Free fall orbit time dilation

[Dale]

These two clocks are not local:

(the clock in flat spacetime in distant space outside the shell and the clock in flat spacetime in flat spacetime inside the shell)

These two clocks are local:

once the radiation had passed the distant clock

 

[Dale]

What energy distribution difference between 1 and 2 were you thinking would explain the affect?

The distribution of the energy in the shell

 

[name123]

Gravitational time dilation is always between two separated clocks. With just one clock there is nothing to explain.

Then consider both the clocks.

1) Clock 1 in the shell in flat spacetime before the shell had radiated, and Clock 2 a light year outside the shell
2) Clock 1 two years after the shell has radiated away, the space time flat as it was in scenario 1, Clock 2 a year after the radiation from the shell had passed it.

 

[name123]

The distribution of the energy in the shell

So the gravitational energy causes the time dilation?

 

[name123]

Consider satellites A and B, going at different velocities, in free fall orbit around a massive body C at different altitudes for a million years, then being brought together and the clocks on them compared. Presumably the bringing them together would become less significant the longer they orbited, and the amount of time dilation due to curvature would be frame of reference independent, but what about the observed velocities from the mass being orbited's perspective? If the satellites were labelled A and B and the mass C then would the clock comparison figure be correctly calculated no matter which you considered at rest?

Yes, but note that the metric will have a different form in these different charts. With a correct expression for the metric, all invariants will be correctly calculated.

If each had a ruler and a clock on board, when they were at rest together could they not all go by their own clock and ruler, and agree that their clocks and rulers were metrically equivalent?

Also regarding A, B and C being at rest, I am not quite clear how A or B could be considered at rest and not rotating, as while C (the mass) could be considered to orbit A or B, if it was so considered, would it not also need to be considered that it was rotating, for an explanation of why the same section of C was not always facing A (or B), and could it not be objectively measured that it was not rotating?

Could you explain this please?

 

[Dale]

So the gravitational energy causes the time dilation?

Also energy due to "stuff" like matter or light or whatever.

 

[A.T.]

So the gravitational energy causes the time dilation?

https://en.wikipedia.org/wiki/Stress–energy_tensor

 

[Dale]

If each had a ruler and a clock on board, when they were at rest together could they not all go by their own clock and ruler, and agree that their clocks and rulers were metrically equivalent?

Yes, completely.

Could you explain this please?

The question is ambiguous. I cannot tell if you are asking about proper rotation or coordinate rotation.

 

[name123]

The question is ambiguous. I cannot tell if you are asking about proper rotation or coordinate rotation.

Proper rotation assuming that it what causes the proper acceleration that can be measured.

 

[Dale]

Proper rotation assuming that it what causes the proper acceleration that can be measured.

Whether or not a given object is undergoing proper rotation is an invariant. It does not depend on the coordinates chosen. You can choose coordinates where a proper-rotating object is at coordinate-rest. In such coordinates there will be "fictitious forces" which will lead to the correct amount of proper rotation.

 

[name123]

Consider satellites A and B, going at different velocities, in free fall orbit around a massive body C at different altitudes for a million years, then being brought together and the clocks on them compared. Presumably the bringing them together would become less significant the longer they orbited, and the amount of time dilation due to curvature would be frame of reference independent, but what about the observed velocities from the mass being orbited's perspective? If the satellites were labelled A and B and the mass C then would the clock comparison figure be correctly calculated no matter which you considered at rest?

Yes, but note that the metric will have a different form in these different charts. With a correct expression for the metric, all invariants will be correctly calculated.

If each had a ruler and a clock on board, when they were at rest together could they not all go by their own clock and ruler, and agree that their clocks and rulers were metrically equivalent?

Yes, completely.

So A would agree with B about the expected differences between their clocks due to gravitational time dilation. And the affect due to bringing the clocks together could be considered insignificant if the orbiting had gone on for long enough.

