Do rotating frames have planes of simultaneity?

[analyst5]

It's a pretty straight-forward question, and it got me confused since most articles on the internet mention planes of simultaneity in the context of inertial frames. So if rotating frames also have planes of simultaneity, what SR says about it and how does it differ from the planes of simultaneity of inertial frames?

Regards

 

[Ben Niehoff]

No, they don't.

If you take a collection of rotating observers, they are accelerating relative to one another, so the notions of simultaneity between nearby observers do not agree.

 

[Dale]

No, they don't.

I would say it a little differently. Simultaneity is a convention and non inertial observers (including rotating ones) don not have a standard convention. But you certainly are free to adopt a non-standard convention for rotating observers, you just have to be explicit in what your convention is since nobody will know it.

 

[WannabeNewton]

To add on to DaleSpam, see the following: http://mathpages.com/rr/s4-08/4-08.htm

 

[analyst5]

I would say it a little differently. Simultaneity is a convention and non inertial observers (including rotating ones) don not have a standard convention. But you certainly are free to adopt a non-standard convention for rotating observers, you just have to be explicit in what your convention is since nobody will know it.

Interesting. So what would happen in a hypotethical situation where, for instance a ball moves with a constant speed, then starts rotating and then gets back to moving with a constant speed.
Would the scenario go something like this 'has a plane of simultaneity - doesn't have a plane of simultaneity - has a plane of simultaneity' in each of those 3 states of motion.

P.S. Thq Wannabe Newton on the link, I'm reading it now!

 

[WannabeNewton]

Are you talking about a geometric "time slice" when you say plane of simultaneity? In other words, if I have a family of observers whose worldlines cover all of Minkowski space-time, do these observers define a "time slice" of Minkowski space-time at each instant of time as read by them? Is that what you are asking?

 

[analyst5]

Are you talking about a geometric "time slice" when you say plane of simultaneity? In other words, if I have a family of observers whose worldlines cover all of Minkowski space-time, do these observers define a "time slice" of Minkowski space-time at each instant of time as read by them? Is that what you are asking?

Yes, exactly that. So I wonder why rotating frames do not have at least some 'point of view' on the events around them, at least some time slice that you referred to. After all, rotating bodies are simultaneous with themselves, as trivial as it sounds, so I wonder from which frame can that fact be deduced since their own frame 'doesn't exist' in this sense.

 

[Dale]

So what would happen in a hypotethical situation where, for instance a ball moves with a constant speed, then starts rotating and then gets back to moving with a constant speed.

The ball is non-inertial so there is no established convention for simultaneity with respect to the ball, but you are free to define one.

 

[DrGreg]

After all, rotating bodies are simultaneous with themselves, as trivial as it sounds...

It may sound trivial, but it's not true. Or rather, it may or may not be true depending on what simultaneity convention you choose to use.

 

[WannabeNewton]

Yes, exactly that

There's a mathematical reason for why a family of observers rotating relative to one another cannot foliate space-time into a family of "time slices" using their worldlines, unlike e.g. a family of inertial observers. This is what Ben was referring to essentially. So yeah if by planes of simultaneity you meant a one-parameter family of "time slices" of space-time orthogonal to the worldlines of the rotating observers in the family then this fails to exist for a mathematical reason (it's easy to see why it fails intuitively and only slightly harder to prove mathematically).

 

[analyst5]

It may sound trivial, but it's not true. Or rather, it may or may not be true depending on what simultaneity convention you choose to use.

What exactly do you mean by simultaneity convention? Something like 'whatever you say about this rotating body is true'? After all, isn't being simultaneous with itself necessary so we can speak about the rotating body, or the set of simultaneous points that together rotate? How to explain that?

