a physical interpretation in what you call "ordinary language" is not a different interpretation - it's a fake difference.

You don't "restate a physical interpretation in ordinary language". You restate the math of a physical theory in ordinary language; this is called an "interpretation". Different restatements in ordinary language of the same theory and the same math are different interpretations. If that's just a "fake difference", then why has this thread gone on for so long?

 

[PeterDonis]

I came to this thread (in post no.3) with the request to map words to calculations for clarification of meaning.

And I've done this, repeatedly, but you keep insisting on reading my posts as proposing different theories (different math), when all I am doing is giving different labels in ordinary language for the same theory (same math).

I'll try once more. First I'll give the math (the theory); then I'll give two different ordinary language descriptions of the math (the two interpretations).

The Math

We have an observer E located at rest on the surface of some planet. (It can't actually be Earth with your original numbers because a 1% redshift factor is way too large for Earth, but that's immaterial.) Observer E has 4-velocity $u_E$. He emits light signals with 4-momentum $p_E$ vertically upward. E measures these light signals to have a frequency $u_E \cdot p_E$ as they are emitted.

We have an observer S located in a spaceship that is hovering at rest vertically above observer E. Observer S has 4-velocity $u'_S$. He receives the light signals from observer E, which have 4-momentum $p'_E$ when they reach him. O measures these light signals to have a frequency $u'_S \cdot p'_E$ as they are received.

Observer S also emits light signals with 4-momentum $p'_S$ vertically downward. These light signals are received by observer E, and they have 4-momentum $p_S$ when they reach him. S measures these light signals to have a frequency $u'_S \cdot p'_S$ as they are emitted. E measures these light signals to have a frequency $u_E \cdot p_S$ as they are received.

By hypothesis, we have $u_E \cdot p_E = u'_S \cdot p'_S$. That is, the emitted frequencies are both the same. Then, as a prediction of GR (and as an experimental fact), we have $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_S > u'_S \cdot p'_S$.

Also, both light sources (at E and at S) are identical, as are both frequency measuring devices.

Interpretation #1: Energy is reduced by gravitational fields

We define the "energy" of a light source as the frequency of light emitted by that source as measured at spaceship S. Since $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_E = u'_S \cdot p'_S$, we have $u'_S \cdot p'_E < u'_S \cdot p'_S$. That is, the measured frequency of observer E's light at spaceship S is less than the measured frequency of spaceship S's light at spaceship S. Therefore, observer E's energy is reduced by being deeper in a gravitational field. We define the "change in the light" as a change in its 4-momentum, compared to parallel transport; but since the 4-momentum of both light signals is parallel transported along their respective worldlines, i.e., $p'_E$ is just $p_E$ parallel transported, and $p_S$ is just $p'_S$ parallel transported, there is no "change in the light" under this interpretation.

Interpretation #2: Gravitational fields affect the propagation of light

We define the "energy" of a light source as the measured frequency of light by the observer that emits the light. So, since $u_E \cdot p_E = u'_S \cdot p'_S$, the energy of observers E and S is identical. But since $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_S > u'_S \cdot p'_S$, the light signals are changed by propagating through the gravitational field; i.e., we define the "change in the light" as the difference between the measured frequency of the light by the observer that emits the light, and the measured frequency of the light by the observer that receives the light. We can also describe this "change in the light" as a "change in energy" of the light--it starts out from the source with a certain quantity of energy, and loses energy as it climbs, or gains energy as it falls. We are just describing the same math in different words.

Interpretation #2 above is what I was referring to all those many posts ago. I understand that you don't like interpretation #2 because you don't like the way it labels things; you prefer the way interpretation #1 labels things. But that doesn't change the fact that both interpretations are describing the same math and the same predictions, just with different ordinary language labelings.

 

[Sonderval]

@Peter
Thanks a lot for stating this so clearly.
So here is a follow-up question (which is in essence the question discussed in the reference I gave in my first post):
Consider a hydrogen atom with a ground state energy corresponding to a frequency ω. Can I use the same two interpretations to describe the behaviour of the atom at the surface of our planet and the spaceship?

 

[harrylin]

"In my classical example only interpretation 1. remains as possibly valid explanation."
Yes, unless we find a way to make explanation 2 valid by having everything in transit being affected in exactly the same way (don't see how this would work for the blips, though).
So let's agree on this point and see where it leads us. [..]