Would not A think, using the metric of A's clock, that B's clock had ticked less than would have been expected (taking gravitational time dilation into account) if it had been in A's rest frame, and B, using the metric of B's clock, think that A's clock had ticked less than would have been expected (taking gravitational time dilation into account) if A had been in B's rest frame?

 

[name123]

Whether or not a given object is undergoing proper rotation is an invariant. It does not depend on the coordinates chosen. You can choose coordinates where a proper-rotating object is at coordinate-rest. In such coordinates there will be "fictitious forces" which will lead to the correct amount of proper rotation.

So with the A, B, and C coordinates it seems arguable that while the maths can model the relative motion regardless of which is thought to be at rest, only one rest frame would give the coordinate rotation which corresponds to the correct measurable proper rotation, without the addition of fictitious forces? Also I have read that fictitious forces are used with non-intertial frames of reference, but would not both A and B at rest both be intertial frames of reference (what intertial frame are they undergoing accceleration relative to)?

 

[Dale]

So A would agree with B about the expected differences between their clocks due to gravitational time dilation. And the affect due to bringing the clocks together could be considered insignificant if the orbiting had gone on for long enough.

Yes, as edited.

I have no idea how to parse the other question. In GR, the proper time on any clock undergoing any motion in any frame in any spacetime is given by $\int \sqrt{ g_{\mu \nu} dx^{\mu} dx^{\nu}}$

 

[name123]

I have no idea how to parse the other question. In GR, the proper time on any clock undergoing any motion in any frame in any spacetime is given by $\int \sqrt{ g_{\mu \nu} dx^{\mu} dx^{\nu}}$

Could you make up some figures for the scenario, and use them in the equation to demonstrate please (just so I can see how it works)?

P.S. Regarding where I had written: "So A would agree with B about the expected differences between their clocks due to gravitational time dilation." I had thought it was invariant how much difference would be expected due to gravitational time dilation. I thought in the Hafele-Keating experiment they had split the affects expected due to gravitational time dilation and kinematic time dilation to come up with their estimates, and so was thinking that you might be able to explain whether the expected gravitational time dilation affects would be same regardless of frame of reference, and if they were, and the coming together was insignificant (given long enough orbit), then simply explain why the kinematic time dilation expectations would be the same from each perspective, or how else they came to the same answer when added to the gravitational time dilation effects.

 

[Dale]

Why don't you work through the first few chapters of Sean Carroll's lecture notes on general relativity first.

 

[name123]

Why don't you work through the first few chapters of Sean Carroll's lecture notes on general relativity first.

Are they free and do they contain a worked example of the equation you quoted? I think, I or other basic level followers could probably follow where the numbers were plugged in, but might not so easily follow how to correctly use the equation in the situation otherwise.

 

[name123]

I hope you do not mind me re-replying to the post but I had asked:

Also I am not quite clear how A or B could be considered at rest and not rotating, as while C (the mass) could be considered to orbit A or B, if it was so considered, would it not also need to be considered that it was rotating, for an explanation of why the same section of C was not always facing A (or B), and could it not be objectively measured that it was not rotating?

and you asked me to clarify what I meant proper rotation or coordinate rotation, I explained that I had meant proper rotation (I was referring to the measurement, not the coordinate rotation that would appear if A or B were considered to be at rest) and you replied:

Whether or not a given object is undergoing proper rotation is an invariant. It does not depend on the coordinates chosen. You can choose coordinates where a proper-rotating object is at coordinate-rest. In such coordinates there will be "fictitious forces" which will lead to the correct amount of proper rotation.

But that still does not seem to answer my question. As I now understand it "fictitious forces" have a special meaning in physics and refer to forces added for explanation when describing motion from a non-inertial frame of reference. But the question was about from the frame of reference A or B which as I understand it are both inertial frames of reference. While they considering them at rest, C would show coordinate rotation (though there would be no measurable rotation). So the measurements would not seem to support how it would be expected to be if A or B were actually at rest relative to C. If they were, and C were actually rotating then you would expect to measure proper rotation on C.

