I What is the angular velocity of a satellite?

[A.T.]

Yes, that is a good point. I was neglecting the lunar gravitational time dilation (pure laziness, no justification).

I don't think that it changes the qualitative answer, that a clock on the Moon runs faster than one on Earth. As the graph shows, the kinematic time dilation approaches zero, so the total time dilation is dominated by the gravitational potential. Near the Moon that gravitational potential drops by only a fraction of the Earth's potential well, so the clocks still run faster than on Earth.

 

[binis]

In #42, @binis said: Conclusively,time is running faster on the moon than the Earth but slower on the satellites.

Time does not run fast or slow.Don't think in terms of time running fast or slow.[/i][/i]

I like this view.Time cannot be extended nor shortened.I prefer to say "apparent" time dilation.

 

[A.T.]

I like this view.Time cannot be extended nor shortened.I prefer to say "apparent" time dilation.

Wordplay will not provide more understanding here. You have to learn how it relates to physical processes.

For example: If you take half of a radioactive rod to the Moon, leave it there for a long time, then bring it back, it will have decayed more than the other half that remained on Earth all the time. The same applies to any other process. So I don't know what is "apparent" about this.

 

[Nugatory]

I like this view.Time cannot be extended nor shortened.I prefer to say "apparent" time dilation.

Better would be to understand the role of relativity of simultaneity (in the simple case of velocity based time dilation and the more complicated case of the twin paradox) and of simultaneity conventions (in the curved spacetime of general relativity).

 

[binis]

If you take half of a radioactive rod to the Moon, leave it there for a long time, then bring it back, it will have decayed more than the other half that remained on Earth all the time.

Have this experiment done on the ISS or on a satellite below 3000 km?

The same applies to any other process.

It is known that gravity affects nuclear decay.Are there any examples for other processes?

 

[jbriggs444]

Have this experiment done on the ISS or on a satellite below 3000 km?

Clearly not. The measurement of nuclear decay rates makes for an insanely clumsy clock. You use it when no other alternative is available (e.g. carbon dating or potassium argon dating). If you want to measure tiny time dilation effects, you want to use good clocks.

For example, if you want to look at gravitational time dilation, something like Pound-Rebka is much more effective than putting one bucket of Uranium on the first floor and another on the second.

There is little point doing expensive and worthless experiments when there are expensive and worthwhile experiments left undone.

 

[valenumr]

The relevant invariant is the vorticity of the congruence of worldlines describing the object.

I think I understand what you mean, but I've thought long and hard in this, WRT Machian principal.

And I recently came to think on it more watching a starlink launch, where the vehicle had to induce spin to create momentum to deploy it's satellites.

Ultimately, I came to the conclusion that angular momentum is explicitly relative to the geodesic. An extended body is non-rotating for local observers if all parts are essentially parallel to the geodesic.

For example, considering a starship of anti-rotating rings along an axis, if energy were just applied to rotate the axis, the local observers would not experience artificial gravity.

However, if energy were applied to rotating the rings, the observes would experience artificial gravity.

Each case is totally distinguishable, merely by the fact that an extended (circular to keep it simple) body will have a velocity at it's extrema that is perpendicular to the motion along the geodesic.

Is that roughly correct? This has been bothering me.

 

[A.T.]

It is known that gravity affects nuclear decay.

It's not the strength of gravity, but the difference gravitational potential.

Are there any examples for other processes?

Yes most clocks use other processes.

 

[PeterDonis]

Ultimately, I came to the conclusion that angular momentum is explicitly relative to the geodesic.

Just to be clear, the "angular momentum" you are talking about is spin, not orbital. You are talking about an object spinning about its own axis, not revolving about some other object. But the OP of this thread was talking about the latter. The former came up in the course of discussion, and is what my post about vorticity that you quoted referred to.

An extended body is non-rotating for local observers if all parts are essentially parallel to the geodesic.

If you make "essentially parallel" mathematically precise, you will find that it means "zero vorticity".

Each case is totally distinguishable, merely by the fact that an extended (circular to keep it simple) body will have a velocity at it's extrema that is perpendicular to the motion along the geodesic.

Here you are talking about 3-velocity, not 4-velocity. Your intuitive idea here actually turns out not to work, for technical reasons that I'm not sure can really be explained at the "I" level. But the upshot is that, as I said above, if you work out the issues and make things precise, you will find that you come up with "nonzero vorticity" as the invariant, precise definition of "rotating" (in the sense of spin).

