Understand Special Relativity and Time paradox

[Dale]

I tend to feel that nature has put the relativity of simultaneity into our physics and into our reality. Nature gave us a speed of light that is the same for all inertial frames. That is something that we experience because nature put in the photon worldlines so as to bisect the angle between X4 and X1 (thus, the Lorentz-Poincare'-Minkowski-Einstein simultaneous spaces).

I still don't like the word "experience" in any of this. What we experience is our past light cone, not our simultaneous spaces. However, if you scrubbed the word "experience" I don't find this off too far. In a reference frame in which the laws of mechanics hold good the speed of light is c, I believe that is indeed a fact of nature and not a matter of convention.

Further, nature manifests the laws of nature through the continuous sequence of simultaneous spaces we experience as we move along our worldines.

This is flat false. Please try to write any of the laws of nature in this form for the traveling twin.

Where gravity is negligible the laws of nature can be written in terms of the continuous sequence of simultaneous spaces for an inertial worldline. The laws of nature cannot be written in that manner at all for the continuous sequence of simultaneous spaces of a non-inertial worldline.

Furthermore even though they can be written in that manner for an inertial observer, they are not required to be written in that manner. The inertial observer can write them in terms of any other inertial observer's sequence of simultaneous spaces, or simply in terms of an inertial frame not corresponding to any observer. Or they can be written in terms which are completely independent of any frame, inertial or not. In fact, where gravity is not negligible the laws of nature can only be written that way, and not at all in terms of the sequence of simultaneous spaces from SR.

 

[PeterDonis]

Isn't it the case that in Special Relativity, any scenario can be fully described and analyzed using any IRF and that you can use the Lorentz Transformation process to get to any other IRF moving with respect to the original one?

Yes. And often it's easier to do it this way. But some people have an apparently unstoppable desire to have some expression of "how things look to observer X" when observer X is not moving inertially all the time. The fact that there is no unique answer to this question, and that all of the possible answers have significant limitations, doesn't stop them from asking it. So the best we can do is to try to talk about the possible answers and their limitations.

 

[Austin0]

Quote by Austin0

I submit that, within certain limitations of acceleration magnitude and spatial extent , a chart correctly constructed with the synchronization of a sequence of CMRFs would be perfectly tractable throughout an extended domain.

I would be very interested in such a transformation, particularly if you can do it without the sketches. Please post the math at your earliest opportunity!

Well i can give you a rough conceptual basis.

Given an extended coordinate frame (v -->+x) of conventionally synched clocks with a range from x= -100 to x=100 with the Traveler at x=0 and the center of resynchronization.

I am assuming that by cosmic coincidence , every succeeding MCRF has the same proper time reading at the Traveler location, so the system can be recalibrated by simply having the traveler observers set their clocks to the time of the proximate CMRF clock throughout the system.
For clarity we can do this at discrete intervals say 20 sec with an acceleration such that this results in a 0.1c velocity change between resynchronizations

So if acceleration begins at T₀ at Traveler x=0

at x=-100,t₀ and x=100, t₀

At T₁=T₀+20... MCRF1 t'=T₀ +20 and also at x=-100,t=t₀+20 and x=100,t=t₀+20

Now we know that at T₀ the target MCRF1 has a relative v of 0.1c
so the synchronization offset is simply vx' or 0.1x' . At x'=-100 this means 0.1(-100)=-10s
behind the traveler clock at that location.

At T₁ the clock at x=-100 still has the synchronization of t₀
so we know that it is 10 sec ahead of the proximate MCRF clock at x'=-100.

So x=-100,t₁=t₀+20 +(-10) =t₀+10

AT x=100,t₁=t₀+20 +(10)=t₀+30

AT T₂=T₀+40

x=-100,t₂=t₁+20 +(-10) =t₀+20

AT x=100,t₂=t₁+20 +(+10) =t₀+60

___________________________________________________________________

T₀
x=-100,t₀ and x=100, t₀

T₁=T₀+20..
x=-100,t₁=t₀+10,,,x=100,t₁=t₀+30

T₂=T₀+40
x=-100,t₂=t₀+20 ,,,x=100,t₂=t₀+60

And so on through whatever course of continuous acceleration in the same direction.

In this case if vx' is less than dT =20 the -x clocks continue to increment forward. At x=-200 they would remain at the same value and beyond that would actually be decrementing back from previous time.

Obviously this is a gross simplification. it ignores differential acceleration and the fact that such a system could not be co-moving with a single MCRF throughout the system. SO at different locations the resynchronization would be with different MCRFs with different velocities and different synchronization. A serious treatment would require these differential complexities. But my purpose was simply to get a picture of what such a system would look like. How the time structure would evolve and it doesn't appear to me that the addition of the velocity differential would change the basic continuous forward progression of resynchronized times anywhere in the system.

Hopefully you will agree that such a chart would be without gap or overlap throughout the defined domain?

 

[Austin0]

Quote by Austin0

Well you hit many salient points but i think I have a different perspective on core issues.
I think that this thread is basically misdirected and is missing the crucial point.
Which is the inherent problem with charting accelerated systems in Minkowski space. So I think that the problem is not with CMRF's and adopting their simultaneity but the fact that a system based such a series of frames is incorrectly charted in such a diagram.

I don't follow what you're saying. 1) I understand Minkowski space to be the flat manifold, independent of any coordinate chart. In SR, it is the only manifold under consideration, and is the only manifold to be charted - in any valid way.

