Frames vs Lines of Simultaneity

[Austin0]

------------Frames and Lines of Simultaneity----------------

Is there any difference between the two?

If there is what is it??

I may be missing something obvious but as far as I can see they are just two ways of graphing and conceptualizing a singular entity.

Thanks

 

[Aaron_Shaw]

A line of simultaneity just links more than one event that are considered simultaneous according to the observer. So on a spacetime diagram they'll be horizontal lines all along the time axis.

Usually they are used to show differences in simultaneity between 2 different observers moving relatively to each other. So in this case the observer who is being considered stationary might plot a line of simultaneity which maps a second, moving, observers notion of simultaneous events. In this case the line will be sloped.

In this case the initial observer can see that what the other observer considers simultaneous events, the first considers to happen one after the other.

 

[Austin0]

A line of simultaneity just links more than one event that are considered simultaneous according to the observer. So on a spacetime diagram they'll be horizontal lines all along the time axis.

Usually they are used to show differences in simultaneity between 2 different observers moving relatively to each other. So in this case the observer who is being considered stationary might plot a line of simultaneity which maps a second, moving, observers notion of simultaneous events. In this case the line will be sloped.

In this case the initial observer can see that what the other observer considers simultaneous events, the first considers to happen one after the other.

Hi Aaron_Shaw
I was not asking about the meaning and application of L's of S.

Regarding the horizontal line of the rest frame. Those lines are purely and simply the rest frame itself. The point used to locate the timeline in the graph is is simply a single point on the extended coordinate frame.

Similarly the sloped timelines of the "moving" frame are simply that frame as an extended coordinate system at different points in time. The point in that frame which is used to locate the worldline is also just a point on the total frame.

The slope of the line is simply a graphic convention. In the real world they are congruent with the path of motion.

The information they contain [relative time relationship and spatial locations] where they intersect another worldline is the same as what would be found if you did a parallel analysis.

For example the train and tracks. The relative simultaneity [relative clock desynchronization] can be directly calculated from the frames coordinates with the Lorentz transform. This will tell the relative time at the points of the lightning flash. The relative simultaneity of clocks at those points and will be exactly consistent with the diagrams for the same point in time.

This will hold true no matter how long the train and tracks are. SO the lines of simultaneity that are sloped in the diagram are just long trains indefintely extended in space.
ANd any points of intersection represent observers and clocks on the train and tracks at that point.

L's of S in the diagrams are simply a wonderful convention telling you the relative clock readings at distant locations without having to do the specific calculations for the point.
Of course to get the location of this colocation in the moving frame you still need to transform with gamma.

So does any of this make sense?

Thanks for your responce

 

[my_wan]

Lines of Simultaneity only have meaning relative to a particular frame. It's hard to call them the same thing though, when spacelike separated frames can agree on certain lines of simultaneity in some cases.

 

[espen180]

Each frame has a plane of simultanety (in 2D space + 1D time) associated with it. The PoS is just a property of the frame, and is only non-trivial when compared to another frame.

Frames which travel in the same direction with the same velocity have parallel PoS's.

Since the PoS is parallel to the spatial axes, their use in spacetime diagrams are to, as you said, find out when an event occurs in a paticular frame.

 

[yossell]

What do you have in mind by a frame? What do you have in mind by a line of simultaneity?

This is complicated to say, but the pictorial and intuitive idea isn't really too hard:

If you have a 4-d Minkowski manifold, 3 of space and 1 of time, then you can naturally partition the points of this manifold into a sequence of 3-d `simultaneity' spaces. Think of these as a kind of slicing up of space-time, into a series of `moments'. Each element of the sequence corresponds to space at a moment in some inertial frame. Lines that are orthogonal to the surfaces are timelike, and represent the inertial paths of things that are stationary with respect to that frame.

Unlike the Newtonian case, where there is an absolute notion of simultaneity, there is no one special way of slicing up Minkowski space-time - different partitions, or slicings, correspond to different ways in which various objects would slice up space time.

In a 2-d Minkowski manifold - which corresponds very closely to a MInkowski diagram which I know you've been focussing on - a frame's `moment' would correpond to a simultaneity line, and the frame could be sliced up (partitioned) into a sequence of such lines. I think this is what you had in mind? For the realistic 3+1 case, we have hyperplanes of simultaneity.

