I What is the angular velocity of a satellite?

[cianfa72]

Requiring a geodesic in the definition is, AFAIK, most important for spacelike separation, since if we are allowed to use arbitrarily curved (non-geodesic) spacelike curves, we can connect any pair of points whatever.

Why ? Take two timelike separated events: does it always exist an arbitrarily curved spacelike path connecting them ?

 

[PeterDonis]

Take two timelike separated events: does it always exist an arbitrarily curved spacelike path connecting them ?

Yes.

 

[cianfa72]

Yes.

Ah right, just to visualize it I made this sketch: events A and B are timelike separated yet there is a spacelike path (in black) connecting them (I believe there is no problem in smoothing out its acute angle)

 

[PeterDonis]

Ah right, just to visualize it I made this sketch: events A and B are timelike separated yet there is a spacelike path (in black) connecting them (I believe there is no problem in smoothing out its acute angle)

View attachment 288181

Not quite. You can't do it in just two spacetime dimensions, because when you smooth out the acute angle, you find that the path becomes timelike for some portion of that region.

If you add one more spatial dimension, however, you can have a helical path in spacetime that is always spacelike (no acute angle, just constant curvature) but connects two timelike separated (in the sense of having a timelike geodesic between them) events.

 

[PeterDonis]

Requiring a geodesic in the definition is, AFAIK, most important for spacelike separation

There is actually another reason for requiring a geodesic in the definition, which applies to any kind of separation. Given an event and a tangent vector at that event (and tangent vectors at a single event will always be definitely timelike, spacelike, or null), a unique geodesic is determined throughout the spacetime. But the uniqueness only holds for geodesics; there are an infinite number of non-geodesic curves that pass through the same event and have the same tangent vector at that event. So for the "separation" of two events to be well defined and unique, the definition has to require geodesics.

 

[cianfa72]

w.r.t. the events of the second subset we can further 'group' them based on axes locally defined in 'space' (e.g. gyroscope axes): events spatially aligned with a such axis are part of a group that -- in the limit of smaller and smaller region around the given event-- basically defines a spacelike direction in spacetime.

So, just to check my understanding, in the case of 1 + 1 curved spacetime in the following local chart

all events in the grayed area count as 'aligned' on the same spatial axis at event A (i.e. made part of the same 'group') and in the limit of smaller and smaller region around A they actually defines the (unique) spacelike direction (unique because the spacetime chosen is just 1 + 1).

 

[binis]

The system attached to my head is inertial because if I let go of a small body it stays at rest with respect to me

? You are also rotating so if you release a button it will fly away from you

 

[Ibix]

? You are also rotating so if you release a button it will fly away from you

You have a funny way of swinging a ball around your head. I usually stand still and whirl my arm, so I am not rotating. I suppose you could hold your arm rigid and do a pirouette, but that was not what I had in mind.

 

[binis]

One reference system was attached to my head and one to the ball.

Nevertheless, why you use two frames instead of one?

You have a funny way of swinging a ball around your head.

Oops! What a misconception! I feel like a jerk.

 

[Ibix]

Nevertheless, why you use two frames instead of one?

Because that's the topic of discussion - you started this thread asking if the angular velocity of a satellite was $d\theta/dt$ or $d\theta/dt'$. You implicitly defined two frames right there.

 

[binis]

You implicitly defined two frames right there.

In post #5 I set the ECI frame. I think it's reasonable to set one frame in order to study a problem.

 

[PeterDonis]

all events in the grayed area count as 'aligned' on the same spatial axis at event A

If "aligned" is taken to mean "aligned in some inertial frame whose origin is event A", then yes. More precisely, if you look at all possible spacelike lines that pass through event A, those lines foliate the grayed area, and each line corresponds to "the x axis" in some inertial frame whose origin is event A. So each line represents "the spatial x direction" in one of those inertial frames. If an observer whose worldline passes through event A was carrying a gyroscope pointed in the spatial x direction, the tangent vector to each line would represent the gyroscope at the instant the observer passes through event A, in one of those inertial frames.

(i.e. made part of the same 'group')

I'm not sure what you mean by "group" here.

in the limit of smaller and smaller region around A they actually defines the (unique) spacelike direction (unique because the spacetime chosen is just 1 + 1).

You don't need to take the limit. You can just use the lines I described above.

 

[cianfa72]

So each line represents "the spatial x direction" in one of those inertial frames. If an observer whose worldline passes through event A was carrying a gyroscope pointed in the spatial x direction, the tangent vector to each line would represent the gyroscope at the instant the observer passes through event A, in one of those inertial frames.

