Rotating Earth as an inertial frame

[Dale]

The original question has not been answered satisfactorily: the rotating Earth is, in fact, an inertial frame under the standard definition: "an inertial frame is a coordinate system tied to the state of the observer."

Ahh, that explains the confusion. This is most certainly not the definition of an inertial frame.

You can use anyone of several equivalent definitions of an inertial reference frame:
1) Newtonian definition: a reference frame where dp/dt=F.
2) Fictitious forces: a reference frame with no fictitious forces.
3) SR - standard form: a reference frame where all laws take take their "textbook" form.
4) GR - accelerometers: an observer* where an attached ideal accelerometer reads 0.

For a rotating reference frame you have:
1) dp/dt = F - 2mω x v - mω x (ω x r) - m dω/dt x r ≠ F
2) the fictitious Coriolis, centrifugal, and Euler forces exist
3) laws don't take their "textbook" form, e.g. see 1) for the form of Newton's 2nd law
4) an ideal accelerometer on the surface of the Earth reads g - 2ω x v - ω x (ω x r)

*Note, in GR there is no distinction between inertial and non-inertial frames (global frames), just between inertial and non-inertial observers (local frames).

 

[matheinste]

Hello Tam Hunt

Rindler describes an inertial frame as non-rotating uniformly moving.

Hartle says inertial frames are non-ritating

Schutz says that in an inertial frame all points are at rest with relation to the origin. This is not true of a rotating frame. (My favourite)

Weinberg (Gravitation) says the surface of the Earth is not an inertial frame.

These are just the better known authors. I could go on but what is the point

When i have some time, if nobody else does it first i will look up a reference for the apparent superluminal movement of stars
Matheinste

 

[RandallB]

... , in GR there is no distinction between inertial and non-inertial frames (global frames), just between inertial and non-inertial observers (local frames).

But that does not clear up my question/point in post #30.
Clearly you can have an accelerating (and therefore non-inertial) observer that does not rotate positioned at any coordinate in a non-rotating frame with rectilinear accelerations and “jerks” (accelerations of acceleration) frame.

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

If that is true – then there is no issue for Tam.
I do not see how GR can use any position on a rotating frame as an obsevation point – but I’m not saying to Tam that this is true and a resolution to his issue
– because I do not know GR that well that is why I ask:

Does anyone here know GR well enough to say if it allows placing a non-inertial observer at any distance on a radial of a rotating frame or does it require it be at the center of rotation.?

 

[Dale]

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

An observer can follow any arbitrary (timelike) worldline in any arbitrary coordinate system in GR. They are not restricted to be stationary in the coordinate system nor are they restricted to a "special" location. As long as you know the metric you can do the physics.

 

[atyy]

I suggest you read up on the International Celestial Reference Frame, for example http://aa.usno.navy.mil/faq/docs/ICRS_doc.php" at iers.org.

Thanks for the links. I remember looking them up once before after reading a previous post of yours, and read something about a specification of a metric tensor. If there's a metric tensor, how can it be an inertial frame? Or is it a Minkowski metric written in some unfamiliar coordinates?

 

[D H]

Thanks for the links. I remember looking them up once before after reading a previous post of yours, and read something about a specification of a metric tensor. If there's a metric tensor, how can it be an inertial frame? Or is it a Minkowski metric written in some unfamiliar coordinates?

The ICRF is a pseudo inertial frame. First off, there is no such thing as a true inertial frame (name one!). Secondly, the frame origin is at the solar system barycenter, so it is an accelerating frame. Thirdly, it is a Cartesian frame. Better stated, the ICRF is a non-rotating frame. The gory details are in IERS Technical Note 29. The five "Comparision of 'Old' and 'New' Concepts" position papers starting on PDF page 25 are particularly relevant. IERS Technical Note 29 can be accessed (no fee) from http://www.iers.org/MainDisp.csl?pid=46-25773. You can download the entire document or just download the relevant sections. The five comparison position papers can be downloaded individually (in which case, ignore the comment about PDF page 25).