 

[WannabeNewton]

Regarding the topic of simultaneity conventions for inertial frames (in particular Einstein's original convention) see here: http://plato.stanford.edu/entries/spacetime-convensimul/

Wiki has a nice entry on the Einstein convention as well: http://en.wikipedia.org/wiki/Einstein_synchronisation

 

[analyst5]

Regarding the topic of simultaneity conventions for inertial frames (in particular Einstein's original convention) see here: http://plato.stanford.edu/entries/spacetime-convensimul/

Wiki has a nice entry on the Einstein convention as well: http://en.wikipedia.org/wiki/Einstein_synchronisation

Thank you for the articles WBN, I've red them but I still don't understand why rotating frames don't at least have a local meaning of simultaneity. By that I mean 'all points taken at once' viewed from their own frame. The previous example you mentioned about the common sense that leads to conclusion that rotating frames don't have simultaneity hyperplanes seems really good and valid, but how can a rotational entity not be simultaneous with itself? I don't understand how this is a matter of conveniton, since we couldn't even define a rotating object without referring to all of its points simultaneously from some frame, in this case its own frame. I hope you see my issue.

 

[WannabeNewton]

The Einstein synchronicity convention is an equivalence relation so that in particular any clock is synchronized with itself. This convention is the standard convention for inertial frames; using this convention we can, if we wish to, build global "time slices" (simultaneity slices or planes of simultaneity) of Minkowski space-time using a canonical global time-function. For non-inertial frames you have to define what convention you are using because there is no standard. The reflexivity found in the Einstein convention may or may not carry over, which is what I think DrGreg was referring to.

 

[analyst5]

The Einstein synchronicity convention is an equivalence relation so that in particular any clock is synchronized with itself. This convention is the standard convention for inertial frames; using this convention we can, if we wish to, build global "time slices" (simultaneity slices or planes of simultaneity) of Minkowski space-time using a canonical global time-function. For non-inertial frames you have to define what convention you are using because there is no standard. The reflexivity found in the Einstein convention may or may not carry over, which is what I think DrGreg was referring to.

Hey, that makes it a little easier to understand. But what would the concrete situation be like in which a rotated object isn't simultaneous with itself? How would we measure it, or even know that's rotating?

 

[WannabeNewton]

I'm not sure if it's possible to find a convention in which a clock isn't synchronized with itself (it doesn't make any sense to me physically). I was just trying to interpret what DrGreg meant in his reply to your post #7. It's hard to intepret what you mean by "object isn't simultaneous with itself" because simultaneity involves different events in space-time as represented in a given frame.

 

[analyst5]

I'm not sure if it's possible to find a convention in which a clock isn't synchronized with itself (it doesn't make any sense to me physically). I was just trying to interpret what DrGreg meant in his reply to your post #7. It's hard to intepret what you mean by "object isn't simultaneous with itself" because simultaneity involves different events in space-time as represented in a given frame.

I mean the same as you. By being simultaneous with itself I mean the same as having a clock that is synchronized with itself. So it doesn't make sense to me, since if we had a clock on a rotating object it would measure the duration of the rotation and for that it seems the clock should be in sync with itself. That's my quasi-logical opinion. I hope Dr Greg or DaleSpam could explain this situation.

 

[Dale]

I hope Dr Greg or DaleSpam could explain this situation.

I don't know why you think I can explain something you said. I don't think that the phrase "bodies are simultaneous with themselves" has any meaning. Simultaneity is a relationship between events, not bodies.

My point is and remains that for any non-inertial object there is no standard convention of simultaneity, but you can certainly define one. You just have to tell people explicitly what convention you have chosen because otherwise they won't know.

You have seemed to completely ignore this point.

 

[Nugatory]

By being simultaneous with itself I mean the same as having a clock that is synchronized with itself. So it doesn't make sense to me, since if we had a clock on a rotating object it would measure the duration of the rotation and for that it seems the clock should be in sync with itself.

"Simultaneous with itself" isn't an especially clear or precise term.. I THINK what you're trying to say is that within a sufficiently small region around the clock and with observers who are at more or less at rest relative to the clock, there's a natural simultaneity convention that we can use. My neighbor and I don't have to employ elaborate relativistic calculations allowing for the rotation and gravitational effects of the Earth before saying things like "I'll be done mowing my lawn in an hour".

However, that local sense of time is just that: local. It cannot be used to define the planes of simultaneity that this thread is about.

It's not not even all that useful for measuring "the duration of the rotation" as you suggest above: First you have to define the start and the end of a rotation and you can't do that locally. (Consider that the duration, as measured by this clock, of one rotation of the Earth about its axis is different in the non-inertial frame in which the sun and the Earth are at rest, and the non-inertial frame in which the Earth is rotating at the origin while the sun orbits the Earth once a year).