2) Against an inertial chart (which covers the complete manifold), any valid alternative chart can be drawn, for whatever region of spacetime such a chart covers.

Do you disagree with any of this?.

I certainly agree with #1
with #2 i have question.
With inertial frames the charts are fundamentally Euclidean and the metrics static so you can superpose the traveler chart with linear , one to one correspondense over an extended time range..
An accelerated chart has a dynamic metric and is in a sense inherently non-Euclidean
It can map (assign coordinates ) unambiguously to a flat manifold but I do not see how an extended time range of such a chart can be linearly mapped to a single uniform orthogonal matrix .
.
In the inertial case the simultaneity line , the tilted x-axis represents a historical record in the chart. Those who are so inclined can choose to consider this a simultaneous moment of the traveler frame but what it explicitly is, is a log of coordinate times and locations, in the rest frame, attached to the traveler clocks with a certain equal time reading . Because the metric is constant this set of events falls on a straight line in the rest frame.

With an accelerated frame it seems to me that with a dynamic metric the set of coordinate events logging a particular time value could not possibly fall on a straight line in the rest frame. Yet this is exactly what is portrayed by a straight line of simultaneity attached to such an accelerated frame.
It charts an implicit assumption that simply because the traveler momentarily adopts the synchronization of a MCRF that this makes the frame congruent with the history of the MCRF.
I think such lines are actually misleading during accleration in both directions but in the case of towards Earth they don't lead to obvious anomalies because when the traveler does go inertial, the traveler frame then does become congruent with the future of the final MCRF .So the intersection of that line with the Earth does agree with the later appearence of the traveler clock there.

Quote by Austin0

I submit that, within certain limitations of acceleration magnitude and spatial extent , a chart correctly constructed with the synchronization of a sequence of CMRFs would be perfectly tractable throughout an extended domain.
That it would produce smooth continuity from a conventional inertial system through acceleration to a final inertial state without overlap or gap over an large spatial area. With no temporal ambiguities and complete agreement on events with Earth and all other inertial frames.That the overlap that occurs in the outward region in a Minkowski chart is neither inherent in nor an accurate representation of such a system but is purely an artifact of Minkowski graphing.

I agree that the sequence of CMRF simultaneity defines a perfectly reasonable chart covering a substantial region of spacetime. However, the region where the surfaces intersect is not an artifact. Two surfaces intersecting is a geometric fact. For this region, you can't simply use these slices to chart that region. Note that a while ago, I noted that you could imagine a (sideways) W shaped path for the traveling twin. For such a path, the CMRF slices would not be valid for covering the complete home twin world line in one coordinate chart.

Well here I must beg to differ.I think that not only is the region from the intersection through divergence not a geometric fact I think it is totally non-existent. It is purely a mental construct produced by falsely assuming the simultaneity planes from a series of MCRFs has any correspondence to the accelerated traveler frame in this region..
I am somewhat surprised by you as you seem to be agreeing with bobc2 on this point. WHile he embraces this interpretation with all its unlikely implications , you want to excise it from the chart for coordinate misbehavior.
but both seem to agree that it does represent the accelerated frame with instantaneous conventional synchronization.
I suggest that it simply has no relation to the accelerated frame or a traveler chart constructed with this convention. That the events portrayed in that region map events in the past of the various MCRFs but do not map any events of the accelerated frame. They would not appear in the traveler chart nor would they appear in the chart of any inertial frame logging the locations and times of the accelerated frame.

Quote by Austin0

There is one resulting condition of such an implementation; the synchronous coordinate time generated would not have a uniform rate throughout the system.

Specifically,,if we assume the traveler location as the point of synchronization for the frame , then the coordinate time on clocks running back toward Earth would be slowed down by increasing degrees relative to the proper rate of a natural traveler clock there. Comparably the coordinate time outward from the traveler would have increasing rates.

I don't understand the rest of your post at all. What would help are either equations for transforming between home twin inertial coordinates and your proposed coordinates (you don't even need to specify the metric; I can figure that if you give the transform). Alternatively, I insist that against a complete chart like the inertial frame, any other coordinate chart can be diagrammed via drawing or charting its coordinate lines. The specification of units on them would be needed to finalize the metric, but I wouldn't need that to understand your proposal - the lines alone determine the metric to within scaling factors..

My understanding was that adopting the synchronization of a series of MCRFs automatically defined the math to be the normal L.T. and metric.
So the difference is only in my approach in applying that math.
That approach was simplistic. Start with a hypothetical physical system of clocks and rulers and then determine what the chart of such a system would look like with continuous resynchronization conforming to the MCRFs.
How the clock readings would evolve over time at various locations within the frame.

With this simple model certain things seem clear.

All inertial frames would chart the physical system proceeding uniformly through time.That this log of positions and times is independent of any clock readings or synch convention occurring within the frame. SO any implemented convention could only change the observed clock readings but not effect any change in the position of the frame as indicated by the rotating x' axes attached to the accelerated frame in the standard chart.

So in my description of the events in the instant turnaround scenario with the traveler clocks being turned back along the line towards Earth (overlapping coordinates) and turned forward outward from the traveler (coordinate gap), I was describing a frame independent reality. The physical event of changing a clock time is invariant. Inertial frames would assign their own coordinates to these events but all frames must agree on the numerical values of the change and where they occurred in the traveler frame. Agreed?.

i think that in the case of less radical acceleration that all frames would also agree on the continuous forward progression of continually resynchronized clock times throughout the system as I outlined .