Here we are assuming a (toy) 1 + 1 spacetime, so there is just one spatial dimension. Take two different observers carrying gyroscopes passing through event A (hence at event A they have different velocities).

Since there exist just one spatial dimension the two gyroscopes axes cannot really point in different spatial directions, even if they actually point in two different spacetime directions.

What do you mean physically with 'an observer carrying a gyroscope pointed in the spatial x direction' in this specific scenario ?

 

[PeterDonis]

Since there exist just one spatial dimension the two gyroscopes axes cannot really point in different spatial directions, even if they actually point in two different spacetime directions.

With the implicit definition of "spatial direction" that you are using here, yes, this is true, since by construction there is only one "spatial direction" at all in this spacetime.

What do you mean physically with 'an observer carrying a gyroscope pointed in the spatial x direction' in this specific scenario ?

See above.

 

[PeterDonis]

Here we are assuming a (toy) 1 + 1 spacetime

You don't need to in order to define the "spatial x direction" if it is also the direction of relative motion between two observers, which is what is assumed when you draw a 1 + 1 spacetime diagram. That doesn't mean there is only one spatial dimension in the spacetime, period; it just means that only one spatial dimension is relevant to the particular problem you are describing.

 

[cianfa72]

That doesn't mean there is only one spatial dimension in the spacetime, period; it just means that only one spatial dimension is relevant to the particular problem you are describing.

I take it as we can drop spatial dimensions not relevant for the problem at hand.

The point I was trying to make is that at event A gyroscopes having their axes pointing in the same given spatial direction but with different relative velocities actually define different spacelike directions from A.

 

[PeterDonis]

I take it as we can drop spatial dimensions not relevant for the problem at hand.

Yes, but it's worth remembering that in the actual world, those other spatial dimensions are still there. Restricting relative motion to one spatial direction does not mean we change the universe to a toy 1+1 spacetime.

The point I was trying to make is that at event A gyroscopes having their axes pointing in the same given spatial direction but with different relative velocities actually define different spacelike directions from A.

They define different spacelike vectors if we assume that the spacelike vectors being defined are orthogonal to the 4-velocities of the gyroscopes, yes. That's equivalent to saying that they are at rest in different inertial frames which have different basis vectors in the "x direction".

 

[vanhees71]

The formal description of the gyroscopes, provided there are no external torques acting on them, is that the spatial basis vectors defined by them are Fermi-Walker transported along the time-like trajectory of this gyroscope. Using enough co-moving gyroscopes you can in this way define rotation free tetrades. If the trajectory of the gyroscopes is a geodesic, i.e., if they are in free fall, you get local inertial frames along this trajectories.

 

[cianfa72]

The formal description of the gyroscopes, provided there are no external torques acting on them, is that the spatial basis vectors defined by them are Fermi-Walker transported along the time-like trajectory of this gyroscope.

The spatial basis vectors you're talking about should be spacelike directions in spacetime.

The formal description of the gyroscopes, provided there are no external torques acting Using enough co-moving gyroscopes you can in this way define rotation free tetrades. If the trajectory of the gyroscopes is a geodesic, i.e., if they are in free fall, you get local inertial frames along this trajectories.

I take it as if we 'disseminate' the space with 'groups' of 3 co-moving gyroscopes having their axes mutually orthogonal in space, we can define in this way a rotation-free tetrad field.

 

[vanhees71]

A torque-free gyro's spin direction defines a spacelike direction in spacetime (where else, there is only spacetime in GR), which is Fermi-Walker transported along its world line (which is also defined in spacetime).

 

[binis]

This was a scenario where I was whirling a ball on the end of a piece of string. One reference system was attached to my head and one to the ball. The system attached to my head is inertial because if I let go of a small body it stays at rest with respect to me while if the ball releases a small mass it will flycaway from it with a time varying distance growth.
This is a slight idealisation where I assume that I am very much more massive than the ball and am in zero g.

This is not happening among earth-ISS.This is not the case of the couple earth-ISS, an almost pure SR problem (#45).In the ISS time is running slower than the earth. But velocity is relative. We can choose to deem either ISS or Earth to be at rest. An observer in the ISS sees the Earth revolve around the ISS. By his frame, is time running slower on the earth?

