Role of distance in defining simultaneity

[analyst5]

Hi guys, I thought lot about this and I need an opinion from people who will be able to give me an appropriate answer that may serve me well. My question is about the consequences of Lorentz transformations. So I hope you'll be able to correct if I'm wrong in some sentences and add what's been missing.

So basically, following Lorentz transformations, all observers that are at rest with respect to the worldtube of an object X, will have the same event on its worldtube as their present, no matter how far away they are from the object speaking.

But, as I've understood, in moving frames with respect to the object, the distant the frame is from the object, the distant the event on the worldtube will be in time. So it may be possible, for instance, that two objecs travel with the same velocity towards an object, but the distant one will have the 'more distant future event' in its present. (future relative to what the stationary observer regards as present of course).

So my question is, if the previous sentences are correct, what happens during acceleration (change of frames)? Is it true that the greater distance between an accelerated frame and the object's worldtube is, the more time will pass relative to accelerated frame.
By that I mean to really ask what does really happen while changing frames, and what role does the distance play here, since it certainly plays a role in defining 'a now moment' relative to an inertial frame.

Sorry for the possible vagueness, if anybody wants a detailed explanation of the things I mean, I'll gladly write it. Thanks for your patience.

 

[ghwellsjr]

Hi guys, I thought lot about this and I need an opinion from people who will be able to give me an appropriate answer that may serve me well. My question is about the consequences of Lorentz transformations. So I hope you'll be able to correct if I'm wrong in some sentences and add what's been missing.

So basically, following Lorentz transformations, all observers that are at rest with respect to the worldtube of an object X, will have the same event on its worldtube as their present, no matter how far away they are from the object speaking.

But, as I've understood, in moving frames with respect to the object, the distant the frame is from the object, the distant the event on the worldtube will be in time. So it may be possible, for instance, that two objecs travel with the same velocity towards an object, but the distant one will have the 'more distant future event' in its present. (future relative to what the stationary observer regards as present of course).

So my question is, if the previous sentences are correct, what happens during acceleration (change of frames)? Is it true that the greater distance between an accelerated frame and the object's worldtube is, the more time will pass relative to accelerated frame.
By that I mean to really ask what does really happen while changing frames, and what role does the distance play here, since it certainly plays a role in defining 'a now moment' relative to an inertial frame.

Sorry for the possible vagueness, if anybody wants a detailed explanation of the things I mean, I'll gladly write it. Thanks for your patience.

I'd sure like a detailed explanation of the things you mean because there is a lot of confusion in your post.

1) If your "question is about the consequences of Lorentz transformations", then you should not be asking about accelerated frames. LT's only work on inertial frames.

2) Frames don't move with respect to objects. We use the coordinates of frames to describe and specify how objects move.

3) We use the LT when we want one frame to move inertially (not accelerated) with respect to another frame. The frames apply to all the objects.

4) If you want to know what happens when transforming from one frame to another, just do the calculations of the LT and it will tell you.

 

[Dale]

So my question is, if the previous sentences are correct, what happens during acceleration (change of frames)?...

if anybody wants a detailed explanation of the things I mean, I'll gladly write it.

Unfortunately, there is no standard meaning for the reference frame of an accelerated (non-inertial) observer. In order to answer your question, it is necessary for you to pick a convention for defining the reference frame of an accelerated observer.

My personal preference is the Dolby and Gull convention:
403 Forbidden
but you may be interested in a different convention and since there is no standard convention you need to specify.

 

[WannabeNewton]

Perhaps I'm misunderstanding the terminology, but the reference frame of an accelerated observer is always well defined (up to a rotation). What I mean by this is if we have an accelerating observer, then at any given event on the observer's worldline, the rest frame of the observer is simply an inertial frame instantaneously comoving with the observer at that event. There are an infinity of such frames, all of them being related by a rotation of the spatial basis vectors but other than that it's pretty standard. I don't see where a simultaneity convention comes into that.

Certainly however, if we have a family of arbitrarily accelerating observers there is no standard simultaneity convention amongst the family as opposed to say a family of inertial observers in Minkowski space-time, wherein the standard convention amongst the family would be that defined by Einstein synchronization.