Some years ago i started a thread attempting to resolve these exact issues but it got bogged down in the same arguments between those that accepted the implications of such simultaneity lines as valid pictures of reality and those who, like you, thought it was simply a matter of striking them from the chart. So I welcome this second opportunity to possibly shed some additional light on the question..

 

[PAllen]

with #2 i have question.
With inertial frames the charts are fundamentally Euclidean and the metrics static so you can superpose the traveler chart with linear , one to one correspondense over an extended time range..
An accelerated chart has a dynamic metric and is in a sense inherently non-Euclidean
It can map (assign coordinates ) unambiguously to a flat manifold but I do not see how an extended time range of such a chart can be linearly mapped to a single uniform orthogonal matrix .

Several issues here: changing coordinates does not change geometry. Changing coordinates with changed metric preserves geometric objects, including the curvature tensor. If a manifold is flat (e.g. SR) it has zero curvature in all coordinates.

Who said linearly? You can draw polar coordinate lines on cartesian coordinates just fine. The transform is non-linear. The Euclidean geometry metric expressed in polar coordinates is no longer diag(1,1), but the curvature tensor is still identically zero everywhere. You don't make a plane curved by drawing different coordinates on it.

I don't mean to be insulting, but have you read any introduction to differential geometry?

My point, intended to be obvious, is that if you have one coordinate chart that covers a complete manifold, and you have any other coordinate chart (which provides one label for every point in the manifold in the manifold that it covers, and is continuous one-one mapping from any other coordinate chart for portions that overlap), then any coordinate chart can be plotted on any chart that covers the whole manifold - as the standard Minkowski coordinates do.

With an accelerated frame it seems to me that with a dynamic metric the set of coordinate events logging a particular time value could not possibly fall on a straight line in the rest frame. Yet this is exactly what is portrayed by a straight line of simultaneity attached to such an accelerated frame.

I never said or implied that another coordinate chart's coordinate lines have to be straight when plotted in an inertial chart. It was Bobc2 who wanted to do this. My point is that straight or not, if two lines intersect, a coordinate change won't make them not intersect. If you are proposing simultaneity surfaces (or line restricted to x-t plane) that curve, you are emphatically not talking about the same simultaneity lines as Bobc2. I have stated a few time that not only is it possible to construct simultaneity lines that agree closely with MCIF near the traveler world line but differ at distances from it such that they never intersect, but that there are uncountably infinite ways of doing this with no clear way to prefer one over the other.

It charts an implicit assumption that simply because the traveler momentarily adopts the synchronization of a MCRF that this makes the frame congruent with the history of the MCRF.
I think such lines are actually misleading during accleration in both directions but in the case of towards Earth they don't lead to obvious anomalies because when the traveler does go inertial, the traveler frame then does become congruent with the future of the final MCRF .So the intersection of that line with the Earth does agree with the later appearence of the traveler clock there.

I completely agree with this.

Well here I must beg to differ.I think that not only is the region from the intersection through divergence not a geometric fact I think it is totally non-existent. It is purely a mental construct produced by falsely assuming the simultaneity planes from a series of MCRFs has any correspondence to the accelerated traveler frame in this region..

That is exactly what Bobc2 was doing. It is not a 'false' way of doing things, just a way that provides limited coordinate coverage. There is no such thing as 'false' coordinates. As for alternatives that don't have this intersection problem for any twin scenario, two that I know of that have names are radar simultaneity and Minguzzi simultaneity. I thought you were claiming that the intersections of MCIF lines could be removed by coordinate transform. That is nonsense. However, it is certainly possible pick different simultaneity lines that don't have intersections (uncountably many ways to do so).

I am somewhat surprised by you as you seem to be agreeing with bobc2 on this point. WHile he embraces this interpretation with all its unlikely implications , you want to excise it from the chart for coordinate misbehavior.

I agree with Bobc2 that it is a possible choice for simultaneity; it is a quite useful one locally. I disagree with Bobc2 that it has any more physical meaning globally than any number of other choices, and that where it has ridiculous implications, that means - mathematically - that it has become an invalid method of mapping spacetime.

but both seem to agree that it does represent the accelerated frame with instantaneous conventional synchronization.

No, I claim there is no preferred synchronization for non-inertial observers. I thought I have explained in great detail that the reason there is one for inertial observers is that any reasonable method of synchronizing separated clocks agrees with any other. For non-inertial observers, essentially every method of synchronizing separated clocks disagrees with all the other methods, so there is no reasonable basis to claim a preference.

I suggest that it simply has no relation to the accelerated frame or a traveler chart constructed with this convention. That the events portrayed in that region map events in the past of the various MCRFs but do not map any events of the accelerated frame. They would not appear in the traveler chart nor would they appear in the chart of any inertial frame logging the locations and times of the accelerated frame.

I agree with this.

My understanding was that adopting the synchronization of a series of MCRFs automatically defined the math to be the normal L.T. and metric.

No, this is not correct. If you adopt the series of MCIF simultaneity lines, parametrized by proper time along a non-inertial path, you get a chart (covering only part of spacetime) with a metric completely different from diag(1,-1,-1,-1). However, the geometry it describes is the same: curvature is still zero; all invariants come out the same.

