# Free fall orbit time dilation

• B
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.

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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?

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Dale
Mentor
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.

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
Mentor
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.

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
Homework Helper
2019 Award
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
Mentor
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 wierd name). The lecture notes describe them in detail.

They are called "Christoffel symbols" (I know, it is a wierd 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
Gold Member
2019 Award
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.

Dale
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
Gold Member
2019 Award
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.

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
Gold Member
2019 Award
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.

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?

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Dale
Mentor
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.

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
Mentor
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

vanhees71
Gold Member
2019 Award
One should note that (3.47) is only proper acceleration if ##\lambda## is normalized such that
$$g_{\mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \lambda} \frac{\mathrm{d} x^{\nu}}{\mathrm{d} \lambda}=1.$$
The formula, i.e., the equation for a geodesic, parametrized in terms of an affine parameter ##\lambda##, is more general. You can also solve it if the tangent vector is null (world lines of massless particles or light rays in the sense of the eikonal approximation) or spacelike.

Dale
stevendaryl
Staff Emeritus
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, two clocks that are in different orbits will show different amounts of elapsed time when they pass each other. You don't need to bring them together; instead you can put A into a circular orbit around C and put B into a very eccentric elliptical orbit. If you arrange things perfectly, then you can make sure that A and B pass each other with some regularity. I haven't done the calculation, but I believe that when they get back together, B will show more elapsed time than A.

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
As I mentioned earlier to Vanhees earlier
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?
Could you possibly just use that equation (which would seem to show no change to the x, y, z coords) using a single coordinate (t=0, x = 1, y = 1, z = 1) at rest with the observer standing on Earth to illustrate how it shows proper acceleration?

One should note that (3.47) is only proper acceleration if ##\lambda## is normalized such that
$$g_{\mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \lambda} \frac{\mathrm{d} x^{\nu}}{\mathrm{d} \lambda}=1.$$
The formula, i.e., the equation for a geodesic, parametrized in terms of an affine parameter ##\lambda##, is more general. You can also solve it if the tangent vector is null (world lines of massless particles or light rays in the sense of the eikonal approximation) or spacelike.
Could you just use the equation with a single coordinate at rest with the observer on Earth to illustrate, as I mentioned in #67?

Yes, two clocks that are in different orbits will show different amounts of elapsed time when they pass each other. You don't need to bring them together; instead you can put A into a circular orbit around C and put B into a very eccentric elliptical orbit. If you arrange things perfectly, then you can make sure that A and B pass each other with some regularity. I haven't done the calculation, but I believe that when they get back together, B will show more elapsed time than A.
I am happy with them being brought together though, else there might be something to do with the elliptical orbit that has not been made clear. As you seemed to be able to conclude that B's clock would have gone faster, but in the bit you had quoted of what I had written, I had not mentioned which was in lower orbit, or their respective velocities. Any affects of bringing together tends to insignificant the longer A and B orbit anyway. If B's clock was in the lower orbit then time dilation due to gravity would slow B's clock relative to A's, and presumably that affect is invariant across frames of reference. So would there not just be the kinematic time dilation left, and 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?