 

[jbriggs444]

This is not happening among earth-ISS.This is not the case of the couple earth-ISS, an almost pure SR problem (#45).In the ISS time is running slower than the earth. But velocity is relative. We can choose to deem either ISS or Earth to be at rest. An observer in the ISS sees the Earth revolve around the ISS. By his frame, is time running slower on the earth?

If we are treating this as an SR problem then the ISS is not continuously at rest in anyone inertial frame. One can pick a pair of inertial rest frames and see a symmetry -- briefly. But that symmetry will disappear if you try to paste a series of inertial frames together to follow a satellite around a complete orbit.

Edit: @binis, you've indicated skepticism. Possibly the Sagnac effect is of interest.

 

[Dale]

This is not the case of the couple earth-ISS, an almost pure SR problem (#45).

Hold on. My post 45 does NOT indicate that the Earth and ISS can be treated as a purely SR problem. It is a fully GR problem! It only says that the overall time dilation can be partitioned into a gravitational and a kinematic part and that the kinematic part is largest. That in no way implies that GR can be neglected.

@jbriggs444 is correct in his reply

 

[Ibix]

This is not the case of the couple earth-ISS, an almost pure SR problem (#45).

The ISS most certainly is not "almost pure SR". Gravity is absolutely critical to explaining why it's going round the Earth.

An observer in the ISS sees the Earth revolve around the ISS. By his frame, is time running slower on the earth?

The problem is that there is not a unique coordinate system in which the ISS is at rest. Depending on how you deal with that, the relative clock rates may vary. However, the rates will average out over one orbit to Earth clocks ticking fast, since Earth clocks see the ISS's clocks ticking slow on that same average.

Note that @jbriggs444's answer is consistent with this. He's defined part of a coordinate system where the Earth clocks tick slow, but notes that this cannot hold over a full orbit.

 

[Ibix]

Why "skeptical" @binis? It should be no surprise that orbits in curved spacetime don't behave like inertial motion in flat spacetime.

 

[binis]

He's defined part of a coordinate system where the Earth clocks tick slow

I had set (#5) one frame to probe the problem.I think this is reasonable, but

One can pick a pair of inertial rest frames and see a symmetry

why you usually use a pair of frames?

 

[jbriggs444]

I had set (#5) one frame to probe the problem.I think this is reasonable, but

why you usually use a pair of frames?

One rest frame for Earth. One rest frame for satellite. If you want to calculate time dilation for one frame's elapsed time ($\Delta t$) compared with another frame's elapsed time ($\Delta t'$), you need two frames.

You could use one frame if you wanted to compare a change in coordinate time with elapsed proper time. But then you'd still need another frame if you wanted to invoke "symmetry".

 

[Ibix]

I had set (#5) one frame to probe the problem.I think this is reasonable, but

why you usually use a pair of frames?

You have a choice.

You can consider a weak field GR scenario where the ISS moves inertially in curved spacetime. In this case, there are no global inertial frames and you cannot use statements like "velocity is relative" to import SR intuitions about how clocks ought to behave. You can, as @jbriggs444 did, approximately define an inertial frame instantaneously covering the ISS and an Earth clock below it- in fact you can define two, one where the Earth clock is at rest and one where the ISS is at rest. In both those frames the at-rest clock will see the other tick slow. However these frames are only valid (even approximately) for very short times compared to an orbit and you cannot construct a string of such frames and chain them together. Attempting to chain them will eventually lead to you trying to set a single clock to two different times because the chaining is inconsistent, and resolving this will lead you to my earlier statement - clock rates depend on how you choose to resolve the inconsistency, but will always average to the Earth clock ticking fast as seen by the ISS and the ISS clocks being seen to tick slow by the Earth.

Your other option is to imagine flat spacetime with the ISS on a string circling an inertial clock. In this case the ISS is clearly not moving inertially. Again, you can define instantaneous inertial rest frames but again you cannot chain them together over an orbit because you'll have clock synchronisation issues if you try. Instantaneous clock rates will depend how you resolve that, but any approach will average over one orbit to the inertial clock ticking fast as measured by the ISS clock and the ISS clock ticking slow as seen by the inertial clock.

I strongly recommend attempting to make sense of the ISS-on-a-string case before involving gravity.

 

[binis]

In this system it could measure the orbital period of my head, and would come out with a value that is lower than my value by a factor of $\gamma$.

clock rates depend on how you choose to resolve the inconsistency. Instantaneous clock rates will depend how you resolve that

Depending on how you deal with that, the relative clock rates may vary.He's defined part of a coordinate system where the Earth clocks tick slow.

Is time frame dependent?