So the difference is only in my approach in applying that math.
That approach was simplistic. Start with a hypothetical physical system of clocks and rulers and then determine what the chart of such a system would look like with continuous resynchronization conforming to the MCRFs.
How the clock readings would evolve over time at various locations within the frame.

With this simple model certain things seem clear.

I don't know what this part means. A fundamental property of non-inertial world lines in SR is:
- rigid rulers are cannot extend very far, even assuming the artifice of Born rigidity
- the Einstein clock synchronization convention disagrees with rigid ruler simultaneity, even where they both apply.

Given this, I truly have no idea what you are describing.

All inertial frames would chart the physical system proceeding uniformly through time.That this log of positions and times is independent of any clock readings or synch convention occurring within the frame. SO any implemented convention could only change the observed clock readings but not effect any change in the position of the frame as indicated by the rotating x' axes attached to the accelerated frame in the standard chart.

So in my description of the events in the instant turnaround scenario with the traveler clocks being turned back along the line towards Earth (overlapping coordinates) and turned forward outward from the traveler (coordinate gap), I was describing a frame independent reality. The physical event of changing a clock time is invariant. Inertial frames would assign their own coordinates to these events but all frames must agree on the numerical values of the change and where they occurred in the traveler frame. Agreed?.

If I understand this, it is complete nonsense. But maybe you have not made your meaning clear.

What each observer sees of the the other clock is continuous forward only movement, always. What they choose to interpret about the relationship between what they see and what is 'now' - which is purely a convention - is up for grabs, but one thing prohibited for a mathematically valid mapping is reversal of causality along a distant world line.

 

[zonde]

This is flat false. Please try to write any of the laws of nature in this form for the traveling twin.

Where gravity is negligible the laws of nature can be written in terms of the continuous sequence of simultaneous spaces for an inertial worldline. The laws of nature cannot be written in that manner at all for the continuous sequence of simultaneous spaces of a non-inertial worldline.

I view accelerating twin as undergoing physical transformation at the moment of acceleration. So we don't have to be able to write consistent laws when assuming that accelerating twin just stays what it is.

But the interesting thing is when we bring GR into the picture. An observer standing on the surface of a gravitating body according to GR is accelerated. And yet all the physical laws we have are developed by such an observer.

 

[SAYANTH K J]

According to me
The S R T says that space time is faster where gravitational force is more comparatively And G F is inversely proportional to distance ]therefore you will be older than your twin brother.

 

[Dale]

So we don't have to be able to write consistent laws when assuming that accelerating twin just stays what it is.

Sure, we don't have to be able to, but we are able to. We are able to write down such laws, but not in the fashion that bobc2 claims that nature gave us.

 

[ghwellsjr]

PAllen and PeterDonis, your guy's discussion of different kinds of frames and coordinates makes me wonder if I'm doing something wrong by emphasizing Inertial Reference Frames (IRF's). It seems so simple to me but all this other talk makes me wonder if I'm just oversimplifying things. Isn't it the case that in Special Relativity, any scenario can be fully described and analyzed using any IRF and that you can use the Lorentz Transformation process to get to any other IRF moving with respect to the original one? I realize that I'm limiting my discussion to the Standard Configuration so I'm not talking about transforms in other directions or where the coordinates don't share a common origin.

Yes. And often it's easier to do it this way. But some people have an apparently unstoppable desire to have some expression of "how things look to observer X" when observer X is not moving inertially all the time. The fact that there is no unique answer to this question, and that all of the possible answers have significant limitations, doesn't stop them from asking it. So the best we can do is to try to talk about the possible answers and their limitations.

Your answer implies that there is something more to be learned, that is, "how things look to observer X", by doing a more complicated analysis because you say "there is no unique answer to this question" and I know that is not what you meant. There is only one answer to the question of "how things look to observer X" and it can be determined in any single Inertial Reference Frame. Transforming to a different IRF will not in any way affect "how things look to observer X". I have given so many examples of this throughout this thread.

In fact, all that can be learned by doing a more complicated analysis, is that no matter how convoluted or how complex or how confusing the analysis, it will not in any way change "how things look to observer X".

Can I hear you say that in no uncertain terms, no equivocation, no ambiguity, no ifs, ands, or buts?

 

[PeterDonis]

There is only one answer to the question of "how things look to observer X" and it can be determined in any single Inertial Reference Frame. Transforming to a different IRF will not in any way affect "how things look to observer X". I have given so many examples of this throughout this thread.

In fact, all that can be learned by doing a more complicated analysis, is that no matter how convoluted or how complex or how confusing the analysis, it will not in any way change "how things look to observer X".

Can I hear you say that in no uncertain terms, no equivocation, no ambiguity, no ifs, ands, or buts?

For the meaning of "how things look to observer X" that you and I are using, yes. That meaning being, I assume, that "how things look to observer X" is determined by invariants that can be calculated using X's 4-velocity and other geometric objects. Invariants are the same in every frame, so you can always calculate them in whatever frame you like, and once you've done it once, doing it again and again in different ways doesn't change the answer. (Though it may be worth doing in a really complicated problem where you want a check on your calculations.)

But other people want to mean something else by "how things look to observer X": for example, they want "how things look to observer X" to be associated with quantities that are *not* invariant, such as particular coordinates in a particular frame. Much of the effort we put forward in these threads is in trying to convince them that trying to assign those other meanings to "how things look to observer X" leads nowhere.