 

[valenumr]

Just to be clear, the "angular momentum" you are talking about is spin, not orbital. You are talking about an object spinning about its own axis, not revolving about some other object. But the OP of this thread was talking about the latter. The former came up in the course of discussion, and is what my post about vorticity that you quoted referred to.If you make "essentially parallel" mathematically precise, you will find that it means "zero vorticity".Here you are talking about 3-velocity, not 4-velocity. Your intuitive idea here actually turns out not to work, for technical reasons that I'm not sure can really be explained at the "I" level. But the upshot is that, as I said above, if you work out the issues and make things precise, you will find that you come up with "nonzero vorticity" as the invariant, precise definition of "rotating" (in the sense of spin).

Yes, about its own axis. I meant parallel WRT forward motion, and perpendicular "rotation" to that axis. I would be happy to have some pointers on the topic. Rotation and angular momentum are a very interesting point for me.

 

[PeterDonis]

I meant parallel WRT forward motion, and perpendicular "rotation" to that axis.

Yes, but as I noted, this turns out not to work as you state it. The problem is that, as I said in a previous post, "parallel" turns out to mean "zero vorticity" when you make it precise, "rotation" turns out to mean "nonzero vorticity" when you make it precise, and nonzero vorticity turns out to be incompatible with "perpendicular". In more technical language, nonzero vorticity means the congruence of worldlines that describes the object is not hypersurface orthogonal; that means it is impossible to define a family of spacelike "slices" of the object (each representing the object "at an instant of its time") that are perpendicular to the worldlines. (This observation has led, in the past, to such apparent "paradoxes" as the Ehrenfest paradox; I believe we have had some good past PF threads on that topic.)

I would be happy to have some pointers on the topic.

This Wikipedia article at least gives a start (note that it talks about all of the components of the kinematic decomposition, not just vorticity):

https://en.wikipedia.org/wiki/Congr...atical_decomposition_of_a_timelike_congruence

Also the references given at the end of that article are good sources.

 

[binis]

One calculation is performed against an inertial frame and one against a non-inertial frame.

Which one and why is the inertial frame?

 

[Ibix]

Which one and why is the inertial frame?

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.

 

[Prishon]

Summary:: Angular speed ω is by definition the runned angle dθ per time dt elapsed: ω=dθ/dt. If the time elapsed in the center of the Earth is dt, the dilated time elapsed on satellite is dt′. What is the satellite's angular speed? Is it dθ/dt or dθ/dt′?

Angular velocity ω is by definition the runned angle dθ per time dt elapsed: ω=dθ/dt. If the time elapsed in the center of the Earth is dt, the dilated time elapsed on satellite is dt′. What is the satellite's angular velocity? Is it dθ/dt or dθ/dt′?

There is an additional general relativistic rotation.

 

[Ibix]

There is an additional general relativistic rotation.

What do you mean?

 

[Prishon]

What do you mean?

Well, that rotation is very tiny but if you parallel tranport a vector around the curved spacetime of the Earth it is rotated a tiny tiny bit. This doesn't happen in Newtonian spacetime.

 

[Ibix]

Well, that rotation is very tiny but if you parallel tranport a vector around the curved spacetime of the Earth it is rotated a tiny tiny bit. This doesn't happen in Newtonian spacetime.

Can I suggest that, given that we finally seem to have persuaded the OP to leave GR out of the problem while they continue to struggle with SR, introducing extra complexity to this problem is probably a mistake?

 

[Prishon]

Can I suggest that, given that we finally seem to have persuaded the OP to leave GR out of the problem while they continue to struggle with SR, introducing extra complexity to this problem is probably a mistake?

Yes you can.

 

[cianfa72]

However, the rate at which an object rotates (and whether it rotates rigidly) is not invariant. Not all inertial frames will agree about the rate of rotation of an unaccelerated object about its center of mass.

In the sentence above I took 'unaccelerated object' as amounting to the worldline of its center of mass is unaccelerated (i.e. the body's center of mass is in free fall - zero proper acceleration).

 

[PeterDonis]

the rate at which an object rotates (and whether it rotates rigidly) is not invariant.

The first of these statements is true, but the second is not. Rigid rotation, meaning whether or not the congruence of worldlines describing the object is Born rigid (has zero expansion and shear), is an invariant.

 

[cianfa72]

How 'proper angular velocity' as in post#17 is related to the vorticity of the congruence of worldlines describing the object ?