 

[JesseM]

The question is; if you have a rotating frame does GR require that center of rotation, which may also move with non-inertial rectilinear accelerations, be the only place a valid GR non-inertial observer can be positioned?

This question doesn't really make sense to me--what do you mean by "observer", exactly? Coordinate systems aren't linked to "observers" at unique positions, you can have a human observer at absolutely any position in any coordinate system, just like you can have an asteroid at absolutely any position in any coordinate system, human observers aren't treated any differently from any other physical object in relativity. Of course if an observer measures the velocity of some nearby object (like a light beam) in his own local neighborhood using a locally inertial coordinate system, the velocity in these locally inertial coordinates may be totally different from the velocity of the same object in a non-inertial coordinate system like a rotating frame (in particular, an observer will always measure light to move past him at c in locally inertial coordinates, even though in a non-inertial frame the light may move past him at some velocity very different from c).

 

[truhaht]

The original question has not been answered satisfactorily
... me, on the surface of the earth. As the stars whip by each night, they are, from my reference frame, moving far faster than the speed of light. No one has satisfactorily answered why this is allowed
... From my point of view, the stars are moving, relative to me, far faster than the speed of light, in a radial fashion.

Sure, I answered it justly in post #25. And you don't mean "radial"; you mean "tangentially", which is quite the opposite. If a star were moving faster than the speed of light radially from you, then that would really be news! That would be a violation. No actual object/body/particle can move toward or away from another at light speed. In the rotating Earth frame, no celestial object moves superluminally with regard to you or with regard to any other actual material thing. That's essential. The superluminal speeds to which you refer are motions of real bodies with respect to the coordinate lines that you've defined in your mind's eye, and those aren't actual physical things. This is the essence -- TRUST me!

 

[RandallB]

An observer can follow any arbitrary (timelike) worldline in any arbitrary coordinate system in GR. They are not restricted to be stationary in the coordinate system nor are they restricted to a "special" location. As long as you know the metric you can do the physics.

but they canot move in the frame they are in.
(Edit; better to state “in” as used here; as the reference frame defined as the “rest Frame” for the observer’s “location”; location understood as a “point particle” location.)
Do they have to be at the center of rotation in their frame?

 

[JesseM]

but they canot move in the frame they are in. Do they have to be at the ceneter of rotation in their frame?

Observers aren't "in" one frame or another, all frames make the same predictions about all physical observations they make. For inertial observers you can talk about their unique rest frame, for non-inertial ones there are an infinite variety of coordinate systems in which they remain at rest.

 

[Dale]

but they canot move in the frame they are in. Do they have to be at the ceneter of rotation in their frame?

No, for example, an observer located on the end of a turbine blade would be at rest in a rotating frame centered on the axle. Such an observer would obviously be non-inertial.

However, I would like you to pay attention to JesseM's point. Observers aren't "in" one frame, they "have" a frame (or rather many frames) where they are at rest. For an observer O this is called "O's rest frame" or simply "O's frame". But they are "in" all frames.

 

[Tam Hunt]

I'm making some progress here, so thanks to everyone for the great information.

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

So, let's dispense with inertial frames and any discussion of SR and focus instead on frames of reference more generally, and GR.

I'm still having trouble understanding how GR explains the apparent superluminal motion of the stars with respect to the rotating Earth as a reference frame (not an inertial frame). JustinLevy hasn't chimed in further re Poincare symmetry, but perhaps DaleSpam or someone else could chime in on this?

Truhaht, thanks for pointing out my error in terminology - you are right that I meant tangential motion, not radial. However, it seems that your response still doesn't explain the apparent superluminal motion. Just as the Earth rotates about the center of gravity of the solar system, we validly describe the fixed stars as rotating around the Earth (regardless of the fact that the Earth is most definitely not the center of gravity of the fixed stars). As such, this apparent motion is, according to the general principle of relativity, as 'real' as the motion of the Earth around the Sun. All motion is relative and, in my hypothetical, I am describing the motion of the fixed stars relative to the rotating earth. And in this description, it seems to me still that superluminal velocities are being observed.