Edit: Also, people want to include things in "how things look to observer X" that shouldn't be in that category at all. For example, they want to include "what is happening in the Andromeda Galaxy *right now*" in "how things look to observer X", and they start obsessing about how X can change "what is happening in the Andromeda Galaxy *right now*" by changing his state of motion, and whether his acceleration affects it, etc., etc. It's hard for many people to accept the real answer, which is simply that questions like "what is happening in the Andromeda Galaxy *right now*?" have no well-defined answer. You can make arbitrary choices that give it an answer, but those are just arbitrary choices with no physical content. We spend a lot of time trying to explain that too.

 

[Alain2.7183]

The Wikipedia page on the twin paradox, in the section on the "viewpoint of the traveling twin", explains the use of "gravitational time dilation" (via the "equivalence principle") to resolve the paradox from the traveler's viewpoint. The result is that, according to the traveler, the home twin's age increases a lot during the traveler's turnaround, enough to more than make up for the home twin's slower aging when the traveler isn't turning around. That result doesn't seem to be presented as "just one of many arbitrary simultaneity conventions". Is that a mistake?

 

[PAllen]

The Wikipedia page on the twin paradox, in the section on the "viewpoint of the traveling twin", explains the use of "gravitational time dilation" (via the "equivalence principle") to resolve the paradox from the traveler's viewpoint. The result is that, according to the traveler, the home twin's age increases a lot during the traveler's turnaround, enough to more than make up for the home twin's slower aging when the traveler isn't turning around. That result doesn't seem to be presented as "just one of many arbitrary simultaneity conventions". Is that a mistake?

You have to read it with appropriate background. It is making analogy to gravitation. However, gravitational time dilation in GR is coordinate dependent in the sense that a different set of coordinates makes the difference in clock rates have a kinematic origin. The key background is that in both SR and GR, all simultaneity conventions are just that - conventions for setting up coordinates. The observables: differential aging, differences in clock rates measured by exchange of signals (determined by the Doppler factor), come out the same for any simultaneity convention.

 

[PeterDonis]

That result doesn't seem to be presented as "just one of many arbitrary simultaneity conventions". Is that a mistake?

Yes. This is only one of many ways of analyzing the twin paradox; it's called "the equivalence principle analysis" in the Usenet Physics FAQ:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

The Wikipedia page seems to cover a lot of the same ground, but it talks about each way of analyzing the scenario as "the" resolution, which obfuscates the point that all of these analyses are valid; there is no single resolution which is "the" resolution. The closest thing to that, IMO, is what the FAQ calls "the spacetime diagram analysis" and the Wikipedia page calls "difference in elapsed times as a result of differences in the twins' spacetime paths". This way of looking at it, as the physics FAQ notes, provides a kind of "lingua franca" where you can see how all of the other analyses work and how they all fit together.

 

[Dale]

Given an extended coordinate frame (v -->+x) of conventionally synched clocks with a range from x= -100 to x=100 with the Traveler at x=0 and the center of resynchronization.
...
In this case if vx' is less than dT =20 the -x clocks continue to increment forward. At x=-200 they would remain at the same value and beyond that would actually be decrementing back from previous time.
...
Hopefully you will agree that such a chart would be without gap or overlap throughout the defined domain?

I do agree if by "defined domain" you specifically mean x=-100 to x=100. It appears that you are applying the usual MCIRF synchronization convention that bobc2 is using, but over a limited spatial domain. That is the correct way to do it. Once you try to extend it into a region with an overlap then you have problems. You are avoiding those problems by limiting the domain, which is a perfectly legitimate thing to do, assuming I understood you correctly.

 

[zonde]

Your answer implies that there is something more to be learned, that is, "how things look to observer X", by doing a more complicated analysis because you say "there is no unique answer to this question" and I know that is not what you meant. There is only one answer to the question of "how things look to observer X" and it can be determined in any single Inertial Reference Frame. Transforming to a different IRF will not in any way affect "how things look to observer X". I have given so many examples of this throughout this thread.

In fact, all that can be learned by doing a more complicated analysis, is that no matter how convoluted or how complex or how confusing the analysis, it will not in any way change "how things look to observer X".

Can I hear you say that in no uncertain terms, no equivocation, no ambiguity, no ifs, ands, or buts?

ghwellsjr, do you really mean "how things look to observer X" or rather "what observer X sees"?
Because while it is true that transforming between different IRF does not change physical facts nonetheless what an observer makes out of these facts could be quite uncertain.

So if by "how things look to observer X" we mean what an observer X makes out of what he sees the answer can be quite ambiguous.

 

[ghwellsjr]

ghwellsjr, do you really mean "how things look to observer X" or rather "what observer X sees"?

I don't see any difference between the two. They look the same to me.

Because while it is true that transforming between different IRF does not change physical facts nonetheless what an observer makes out of these facts could be quite uncertain.

Not if he understands Special Relativity.

So if by "how things look to observer X" we mean what an observer X makes out of what he sees the answer can be quite ambiguous.

Look, it's not anything about observer X that makes any difference. It's the frame that any observer chooses to use but the real big insurmountable problem that makes this all so much nonsense is that no observer can see anything beyond his own local experience. When I draw my diagrams of the very simple Twin situation, neither twin can have any awareness of what is going on with the other twin until some time later, when the light signal reaches them--in other words, what the Doppler analysis indicates. At that time, if they want, they can construct a partial frame to assign Time Dilation or Simultaneity or Length Contraction in any way that is consistent with what they have seen and according to any frame they choose. If they understand what they are doing, it won't be ambiguous. But if they expect to gain some additional insight into what already happened, then who knows what confusion they are in for?