 

[PeterDonis]

How 'proper angular velocity' is related to the vorticity of the congruence of worldlines describing the object ?

The magnitude of the vorticity is the proper angular velocity.

 

[ergospherical]

Vorticity is a 2-form $\boldsymbol{\Omega} = \mathrm{d} \mathbf{u}$ and can be turned into a vector by Hodging and then taking the metric dual, $\boldsymbol{\omega} = (\star \boldsymbol{\Omega})^{\sharp} = \nabla \times \mathbf{u}^{\sharp}$.

The magnitude of the vorticity is the proper angular velocity.

Also the angular velocity of a material element is half the vorticity.

 

[cianfa72]

Related to this topic I would like to ask: in the context of SR and GR what is the proper velocity of an object from a (spacetime) geometrical point of view ?

I'm aware of object's proper acceleration (as measured by an accelerometer attached on it) corresponds in the mathematical model of spacetime (Lorentzian manifold) to the path curvature of the worldline representing it.

What about proper velocity (if any ) ? Thanks.

 

[DrGreg]

Proper velocity is the 3-vector $$ \left( \frac{\text{d} x^1}{\text{d} \tau} , \frac{\text{d} x^2}{\text{d} \tau}, \frac{\text{d} x^3}{\text{d} \tau} \right) $$ which is the spatial component of the 4-velocity. It is also known as "celerity", a name I much prefer because "proper velocity" doesn't really behave like other "proper" things in relativity. As it's a 3-vector, not a 4-vector, it is coordinate-dependent and so doesn't really have a geometrical spacetime interpretation.

To get a geometrical interpretation related to velocity, think of the angle between two intersecting worldlines, the worldline of the observer and the worldline of the object being measured (or, strictly, hyperbolic angle). It's called "rapidity". The speed of one relative to the other is the "slope" of one worldline relative to the other worldline.

 

[cianfa72]

Proper velocity is the 3-vector $$ \left( \frac{\text{d} x^1}{\text{d} \tau} , \frac{\text{d} x^2}{\text{d} \tau}, \frac{\text{d} x^3}{\text{d} \tau} \right) $$ which is the spatial component of the 4-velocity. It is also known as "celerity", a name I much prefer because "proper velocity" doesn't really behave like other "proper" things in relativity. As it's a 3-vector, not a 4-vector, it is coordinate-dependent and so doesn't really have a geometrical spacetime interpretation.

From above the four-velocity, instead, as a 4-vector should be geometrically well defined -- namely the tangent vector to the object worldline (tangent vector as an element of the tangent space defined at each point along the object worldline).

So the claim that velocity is not frame-invariant does not apply to four-velocity, I believe...

 

[PeterDonis]

So the claim that velocity is not frame-invariant does not apply to four-velocity, I believe...

That's correct; the 4-velocity of a given object at a given point is an invariant.

 

[cianfa72]

That's correct; the 4-velocity of a given object at a given point is an invariant.

As for the object's proper acceleration (physically measured by an accelerometer attached to it) there is the analogue for the object's 4-velocity (since we said it is frame-independent - i.e. each frame assign to the given object at a given point the same 4-velocity vector) ?

 

[PeterDonis]

As for the object's proper acceleration (physically measured by an accelerometer attached to it) there is the analogue for the object's 4-velocity (since we said it is frame-independent - i.e. each frame assign to the given object at a given point the same 4-velocity vector) ?

If you are asking if there is a way to directly measure an object's 4-velocity, the way an accelerometer can directly measure the object's 4-acceleration, first we need to ask what, exactly, we measure with an accelerometer.

The simplest kind of accelerometer (such as a bathroom scale) just measures the magnitude of an object's proper acceleration. The magnitude of an object's 4-velocity is always $1$, by definition (or $c$ if you're using conventional units), so there's nothing to measure in that respect.

More complicated accelerometers (like the ones used in inertial navigation systems) include gyroscopes to measure the direction of proper acceleration. But this "direction" is a direction in space and requires a reference system. We can do the same with velocity, but in both cases the direct measurement we make is not quite the same as the direction in spacetime of the corresponding 4-vector (4-acceleration or 4-velocity). We have to calculate that direction in spacetime from the measurement data if that's what we want to know. So in this sense we don't directly measure 4-acceleration (the full 4-vector); we measure things from which we can indirectly calculate 4-acceleration. We can do the same for 4-velocity.