The Stanford Enc. entry suggests at the end that GR describes space as not a fixed coordinate system, as in SR or classical mechanics, but a variable spacetime system deformed by matter and energy throughout. As such, is this what JustinLevy has described as the distinction between local poincare symmetry and global poincare symmetry? If so, I'd like to know how the distinction is made. At what extent does local give way to global? It seems that such a distinction must necessarily be rather arbitrary and thus rather dubious epistemologically and ontologically.

 

[truhaht]

Truhaht, thanks for pointing out my error in terminology - you are right that I meant tangential motion, not radial. However, it seems that your response still doesn't explain the apparent superluminal motion. Just as the Earth rotates about the center of gravity of the solar system, we validly describe the fixed stars as rotating around the Earth (regardless of the fact that the Earth is most definitely not the center of gravity of the fixed stars). As such, this apparent motion is, according to the general principle of relativity, as 'real' as the motion of the Earth around the Sun.

Real, sure, I suppose, just not superluminal. But if you're so convinced it is real then why did you just then above refer to them as merely "apparent motion", hmmm?

All motion is relative and, in my hypothetical, I am describing the motion of the fixed stars relative to the rotating earth. And in this description, it seems to me still that superluminal velocities are being observed.

Well then you're wrong because superluminal velocities are NOT being observed and in general, superluminal velocities (obviously) CANNOT be observed. Superluminal velocities will not be observed by anyone at any time. You can define your observing frame as the rotating Earth surface alright, but those stars aren't moving superluminally "relative to the rotating earth" (your words). Inertial frame SR math is simple and so it can readily be applied to the entirety of the observer's X-Y-Z space. From an inertial frame observatory, if something moves transverse to your line of sight, then that viewed object's speed is still computed as being relative to your defined coordinate axes and the speed will not exceed light speed. But in actuality, if you can just twist your neck and thereby follow the movement of that transverse-moving body, then its speed approximates zero, relative to your twisting (rotating) head. In order to address the hard core crux of the superluminal prohibition without such ambiguities, one must talk about the most direct case, and that is the case of a body moving straight toward or straight away from the observer. That speed CANNOT be superluminal. And since any tangible body/object/particle can potentially be an observatory, you will never ever encounter superluminal speed of one object directly receding from or directly closing in on another material object.

All the above is no different from what I asserted in my other two posts: a mere cerebral concoction of coordinate lines in superluminal motion with respect to the paths of tangible bodies does not a violation make.

 

[JesseM]

I'm making some progress here, so thanks to everyone for the great information.

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

There certainly is such a thing as an inertial frame in the flat spacetime SR. And in GR there are local inertial frames, meaning that if you zoom in on an arbitrarily small neighborhood of a point in curved spacetime, the spacetime in that neighborhood gets arbitrarily close to flat. This is the basis for the equivalence principle.

I'm still having trouble understanding how GR explains the apparent superluminal motion of the stars with respect to the rotating Earth as a reference frame (not an inertial frame). JustinLevy hasn't chimed in further re Poincare symmetry, but perhaps DaleSpam or someone else could chime in on this?

Again, light only is guaranteed to travel at c in inertial frames, and a rotating frame is not one. In the local inertial frame of an observer who passes right next to a light beam, that light beam is moving at c.

The Stanford Enc. entry suggests at the end that GR describes space as not a fixed coordinate system, as in SR or classical mechanics, but a variable spacetime system deformed by matter and energy throughout. As such, is this what JustinLevy has described as the distinction between local poincare symmetry and global poincare symmetry? If so, I'd like to know how the distinction is made. At what extent does local give way to global? It seems that such a distinction must necessarily be rather arbitrary and thus rather dubious epistemologically and ontologically.

My understanding is that global Poincare symmetry only applies in the case of flat SR spacetime, which is what GR would predict in the case of a universe with no mass or energy anywhere (or infinitesimal amounts). And local Poincare symmetry is the idea I described above that GR reduces to SR in the limit as you pick smaller and smaller neighborhoods of a point in spacetime (the equivalence principle article I linked to may help you to understand this idea, I recommend reading it all the way through).