 

[Dale]

Perhaps I'm misunderstanding the terminology, but the reference frame of an accelerated observer is always well defined (up to a rotation). What I mean by this is if we have an accelerating observer, then at any given event on the observer's worldline, the rest frame of the observer is simply an inertial frame instantaneously comoving with the observer at that event.

That is the momentarily co-moving inertial frame. That is not the same as a non inertial observer's reference frame.

I.e. An observer's frame (loosely speaking) is a single frame which describes all of physics "from the observers perspective". There is a different MCIF for each point along a non inertial observers worldline, not one single overall frame.

 

[WannabeNewton]

That is how MTW (section 6.2, p.166) defines the rest frame of the accelerating observer at any given instant.

 

[Dale]

Interesting. The MCIF is usually not considered to be the "non-inertial observers frame" since they are only momentarily at rest in it. However, MTW are considered authoritative.

 

[WannabeNewton]

I.e. An observer's frame (loosely speaking) is a single frame which describes all of physics "from the observers perspective". There is a different MCIF for each point along a non inertial observers worldline, not one single overall frame.

Yes but by having a continuous one-parameter family of MCIFs (the parameter being the proper time along the accelerating observer's worldline) one can construct a coordinate system that is always comoving with the observer (i.e. such that the observer is always at the spatial origin of the coordinates) and this coordinate system allows one to describe all of the physics from the perspective of the observer. See for example section 13.6 (p.327) in MTW. They call this the "proper reference frame" of the accelerating observer.

The problem of course is when we have a family of such non-inertial observers. If we have something like a family of rotating observers in flat space-time, then in the "proper reference frame" of a given observer in the family, nearby observers in the same family will not be at rest but will be rotating (relative to three mutually perpendicular gyroscopes carried by the given observer) so the standard simultaneity convention for inertial observers won't really be of much help here since it will be path-dependent; one has to choose a non-standard but useful convention. This is of course exactly what you have stated here and elsewhere; I'm just trying to make clear my own point so that nothing is ambiguous with regards to my posts :)

 

[Dale]

one can construct a coordinate system that is always comoving with the observer (i.e. such that the observer is always at the spatial origin of the coordinates) and this coordinate system allows one to describe all of the physics from the perspective of the observer.

Yes, one CAN do so, but one CAN also construct other different coordinate systems that are comoving with the observer and also allow one to describe all of the physics "from the perspective of the observer". There are many such possible coordinates, which is why you need to specify the convention you mean.

 

[WannabeNewton]

Certainly, I don't disagree with that and perhaps this is all just a point about terminology but I can't see how that choice of a coordinate system/frame for a single non-inertial observer is the same as a simultaneity convention amongst a family of non-inertial observers.

 

[Dale]

I don't know what the relevance is of the family of observers. Even a single observer still needs to define a simultaneity convention to establish a coordinate system. For a non inertial observer that convention is not unambiguous.

 

[WannabeNewton]

I don't know what the relevance is of the family of observers. Even a single observer still needs to define a simultaneity convention to establish a coordinate system. For a non inertial observer that convention is not unambiguous.

Certainly, I don't disagree with this at all. I was speaking of families because that's how I interpreted the OP's statement "all observers that are at rest with respect to the worldtube of an object".

 

[Dale]

Hmm, very good point. I may not have been responsive to the OPs question.

 

[Dale]

For a non inertial observer that convention is not unambiguous.

Ugh, what a convoluted sentence. My apologies.

 

[WannabeNewton]

Ugh, what a convoluted sentence.

I can very strongly relate to that haha. For some reason I always use "not unambiguous" instead of "ambiguous".

 

[pervect]

Hi guys, I thought lot about this and I need an opinion from people who will be able to give me an appropriate answer that may serve me well. My question is about the consequences of Lorentz transformations. So I hope you'll be able to correct if I'm wrong in some sentences and add what's been missing.

So basically, following Lorentz transformations, all observers that are at rest with respect to the worldtube of an object X, will have the same event on its worldtube as their present, no matter how far away they are from the object speaking.