 

[cianfa72]

The simplest kind of accelerometer (such as a bathroom scale) just measures the magnitude of an object's proper acceleration. The magnitude of an object's 4-velocity is always $1$, by definition (or $c$ if you're using conventional units), so there's nothing to measure in that respect.

ok, got it thanks.

More complicated accelerometers (like the ones used in inertial navigation systems) include gyroscopes to measure the direction of proper acceleration. But this "direction" is a direction in space and requires a reference system.

Can you elaborate this point please ?

We have to calculate that direction in spacetime from the measurement data if that's what we want to know. So in this sense we don't directly measure 4-acceleration (the full 4-vector); we measure things from which we can indirectly calculate 4-acceleration. We can do the same for 4-velocity.

How can we actually 'rebuild' the full 4-acceleration vector (or the full 4-velocity vector) from those measurements ?

 

[PeterDonis]

Can you elaborate this point please ?

Consider, say, an inertial navigation system for an airplane on Earth. It will have gyroscopes oriented North-South, East-West, and up-down. The direction it assigns to the instantaneous velocity of the plane will be with respect to those axes; it is a direction in space, where "space" is defined by those axes.

How can we actually 'rebuild' the full 4-acceleration vector (or the full 4-velocity vector) from those measurements ?

You would need a clock traveling along the same worldline as the accelerometer (or inertial navigation system) to provide the proper time. Then you would have to use the clock and the spatial axes of your reference system to construct what amount to Fermi normal coordinates centered on the chosen worldline. You can then treat the spatial 3-vectors from your measurements as the "space" components of 4-vectors, with your clock providing the "time" component, in those coordinates.

 

[vanhees71]

In other words, a smartphone will do. It has a clock as well as an accelerometer

 

[PeterDonis]

In other words, a smartphone will do. It has a clock as well as an accelerometer

If you add the app to do all the 4-vector calculations, sure.

 

[vanhees71]

Well, yes, I only think that neither the clock nor the accelerometer are accurate enough to resolve the relativistic effects. Nevertheless it does pretty well for all kinds of experiments, but that's a bit off-topic in this thread:

https://phyphox.org/

 

[PeterDonis]

I only think that neither the clock nor the accelerometer are accurate enough to resolve the relativistic effects

With current smartphones, yes, I would agree. But someday someone might sell one that has an atomic clock and a high precision inertial navigation system in it.

 

[vanhees71]

Well, maybe then I'll finally buy one!

 

[cianfa72]

Consider, say, an inertial navigation system for an airplane on Earth. It will have gyroscopes oriented North-South, East-West, and up-down.

Sorry for the stupid question: the inertial navigation system you were talking of is actually onboard on the airplane I guess...

 

[PeterDonis]

the inertial navigation system you were talking of is actually onboard on the airplane I guess...

Yes.

 

[cianfa72]

You would need a clock traveling along the same worldline as the accelerometer (or inertial navigation system) to provide the proper time. Then you would have to use the clock and the spatial axes of your reference system to construct what amount to Fermi normal coordinates centered on the chosen worldline. You can then treat the spatial 3-vectors from your measurements as the "space" components of 4-vectors, with your clock providing the "time" component, in those coordinates.

So basically the inertial navigation system (with its gyroscopes oriented North-South, East-West, and up-down, plus a clock attached to it) physically 'implements' the Fermi-Walker transport of the tetrad defined by its four axes (one timelike and three spacelike).

 

[Ibix]

So basically the inertial navigation system (with its gyroscopes oriented North-South, East-West, and up-down, plus a clock attached to it) physically 'implements' the Fermi-Walker transport of the tetrad defined by its four axes (one timelike and three spacelike).

Yes.

 

[cianfa72]

I asked that because I'm often in trouble with the difference (if any) between physics and the mathematical model we use to represent it.

For example we were talking here about timelike and spacelike directions in spacetime. Are we really talking about physics or just of the mathematical model we use to represent it (namely spacelike and timelike vectors elements of the tangent space at each point of the Lorentzian manifold representing the spacetime) ?

I hope my doubt really makes sense...

 

[PeterDonis]

So basically the inertial navigation system (with its gyroscopes oriented North-South, East-West, and up-down, plus a clock attached to it) physically 'implements' the Fermi-Walker transport of the tetrad defined by its four axes (one timelike and three spacelike).

Yes.