 

[D H]

From the various posts and from some additional research (Stanford Enc. of Philosophy has a great entry on inertial frames), I see now that there is no such thing as an inertial frame. Einstein actually pointed this out in his 1938 book The Evolution of Physics, but I didn't fully internalize the reasoning.

Whoa now. You are once again using faulty references and taking those faulty references out of context. The Stanford Encyclopedia of Philosophy is a great resource -- if you want to study philosophy, that is. Using it as the basis for reasoning about physics is faulty. The basis for your claim "there is no such thing as an inertial frame" is apparently this from http://plato.stanford.edu/entries/spacetime-iframes/: "the variable curvature of spacetime makes the imposition of a global inertial frame impossible". You omitted the key word global. Moreover, the referenced page does not address the issue of rotation whatsoever, so it is quite out-of-context.

Cherry-picking, quoting out of context, use of faulty sources, and use of logical fallacies might be acceptable in environmental law, it is not acceptable in physics. The preferred language for physics is mathematics. The mathematics of general relativity is embodied in the metric tensor. The metric tensor for a non-rotating frame differs from that for versus rotating frames versus a rotating frame are distinguishable. All local Lorentz frames are non-rotating and have an origin that follows a geodesic: They are a local inertial frame. In such a frame the distant stars will not have superluminal velocity.

 

[JustinLevy]

The preferred language for physics is mathematics. The mathematics of general relativity is embodied in the metric tensor. The metric tensor for a non-rotating frame differs from that for versus rotating frames versus a rotating frame are distinguishable. All local Lorentz frames are non-rotating and have an origin that follows a geodesic: They are a local inertial frame.

Please be careful what you say here.
Locally they are the same. If a local lorentz frame has a metric with diagonal -1,1,1 at the origin, then so too does the origin in a rotating system defined with the origin following a geodesic.

In such a frame the distant stars will not have superluminal velocity.

Tam,
please note that in the strict sense this is correct. Nothing moves faster than literally what light travels at that location and in that direction. This does not mean the coordinate velocity of the stars is restricted to be less than c. Do you understand the difference?

Much of your problem seems to stem from an overly physical interpretation of coordinate systems. Since you are having trouble with GR since it requires reducing many arguments to local arguments, I feel some of the essence is getting lost in the mix here.

So let me give you an example, and in SR (flat spacetime). Consider a marble free floating in space. Let's choose this as the origin for our coordinate system. Let's label all spatial locations using standard rulers measuring from that origin. Let's also label all times using clocks at the event being labelled, and the clocks will be stationary with respect to the marble.

Sounds like an inertial coordinate system right?
Well, an inertial coordinate system would indeed fit that description.
However, coordinate systems in which the coordinate speed of light is not constant (changes depending on direction) are also possible which fit that description. This can be done by merely changing our synchronization convention. So even in SR, in flat spacetime, and even restricting ourselves to labelling time coordinates with clocks and spatial coordinates with rulers ... the coordinate speed of light need not be c.


If you understand this, you will understand why parts of the universe can "expand" away from us at faster than c, or can stars move faster than c in a rotating frame ... and yet nothing ever moves faster than the literal speed of light at that location and in that direction.

Just because you found a coordinate system where the coordinate velocity of an object is greater than c, does not mean you found a problem with relativity.

 

[D H]

Please be careful what you say here.
Locally they are the same. If a local lorentz frame has a metric with diagonal -1,1,1 at the origin, then so too does the origin in a rotating system defined with the origin following a geodesic.

A caveat: I am a GR potzer. That said, what you just said appears to conflict with my understanding of the Born metric for a rotating frame. For example, see http://arxiv.org/abs/gr-qc/0305084 equation (43) (pdf page 22).

 

[Dale]

I'm still having trouble understanding how GR explains the apparent superluminal motion of the stars with respect to the rotating Earth as a reference frame (not an inertial frame).