What's a frame? If a frame is an inertial rectangular coordinate system, then it's not quite true that frames and simultaneity partitions correspond. In a coordinate system, we have to choose an origin, set t = 0, focus on a particular inertial line, whereas a slicing gives us a whole set of parallel inertial lines. However, we of course feel that choice of origin is a purely arbitrary choice, that this kind of difference is of only mathematical rather than physical importance. If we get rid of such arbitrary differences between coordinate systems and consider instead equivalence classes of coordinate systems, coordinate systems that differed only by choice of origin, then, I think, there is a 1-1 correspondence between such sets of frames and a partitioning of space-time into a sequence of simultaneity hyperplanes.

So - the intuition that, in Minkowski space time, there's not much real difference between an inertial frame and a way of partitioning space-time into simultaneity equivalence classes, you'd more or less be right. There's a very natural correspondence between them.

That took way longer to explain than it should have. I blame myself.

 

[yossell]

Each frame has a plane of simultanety (in 2D space + 1D time) associated with it.

Wouldn't a frame have a sequence of planes of simultaneity associated with it? Not just one?

Frames which travel in the same direction with the same velocity have parallel PoS's.

Wouldn't they have the same planes of simultaneity?

 

[espen180]

Wouldn't a frame have a sequence of planes of simultaneity associated with it? Not just one?

Yes, but what I had in mind was a plane moving with time along the time axis.

Wouldn't they have the same planes of simultaneity?

Yes, I would think so.

 

[Austin0]

Each frame has a plane of simultanety (in 2D space + 1D time) associated with it. The PoS is just a property of the frame, and is only non-trivial when compared to another frame.
Frames which travel in the same direction with the same velocity have parallel PoS's.

Since the PoS is parallel to the spatial axes, their use in spacetime diagrams are to, as you said, find out when an event occurs in a paticular frame.

Exactly! But doesn't the PoS have an exact one to one mapping to the actual coordinate plane it is a property of?

Isn't the term PoS simply a useful semantic distinction with no real difference or meaning?

Don't misunderstand me ; I understand the usefullness and am not suggesting that they should be called by the same name.

I am simply interested in finding out if there is a real difference that I have missed.

Thanks

 

[Austin0]

Wouldn't a frame have a sequence of planes of simultaneity associated with it? Not just one?.

=espen180;2796970]Yes, but what I had in mind was a plane moving with time along the time axis..

Exactly my understanding and conceptualization

Wouldn't they have the same planes of simultaneity?

.

Yes, I would think so.

Me too

Since the PoS is parallel to the spatial axes, their use in spacetime diagrams are to, as you said, find out when an event occurs in a paticular frame.

Wouldn't this only apply to the rest frame. The lines of S of the moving system in the diagram are sloped relative to the spatial axis??

Isn't the PoS actually congruent to the spatial axis at any specific point on the time line?

 

[Austin0]

=yossell;2796965]What do you have in mind by a frame? What do you have in mind by a line of simultaneity?[\QUOTE]
My long held understanding is totally constient with your descriptions below.

If you have a 4-d Minkowski manifold, 3 of space and 1 of time, then you can naturally partition the points of this manifold into a sequence of 3-d `simultaneity' spaces. Think of these as a kind of slicing up of space-time, into a series of `moments'.

But don't these partitions exactly coorespond to the frame itself in these series of moments?

In a 2-d Minkowski manifold - which corresponds very closely to a MInkowski diagram which I know you've been focussing on - a frame's `moment' would correpond to a simultaneity line, and the frame could be sliced up (partitioned) into a sequence of such lines. I think this is what you had in mind? For the realistic 3+1 case, we have hyperplanes of simultaneity

.

Is it not the hyperframe , the continuous extended world line of the frame, that is being sliced up rather than the frame itself??

Isn't the fundamental definition of the coordinate system that it is a 3-D grid of spatial locations with clocks at those locations?

It is commonly considered that we can assume observers at all points also ?

In this context isn't the (3-d)-sphere, or (2d)- plane, or (1-d)- line of simulatanity also a system of locations and clocks that are exactly the same at any moment and colocated with the frame?

With virtual observers??

What's a frame? If a frame is an inertial rectangular coordinate system, then it's not quite true that frames and simultaneity partitions correspond.

But outside of difference in name , as rectangular coordinate systems how do they not correspond?

In a coordinate system, we have to choose an origin, set t = 0, focus on a particular inertial line, whereas a slicing gives us a whole set of parallel inertial lines.

Doesn't the Plane of S have the same origen and t-0

So - the intuition that, in Minkowski space time, there's not much real difference between an inertial frame and a way of partitioning space-time into simultaneity equivalence classes, you'd more or less be right. There's a very natural correspondence between them.

But is there any difference?
Thanks

 

[espen180]

Exactly! But doesn't the PoS have an exact one to one mapping to the actual coordinate plane it is a property of?