For example we were talking here about timelike and spacelike directions in spacetime. Are we really talking about physics or just of the mathematical model we use to represent it

Both. Obviously the mathematical model has spacelike and timelike (and null) vectors in it, but those features of the model directly correspond to physical features of the world--for example, we measure spacelike intervals with rulers but we measure timelike intervals with clocks.

 

[cianfa72]

Both. Obviously the mathematical model has spacelike and timelike (and null) vectors in it, but those features of the model directly correspond to physical features of the world--for example, we measure spacelike intervals with rulers but we measure timelike intervals with clocks.

So, from a physical point of view a spacelike direction at a given event is obtained from a limiting procedure of a set of events spacelike separated w.r.t. the given event, while a timelike direction from a limiting procedure of a set of events timelike separated w.r.t. the given event.

 

[PeterDonis]

So, from a physical point of view a spacelike direction at a given event is obtained from a limiting procedure of a set of events spacelike separated w.r.t. the given event, while a timelike direction from a limiting procedure of a set of events timelike separated w.r.t. the given event.

Not really, because physically you can't perform such a procedure. You can't sit at a particular event in spacetime indefinitely while you extend smaller and smaller rulers from it or measure smaller and smaller clock intervals from it.

If you want to physically realize directions in spacetime, as opposed to intervals, then the direction in which your worldline points at a given event (which will depend on your state of motion at that event) is a timelike direction, and you can realize spacelike directions with gyroscopes. (Note that strictly speaking, you can't realize spacelike directions by pointing at distant objects like stars, because the light coming from those stars, which is what you're actually pointing at, is coming from a null direction, not a spacelike direction.)

 

[cianfa72]

If you want to physically realize directions in spacetime, as opposed to intervals, then the direction in which your worldline points at a given event (which will depend on your state of motion at that event) is a timelike direction, and you can realize spacelike directions with gyroscopes. (Note that strictly speaking, you can't realize spacelike directions by pointing at distant objects like stars, because the light coming from those stars, which is what you're actually pointing at, is coming from a null direction, not a spacelike direction.)

Let me use non technical language to describe my point of view to help intuition.

Just to 'visualize' spacetime events near an object's worldline consider a region of spacetime around it disseminated with exploding firecrackers. W.r.t. a given event on the object's worldline (say event A) we can split up events in the region around it (i.e. exploding firecrackers events) in two subset: the first one includes events can be 'reached' from (to) the given event A by massive objects or light rays (i.e. events inside the light-cone at event A); the second subset is the complement of the first one in that spacetime region.

The first subset defines events timelike separated w.r.t. the event A while the second subset events spacelike separated from it.

Now in order to physically define directions in spacetime at event A we can proceed as follows:

w.r.t. the first subset each path followed by a massive object through A with a different velocity defines a timelike direction in spacetime.

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.

 

[PeterDonis]

The first subset defines events timelike separated w.r.t. the event A

Or lightlike separated, since you've included light rays in the definition.

w.r.t. the first subset each path followed by a massive object through A with a different velocity defines a timelike direction in spacetime.

Yes. And similarly, each path followed by a distinct light ray through A defines a lightlike direction in spacetime.

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.

Yes.

 

[cianfa72]

Or lightlike separated, since you've included light rays in the definition.

Yes. And similarly, each path followed by a distinct light ray through A defines a lightlike direction in spacetime.

Sure, of course

 

[cianfa72]

I was thinking about this other topic: take two timelike separated events in the context of GR.

As definition of timelike separated events take the following: two events are timelike separated if there is at least a timelike path between them (it seems to me a reasonable definition).

My question is: Does that definition amount to say there is a timelike geodesic joining them ?

Thanks for you time !

 

[PeterDonis]

As definition of timelike separated events take the following: two events are timelike separated if there is at least a timelike path between them (it seems to me a reasonable definition).

The usual definition is that there must be a timelike geodesic connecting them. And similarly for null and spacelike separated. 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.

 

[PeterDonis]

As definition of timelike separated events take the following: two events are timelike separated if there is at least a timelike path between them (it seems to me a reasonable definition).

My question is: Does that definition amount to say there is a timelike geodesic joining them ?

We can rephrase the question slightly so that it doesn't depend on a particular definition of "timelike separated": does the existence of any timelike curve between two events necessarily imply the existence of a timelike geodesic between those events?

I think the answer is yes for globally hyperbolic spacetimes, but I'm not sure it is yes for spacetimes that aren't. (For example, I'm not sure the answer is yes for Godel spacetime, which is not globally hyperbolic--it has closed timelike curves.)