 

[Ibix]

Is time frame dependent?

Coordinate time is coordinate dependent. Proper time is not. Relative clock rates may or may not be, depending on circumstances and what exactly you are measuring.

What are you hoping to get out of this thread? To me, we don't seem to be advancing anywhere. We're six pages in and still going round and round the fact that "time" doesn't have one unique meaning in relativity. It's two or three distinct concepts, some of which are frame dependent so mean different things to different people. And you still seem to be asking questions looking for general truths that simply aren't there in relativity in the form your questions suppose. You always need to specify who is measuring something and at least the basics of how they do it.

What understanding are you trying to get here? Because I suspect that this thread isn't getting you there.

 

[binis]

You have a choice.

I choose the ECI frame.

Your other option is to imagine flat spacetime with the ISS on a string circling an inertial clock.

"Imagination is more important than thought". I can imagine curved spacetime with a geosynchronous satellite on a rope circling the earth. Or, better, on a pillar. From my point of view, a clock on the ground floor and another clock on the top of a skyscraper are stationary each to the other. A clock on the base of the pillar and another clock on the satellite are immobile each to the other. So,my OP question becomes a pure GR problem. Do you agree?

 

[jbriggs444]

I choose the ECI frame.

"Imagination is more important than thought". I can imagine curved spacetime with a geosynchronous satellite on a rope circling the earth. Or, better, on a pillar. From my point of view, a clock on the ground floor and another clock on the top of a skyscraper are stationary each to the other. A clock on the base of the pillar and another clock on the satellite are immobile each to the other. So,my OP question becomes a pure GR problem. Do you agree?

On the one hand you are choosing special relativity and the ECI frame.

On the other hand, you have characterized an earth-bound clock and a clock on a pillar to be "stationary relative to each other". This means that you have chosen two more frames (one base-centered, one perch-centered and both rotating) and then claimed that this converts the situation to one involving GR.

Stationary relative to is a declaration that the one object defines a frame of reference against which the position of the other is measured and found to be constant. Stationary relative to each other would further declare that the situation is symmetric. The only way to make this fit the scenario you are proposing about a rotating pair of objects is if both frames are rotating.

That sounds like four choices. (Earth-centered inertial, base-centered rotating, perch-centered rotating, coordinate-free GR).

You are free to choose. But you should actually choose.

In addition, invoking the ECI frame in one breath and a curved spacetime in the next is a contradiction. There are no inertial frames that cover a curved region of spacetime. You can choose to put the satellite on a string and use inertial frames and SR. Or you can choose to untether the satellite and use the curved spacetime of GR. But you have to actually choose.

 

[Ibix]

I can imagine curved spacetime with a geosynchronous satellite on a rope circling the earth.

Of course you can. The question is whether your imagination matches GR, and the evidence would suggest not.

From my point of view, a clock on the ground floor and another clock on the top of a skyscraper are stationary each to the other. A clock on the base of the pillar and another clock on the satellite are immobile each to the other. So,my OP question becomes a pure GR problem. Do you agree?

This is all rather confusing. I think you are trying to ask about a clock at the base of a tower compared to a clock at the top of a tower. This is not the same as a satellite, except in the special case of a satellite in a geosynchronous orbit - but I will assume that we are talking about the tower case.

No, this is not a pure gravitational time dilation case, because the Earth is spinning. The clock at the top of the tower is moving in a larger circle than the one at the bottom of the tower, so the tick rate ratio will not be the same as it would be if the Earth were not rotating (a tower on a non-rotating Earth would be a pure gravitational time dilation case). For practical purposes the kinematic time dilation difference due to the spin of the Earth is negligible, but it is there in principle.

Despite this, both clocks could exchange radar pulses and confirm that the distance to the other one is constant. Furthermore, if they emit radar pulses once per second by their own time, they can directly compare their tick rates to the received pulse rate from the other. They will agree that the higher clock ticks faster. On a rapidly rotating Earth they could find otherwise (I'd have to do more maths than I want to do to work out when that is - not sure about how fast Earth has to spin before the Schwarzschild metric isn't really appropriate).

Note how this differs from a general satellite - the distance between the clocks is unchanging, and the flight time of the radar pulses is unchanging, so the clocks are in some sense doing the same thing all the time. For a satellite and ground station the relative positions would be changing and there is a lot more flexibility in how one interprets the changing flight time and Doppler of the radar pulses. That's where the "it depends what coordinates you are using" comes in unless you do something like take an average over one orbit, in which case there is a clear answer.