Isn't the term PoS simply a useful semantic distinction with no real difference or meaning?

Don't misunderstand me ; I understand the usefullness and am not suggesting that they should be called by the same name.

I am simply interested in finding out if there is a real difference that I have missed.

Thanks

In this Minowski diagram, the thin blue lines show the blue S' frame's simultaneity planes. The thin black lines show the black S frame's simultaneity planes.

In a Minowski diagram, the usual rules of coordinates apply. If you have x'=3 at t=t'=0 and want to find the location of x'=3 in S at time t', you simply translate the point (3,0)' along the t' axis. In this sense, the simultaneity planes are copies of the spatial coordinate plane, or at least a temporal extension, however you want to define it.

 

[matheinste]

We could describe lines, planes and hypersurfaces of simultaneity as (loosely speaking) subspaces of frames of reference.

A frame of reference is an imagined system of rigid rods forming a grid throughout all space and ideal synchronized clocks attached to points of the grid. We can attach to this reference frame a system of coordinates of our choice. For an observer at rest in any particular frame of reference his line/plane/hypersurface of simultaneity is just the set of points, selected by some convention, which he considers to be his now. For a line of simultaneity on a two dimensional spacetime diagram the line of simultaneity corresponds to his spatial axis. The same applies, mutatis mutandis, to planes and hypersurfaces of simultaneity.

Matheinste.

 

[Austin0]

=Austin0;2797918

Isn't the fundamental definition of the coordinate system that it is a 3-D grid of spatial locations with clocks at those locations?

We could describe lines, planes and hypersurfaces of simultaneity as (loosely speaking) subspaces of frames of reference.

A frame of reference is an imagined system of rigid rods forming a grid throughout all space and ideal synchronized clocks attached to points of the grid. We can attach to this reference frame a system of coordinates of our choice. For an observer at rest in any particular frame of reference his ((1))... line/plane/hypersurface of simultaneity is just the set of points, selected by some convention, which he considers to be his now For a line of simultaneity on a two dimensional spacetime diagram the line of simultaneity corresponds to his spatial axis ]. The same applies, mutatis mutandis, to planes and hypersurfaces of simultaneity.

Matheinste.

From this can I infer that there is no disagreement on our definitons and conceptions of the coordinate systems in question?

But why a subspace? Why would the PoS be less indefinitely extended than the frame?

((1)) Is n't this exactly equivalent as applied to the frame itself? The spatial coordinates and clocks as selected by convention which is then considered "now"?

Thanks

 

[espen180]

From this can I infer that there is no disagreement on our definitons and conceptions of the coordinate systems in question?

But why a subspace? Why would the PoS be less indefinitely extended than the frame?

((1)) Is n't this exactly equivalent as applied to the frame itself? The spatial coordinates and clocks as selected by convention which is then considered "now"?

Thanks

Because a simultaneity space doesn't have a temporal extension, it is a subspace of the 4D frame which has.

 

[Austin0]

From this can I infer that there is no disagreement on our definitons and conceptions of the coordinate systems in question?

But why a subspace? Why would the PoS be less indefinitely extended than the frame?

((1)) Is n't this exactly equivalent as applied to the frame itself? The spatial coordinates and clocks as selected by convention which is then considered "now"?

Because a simultaneity space doesn't have a temporal extension, it is a subspace of the 4D frame which has.

I am not sure I understand you. At an instant of time the 4-D frame is indefinitely extended in space but would seem to be unextended in time , yes?
This would be exactly the same for the 4-DofS wouldn't it??
They are both only extended temporally as they move along the worldline, no?

 

[Austin0]

In this Minowski diagram, the thin blue lines show the blue S' frame's simultaneity planes. The thin black lines show the black S frame's simultaneity planes.

In a Minowski diagram, the usual rules of coordinates apply. If you have x'=3 at t=t'=0 and want to find the location of x'=3 in S at time t', you simply translate the point (3,0)' along the t' axis. In this sense, the simultaneity planes are copies of the spatial coordinate plane, or at least a temporal extension, however you want to define it.

Thanks for the diagram

 

[espen180]

I am not sure I understand you. At an instant of time the 4-D frame is indefinitely extended in space but would seem to be unextended in time , yes?
This would be exactly the same for the 4-DofS wouldn't it??
They are both only extended temporally as they move along the worldline, no?

No, moving along the temporal axis is not equivalent to being extended along that axis. The simultaneity space doesn't have temporal extension. Why would it? The simultaneity space has the same amount of dimensions as the number of spatial axes. That's why we have simultaneity lines and not simultaneity surfaces in 2D minowski diagrams.