Hi Tam, this question was already answered back in post #27 by Justin.

Yes, the stars in a rotating frame are traveling faster than c. This does not violate relativity. The points on the world-line of the star are still time-like separated, just like they were according to the inertial frame.

Based on your responses I am going to assume that you may not have been introduced to metrics and spacetime intervals and therefore don't understand the geometric distinction that is being made here. My apologies if I am being overly pedantic.

First, from SR I am sure that you are aware that time is dilated and lengths are contracted in a moving frame. However, there is a "distance" that is invariant, i.e. its value is the same in all reference frames. This is the http://en.wikipedia.org/wiki/Spacetime#Basic_concepts". The equation which describes the spacetime interval in a particular coordinate system is called the "metric". For a traditional SR inertial reference frame the metric is ds²=c²dt²-dx²-dy²-dz². In other coordinate systems the metric takes a different form, but all coordinate systems agree on the value of the metric along any path.

Now, starting from any arbitrary event the metric divides spacetime into three regions, the region where the interval ds²>0 (aka timelike), the region where ds²<0 (aka spacelike), the region where ds²=0 (aka lightlike or null). Since all different coordinate systems agree on the interval they will also all agree on this division of spacetime. Geometrically, this is the same as placing a light cone centered on the event, the timelike region is the set of all events inside the light cone, the spacelike region is the set of all events outside the light cone, and the lightlike region is the set of all events on the light cone.

In a traditional SR inertial reference frame the coordinate velocity of an object can be written v=sqrt(dx²+dy²+dz²)/dt. Different inertial frames will disagree on this quantity, but as long as this frame-variant coordinate expression gives v<c everywhere on the object's worldline then the frame-invariant spacetime interval will always be timelike. This is what is meant by a "timelike worldline", and geometrically it means that the worldline always remains inside the light cone.

Although not all coordinate systems agree on v they do all agree if a worldline is timelike. In fact, for many non-inertial coordinate systems it is not even possible to uniquely define a meaningful coordinate velocity, but it is always possible to classify a worldline as timelike or not. So basically, the bottom line is that the coordinate-dependent statement that v<c is only equivalent to the geometric statement that the worldline is timelike for traditional SR inertial frames. In other frames timelike worldlines may have v>c and in still other frames there may not even be a suitable coordinate velocity v.

I hope that helps you understand Justin's answer to your question.

 

[Tam Hunt]

Thanks again Truhaht. I'm not trying to be difficult, but I don't think you've answered the question still. You state that objects simply cannot exceed the speed of light, but this is a consequence of SR and GR theory, not something that is necessarily written into the laws of the universe. As we've discussed in this thread, there are in fact some exceptions to this limit in GR (not in SR), but they don't apply to my hypothetical.

And theories change. There are a number of challengers to GR today (MOND, MOG, process physics), developed due to the physical anomalies that haven't been adequately explained by GR, such as the accelerating expansion of the universe, Pioneer anomalies, rotation of galaxies, etc.

Re turning one's head and changing the frame of reference, this is in fact a legitimate frame of reference in GR. It's quite clear that ANY frame of reference is equivalent for describing physical phenomena. So, yes, assigning the center of one's head or the tip of one's nose to the origin of a coordinate system is legitimate, with all the consequences that follow. It seems, however, that perhaps the "solution" here is what the Stanford Enc. suggests: the mistake is thinking that any assigned coordinate system is rigid. But then we're back to my previous question about where local and global separate and how such distinctions are made.

 

[The Dagda]

There are no discreet frames of reference. This says all you need to know about frames of reference, and it also implies that there are no stationary points in the Universe, which if you think about expansion must be true except at the centre, wherever that is. No mass object can travel faster than c, photons cannot propagate at less than c, no hypothetical FTL objects can travel slower than c. This is implied by experiment and mathematics of SR which are derived from Lorentz transforms in terms of c being the speed limit of the universe. This shows there is no inertial frame, time dilation and contraction and everything else fairly neatly. The OP is a misinterpretation of theory.