OK :-) As we are now on the same page, it's time to get back to your example. BTW I see that peter now proceeded to clarify his post #2 more precisely, in the way I requested, and I'll have a careful look at that next.

[..]
I still don't see how the number of cycles argument works. Let's say we have two observers at different heights; the lower one transmitting 100 light signals at a frequency of 1Hz. For the upper observer, the time elapsed will be longer than 100s, so she will see a lower frequency. The number of cycles is constant, but the frequency is not. [..]
Let's have the lower observer send out a continuous em wave with a frequency of 1Hz. At each crest (peak?) of the wave, he also sends out a very short light pulse (of, say 1microsecond duration), so that each wavecrest is accompanied and marked by a "blip".
Both crests and "blips" should arrive at the upper observer still completely in phase, but the time gap between the blips will have increased. [..]
I still don't see how the number of cycles argument works. Let's say we have two observers at different heights; the lower one transmitting 100 light signals at a frequency of 1Hz. For the upper observer, the time elapsed will be longer than 100s, so she will see a lower frequency. The number of cycles is constant, but the frequency is not.
[..] I would use local stationary observers with identical clocks (so that clocks in greater height are not artifcially slowed down in any way and thus "run faster").

OK, here's the same analysis as before but with your numbers and without correcting their frequency in orbit; and with some elaboration about the number of cycles and transit time.

The calculation with the conservation of cycles works for this case very similar to the example with the warmer clocks:

A radio station emits for 100 seconds a 1Hz EM wave with 1 Hz light blips upwards to a geostationary satellite that remains at constant height, and we neglect the motion of the Earth. As before let's pretend that the time dilation is 1%.

It makes little difference if we now calculate for the EM waves or the light blips, as they must remain in phase; let's take the blips and neglect their duration. I take the liberty to include one start blip and one end blip (101 blips in total for 100s of signal).

The signal is thus emitted as 101 blips of 1.00Hz, with a duration of 100s of Earth time, starting from t=0. It will be detected by means of satellite time as 101 blips of 0.99Hz, with a duration of 101s.

This necessarily takes place during 100 seconds of Earth time, because the distance is held constant and reflected blips must be able to all get back in time.
For example, the time interval for the first blip to be sent out, reflected on the satellite and received back is 1 ms. It doesn't matter what the speed of the light is or how big the distance is: as long as the case is stationary, this time interval does not change. It must therefore also be 1 ms for the last blip, so that this last blip must be received back on Earth at t=100.001 s of Earth time.
In other words, in this example 100 s Earth time = 101 s satellite time.

Elaboration: If one thinks that perhaps the satellite clock was in fact ticking at the same rate as the Earth clock and the effect was instead due to the light, then the last blip would be reflected after 101.000 s of Earth time; it can then not make it back on Earth after only 100.001 s.

 

[PeterDonis]

Consider a hydrogen atom with a ground state energy corresponding to a frequency ω.

This frequency isn't a direct observable. The observable is the frequency of light absorbed when the atom transitions from the ground state to an excited state, or emitted when it transitions from an excited state to the ground state.

Can I use the same two interpretations to describe the behaviour of the atom at the surface of our planet and the spaceship?

For measurements of the frequency of light associated with an energy level transition, yes. For example, say observers E and S each have a hydrogen atom. E's atom is in the first excited state, and S's atom is in the ground state. E's atom emits a photon and transitions to the ground state, and this photon climbs up from E to S. When it reaches S, it will not be able to induce a transition in S's hydrogen atom from the ground state to the first excited state; its frequency will not be right (it will be too low).

 

[harrylin]

And I've done this, repeatedly, but you keep insisting on reading my posts as proposing different theories (different math), when all I am doing is giving different labels in ordinary language for the same theory (same math).

It's an issue of meaning of words, as all too often. Most people call different physical explanations that pretend to relate to the same mathematical predictions different interpretations, and so do I. Think of Lorentz ether vs block inverse, and Copenhagen interpretation vs multi world interpretation. However, there is some dispute if the Copenhagen interpretation is really an interpretation, or effectively a "non-interpretation".
Now, looking below at your clarifications, for sure you present different descriptions. And it appears to me that your "second interpretation" is an alternative description that resembles Copenhagen in its apparent absence of interpretation.