 

[yossell]

Is it not the hyperframe , the continuous extended world line of the frame, that is being sliced up rather than the frame itself??

You're right here - I shouldn't have used the word frame that second time - I meant to say it's the space-time that's being partitioned. But I wasn't talking about the extended worldline.

But is there <b>any</b> difference?

Unfortunately, because I'm still not sure how you're using the concepts, I don't know how to answer this without repeating my last answer.

I think espen's diagram says it all - but notice that on the lines and planes he's drawn no *numbers* are attached. Coordinate systems typically involve actually assigning numbers to events, but since this obviously involves nothing more than pure conventional choice of unit, we tend to forget about this.

I prefer to think of splitting up spacetime into a sequence of simultaneity (hyper)planes, rather than thinking of this as a single plane which `moves' through an observer's time - we can treat Newtonian space time as a sequence of copies of the very same plane at different times, because there's an absolute notion of sameness at place at different times, i.e absolute rest, in his theory. But provided we don't push the heuristic too far, and we understand there's no true identity, we can talk of various planes at different times as another way of thinking about the sequence of simultaneity planes.

But outside of difference in name , as rectangular coordinate systems how do they not correspond?

I thought I'd explained this - but maybe I don't get your 'outside of difference in name'. A plane of simultaneity is just a plane of simultaneity. Think of it purely geometrically. There's no number associated with it. That requires a choice - setting the clocks at zero rather than 3. It has no origin either. To get a coordinate system going, numbers associated with points, we have to choose spatial axes too. But I doubt that this kind of thing really matters for any conceptual/physical purpose and can be counted as a mathematical nicety (though, for my own part, I am partial to a few mathematical niceties!)

Doesn't the Plane of S have the same origen and t-0

As I've been using it, a plane of spacetime has no origin, just as the lines on espen180's diagram have no numbers attached to them.

I think most of us agree that any two observers traveling at the same velocity will partition the planes of simultaneity in the same way; that any partition of the planes of simultaneity will have associated with it a set of time like lines that correspond to inertial observers with the same velocity; etc etc, and so tend to think of these as different ways of describing the same situation.

The only thing that you've said that concerned me was this:

Isn't the term PoS simply a useful semantic distinction with no real difference or meaning??

I don't know what you're contrasting this to, but I would urge that a sequence of simultaneity classes corresponds to genuine geometric structure on the Minkowski space-time: they're the surfaces orthogonal to inertial lines. Minkowski space-time privileges no unique sequence of simultaneity classes - in this respect, it is very different from Newtonian space-time, but their existence is not a purely semantic issue.

 

[Austin0]

No, moving along the temporal axis is not equivalent to being extended along that axis. The simultaneity space doesn't have temporal extension. Why would it? The simultaneity space has the same amount of dimensions as the number of spatial axes. That's why we have simultaneity lines and not simultaneity surfaces in 2D minowski diagrams.

Isn't the reason we have lines instead of surfaces is simply the limitations of graphing a 4-D reality onto 2D?? The frames also appear as lines.

Maybe if you explained how you see a frame as having temporal extension I could get it?

 

[espen180]

Isn't the reason we have lines instead of surfaces is simply the limitations of graphing a 4-D reality onto 2D?? The frames also appear as lines.

Maybe if you explained how you see a frame as having temporal extension I could get it?

The frames are represented (in 2D minowski diagrams) as surfaces, two-dimensional manifolds, with a spatial and temporal extension.

It has nothing to do with limitations. In the case of inertial parallel/antiparallel motion, you can always allign your coordinate axes such that all motion is along one axis.

 

[Austin0]

The frames are represented (in 2D minowski diagrams) as surfaces, two-dimensional manifolds, with a spatial and temporal extension.

It has nothing to do with limitations. In the case of inertial parallel/antiparallel motion, you can always allign your coordinate axes such that all motion is along one axis.

I meant limitations in the sense that obviously all 4 dimensions cannot be graphed on a 2 D coordinate matrix.
I understand how you can view the spatial x dimension as extended horizontally but still don't see how the temporal dimension is extended vertically unless you are talking about past and future points on the timeline??
Sorry if I am being obtuse here.

 

[espen180]

I meant limitations in the sense that obviously all 4 dimensions cannot be graphed on a 2 D coordinate matrix.
I understand how you can view the spatial x dimension as extended horizontally but still don't see how the temporal dimension is extended vertically unless you are talking about past and future points on the timeline??
Sorry if I am being obtuse here.

How can I explain it clearer? The simultaneity space is a 3-dimensional cross-section of a 4-dimensional frame at a paticulat instant.