At the big bang the expansion of the Universe is suspected to have at some point been faster than c, this does not contradict special relativity as time and space are the co-ordinate system itself not an object in it.

 

[JesseM]

Thanks again Truhaht. I'm not trying to be difficult, but I don't think you've answered the question still. You state that objects simply cannot exceed the speed of light

In inertial frames! There is no law that objects "simply cannot exceed the speed of light" in non-inertial frames.

Re turning one's head and changing the frame of reference, this is in fact a legitimate frame of reference in GR. It's quite clear that ANY frame of reference is equivalent for describing physical phenomena. So, yes, assigning the center of one's head or the tip of one's nose to the origin of a coordinate system is legitimate, with all the consequences that follow. It seems, however, that perhaps the "solution" here is what the Stanford Enc. suggests: the mistake is thinking that any assigned coordinate system is rigid. But then we're back to my previous question about where local and global separate and how such distinctions are made.

"Local" means infinitesimally small--if you're familiar with limits in calculus, it's in the limit as the size of the spacetime region you're considering goes to zero. In this limit the laws of physics can be said to approach those of SR in certain ways which is what's meant by the "equivalence principle" (there are some technicalities, see this thread), and one of these ways is that the speed of light is guaranteed to be c in a "locally inertial frame" in this region.

 

[truhaht]

Thanks again Truhaht.
...You state that objects simply cannot exceed the speed of light, but this is a consequence of SR and GR theory, not something that is necessarily written into the laws of the universe.

I don't like to overuse the term "laws" but sure, the constraint is hard-coded into our universe, presuming only that relativity theory is proven to be correct. If that's too big a presumption for you then so be it, but it sits fine with me.

You seem to be seeking the comfort and security of a GR formulation that helps to elucidate the non-violation of your scenario. That's just dandy. For myself though, I much prefer to cut through all the hairy math and know that sure yea, the lightspeed prohibition is an inviolate feature of our World.

 

[RandallB]

No, for example, an observer located on the end of a turbine blade would be at rest in a rotating frame centered on the axle. Such an observer would obviously be non-inertial.

However, I would like you to pay attention to JesseM's point. Observers aren't "in" one frame, they "have" a frame (or rather many frames) where they are at rest. For an observer O this is called "O's rest frame" or simply "O's frame". But they are "in" all frames.

I edited "in" - post 44.

Your example does not address the points I made in Posts # 30, 38 & 44.

Using your turbine example the “point particle” location of an observer at of a turbine blade can define several “at rest frames of reference”:
One of those “rest frames” could be a non-rotating rectilinear frame with non-inertial accelerations moving the observer point location on a “world line” that takes it in a circular orbit around the axis of the turbine POV. (From the observer POV the axis moves in a circle around it.
Likewise assuming our turbine is part of a jet engine:
A passenger seated in the jet will see the observer moving in a circle displaced some distance while the axis point does not move only turns. But the blade end point observer will also see the passenger moving in a displaced circle.

I do not see where GR allows picking a rotating frame for an observer except that the rotation be centered on the point location of the observer.
Additionally if the observer selects a rotating rest frame that allows the axis point of the turbine to remain at a stationary non-rotating fixed distance. This rotating frame cannot be expected to be preferred over another frame that rotates wrt it, such that the movement of the passenger is observed as a displaced circular orbit.

IMO; Under these conditions/rules the problems Tam describes do not exist, much as simultaneity in SR defines that a preferred frame cannot be defined there.

 

[George Jones]

Al, how are you distinguishing coordinate velocity and relative velocity? Einstein's version of the principle of relativity is that any frame is as good as any other frame for describing phenomena AND that the laws of physics are valid in all frames. If this is the case, then it seems that the rotating Earth's frame would also require that all velocities of objects in that frame cannot exceed c, which is, according to everything I have read on this topic, the upper boundary speed limit as a consequence of the basic equations of relativity (mass goes to infinity as velocity approaches c).