My continued, repeated, and, so far, unanswered question for those of you who insist on promoting a preferred frame--the so-called rest frame for each observer--why? Why are you subjecting yourself to such torture? What do you expect to learn from such an exercise? What do you think observer X is going to learn from doing such an exercise? Is he going to say, "did I just see what I thought I saw?" Will it cause him to change his mind and reinterpret whatever he saw?

 

[ghwellsjr]

Your answer implies that there is something more to be learned, that is, "how things look to observer X", by doing a more complicated analysis because you say "there is no unique answer to this question" and I know that is not what you meant. There is only one answer to the question of "how things look to observer X" and it can be determined in any single Inertial Reference Frame. Transforming to a different IRF will not in any way affect "how things look to observer X". I have given so many examples of this throughout this thread.

In fact, all that can be learned by doing a more complicated analysis, is that no matter how convoluted or how complex or how confusing the analysis, it will not in any way change "how things look to observer X".

Can I hear you say that in no uncertain terms, no equivocation, no ambiguity, no ifs, ands, or buts?

For the meaning of "how things look to observer X" that you and I are using, yes. That meaning being, I assume, that "how things look to observer X" is determined by invariants that can be calculated using X's 4-velocity and other geometric objects. Invariants are the same in every frame, so you can always calculate them in whatever frame you like, and once you've done it once, doing it again and again in different ways doesn't change the answer. (Though it may be worth doing in a really complicated problem where you want a check on your calculations.)

Saying

"how things look to observer X" is determined by invariants

is to reopen the can of worms that occupied so many pages on this thread. I would rather say "how things look to observer X" is the raw data that any theory must conform to and they are what determine what invariants must be in any viable theory. In fact that is the point I was making on the second page of this thread to LastOneStanding but if you look at his reaction, he couldn't understand what I was saying. Neither could bobc2 when he took up the cause when LastOneStanding became FirstOneFalling. These people denigrate the Doppler Analysis as a mere curiosity having no real significance to them. To them, something else is real.

But other people want to mean something else by "how things look to observer X": for example, they want "how things look to observer X" to be associated with quantities that are *not* invariant, such as particular coordinates in a particular frame. Much of the effort we put forward in these threads is in trying to convince them that trying to assign those other meanings to "how things look to observer X" leads nowhere.

Edit: Also, people want to include things in "how things look to observer X" that shouldn't be in that category at all. For example, they want to include "what is happening in the Andromeda Galaxy *right now*" in "how things look to observer X", and they start obsessing about how X can change "what is happening in the Andromeda Galaxy *right now*" by changing his state of motion, and whether his acceleration affects it, etc., etc. It's hard for many people to accept the real answer, which is simply that questions like "what is happening in the Andromeda Galaxy *right now*?" have no well-defined answer. You can make arbitrary choices that give it an answer, but those are just arbitrary choices with no physical content. We spend a lot of time trying to explain that too.

Yes, we do. Look at how much resistance is met by these people when it comes to explaining that Einstein's clock synchronization convention is arbitrary and a definition we put into nature rather than one we derive from nature. Can we please hit the nail on the head and state clearly that "how things look to observer X" is the reality to which Special Relativity must conform? It's what Einstein meant when he called his theory "consistent" in his 1905 paper.

 

[PeterDonis]

I would rather say "how things look to observer X" is the raw data that any theory must conform to and they are what determine what invariants must be in any viable theory.

I think we're saying the same thing. The "raw data" *are* the invariants. They are things like "the Doppler shift measured by observer X for light beam L" or "the proper time experienced by observer X between events A and B on his worldline". These are things that X can observe directly, *and* they are the things that are modeled in the theory as invariant scalar quantities. That's the whole point: once you understand that "how things look to observer X" is *entirely* specified by invariants that express X's direct observables, a lot of questions are simply dissolved and it gets a lot easier to analyze scenarios.

These people denigrate the Doppler Analysis as a mere curiosity having no real significance to them.

I agree; but what they're missing is precisely that the Doppler Analysis is *entirely* in terms of invariants--direct observables. You can do the entire analysis without ever talking about *anything* that isn't directly observed--you don't need any coordinates, you don't need any "frames", you don't need any simultaneity conventions, you don't need *any* of that. That's the point.

To them, something else is real.

It appears so, but I think it's because they (or at least bobc2, who has expressed this explicitly) are so worried about not being "positivists" that they end up actually giving direct observables *less* weight than abstractions. That's not what a "realist" is supposed to do. Direct observables are not infallible, certainly, and in order to make sense of them we do end up with no real choice but to believe in things we can't directly observe. But direct observables are where you start from: without those there is nothing to explain and nothing to anchor anything else to.

Can we please hit the nail on the head and state clearly that "how things look to observer X" is the reality to which Special Relativity must conform?

Again, I think we're saying the same thing. See above.

 

[bobc2]

Maybe we are not understanding each other’s views on this. Can we agree on the following: We shall adopt the modern view(largely due to Einstein) that a physical theory is an abstract mathematical model (much like Euclidean geometry) whose applications to the real world consist of correspondences between a subset of it and a subset of the real world. In line with this view, special relativity is the theory of an ideal physics referred to an ideal set of infinitely extended gravity-free inertial frames.