 

[Austin0]

=yossell;2798041] I meant to say it's the space-time that's being partitioned. But I wasn't talking about the extended worldline.?

=Austin0;2797918
But don't these partitions exactly coorespond to the frame itself in these series of moments?

Austin0

In this context isn't the (3-d)-sphere, or (2d)- plane, or (1-d)- line of simulatanity also a system of locations and clocks that are exactly the same at any moment and colocated with the frame?

]=yossell;2] I prefer to think of splitting up spacetime into a sequence of simultaneity (hyper)planes, rather than thinking of this as a single plane which `moves' through an observer's time - we can treat Newtonian space time as a sequence of copies of the very same plane at different times, because there's an absolute notion of sameness at place at different times, i.e absolute rest, in his theory. But provided we don't push the heuristic too far, and we understand there's no true identity, we can talk of various planes at different times as another way of thinking about the sequence of simultaneity planes.

OK I get both views
____________________________________________________________________

Austin0-- But outside of difference in name , as rectangular coordinate systems how do they not correspond?

]=yossell] I thought I'd explained this - but maybe I don't get your 'outside of difference in name'. A plane of simultaneity is just a plane of simultaneity. Think of it purely geometrically. There's no number associated with it. That requires a choice - setting the clocks at zero rather than 3. It has no origin either. To get a coordinate system going, numbers associated with points, we have to choose spatial axes too. But I doubt that this kind of thing really matters for any conceptual/physical purpose and can be counted as a mathematical nicety (though, for my own part, I am partial to a few mathematical niceties!)

I have been thinking of it geometrically from the beginning and I follow your conception of coordinate systems being a blank abstraction that must be assigned values by convention.


austin0

Doesn't the Plane of S have the same origen and t-0

]=yossell]As I've been using it, a plane of spacetime has no origin, just as the lines on espen180's diagram have no numbers attached to them.

In this context when you do actually assign coordinates wouldn't they be exactly the same??
One to one correspondence??

]=yossell]The only thing that you've said that concerned me was this:

austin0

But is there any difference?

=yossellI don't know what you're contrasting this to, but I would urge that a sequence of simultaneity classes corresponds to genuine geometric structure on the Minkowski space-time: they're the surfaces orthogonal to inertial lines. Minkowski space-time privileges no unique sequence of simultaneity classes - in this respect, it is very different from Newtonian space-time, but their existence is not a purely semantic issue.[/ QUOTE]
I was not suggesting that their existence was a matter of semantics.
I have always assumed their geometric existence was a real as the geometric exisence of a frames coordinate structure. I am asking, if as geometric coordinate structures, they aren't in fact the same .
Using Matheniste's rigid rods and clocks. in a moment in time ,wouldn't the simultaneity coordinate structure be indentical.
The difference is the conventional definition of simultaneity; occurring at equal distance from a point as measured by light path.
This convention is applied to every spatial point in the coordinate frames.
This results in every point being in the center of a spherical onion of different layers extending indefinitely into space. All events located on a particular skin are simultaneous.

But this is not really a subset of the points in the frame as the complete onion corresponds and is colocaterd to all points in the frame. ANd the clocks and spatial coordinates are the same as the frame.
The relationship between different overlapping frames >interpenetration of the coordinate frames results in a hypersphere of relative simultaneity. Or in lesser dimensions a hypersurface or plane. In 3D the intersection of the overlapping circles.
Looking at a 3D Minkowski drawing from a perspective view the planes of S as depicted in the drawing are simply the coordinate frames . The circles of simultaneity are on the coordinate planes but are not depicted in the draweing.The hypersurface of simultaneity is the intersection of these concentric circles. In the rest frames coordinates the circles of the moving frame are ellipses of simultaneity.
SO all my commentary has been directed at the actual planes of simultaneity [which as far as I can see are the frame coordinate planes] as it appears in minkowski drawings.
Does any of this make sense or clarify things?

 

[yossell]

I have been thinking of it geometrically from the beginning and I follow your conception of coordinate systems being a blank abstraction that must be assigned values by convention.

But then you had written:

Doesn't the Plane of S have the same origen and t-0

But the geometric plane doesn't have an origin, only once it is coordinatized does it have an origin. So I have a difficulty - although you keep saying that what I write is what you have always thought, what you often write suggests a different conception.

Now, let's restrict our attention to coordinate systems that represent inertial frames, the t axes representing the frame's time, the others representing space. Then:

Given a partitioning of Minkowski space into a sequence of simultaneity hypersurfaces, and given that a coordinate system respects the structure of this partitioning, there still remain *many* coordinate systems that correspond to this partitioning. So, not 1-1. However, I think that they all differ in merely conventional ways: choice of unit, origin, spatial axes - so also given choice of unit etc. then (I think) YES: one coordinate system.