 

[analyst5]

"Simultaneous with itself" isn't an especially clear or precise term.. I THINK what you're trying to say is that within a sufficiently small region around the clock and with observers who are at more or less at rest relative to the clock, there's a natural simultaneity convention that we can use. My neighbor and I don't have to employ elaborate relativistic calculations allowing for the rotation and gravitational effects of the Earth before saying things like "I'll be done mowing my lawn in an hour".

However, that local sense of time is just that: local. It cannot be used to define the planes of simultaneity that this thread is about.

It's not not even all that useful for measuring "the duration of the rotation" as you suggest above: First you have to define the start and the end of a rotation and you can't do that locally. (Consider that the duration, as measured by this clock, of one rotation of the Earth about its axis is different in the non-inertial frame in which the sun and the Earth are at rest, and the non-inertial frame in which the Earth is rotating at the origin while the sun orbits the Earth once a year).

You're close to what I'm trying to say, and thanks for your explanation of some terms. I know simultaneous with itself doesn't have a clear meaning, what I mean was really if we take a non-inertial frame and analyze it, we can measure at least some sort of local time in that frame.. That's the issue. How to define a non-inertial frame if not by taking a set of simultaneous points that are rotating/accelerating?

 

[Nugatory]

How to define a non-inertial frame if not by taking a set of simultaneous points that are rotating/accelerating?

A frame is nothing more than a convention for assigning coordinates to events. Pick a convention for assigning coordinates and you've just defined a frame. Only after you've done this can can you even start to talk about whether events are simultaneous (loosely speaking, have the same time coordinate in whatever coordinate system you've chosen). You don't need simultaneity to define a frame, it's the other way around.

 

[ghwellsjr]

There is a difference between a clock and time. When we talk about time in Special Relativity, we are talking about Coordinate Time at a specific Coordinate Location, which together are the coordinates of an event. Both the Coordinate Time and the Coordinate Location have no interval, that is, the location is a point in space with no size and the time is an instant with no duration. There is no clock that can occupy zero expanse in space and therefore there can be no clock that marks Coordinate Time.

Take the example of a simple light clock consisting of a photon reflecting between two mirrors. The mirrors obviously cannot be at the same point in space, they have to be separated. We could (conceptually) define the time of the light clock as being indicated by the reflection of the photon off of one of the mirrors but the time indicated by the reflection of the photon off the other mirror is at a different location in space and requires a convention to establish the relationship between the two times. If you claim that the photon takes the same amount of time to traverse in both directions between the mirrors and therefore opposite ticks occur half way between the others, then you have used Einstein's convention and you can say that both parts of the light clock are synchronized or that the "clock is simultaneous with itself".

However, this is just a convention, and by that we mean it is just an arbitrary definition that gives meaning to the concept of simultaneity or synchronicity or remote time. It is not something that "nature" provides. We cannot tell if the photon "actually" takes the same amount of time to get from one mirror to the other as it does in the opposite direction.

In fact, if you consider what we do in Special Relativity, a simple light clock is only "simultaneous with itself" in its own inertial rest frame. In some other Inertial Rest Frames, the clock is not "simultaneous with itself". But we can only discuss this issue because of the definitions (postulates) provided by SR for Inertial Reference Frames. If you want to deviate to non Inertial Reference Frames, then you have to do an equivalent process of defining, that is, giving meaning to what simultaneity is.

I get the impression that you think that "planes of simultaneity" are intrinsic to nature and that we can discover them by some experiment rather than that we have to create the concept of "planes of simultaneity" in some arbitrary way. Einstein's way is not the only way, but it's the simplest that anyone has come up with so far. You could invent your own method of establishing simultaneity even for inertial situations. But since there is no standard method for non inertial situations, you have to either make up your own or reference one that someone else has established.

 

[ghwellsjr]

You're close to what I'm trying to say, and thanks for your explanation of some terms. I know simultaneous with itself doesn't have a clear meaning, what I mean was really if we take a non-inertial frame and analyze it, we can measure at least some sort of local time in that frame.. That's the issue. How to define a non-inertial frame if not by taking a set of simultaneous points that are rotating/accelerating?

