I What is the angular velocity of a satellite?

[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.)

 

[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).