Moreover, each coordinate system corresponds to 1 partition of Minkowski space-time, the hyperplanes those planes that all share the same t coordinate of the coordinate system.

Using Matheniste's rigid rods and clocks. in a moment in time ,wouldn't the simultaneity coordinate structure be indentical.

What does `simultaneity coordinate structure' mean? Guessing: The coordinate system should give any two points of the hypersurface in the simultaneity class the same t value.

The difference is the conventional definition of simultaneity; occurring at equal distance from a point as measured by light path.

This does not make sense to me - the difference between what?

This convention is applied to every spatial point in the coordinate frames.

What convention? What's a spatial point of the coordinate frame? The coordinate system is a set of 4 numbers, representing space-time points.

This results in every point being in the center of a spherical onion of different layers extending indefinitely into space. All events located on a particular skin are simultaneous.

I do not know what you have in mind. What spherical onion? There's nothing particularly spherical about a hypersurface - it sounds as if you're partitioning spacetime in a way I don't understand.

So, no, I'm afraid your comments don't clarify things. In fact, they tend to make me worry that we're not really on the same page at all.

 

[espen180]

Just so you are aware, Austin0, the relativity of simultaneity, time dilation etc. are effects which are what is left after you correct for things like signal time and doppler shift.

 

[matheinste]

Hello Austin0.

In spacetime which is four dimensional, the surface of a three dimensional sphere is not a hypersurface of simultaneity as you seem to think. For a particular observer the whole of his three dimensional space is, for him, a hypersurface of simultaneity. Different observers effectively partition spacetime into their own individual Newtonian universes. The usually defined reference frame of a set of rigid rods and ideal synchronized clocks, showing the same time at every spatial point show this quite well.

Matheinste

 

[Austin0]

But then you had written:

Originally Posted by Austin0
Doesn't the Plane of S have the same origen and t-0?

But the geometric plane doesn't have an origin, only once it is coordinatized does it have an origin. So I have a difficulty - although you keep saying that what I write is what you have always thought, what you often write suggests a different conception.

_________________________________________________________________________
__________________________________________________________________________________

Now, let's restrict our attention to coordinate systems that represent inertial frames, the t axes representing the frame's time, the others representing space. Then:

Given a partitioning of Minkowski space into a sequence of simultaneity hypersurfaces, and given that a coordinate system respects the structure of this partitioning, there still remain *many* coordinate systems that correspond to this partitioning. So, not 1-1. However, I think that they all differ in merely conventional ways: choice of unit, origin, spatial axes - so also given choice of unit etc. then (I think) YES: one coordinate system.

Moreover, each coordinate system corresponds to 1 partition of Minkowski space-time, the hyperplanes those planes that all share the same t coordinate of the coordinate system.

_________________________________________________________________________
____________________________________________________________________________
Originally Posted by Austin0
Using Matheniste's rigid rods and clocks. in a moment in time ,wouldn't the simultaneity coordinate structure be indentical..

What does `simultaneity coordinate structure' mean? Guessing: The coordinate system should give any two points of the hypersurface in the simultaneity class the same t value

__________________________________________________________________________
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Originally Posted by Austin0
The difference is the conventional definition of simultaneity; occurring at equal distance from a point as measured by light path.

This does not make sense to me - the difference between what?
What convention? What's a spatial point of the coordinate frame? The coordinate system is a set of 4 numbers, representing space-time points.[\QUOTE]

_________________________________________________________________________

I do not know what you have in mind. What spherical onion? There's nothing particularly spherical about a hypersurface - it sounds as if you're partitioning spacetime in a way I don't understand.

So, no, I'm afraid your comments don't clarify things. In fact, they tend to make me worry that we're not really on the same page at all.

Hi yossell

Another attempt.

Starting with an abstract (blank) coordinate frame:

If you assume orthogonality, and spatial and temporal metrics you have created a fully functional Newtonian coordinate system aka Inertial Frame (NF)

XZ Plane,etc.,etc.

But this is not yet a SImultaneous Frame as defined in SR.

To achieve this you must apply The Synchronization Procedure
In principle you could pick any arbitrary point and proceed however you chose but rationally you would start at the origen with a lot of mirrors and proceed systematically in progressively expanding spheres throughout the total space. Or better; pick an arbitrary time and emit an isometric spherical light burst and all clocks throughout the system would be set on the basis of radial distance/c .