As others have said (I think), there is nothing in either special or general that prohibits coordinate speeds from being greater than $c$ other than misunderstanding, and rhetorical skills do not change this elementary fact.

Consider rotating coordinates defined form standard inertial coordinates by

$$\begin{equation*} \begin{split} x' &= x \cos \left(\omega t \right) - y \sin \left( \omega t \right) \\ y' &= x \sin \left(wt \right) + y \cos( \omega t) \\ z' &= z \\ t' &= t. \end{split} \end{equation*}$$

Then, where it looks like coordinate speeds exceed $c$, $t'$ is not even a timelike coordinate, i.e., $t'$ is a spacelike coordinate! Can anyone see why?

 

[JesseM]

I do not see where GR allows picking a rotating frame for an observer except that the rotation be centered on the point location of the observer.

What does it mean to pick a frame "for" an observer? If you just want to make physical predictions concerning the observer like what he'll see visually, you can do this from the perspective of any frame whatsoever, there's no need to pick one where the observer is at rest.

 

[Dale]

Your example does not address the points I made in Posts # 30, 38 & 44.

I'm sorry; it appears that we are having a miscommunication because I thought I did exactly that. Could you please restate your question in the most clear and concise manner possible (e.g. 2 sentences or less) and without reference to any previous posts.

 

[truhaht]

$t'$ is a spacelike coordinate! Can anyone see why?

Um, is it because its variance results in accelerations, which are +1 higher order spatial flux then the linear time line progress?

Tell me why then.

I like your post too

 

[The Dagda]

As others have said (I think), there is nothing in either special or general that prohibits coordinate speeds from being greater than $c$ other than misunderstanding, and rhetorical skills do not change this elementary fact.

Consider rotating coordinates defined form standard inertial coordinates by

$$\begin{equation*} \begin{split} x' &= x \cos \left(\omega t \right) - y \sin \left( \omega t \right) \\ y' &= x \sin \left(wt \right) + y \cos( \omega t) \\ z' &= z \\ t' &= t. \end{split} \end{equation*}$$

Then, where it looks like coordinate speeds exceed $c$, $t'$ is not even a timelike coordinate, i.e., $t'$ is a spacelike coordinate! Can anyone see why?

Because time an space are intrinsically linked, thus time/space, t', and it's dimensions are one and the same thing. Do I win a prize for stating the obvious?

Time/space in a non euclidian frame is a rotation about an axis, it's as simple as that.

 

[JesseM]

Because time an space are intrinsically linked, thus time/space, t', and it's dimensions are one and the same thing. Do I win a prize for stating the obvious?

Time/space in a non euclidian frame is a rotation about an axis, it's as simple as that.

In relativity there is a clear difference between paths that are timelike, spacelike, and lightlike, and which category a path falls into is a coordinate-independent fact (physically, a timelike path is the worldline of an object moving slower than light, a lightlike path is the worldline of a light beam, and a spacelike path cannot be treated as any actual object's worldline unless we allow FTL particles). The metric gives a coordinate-invariant notion of the "spacetime distance" along a path ds^2 (which for timelike paths is just -c^2 times the proper time along the path), in much the same way that we can talk about the geometric distance along a path on a curved 2D surface like a sphere even though there are different possible coordinate systems you could place on that surface. And ds^2 is negative for timelike paths, 0 for lightlike paths, and positive for spacelike paths. So to say a coordinate is "timelike" at a point means that if you consider the infinitesimal path created by varying that coordinate an infinitesimal amount from the point while keeping all the other coordinates constant, this path would be a timelike one.

 

[atyy]

Then, where it looks like coordinate speeds exceed $c$, $t'$ is not even a timelike coordinate, i.e., $t'$ is a spacelike coordinate! Can anyone see why?

I suppose you are asking for a non-brute force way to see this, since one should always guess before calculating: t' points along the worldline that is an upward spiral around the t axis, as x' gets bigger and bigger the spiral gets shallower so that t' eventually ends up less parallel to the t axis and more parallel to the x axis?