But, as I've understood, in moving frames with respect to the object, the distant the frame is from the object, the distant the event on the worldtube will be in time. So it may be possible, for instance, that two objecs travel with the same velocity towards an object, but the distant one will have the 'more distant future event' in its present. (future relative to what the stationary observer regards as present of course).

So my question is, if the previous sentences are correct, what happens during acceleration (change of frames)? Is it true that the greater distance between an accelerated frame and the object's worldtube is, the more time will pass relative to accelerated frame.
By that I mean to really ask what does really happen while changing frames, and what role does the distance play here, since it certainly plays a role in defining 'a now moment' relative to an inertial frame.

Sorry for the possible vagueness, if anybody wants a detailed explanation of the things I mean, I'll gladly write it. Thanks for your patience.

We should probably make a distinction, which might not be obvious, between frames and coordinate systems here.

There isn't any problem with defining an instantaneous frame of an accelerated observer, but when you try to "stitch" all the frames together into a coordinate system, you do indeed run into difficulties.

Let me also remark that the distinction I'm drawing here between frames and coordinate systems isn't always rigorously adhered to. But one needs to disambiguate two closely related, but distinct, concepts, in order to talk about the issues.

Basically what you find is that there is a limit on the "size" of an accelerated coordinate system. There are a couple of ways of looking at this, One way is to say that the limit is physically imposed by the fact that you can "outrun" light signals if you accelerate at a continuous constant rate - while you are always going slower than light, and the light is always catching up to you, it never actually manages to reach you. So it becomes difficult to assign coordinates to events that you can never receive signals to, or events you can never send signals to.

The other way of saying this is that the different lines of simultaneity from the diffrerent instantaneous "frames" you are trying to combine wind up "crossing". And when this happens, you have multiple times assigned to one event. This makes the resulting coordinate system ill behaved.

This would be a lot clearer with the diagram from the text, but alas I don't have time to sketch one at the moment. I think I've posted before on this topic, but I don't have time to track it down.

Anyway, there are some issues, and I hope the basic idea helps (that there you can define frames easily enough, but stiching them together is harder than it looks).

That is how MTW (section 6.2, p.166) defines the rest frame of the accelerating observer at any given instant.

But MTW also says this ($6.3, Constraints on size of an accelerated frame)

It is very easy to put together the words "the coordinate system of an accelerated observer" but it is muh harer to find a concept these words might refer to. The most useful first remark one can make about these words is that, if taken seriously, they are self-contradictory. The define article "the" in the prhase suggests that one is thinking of some unique coordinae system natrually associated with some specified acceleated observer.

...

But from figure 6.1 it is clear that the events comprising one quarter of all spactime can neither send light signals to, nor recive light signals from the specified observer.

So there are some difficulties here, the difficulties aren't in the concept of a frame, but rather when one tries to combine the different frames together into a coordinate system.

 

[ghwellsjr]

I think all you guys have totally misread the OP's question (which is very easy to do since his sentences are quite convoluted). He's not really asking about accelerated frames. He's thinking that if you start with one frame and then you use the LT to get to a frame that is moving with respect to the first one, then the original frame has accelerated to the second frame, just as if you were talking about an object that changes its speed, it has accelerated.

His question, as his title states, is how the LT results in more distant objects being affected more by time compared to close ones and how this affects the new aspects of simultaneity. Of course, the answer to his question is simple: do the LT and it will tell you.

 

[Dale]

We should probably make a distinction, which might not be obvious, between frames and coordinate systems here.

That is a very good point. It is important to make that distinction, and I rarely do so. My remarks above refer to coordinate systems rather than reference frames (a.k.a. vierbeins or tetrads).

The reason is that a vierbein/tetrad/reference frame doesn't necessarily define a notion of simultaneity, and the OP was asking about simultaneity. For example, it is easy to set up a vierbein for a family of observers on a Born-rigid rotating disk, but you cannot take integral curves of the spacelike vectors to get sensible hypersurfaces of simultaneity. So, I assumed that the OP was actually interested in coordinate systems rather than actual reference frames, but as ghwellsjr suggests, I may be misreading their intentions.