You now have a simultaneous coordinate system, Simultaneity frame (SF)
which is also a fully realized SR frame (SRF), [the only difference between this and the initial NF is the application of the convention]

Aren't these two frames, not simply isomorphic, but actually totally identical?
Same set of 4 numbers for any point in spacetime??

The SRF xz plane is the SF xz Hyperplane??

The only difference being semantic...XZ Plane AKA XZ Hyperplane.

I am not questioning that this semantic distinction may be useful but it seems to me that it also creates confusion , a false impression that there is a significant difference.

Is there any difference between spacetime slices or partitioning and a Euclidean slice at the same point?
____________________________________________________________________________
Originally Posted by Austin0
Using Matheniste's rigid rods and clocks. in a moment in time ,wouldn't the simultaneity coordinate structure be indentical..

What does `simultaneity coordinate structure' mean? Guessing: The coordinate system should give any two points of the hypersurface in the simultaneity class the same t value

Isn't a Hyperplane simply a slice of the Simultaneity coordinate structure just as the SRF xz plane is a slice of the SRF coordinate sturcture. AS such, of course they share the same t value as they do at every point in both frames??

yossell "each coordinate system corresponds to 1 partition of Minkowski space-time"

They are not really coordinate systems themselves are they ,,,any more than the SRF xz plane is a separate coordinate system??

Couldn't you consider the Hyperplane a slice of the SRF coordinate frame with no functional difference?

SO what do you think , are we on the same page at all?

Thanks

 

[Austin0]

Using Matheniste's rigid rods and clocks. in a moment in time ,wouldn't the simultaneity coordinate structure be indentical.
The difference is the conventional definition of simultaneity; occurring at equal distance from a point as measured by light path. This convention is applied to every spatial point in the coordinate frames.
This results in every point being in the center of a spherical onion of different layers extending indefinitely into space. All events located on a particular skin are simultaneous.

Just so you are aware, Austin0, the relativity of simultaneity, time dilation etc. are effects which are what is left after you correct for things like signal time and doppler shift.

Of course. ..... I assume you are referring to the above post.

I may have been unclear in my description.

I was referring to the fundamental definition of simultaneity. The basis of the derived synchronization convention.

I was not suggesting that after the clocks were thus synchronized only events occurring at equal distance would be considered simultaneous . Under the convention all events that occur at the same local clock time are considered simultaneous.
But as you pointed out this is a matter of calculation and compensation for signal time.

IMHO Einstein's brilliant perception , recognition of the fundamental basis of simultaneity do

not only applied to synch conventions,clocks and observations within frames, which he also pointed out are purely relative and thus without physical implications or any real or absolute meaning. Operational assumptions with no interpretation of actual simultaneity.

But IMHO they also apply to the actual physics of the universe. That the sphere (ellipsoid) is a fundamental if not the fundamental geometrical form of spacetime. Relevant to gravitation , EM fields and electrostatics and perhaps other fundamental relationships in QM or yet undiscovered.

Looking at an electron at rest in an electrodynamic field. Considering only the other electrons in a limited spatial range:

There current positions are simultaneous according to equal readings on local clocks by definition,, regardless of their distances. SO also are the events of emission of energy at those points.

But the immediate field potential of the rest electron is not a consequence of those positions. It is dependant on the energy arriving simultaneously at that instant.

Consider a 3 D space (t,x,z) Looking at two electrons A and B traveling at different speeds on paths tangent to a circle centered on rest electron R and intersecting that circle at t=0 At t=3 they have different distances from R but their local positions are still simultaneous according to local clocks; but the energy emitted from those positions will not reach R simultaneously. Only the energy emitted at t=0 is simultaneous in this sense.
When they were equidistant from R.

Extrapolating:
looking at other pairs of electrons etc etc you arrive at a series of concentric circles in space but extended in time. What is arriving simultaneously is also a function of the particular radius, as the radius increases the path length increases and so must have been emitted at an earlier time.

This would apply equally to all the electrons regardless of there velocity but
as "measured " or calculated in the frame of the rest electron the circles of other electrons would be ellipses.

As the idea of a rest electron is not realistic , as measured in the lab frame all circles would be ellipses.

I don't know if this might make things better or worse regarding understanding.

I am fully prepared to get clobbered on any number of accounts.

Thanks

 

[Austin0]

Hello Austin0.

In spacetime which is four dimensional, the surface of a three dimensional sphere is not a hypersurface of simultaneity as you seem to think. For a particular observer the whole of his three dimensional space is, for him, a hypersurface of simultaneity. Different observers effectively partition spacetime into their own individual Newtonian universes. The usually defined reference frame of a set of rigid rods and ideal synchronized clocks, showing the same time at every spatial point show this quite well.