 

[The Dagda]

In relativity there is a clear difference between paths that are timelike, spacelike, and lightlike, and which category a path falls into is a coordinate-independent fact (physically, a timelike path is the worldline of an object moving slower than light, a lightlike path is the worldline of a light beam, and a spacelike path cannot be treated as any actual object's worldline unless we allow FTL particles). The metric gives a coordinate-invariant notion of the "spacetime distance" along a path ds^2 (which for timelike paths is just -c^2 times the proper time along the path), in much the same way that we can talk about the geometric distance along a path on a curved 2D surface like a sphere even though there are different possible coordinate systems you could place on that surface. And ds^2 is negative for timelike paths, 0 for lightlike paths, and positive for spacelike paths. So to say a coordinate is "timelike" at a point means that if you consider the infinitesimal path created by varying that coordinate an infinitesimal amount from the point while keeping all the other coordinates constant, this path would be a timelike one.

Can anything move only in time or only in space?

And how would you describe it?

 

[JesseM]

Can anything move only in time or only in space?

Not in any absolute, coordinate-independent way. Relative to a particular coordinate system, I suppose you could say that an object with a timelike worldline is "moving only in time" if its coordinate position remains constant, but obviously other coordinate systems would disagree. In contrast, the issue of whether a given worldline is timelike, lightlike or spacelike is a coordinate-independent fact that everyone can agree on.

 

[Dale]

I suppose you are asking for a non-brute force way to see this, since one should always guess before calculating: t' points along the worldline that is an upward spiral around the t axis, as x' gets bigger and bigger the spiral gets shallower so that t' eventually ends up less parallel to the t axis and more parallel to the x axis?

To follow up with the brute force approach. The metric in George's rotating frame is:
$$ds^2=dt'^2 \left(c^2 - \omega ^2 (x'^2 + y'^2) \right)+2 \omega dt' (dy' x' -dx' y') -dx'^2-dy'^2-dz'^2$$

For an object "at rest" in this frame (dx'=dy'=dz'=0) this simplifies to:
$$ds^2=dt'^2 \left(c^2 - \omega ^2 (x'^2 + y'^2) \right)$$

Which is clearly spacelike for any $$\omega ^2 (x'^2 + y'^2) > c^2$$

 

[atyy]

To follow up with the brute force approach. The metric in George's rotating frame is:
$$ds^2=dt'^2 \left(c^2 - \omega ^2 (x'^2 + y'^2) \right)+2 \omega dt' (dy' x' -dx' y') -dx'^2-dy'^2-dz'^2$$

For an object "at rest" in this frame (dx'=dy'=dz'=0) this simplifies to:
$$ds^2=dt'^2 \left(c^2 - \omega ^2 (x'^2 + y'^2) \right)$$

Which is clearly spacelike for any $$\omega ^2 (x'^2 + y'^2) > c^2$$

With this in hand, would you like to comment on my guess? I suspect my guess wasn't right, because to answer Tam Hunt's question, shouldn't the worldline at large radii be timelike?

 

[RandallB]

I'm sorry; it appears that we are having a miscommunication because I thought I did exactly that. Could you please restate your question in the most clear and concise manner possible (e.g. 2 sentences or less) and without reference to any previous posts.

Option one:
As best as I can Tell GR requires that an Observer that observes rotations; can use any rotating frame of reference with themselves at the center of rotation they Like to simplify how something is observed. But cannot be required to hold such a selection as “the preferred” rotating frame of reference. Thus if an alternate item of is considered for observation and the same “non-preferred” frame of rotation is continued to be used – then FTL “coordinate” violations should be expected as possible.

Option two:
I cannot tell if you are saying GR is allowing the defining of a “preferred frame of rotation”.
If so it would seem an observer could be defined not only at the center of rotation but also at any radius from the center of such a preferred frame of rotation. As a “preferred frame” it should not see FTL events due to rotation.

Under the Option one understanding of GR the OP question has no standing.

Under the Option two understanding of GR I do not see how anyone has yet to resolve the OP question.

I don't know how to make the question any simpler.