Further, the laws of physics are identical in all inertial coordinate systems, or, equivalently, the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial coordinate system.

 

[PAllen]

Maybe we are not understanding each other’s views on this. Can we agree on the following: We shall adopt the modern view(largely due to Einstein) that a physical theory is an abstract mathematical model (much like Euclidean geometry) whose applications to the real world consist of correspondences between a subset of it and a subset of the real world. In line with this view, special relativity is the theory of an ideal physics referred to an ideal set of infinitely extended gravity-free inertial frames.

Further, the laws of physics are identical in all inertial coordinate systems, or, equivalently, the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial coordinate system.

You haven't specified the part that corresponds to the real world (invariants). Also, one our our disagreements, is that, contrary to your phrasing your second pargraph above, you want to analyze one experiment with a different inertial frame at each moment. That is a whole different thing than anyone inertial frame (and leads to all the complications of this thread - because, like it or not, you are constructing a non-inertial coordinate system when you do that).

[edit: I think another related issue, is 'infinitely extended'. Since the topic is spacetime, infinitely exended means all space and all time. Once you use one 3-d slice of an inertial frame, it isn't an inertial frame anymore - it is an arbitrary slice of spacetime.]

 

[PeterDonis]

Can we agree on the following: We shall adopt the modern view(largely due to Einstein) that a physical theory is an abstract mathematical model (much like Euclidean geometry) whose applications to the real world consist of correspondences between a subset of it and a subset of the real world.

No problem here. The only thing I would add is that the correspondence is always approximate; we don't have any physical theories for which a subset of the model corresponds exactly to a subset of the real world. There is always some error involved.

In line with this view, special relativity is the theory of an ideal physics referred to an ideal set of infinitely extended gravity-free inertial frames.

As PAllen pointed out, this doesn't even talk about the subset of the model--the invariants--that corresponds to a subset of the real world. There are no inertial frames in the real world, any more than there are grid lines on the Earth marking latitude and longitude, or little arrows at a given point on the Earth marking off the vectors that point along great circles.

 

[Dale]

Maybe we are not understanding each other’s views on this.

I don't think that the problem is a lack of understanding each other's views. I think that each of us understand the other's view perfectly well.

What I think is not understood is the math.

Can we agree on the following: We shall adopt the modern view(largely due to Einstein) that a physical theory is an abstract mathematical model (much like Euclidean geometry) whose applications to the real world consist of correspondences between a subset of it and a subset of the real world.

I would agree with that, although I would probably make the predicted correspondences part of the theory.

In line with this view, special relativity is the theory of an ideal physics referred to an ideal set of infinitely extended gravity-free inertial frames.

That is certainly a subset of the mathematical model. The mathematical model also includes non-inertial frames (still gravity-free) as mentioned by PAllen as well as the invariants mentioned by PeterDonis.

The invariants are particularly important since they are the subset of the mathematical model which is predicted to correspond to the appropriate subsets of the real world.

Further, the laws of physics are identical in all inertial coordinate systems, or, equivalently, the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial coordinate system.

Yes.

 

[bobc2]

I'll try taking this one step at a time. Let me pose the questions:

1) After the twins reunite (after they've moved say about a million miles together--in terms of ct), would you say that they then are sharing the same inertial frame?

2) Is the traveling twin at rest in the stay-at-home twin's rest frame?

3) When the traveling twin has momentarily decelerated to zero velocity in the stay-at-home (he stops at the turnaround then heads back toward home), is the traveling twin at that event momentarily at rest in the stay-at-home frame?

 

[PAllen]

I'll try taking this one step at a time. Let me pose the questions:

1) After the twins reunite (after they've moved say about a million miles together--in terms of ct), would you say that they then are sharing the same inertial frame?

everything is in every inertial frame all the time. Each inertial frame is just another way of mapping 'existence'. Observers or bodies don't 'own' frames, or 'have frames'. What I would say is that when the traveling and home twin are at rest relative to each other, they are at rest in the same set of inertial frames. Sounds tautological? It is.

2) Is the traveling twin at rest in the stay-at-home twin's rest frame?

He and the stay at home twin are at rest in the same inertial frame.

3) When the traveling twin has momentarily decelerated to zero velocity in the stay-at-home (he stops at the turnaround then heads back toward home), is the traveling twin at that event momentarily at rest in the stay-at-home frame?

The traveling twin is in every inertial frame all the time. He is at rest in the same inertial frame in which the home twin is at rest when their relative motion is zero. So yes, for moment at turnaround they are at rest in the same frame.

Now let's see where it goes from here.

 

[SysAdmin]

I think in OP case, the one that need to be change is the perspective. Let the twin never meet again. Each of them stay in different galaxy that moving away at nearly the speed of light. Each of them see through telescope and find out his brother (and his galaxy) nearly not moving at all.

In reality, both of them grow old, and both of their galaxy just moving fine. The nearly same effect that we see right now. We see galaxy at distant in very young condition, but actually, it already evolve for trillion years, we just don't know what is the current look like. Now add that that somehow that galaxy is moving away from us at the speed near the speed of the light. We will see that the galaxy is stay still. We will never know what its current look like in double impossible way.

 

[Dale]

Why don't you just learn the math?

