Time dilation in the field interpretation of GR

[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?

 

[PeterDonis]

I assume that my example in itself is not wrong, then?

It's fine as far as it goes, but as I said, your description only applies in one particular frame.

So you mean I should be using a coordinate system moving relative to the wheel?

Yes.

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?

I think so, yes.

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

Will you? What other way do you have of observing the gravity wave, besides its effect on the oscillations of 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?

Sort of. As I said above, the only way you have of observing the gravity wave (i.e., the field) is through its effects on the relative motion of the wheel spokes and the test particles. So it's not that the wheel oscillations will be out of sync with the wave; it's that oscillations of different parts of the wheel will be "out of sync" with each other, if you don't include an effect of the field on the rate of the oscillations. I put "out of sync" in quotes because it's not as simple as, for example, all of the test particles being closest to the hub of the wheel at the same time; you have to also include relativity of simultaneity in the analysis. But if you include an effect of the field on the length of the spokes in the wheel's rest frame, and then transform that effect into the moving frame, you will also have to include an effect of the field on the oscillation rate, or the events won't match up right in the moving frame.

 

[Sonderval]

@Peter
Thanks a lot for elaborating.

As I said above, the only way you have of observing the gravity wave (i.e., the field) is through its effects on the relative motion of the wheel spokes and the test particles.

But I could carry another ring of test particles in my own rest frame to study the wave, couldn't I?

But if you include an effect of the field on the length of the spokes in the wheel's rest frame, and then transform that effect into the moving frame, you will also have to include an effect of the field on the oscillation rate, or the events won't match up right in the moving frame.

I see - it's not quite as simple as I thought to actually work out what happens.

 

[PeterDonis]

I could carry another ring of test particles in my own rest frame to study the wave, couldn't I?

Test particles aren't "in" a particular frame. If you mean another wheel and spokes with test particles at rest relative to you but moving relative to the original wheel and spokes, yes, you could do that. But that second wheel and spokes setup would be moving relative to the wave in a way the original set was not. And you would still be using a wheel and spokes setup to observe the wave; you wouldn't be observing the wave without any wheel or spokes at all.

 

[Sonderval]

But that second wheel and spokes setup would be moving relative to the wave in a way the original set was not.

Yes, you are right.

In principle, there is no way to observe a gravity wave without some kind of test particles (same as for an electrical field), or is there?

 

[PeterDonis]

there is no way to observe a gravity wave without some kind of test particles (same as for an electrical field), or is there?

No, there isn't.

 

[Sonderval]

Thanks, at least some of my intuition is not wrong...

 

[anorlunda]

Thanks, at least some of my intuition is not wrong...

Let me recommend Relativlty by Albert Einstein. It is a very thin book, and very understandable by anyone who unserstands elementary algegra. Einstein not only explains SR, but also the train of reasoning that led him to it.

https://www.amazon.com/dp/1619491508/?tag=pfamazon01-20

 

[harrylin]

Let me recommend Relativlty by Albert Einstein. It is a very thin book, and very understandable by anyone who unserstands elementary algegra. Einstein not only explains SR, but also the train of reasoning that led him to it.

https://www.amazon.com/dp/1619491508/?tag=pfamazon01-20

You mean GR of course. That book is also online, and indeed it provides a good context for his more detailed discussion to which I referred earlier.

Thus :
- https://en.wikisource.org/wiki/Rela.../Part_II#Section_19_-_The_Gravitational_Field
(and further) is a good complement to §22 of:
- https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity