Rotating Earth as an inertial frame

[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

<br /> \begin{equation*}<br /> \begin{split}<br /> x&#039; &amp;= x \cos \left(\omega t \right) - y \sin \left( \omega t \right) \\<br /> y&#039; &amp;= x \sin \left(wt \right) + y \cos( \omega t) \\<br /> z&#039; &amp;= z \\<br /> t&#039; &amp;= t.<br /> \end{split}<br /> \end{equation*}<br />

Then, where it looks like coordinate speeds exceed c, t&#039; is not even a timelike coordinate, i.e., t&#039; 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&#039; 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

<br /> \begin{equation*}<br /> \begin{split}<br /> x&#039; &amp;= x \cos \left(\omega t \right) - y \sin \left( \omega t \right) \\<br /> y&#039; &amp;= x \sin \left(wt \right) + y \cos( \omega t) \\<br /> z&#039; &amp;= z \\<br /> t&#039; &amp;= t.<br /> \end{split}<br /> \end{equation*}<br />

Then, where it looks like coordinate speeds exceed c, t&#039; is not even a timelike coordinate, i.e., t&#039; 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&#039; is not even a timelike coordinate, i.e., t&#039; 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&#039;^2 \left(c^2 - \omega ^2 (x&#039;^2 + y&#039;^2) \right)+2 \omega dt&#039; (dy&#039; x&#039; -dx&#039; y&#039;) -dx&#039;^2-dy&#039;^2-dz&#039;^2

For an object "at rest" in this frame (dx'=dy'=dz'=0) this simplifies to:
ds^2=dt&#039;^2 \left(c^2 - \omega ^2 (x&#039;^2 + y&#039;^2) \right)

Which is clearly spacelike for any \omega ^2 (x&#039;^2 + y&#039;^2) &gt; c^2

 

[atyy]

To follow up with the brute force approach. The metric in George's rotating frame is:
ds^2=dt&#039;^2 \left(c^2 - \omega ^2 (x&#039;^2 + y&#039;^2) \right)+2 \omega dt&#039; (dy&#039; x&#039; -dx&#039; y&#039;) -dx&#039;^2-dy&#039;^2-dz&#039;^2

For an object "at rest" in this frame (dx'=dy'=dz'=0) this simplifies to:
ds^2=dt&#039;^2 \left(c^2 - \omega ^2 (x&#039;^2 + y&#039;^2) \right)

Which is clearly spacelike for any \omega ^2 (x&#039;^2 + y&#039;^2) &gt; 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.

 

[JesseM]

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.

What do you mean by "can use"--are you implying the observer can't use other coordinate systems where they aren't the center of rotation? As I've asked before, do you understand that as far as making physical predictions goes, anyone can use absolutely any coordinate system whatsoever? For example, an inertial observer in SR has no obligation to use their own rest frame when making predictions, they could just as easily use a frame where they are moving at 0.99c.

 

[Dale]

Option one:
As best as I can Tell GR requires that an Observer that observes rotations; can use any rotating frame of reference

Yes, in GR you can use absolutely any coordinate system you choose.

with themselves at the center of rotation they Like to simplify how something is observed.

GR does not require an observer to be at the center of rotation.

But cannot be required to hold such a selection as “the preferred” rotating frame of reference.

Correct, no coordinate system (frame of reference) is preferred over any other. That is why you can use anyone you choose.

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.

Yes, timelike worldlines may have coordinate speeds > c in some coordinate systems.

Option two:
I cannot tell if you are. saying GR is allowing the defining of a “preferred frame of rotation”.

No, there is no preferred coordinate system.

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.

I don't follow. Why would the freedom to choose an observer's distance from the center of rotation imply that a frame is prefered?

If anything, I would think the opposite would be true: any required restriction to a coordinate system would define a preferred set of frames - those that fulfilled the requirements. I think you have this backwards.

As a “preferred frame” it should not see FTL events due to rotation. .

There is no preferred frame in GR.

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 would say that your option 1 is closer to correct, with the only modification being that there is no restriction on coordinate systems in GR, including no restriction on the location of observers.

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

Thanks for the attempt. I hope my responses answered your question.

 

[RandallB]

There is no preferred frame in GR.
I would say that your option 1 is closer to correct,
with the only modification being that there is no restriction on coordinate systems in GR, including no restriction on the location of observers.

Thanks for the attempt. I hope my responses answered your question.

Well I guess that is close enough for me. As I could not see how option 2 could be the case. And certainly under the option 1 (mine or as refined by you) it is clear the OP problem is not a issue for GR.

The only part that I do not quite get is how a GR observer can assume a frame rotation centered somewhere else than at their own location.
1) It seems like requiring information from a second observer – where SR; for example, requires using just one frame of referance at a time.
2) Plus what little math I've seen on rotating GR applications do not seem to use rotations dislocated from the observer in used (seems like defining two points of rotation with one at the observer point holding an alignment to the a distant point of rotation).

But as I said before I don't know GR applications that well especially related to rotating systems – not important that my understanding get that advanced, it may be buried inside background independence anyway.
Thanks for your input; I think we have over killed the OP issue here.

 

[JesseM]

The only part that I do not quite get is how a GR observer can assume a frame rotation centered somewhere else than at their own location.
1) It seems like requiring information from a second observer – where SR E.G. requires using just one.

Once again, do you understand that even in SR it is merely a matter of linguistic convention that we refer to an observer's rest frame as "their" frame, that there is absolutely nothing stopping an inertial observer from using an inertial frame where they are moving at 0.99c in order to make physical predictions?

 

[Dale]

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?

Your guess was exactly correct. The time coordinate is spacelike at large radii, and a massive object's worldline must always be timelike. Therefore, no object can be "at rest" in these coordinates at large radii.

 

[JustinLevy]

[quote="JustinLevy]Please be careful what you say here.
Locally they are the same. If a local lorentz frame has a metric with diagonal -1,1,1,1 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).[/QUOTE]
Well, yes, due to using non-rectilinear coordinates the metric isn't diagonal -1,1,1,1. I guess I should have just said the metric is the same locally at r=0 for that rotating frame and an inertial frame.

For instance if you look at the equation you referenced, setting \Omega=0 would be an inertial frame. At r=0, the metric is the same regardless of the value of omega. So the inertial frame and rotating frame are the same locally.

If I'm somehow misunderstanding something, please do let me know.

 

[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).

Well, yes, due to using non-rectilinear coordinates the metric isn't diagonal -1,1,1,1. I guess I should have just said the metric is the same locally at r=0 for that rotating frame and an inertial frame.

Thanks. I wasn't talking about r=0. I was implicitly talking about non-zero distances, such as the distance to a quasar. Or at least the radius of a ring laser gyroscope. Speaking of which,

DH, I'm going a little beyond Wikipedia here. How does one distinguish between an inertial frame and a rotating frame of reference? And please don't appeal to the fixed stars, which suggests instantaneous action at a distance.

A Foucault pendulum or a even better, a set of three orthogonal ring laser gyros will do exactly what you asked. In fact, spacecraft do exactly this. One of the myriad pre-launch checks performed by spacecraft avionics is to answer the question "do my inertial measurement units report that the vehicle is in the expected non-inertial frame?" The accelerometers should report that the vehicle is accelerating upward at about 9.8 meters/second² and the gyros should report that the vehicle is rotating at 2*pi/sidereal day about the Earth's rotation axis. Failure to detect these known conditions scrubs the mission.