Also from the perspective of the inertial frames A and B if there were distant stars, wouldn't they appear to be moving faster than the speed of light?

 

[Dale]

Yes, they are free
https://arxiv.org/abs/gr-qc/9712019

Working an example without the background would require a lot of effort on my part and result in very little gain for you. But reading the first couple of chapters of the lecture notes will result in a lot more gain for you.

 

[name123]

Yes, they are free
https://arxiv.org/abs/gr-qc/9712019

Working an example without the background would require a lot of effort on my part and result in very little gain for you. But reading the first couple of chapters of the lecture notes will result in a lot more gain for you.

Thanks for the link, and for the help so far. I don't know whether you noticed, but I posted a new reply #52 to one of your earlier replies.

 

[A.T.]

Also from the perspective of the inertial frames A and B if there were distant stars, wouldn't they appear to be moving faster than the speed of light?

Which is a good hint that they aren’t inertial.

 

[Dale]

But the question was about from the frame of reference A or B which as I understand it are both inertial frames of reference.

They are only locally inertial. And even locally it is only inertial if the object is not undergoing any proper rotation.

Also from the perspective of the inertial frames A and B if there were distant stars, wouldn't they appear to be moving faster than the speed of light?

Neither the stars nor the other objects are local.

 

[name123]

They are only locally inertial.

So I then assume A relative to B and C in the scenario should be considered as non-inertial even though an accelerometer at rest with respect to A would measure no acceleration?

If so then with A being non-inertial a fictitious force would be added to describe the coordinate rotation of C from A's rest frame. What is the fictitious force that would explain it?

When the considerations of A, B, and C being at rest are compared, does not only the consideration of C being at rest give a proper rotation for C in line with its coordinate rotation?

 

[jbriggs444]

reference A or B which as I understand it are both inertial frames of reference

In curved space-time there is no such thing as a global inertial frame of reference.

 

[A.T.]

So I then assume A relative to B and C in the scenario should be considered as non-inertial even though an accelerometer at rest with respect to A would measure no acceleration?

Rotating frames are not inertial.

 

[Dale]

So I then assume A relative to B and C in the scenario should be considered as non-inertial even though an accelerometer at rest with respect to A would measure no acceleration?

Yes, if you want to make any non-local measurements then you need to consider them to be non inertial.

What is the fictitious force that would explain it?

They are called "Christoffel symbols" (I know, it is a weird name). The lecture notes describe them in detail.

 

[name123]

They are called "Christoffel symbols" (I know, it is a weird name). The lecture notes describe them in detail.

I've spotted them in the notes, but they are quite a few pages in. I have looked them up elsewhere and it is mentioned that they are used in the geometry. Do they offer a force that explains the lack of measurement of proper acceleration in an object showing coordinate acceleration though, or when the considerations of A, B, and C being at rest are compared, does only the consideration of C being at rest give a proper rotation for C in line with its coordinate rotation?

 

[vanhees71]

I can only also recommend to read a bit about differential geometry in Carrol's Lecture notes first. You just need Secs. 2 and 3 to answer all these questions.

 

[name123]

I can only also recommend to read a bit about differential geometry in Carrol's Lecture notes first. You just need Secs. 2 and 3 to answer all these questions.

Does proper acceleration appear in the notes at all, the term does not seem to be in them, and when acceleration is mentioned I am not clear that it is not referring to relative / coordinate acceleration.

 

[vanhees71]

I'm not sure that I understand what you need that for to learn the basic principles of pseudo-Riemannian (Lorentzian) differential geometry, but I'd define proper acceleration as
$$a^{\mu}=\frac{\mathrm{D} u^{\mu}}{\mathrm{D} \tau},$$
where
$$u^{\mu}=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}$$
is the four velocity and $\tau$ the proper time (I assume you have a massive particle here; for massless particles the issue is a bit more complicated).

Written out the proper acceleration reads
$$a^{\mu} = \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} + {\Gamma^{\mu}}_{\rho \sigma} \frac{\mathrm{d} x^{\rho}}{\mathrm{d} \tau} \frac{\mathrm{d} x^{\sigma}}{\mathrm{d} \tau},$$
where ${\Gamma^{\mu}}_{\rho \sigma}$ are the connection coefficients (Christoffel symbols) of spacetime.

 

[name123]

I'm not sure that I understand what you need that for to learn the basic principles of pseudo-Riemannian (Lorentzian) differential geometry, but I'd define proper acceleration as
$$a^{\mu}=\frac{\mathrm{D} u^{\mu}}{\mathrm{D} \tau},$$
where
$$u^{\mu}=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}$$
is the four velocity and $\tau$ the proper time (I assume you have a massive particle here; for massless particles the issue is a bit more complicated).

Written out the proper acceleration reads
$$a^{\mu} = \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} + {\Gamma^{\mu}}_{\rho \sigma} \frac{\mathrm{d} x^{\rho}}{\mathrm{d} \tau} \frac{\mathrm{d} x^{\sigma}}{\mathrm{d} \tau},$$
where ${\Gamma^{\mu}}_{\rho \sigma}$ are the connection coefficients (Christoffel symbols) of spacetime.

Won't any change in x depend on what coordinate system (what frame of rest) you are using?

 

[vanhees71]

The $x^{\mu}$ are some coordinates, and all components given in my previous posting are with respect to the corresponding holonomous basis of the tangent spaces of the manifold, i.e., $u^{\mu}$ and $a^{\mu}$ are vector components with respect to the holonomous basis of the tangent space at the position of the point particle under consideration.

 

[name123]

The $x^{\mu}$ are some coordinates, and all components given in my previous posting are with respect to the corresponding holonomous basis of the tangent spaces of the manifold, i.e., $u^{\mu}$ and $a^{\mu}$ are vector components with respect to the holonomous basis of the tangent space at the position of the point particle under consideration.

So if an observer was standing on the Earth and was using a coordinate system where they were considered at rest, then what would be the change in the x,y or z part of any coordinate point (at rest with respect to the observer) on the Earth over a period of time. Wouldn't only the time part of the coordinate be changing? So where would the acceleration of those points be using your equations? It seems to me that there would not be any, as they seem to represent coordinate acceleration. But (as I understand it) proper acceleration would be measured at any of those points on the Earth.

If I have misunderstood (sorry my maths is quite poor) then perhaps you could illustrate using a single coordinate ct = 0 x = 1, y = 1, z= 1 in your equations to show how it ends up with the measurable proper acceleration over 10 seconds perhaps?

 

[Dale]

I've spotted them in the notes, but they are quite a few pages in.

Yes. Those intervening pages are important. I really think that you need to go through it. You are asking very haphazard questions because you need a systematic introduction.

Please don't try to skip ahead, but go through the material step by step.

 

[name123]

Yes. Those intervening pages are important. I really think that you need to go through it. You are asking very haphazard questions because you need a systematic introduction.

Please don't try to skip ahead, but go through the material step by step.

Does proper acceleration appear in the notes at all, the term does not seem to be in them, and when acceleration is mentioned I am not clear that it is not referring to relative / coordinate acceleration?

 

[Dale]

Does proper acceleration appear in the notes at all, the term does not seem to be in them, and when acceleration is mentioned I am not clear that it is not referring to relative / coordinate acceleration?

He does not appear to use that term, but the quantity $$\frac{d^2}{d\tau^2}x^{\mu}(\tau)$$ in equation 1.102 is the proper acceleration in flat spacetime.

And the proper acceleration in curved spacetime is given by the left hand side of equation 3.47