Matheinste

Hi Matheinste.

Since this thread has been left somewhat unresolved I have a question:

Do you think I can assume and state with a degree of confidence that there is no significant difference between a hyperplane and the plane of the frame itself?

Thanks and also for your other input.

 

[matheinste]

Hi Matheinste.

Since this thread has been left somewhat unresolved I have a question:

Do you think I can assume and state with a degree of confidence that there is no significant difference between a hyperplane and the plane of the frame itself?

Thanks and also for your other input.

I don't understand what you mean by the plane of the frame!

Matheinste.

 

[Austin0]

Hi Matheinste.

Since this thread has been left somewhat unresolved I have a question:

Do you think I can assume and state with a degree of confidence that there is no significant difference between a hyperplane and the plane of the frame itself?

Thanks and also for your other input.

I don't understand what you mean by the plane of the frame!

Matheinste.

The x lines for the rest frame in a 2D Minkowski diagram represent the x-axis or xz plane if you extend it mentally .

Using your rods and clocks analogy [which I use all the time in my mind and have used myself in previous threads] This line represents a line of rods and clocks and virtual observers.
For any point on the worldline this line or plane is limited to the spatial dimension(s)
The clocks all frozen at the same reading. The hyperline (plane) at this moment in time is the exact same rods and clocks. There are no others are there??

This is exactly the same for the sloped hyperline of the moving frame. The same set of rods and clocks as the x'z' plane of the frame.

This has been my understanding and operative assumption for a long time but I did not want to be making statements based on this without checking to see if there was something I was missing or some real difference I was unaware of. SInce so many people seemed to assume they were different I posted this thread to get feedback and other perspectives.

Clearer? Thanks for the feedback it is appreciated

 

[matheinste]

The x lines for the rest frame in a 2D Minkowski diagram represent the x-axis or xz plane if you extend it mentally .

Using your rods and clocks analogy [which I use all the time in my mind and have used myself in previous threads] This line represents a line of rods and clocks and virtual observers.
For any point on the worldline this line or plane is limited to the spatial dimension(s)
The clocks all frozen at the same reading. The hyperline (plane) at this moment in time is the exact same rods and clocks. There are no others are there??

This is exactly the same for the sloped hyperline of the moving frame. The same set of rods and clocks as the x'z' plane of the frame.

This has been my understanding and operative assumption for a long time but I did not want to be making statements based on this without checking to see if there was something I was missing or some real difference I was unaware of. SInce so many people seemed to assume they were different I posted this thread to get feedback and other perspectives.

Clearer? Thanks for the feedback it is appreciated

Having given it more thought I will stick my neck out and say that a network consisting of rods and synchronized clocks with respect to which you are at rest is the same as the hypersurface of simultaneity for the frame of reference in which you are at rest, although I have never seen it described as such and I may receive some adverse comments. However lines and planes of rods and synchronized clocks, although lines ad planes of simultaneity are not the same as the frame of reference but are sections through it.

Matheinste.

 

[Austin0]

Having given it more thought I will stick my neck out and say that a network consisting of rods and synchronized clocks with respect to which you are at rest is the same as the hypersurface of simultaneity for the frame of reference in which you are at rest, although I have never seen it described as such and I may receive some adverse comments. However lines and planes of rods and synchronized clocks, although lines ad planes of simultaneity are not the same as the frame of reference but are sections through it.
Matheinste.

Hi Matheinste Thanks for sticking your neck out and giving me a straight answer.

regarding the second part: Do you mean they are not the reference frame in any sense different from; the lines and planes of that frame as depicted in a diagram are sections through it and not the frame??

Or alternately: Outside the limitations of 2d drawings , as fully 4D constructs would there be any difference in this regard??

Thanks again

 

[matheinste]

regarding the second part: Do you mean they are not the reference frame in any sense different from; the lines and planes of that frame as depicted in a diagram are sections through it and not the frame??

Or alternately: Outside the limitations of 2d drawings , as fully 4D constructs would there be any difference in this regard??

Thanks again

They are not reference frames in the usually defined way as reference frames are usually understood to extend indefinitely in every spatial direction.

In answer to the second part the lower dimensional drawings are limited as you say and so cannot represent reference frames in the usual sense of the word. A reference frame is not a four dimensional construct but three spatial dimensions with clocks added. A snapshot or drawing of a reference frame in three dimensions is a snapshot of the entire spatial extent of space at one particular time. This is a sort of how a reference frame is loosely defined.

Matheinste.