Time dilation in the field interpretation of GR

[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