I'll try once more. First I'll give the math (the theory); then I'll give two different ordinary language descriptions of the math (the two interpretations).

Thanks Peter! Now it is really clear what you meant.

[..] as a prediction of GR (and as an experimental fact), we have $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_S > u'_S \cdot p'_S$.

Also, both light sources (at E and at S) are identical, as are both frequency measuring devices.

Interpretation #1: Energy is reduced by gravitational fields

We define the "energy" of a light source as the frequency of light emitted by that source as measured at spaceship S. Since $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_E = u'_S \cdot p'_S$, we have $u'_S \cdot p'_E < u'_S \cdot p'_S$. That is, the measured frequency of observer E's light at spaceship S is less than the measured frequency of spaceship S's light at spaceship S. Therefore, observer E's energy is reduced by being deeper in a gravitational field. We define the "change in the light" as a change in its 4-momentum, compared to parallel transport; but since the 4-momentum of both light signals is parallel transported along their respective worldlines, i.e., $p'_E$ is just $p_E$ parallel transported, and $p_S$ is just $p'_S$ parallel transported, there is no "change in the light" under this interpretation.

Interpretation #2: Gravitational fields affect the propagation of light

We define the "energy" of a light source as the measured frequency of light by the observer that emits the light. So, since $u_E \cdot p_E = u'_S \cdot p'_S$, the energy of observers E and S is identical. But since $u'_S \cdot p'_E < u_E \cdot p_E$, and $u_E \cdot p_S > u'_S \cdot p'_S$, the light signals are changed by propagating through the gravitational field; i.e., we define the "change in the light" as the difference between the measured frequency of the light by the observer that emits the light, and the measured frequency of the light by the observer that receives the light. We can also describe this "change in the light" as a "change in energy" of the light--it starts out from the source with a certain quantity of energy, and loses energy as it climbs, or gains energy as it falls. We are just describing the same math in different words.

Interpretation #2 above is what I was referring to all those many posts ago. I understand that you don't like interpretation #2 because you don't like the way it labels things; you prefer the way interpretation #1 labels things. But that doesn't change the fact that both interpretations are describing the same math and the same predictions, just with different ordinary language labelings.

Indeed, I prefer the descriptive language #1 for reasons I already gave, and for consistency of measurement. In the light of my last post, I get the impression that description #2 is a bit problematic. Suppose that the clocks in post #46 are synchronized by means of "Einstein synchronisation", so that t=t'=0 .
How would you describe the reflection of the last blip as function of time? It seems difficult to avoid saying that the blip travels "back in time".

 

[PeterDonis]

If one thinks that perhaps the satellite clock was in fact ticking at the same rate as the Earth clock and the effect was instead due to the light

That's not what the alternate interpretation I proposed ("light loses energy as it climbs in a gravitational field") says. Remember that it's not an alternate theory; it's just a different labeling with ordinary language terms of the same math. Let me do a similar description to post #44 for this case; first the math, then the two interpretations.

The Math

Observer E emits 101 blips vertically upward, one per second by his clock. Each blip takes 1 ms by his clock to make the round trip from him to observer S. Observer S receives and reflects 101 blips from E. They arrive at intervals of 1.01 second by his clock.

Once the above is finished and everyone catches their breath, observer S emits 101 blips vertically downward, one per second by his clock. Each blip takes 1.01 ms by his clock to make the round trip from him to observer E. Observer E receives and reflects 101 blips from S; they arrive at intervals of 0.99 second by his clock.

Interpretation #1: Energy is reduced by gravitational fields

Once again, we define "energy" as "the frequency of blips as measured at spaceships S". E's blips have a lower frequency (1 blip per 1.01 second instead of 1 blip per second), so E has lower energy. We have no separate observable corresponding to "change in the light" in this scenario; we just work by elimination--since all of the change in blip frequency between E and S is attributed to the reduction in E's energy, there is none left over to be attributed to any change in the light itself.

Interpretation #2: Gravitational fields affect the propagation of light

Once again, we define "energy" as "the frequency of blips as measured by the observer emitting the blips". E and S both emit blips with the same frequency, 1 blip per second, by their own clocks. So they have the same energy. Therefore, since the blip frequency received by S is lower than the blip frequency emitted by E, the change must be due to a change in the light itself as it climbs.

Notice that in both interpretations, the two clocks, E and S, tick at different rates; interpretation #2 does not require that the two clocks tick at the same rate. Such a claim would be a different (falsified) theory, not a different interpretation of the same theory.

 

[Sonderval]

@harrylin

In other words, in this example 100 s Earth time = 101 s satellite time.

Yes - I don't think anyone would disagree.
But as far as I can see, this does not invalidate Peter's two interpretations, or does it?

@Peter
Here is another scenario on top of the H-atom: Let's create a clock that is based on radioactive decay. Would it be correct to say that in the field interpretation, the clock is slowed down because the energy of the atom is lowered, so the energy uncertainty becomes smaller, so the half time increases (by virtue of ΔE Δt≥ħ/2)?

 

[PeterDonis]

How would you describe the reflection of the last blip as function of time?

As a function of whose time? You keep on using the word "time" as if it were absolute, even though I have pointed out repeatedly that it is not. No blip ever moves "backward in time" according to anyone's actual clock. It only appears to move "backwards in time" if you switch the definition of "time" in mid-stream, from "time by S's clock" to "time by E's clock". But that just means your definition of "time" is inconsistent.

 

[harrylin]

As a function of whose time? You keep on using the word "time" as if it were absolute, even though I have pointed out repeatedly that it is not. No blip ever moves "backward in time" according to anyone's actual clock. It only appears to move "backwards in time" if you switch the definition of "time" in mid-stream, from "time by S's clock" to "time by E's clock". But that just means your definition of "time" is inconsistent.

I asked about the definition of time in your alternative description. I may have misunderstood, but that description appears to switch definition of time from "time by S's clock" to "time by E's clock" for the light wave in flight.

On top of that, you argued in post #49:

Once again, we define "energy" as "the frequency of blips as measured by the observer emitting the blips". E and S both emit blips with the same frequency, 1 blip per second, by their own clocks. So they have the same energy. [..]
Notice that in both interpretations, the two clocks, E and S, tick at different rates

Those two statements are clearly inconsistent. The clock rates of atomic clocks E and S are proportional to the locally emitted frequencies. It could be even the same light signal that steers the clock and is sent out, in which case we have a pure contradiction I'm afraid.

 

[PeterDonis]

Let's create a clock that is based on radioactive decay. Would it be correct to say that in the field interpretation, the clock is slowed down because the energy of the atom is lowered, so the energy uncertainty becomes smaller, so the half time increases (by virtue of ΔE Δt≥ħ/2)?

First of all, $\Delta E \Delta t \ge \hbar /2$ is not really a valid statement of the uncertainty principle, because time is not an observable in quantum mechanics, it's a parameter. (A more detailed discussion of that would belong in a new thread in the Quantum Physics forum.)

Second, from the point of view of relativity, the detailed mechanism by which radioactive decay works is not really modeled. Radioactive decay is just treated as a function of proper time; i.e., the half-life of any radioactive substance is measured in units of that substance's proper time. So the half-life of any given radioactive substance will always be the same as measured by a clock co-located with the substance. In the context of relativity, that's just a fact about radioactive substances; to explain why it's true, you need a quantum mechanical model of the radioactive substance, and that model will not take into account any effects of a gravitational field (from the point of view of relativity, it will be done in flat spacetime).

So no, you can't really link the half-life with the "energy of the atom" the way you are trying to do.

 

[Sonderval]

But isn't half-life usually linked to an energy uncertainty, for example in particle physics as explained here:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parlif.html
?

So no, you can't really link the half-life with the "energy of the atom" the way you are trying to do.

But does this not imply that there is no way to formulate the difference observed by the two observers (observer S looks down on the radioactive clock of observer E and sees that it runs slower than her clock) in the field interpretation (where everything happens on a flat Minkowski background, see MisnerThorne Wheeler, box 18.1)? (Basically, I'm still trying to grasp how this field interpretation can work out consistently).

As you see, I'm still very much confused (and thanks once again for your patience in explaining things).

 

[PeterDonis]

But isn't half-life usually linked to an energy uncertainty, for example in particle physics as explained here:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parlif.html
?