1) After the twins reunite (after they've moved say about a million miles together--in terms of ct), would you say that they then are sharing the same inertial frame?

No, I wouldn't say that. I would say "they are at rest in the same frame".

2) Is the traveling twin at rest in the stay-at-home twin's rest frame?

Yes. (I assume you mean after they reunite).

3) When the traveling twin has momentarily decelerated to zero velocity in the stay-at-home (he stops at the turnaround then heads back toward home), is the traveling twin at that event momentarily at rest in the stay-at-home frame?

Yes, by definition. The term "at rest" means "zero velocity".

 

[SysAdmin]

if the OP want to change more perspective, let say that the galaxy is in pacman universe, if we exit from one side, we will enter from the other side.

From one side, the twin will see his brother nearly stay still, while from the other side, what will he see?

I think, the next photon will nearly never arrive also from both point of view, so from both side the other twin galaxy is look like nearly stay still.

 

[bobc2]

Based on my sketch below, would you say that my Minkowski diagram is a reasonable representation of the selected momentary lengths of the traveling twin's rocket as presented in the stay-at-home frame coordinates (the curves represent front and aft ends of the rocket during constant deceleration-acceleration, analyzed as hyperbolae)? From your previous responses I take it that you would agree that the lengths are the same before start of trip as compared to the momentary midpoint.

 

[zonde]

I don't see any difference between the two. They look the same to me.

Well, still learning the language

My continued, repeated, and, so far, unanswered question for those of you who insist on promoting a preferred frame--the so-called rest frame for each observer--why?

Classical laws of physics work in observer's rest frame.

 

[zonde]

Look at how much resistance is met by these people when it comes to explaining that Einstein's clock synchronization convention is arbitrary and a definition we put into nature rather than one we derive from nature.

Einstein's clock synchronization convention is not arbitrary given classical laws of physics.

 

[ghwellsjr]

My continued, repeated, and, so far, unanswered question for those of you who insist on promoting a preferred frame--the so-called rest frame for each observer--why?

Classical laws of physics work in observer's rest frame.

Only if the observer is inertial. The Stay-At-Home twin is always inertial. The traveling twin is not always inertial and especially not during the turn-around process.

 

[PAllen]

Based on my sketch below, would you say that my Minkowski diagram is a reasonable representation of the selected momentary lengths of the traveling twin's rocket as presented in the stay-at-home frame coordinates (the curves represent front and aft ends of the rocket during constant deceleration-acceleration, analyzed as hyperbolae)? From your previous responses I take it that you would agree that the lengths are the same before start of trip as compared to the momentary midpoint.

Yes, that's a perfectly good diagram in the indicated inertial frame.

Note that if you want to talk about measuring that instant length at turnaround you are talking about multiple measurements, taken in different places, using mutually at rest clocks synchronized in a particular way. This is all relatively straightforward because we can imagine these instruments to be mutually at rest long enough to accomplish synchronization and measurement of the passing rocket. Where you will run into complications is defining a corresponding process that the rocket could use.

 

[ghwellsjr]

Einstein's clock synchronization convention is not arbitrary given classical laws of physics.

Einstein thought it was arbitrary.

If it was not arbitrary, then there would be only one frame in which light propagated at c. Instead there are an infinite number of equally legitimate frames, each having a different clock synchronization based on light traveling at c in each one of them. That's why time is included in the Lorentz Transformation process.

Think about this. In a frame in which an inertial observer is moving, light does not propagate at c relative to him.

 

[PeterDonis]

Based on my sketch below, would you say that my Minkowski diagram is a reasonable representation of the selected momentary lengths of the traveling twin's rocket as presented in the stay-at-home frame coordinates (the curves represent front and aft ends of the rocket during constant deceleration-acceleration, analyzed as hyperbolae)?

I'm not sure "momentary lengths" is a good term, but I agree that the line segments you've drawn are the spacelike lines corresponding to the intersection of the rocket's "world tube" with slices of constant time in the stay-at-home twin's rest frame.

the lengths are the same before start of trip as compared to the momentary midpoint.

Only if the rocket is accelerated in a very special way. I don't want to start another long thread on Born rigid motion, but that's what's required here, and it's *not* what would be realized with an ordinary rocket with an engine at the rear. You would need thrust applied all along the length of the rocket, and in just the right proportions. Since this is a thought experiment, we can gloss over such details and assume that this is possible "in principle"; but it's worth noting that such a thing would be extremely unlikely to be realized in practice.

Also, I assume you realize that (given Born rigid motion as above) the momentary midpoint of the traveling twin's trip is the *only* one of the line segments you've drawn that will be the same length as the one before the start of the trip; the others will all be shorter.

 

[zonde]

Einstein's clock synchronization convention is not arbitrary given classical laws of physics.

Einstein thought it was arbitrary.

If it was not arbitrary, then there would be only one frame in which light propagated at c. Instead there are an infinite number of equally legitimate frames, each having a different clock synchronization based on light traveling at c in each one of them. That's why time is included in the Lorentz Transformation process.

Think about this. In a frame in which an inertial observer is moving, light does not propagate at c relative to him.

This is a mess.

Let me try it that way:
We implement Einstein's clock synchronization convention in particular inertial frame. In every frame we implement it the same way.

If you say that Einstein's clock synchronization convention is arbitrary then I can change it and try to implement it in particular inertial frame. As a result I won't get classical physical laws in that inertial frame. And I won't get one way speed of light c.