Your comment is why I think you believe that we can discover what "nature" means by simultaneous. We can't even measure some sort of local time in an inertial situation, let alone, a non inertial one. We define it and then whatever "measurements" we make are merely confirmations of our own arbitrary definition. We have no other choice. We cannot learn from nature about simultaneity.

 

[harrylin]

[..] rotating bodies are simultaneous with themselves, as trivial as it sounds, so I wonder from which frame can that fact be deduced since their own frame 'doesn't exist' in this sense.

It may sound trivial, but it's not true. Or rather, it may or may not be true depending on what simultaneity convention you choose to use.

[..] By being simultaneous with itself I mean the same as having a clock that is synchronized with itself. So it doesn't make sense to me, since if we had a clock on a rotating object it would measure the duration of the rotation and for that it seems the clock should be in sync with itself. That's my quasi-logical opinion. I hope Dr Greg or DaleSpam could explain this situation.

The way I read your statement at first, made me think of the rotating Earth; and your last remark of "clocks on a rotating object" suggests such a case. Points on the surface are simultaneous in the Earth Centered Inertial frame and not normally in any rotating frame. However, you could think (and maybe you also did) of a clock that is itself rotating. That is a single time keeper - but in such a case there is no synching at all. You may say of course that that clock is by definition synchronous with the time at the origin of the co-rotating reference system in which it is located at that origin.

PS There is the concept of "proper time" - and it sounds as if that is what you have in mind. However proper time refers to the time lapse of such a clock over a trajectory, and not to a frame.
- see https://en.wikipedia.org/wiki/Proper_time

 

[Dale]

what I mean was really if we take a non-inertial frame and analyze it, we can measure at least some sort of local time in that frame.. That's the issue.

Do you understand what it means for something to be a "convention"? You don't measure conventions, you define them.

For inertial frames there is a standard convention for simultaneity, but for non inertial frames there is no standard convention. You can define your own convention.

I don't know why you keep trying to complicate this.

 

[DrGreg]

You're close to what I'm trying to say, and thanks for your explanation of some terms. I know simultaneous with itself doesn't have a clear meaning, what I mean was really if we take a non-inertial frame and analyze it, we can measure at least some sort of local time in that frame.. That's the issue. How to define a non-inertial frame if not by taking a set of simultaneous points that are rotating/accelerating?

In an inertial frame, we synchronise two clocks A and B by sending a light signal from A to a mirror at B and reflect it back to A. A measures the start and end times of the journey, and for B to be synchronised to A it has to be adjusted so that the time recorded by B for the reflection should be exactly half way between the two times recorded by A. That's the standard convention for inertial frames.

Suppose we try the same thing in a rotating frame. And suppose we have lots of clocks arranged in a circle. We use the above method to sync B to A, then C to B, then D to C, etc, all round the circle. What you will find that the last clock Z is not synchronised to the first clock A. That's always the case no matter how many clocks there are and how close they are to each other. So that method of sync doesn't work in rotating frames. There are other ways to choose how to sync clocks, but the most obvious method doesn't work.

 

[WannabeNewton]

analyst, see the following to get some idea (and keep in mind that the semicolon ';' used in the paper represents the covariant derivative): http://arxiv.org/abs/gr-qc/0506127

 

[analyst5]

Do you understand what it means for something to be a "convention"? You don't measure conventions, you define them.

For inertial frames there is a standard convention for simultaneity, but for non inertial frames there is no standard convention. You can define your own convention.

I don't know why you keep trying to complicate this.

Hey DaleSpam, I apologize if I misused some terms or complicated some things. I think I got your point.

https://en.wikipedia.org/wiki/Proper_time[/url]

Yep, I was also thinking about this, I had it in mind. So we may measure the proper time between events on a worldtube of a rotating or accelerating object despite the fact that non-inertial frames don't have a standard convention of simultaneity. Right?


P.S. WBN thanks for the link. I'll start reading right now.

 

[Dale]

So we may measure the proper time between events on a worldtube of a rotating or accelerating object despite the fact that non-inertial frames don't have a standard convention of simultaneity. Right?

Yes, definitely.

 

[Nugatory]

may measure the proper time between events on a worldtube of a rotating or accelerating object despite the fact that non-inertial frames don't have a standard convention of simultaneity. Right?

Right... But do remember that the proper time between two events depends not only on the two events but also on the path you follow to get from one to the other.