In quantum physics, yes, that's one way of formulating it. But in GR, there is no such thing as "energy uncertainty". GR is not a quantum theory; it's a classical theory. In GR, half-life is just a constant in the function that describes radioactive decay as a function of proper time. How that constant arises from quantum physics is outside the purview of GR.

does this not imply that there is no way to formulate the difference observed by the two observers (observer S looks down on the radioactive clock of observer E and sees that it runs slower than her clock) in the field interpretation (where everything happens on a flat Minkowski background, see MisnerThorne Wheeler, box 18.1)?

Not at all. The radioactive clock is just a clock, and it works like any other clock. The radioactive decay is a function of proper time, as I've said. That's all there is to it. So if the field interpretation works for other clocks, it works for the radioactive clock the same way.

 

[Sonderval]

GR is not a quantum theory; it's a classical theory.

If people say that GR is the classical limit of a spin-2 theory in QFT (see MTW box 18.1), somehow it should be possible to link the action of the spin-2 field to things like half-lifes.

So if the field interpretation works for other clocks,

That's exactly the question I started with: How exactly does the field interpretation work for any clock? The paper I quoted in my first post calculates this for the case of the Rydberg frequency of an H-atom, but I do not fully grasp that calculation (especiall eq. 41 - where does this come from?) and I do not see how this works out consistently for any clock.
How do I get from a Minkowskian spacetime to a spacetime where clock speed depends on a local field consistently (so that I can re-interpret this as curved spacetime)?

 

[PeterDonis]

How exactly does the field interpretation work for any clock?

If you already accept that the field affects the lengths of rulers, and you know that in relativity, space and time are interconnected, then the field must also affect the rates of clocks, correct? Otherwise you would violate the principle of relativity.

 

[Sonderval]

Yes, that's what I keep telling myself (actually, that was about the first thing I wrote down when I started to make notes about my thoughts some days ago...), but somehow I'm not wholly satisfied.

For space affecting the length of rulers, I can imagine something like the following situation:
A gravity wave impinges on a wheel with spokes. (I use a gravity wave because it only distorts space, but not time.) Test particles sit at the end of the spokes and can slide along the spokes without friction. If a gravity wave hits the wheel, I can interpret what happens in two ways:
1. Space gets distorted. The test particles follow geodesics because they are free in radial direction, since space is distorted, some slide inwards, some slide outwards, forming an ellipse. (as in this animation http://www.einstein-online.info/elementary/gravWav/rhythm) The spokes of the wheel do not follow the geodesics anymore because they would be compressed or stretched. End result: there is a stress in the spokes and the particles slide inwards.
2. Space does not get distorted, but the field changes the length of rulers. The distance between the test particles is not affected, but the field stretches/shrinks the spokes of the wheel. Again, the particles slide inwards/outwards on the spokes and the spokes are stressed. (Signs are opposite to what they were before, where the distance between particles decrease in interpretation 1, the spoke gets stretched, so that in both cases the particle slides inwards - and vice versa)
Do you think this example is correct?

Basically, what I am looking for is a similar example where I can see how things work out in the two interpretations with respect to time.

 

[PeterDonis]

(I use a gravity wave because it only distorts space, but not time.)

Only if you choose your coordinates in a very particular way. If you change coordinates, the wave distorts both space and time. So in the field interpretation, in order to keep all observables the same when you change coordinates, the field must affect clocks as well as rulers.

Try analyzing the gravity wave in a frame in which the wheel with the spokes and test particles is moving. You will see that if you try to use the periodicity of the oscillations of the test particles relative to the spokes as a clock, it must be time dilated just as SR would predict.

 

[Sonderval]

@Peter
Thanks again. I assume that my example in itself is not wrong, then?

If you change coordinates, the wave distorts both space and time. ...
Try analyzing the gravity wave in a frame in which the wheel with the spokes and test particles is moving.

So you mean I should be using a coordinate system moving relative to the wheel?
If I move with constant speed in the direction of the wave (that seems to be the simplest scenario), I will observe a change in the frequency of the gravity wave (like a Doppler shift) and a corresponding change (due to time dilation) in the frequency of the wheel's oscillations; as far as I can see, there is no additional influence. Is this correct?

If I move perpendicular to the direction of the wave (in the plane of the wheel), I will observe the same frequency of the gravity wave as an observer at the wheel, but there will be time dilation between me and the wheel, so there has to be an additional effect, otherwise I would observe the wheel's oscillations being "out of tune" with the wave. Is this what you mean?