Understand Special Relativity and Time paradox

[bobc2]

Which is precisely why that particular simultaneity convention is not valid in that region.

After the twins are reunited and both at rest in the original stay-at-home rest system, do you consider both of them to share the same simultaneous space?

 

[Dale]

After the twins are reunited and both at rest in the original stay-at-home rest system, do you consider both of them to share the same simultaneous space?

Simultaneity is a matter of convention. You could pick a convention where they do, or you could pick a convention where they do not.

 

[Austin0]

Quote by bobc2

Quote by bobc2

However, as the blue guy moves along his worldline, the event B is presented to his outgoing simlultaneous space first (right at the start of the outgoing trip). Then. event A is presented to blue's simultaneous space just after blue completes his turnaround.

Also, just before blue enters his turnaround path, event C is presented to blue's simlultaneous space. Then, event A is not presented to the blue simultaneous space until after the turnaround is complete.

And this is the core of disagreement. You speak of blue's simultaneous space as if it has some physical meaning. Further, since blue, after turnaround, has a different past than the past of the post turnaround inertial frame, any physical procedure defining simultaneity will come out different for the blue observer than for an observer always at rest in the post turnaround inertial frame. Finally, even as a mathematical convention, talking about blue's simultaneous spaces does imply an overall simultaneity convention for the blue world line. For this, there are mathematical requirements - any region where a proposed simultaneity convention for blue has intersecting surfaces is outside the domain of that convention. If you want to talk about a blue simultaneity for such a region, you must adopt a different convention that does not have intersecting surfaces - of which there are many.

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 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.

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.

Although the time rate would slow down towards Earth it would still proceed forward out to a distance dependent on the acceleration rate. Beyond that it would actually increment backwards. At low accelerations this would occur at very large distance which would decrease as acceleration increased.
Finally at maximal, instant acceleration, the extent of continuity would reduce to the single traveler point. With overlap increasing toward Earth as clocks were set back to a previous reading and an increasing gap outward as the coordinate time was suddenly set forward.

Compare with the Minkowski diagram.

In this the traveler frame is portrayed as rotating clockwise, Resulting in temporal displacement. Into the Earth frame future back at Earth and into the Earth frame past outward from the traveler. With the resulting anomalies where the outward traveler frame intersects and overlaps itself , while the inward region jumps forward

I think the actuality is the opposite. The Earth line of equal time is rotated counter-clockwise. Into the traveler coordinate past at Earth and into the coordinate future outward from the traveler.

The difference is that the minkowski diagram produces a literally impossible picture which could not have frame agreement with inertial observers. I.e. NO inertial observer could be co-located with a traveler with a post turnaround clock reading and an Earth clock with a pre-turnaround Earth time reading at point A And no physical system could meet and overlap itself.

The second picture has a coordinate time discontinuity outward but no overlap , while the overlap actually occurs towards Earth but has no temporal implications whatsoever and is clearly simply a coordinate issue.
As far as that goes , while we prefer that coordinates are smoothly continuous this is not really a serious matter. it can be accommodated with a little relabeling,perhaps PS for post synchronization attached to the redundent times readings toward Earth A little calculational stitching yes?
Forgive me if I have run on. I wanted to keep it as simple as possible but it may have gotten away from me ;-0
So i think the root of the problem is that the direct Minkowki graphing implements an implicit assumption of actual simultaneity within a frame at equal time readings I also think I can pinpoint how this is implemented and why it produces the incorrect results but this is too long already.

 

[Dale]

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!

As far as that goes , while we prefer that coordinates are smoothly continuous this is not really a serious matter.

Well, they have to be continuous, but I agree that you can relax the requirement on smoothness.

 

[PAllen]

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. 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.

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 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.

[This brings up and option I have discussed on other threads, but didn't want to further complicate this thread: It is perfectly routine to use different, overlapping coordinate charts on a manifold. You just specify the mapping that identifies the same events for the overlapping region(s). Of course, this approach gives up on the basically meaningless question of what is 'the' simultaneity map between the stay at home world line and the traveling world line, from 'the point of view' of the traveler. It leaves you with: In patch 1, there is a partial mapping; in patch two there is another partial mapping that isn't and has no need to consistent with the other patch for the overlapping region.]

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.

...

Forgive me if I have run on. I wanted to keep it as simple as possible but it may have gotten away from me ;-0
So i think the root of the problem is that the direct Minkowki graphing implements an implicit assumption of actual simultaneity within a frame at equal time readings I also think I can pinpoint how this is implemented and why it produces the incorrect results but this is too long already.

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.

 

[ghwellsjr]

After the twins are reunited and both at rest in the original stay-at-home rest system, do you consider both of them to share the same simultaneous space?

Before I answer that question, let's consider a different issue: look at the first of my three diagrams in post #209. There you will see event A having a coordinate time of 4 years and being simultaneous with the Earth twin when his clock reads 4 years, assuming that his clock read zero when his twin started on his trip. But even if it didn't, event A would still be simultaneous with the event of the Earth twin four years after the start of the scenario. This is because simultaneity is defined for the coordinate system, not for any particular observers.

We often will talk about an observer being at rest in a particular Inertial Reference Frame (IRF) and we usually mean that his clock is synchronized to the coordinate time and it's in this sense that when we talk about the classic Twin Paradox, we assume that both of their clocks read zero when the one twin departs. And we can assume that prior to that time, both clocks and the coordinate time were all in sync with negative times on them.

So now we consider what happens after the twins are reunited. In this particular scenario, the time on the traveling twin's clock will read 10 years when the Stay-At-Home (SAH) twin's clock reads 13 years and also when the coordinate time is 13 years. So do the twins share the same simultaneous space? I would say yes, because as I said before, simultaneity is defined for the coordinate system, not for any particular observers. But since you asked the question, you probably are using a different definition of simultaneous space that is defined for observers and not for coordinate systems and because their clocks have different times on them, maybe you'll say no. What do you say?

But while we're on the subject, let's think about another issue: consider the question of the simultaneity between the traveling twin's turn-around event and the SAH twin. In the first IRF, this happens for the SAH twin when his clock reads 6.5 years (assuming zero at the start). But if we look at the next IRF that I drew, it happens at around 4.9 years and for the last IRF, it happens at around 9.1 years. So we see that the issue of simultaneity is IRF dependent.

However, if we ask a different question, namely when will the SAH twin see the traveling twin turn around, we can get the answer in the following way:

Look again at the first diagram from post #209:

Note that at the moment of turn-around, the traveling twin is about 4.1 light-years away from the SAH twin. Therefore, we conclude that it will take 4.1 years for the light from the turn-around event to reach the SAH twin and since his clock read 6.5 years at the moment of the turn-around event, he will see his twin turn around when his own clock reads 10.6 years. I have drawn in the blue signal going from the turn-around event to the SAH twin to illustrate this:

But here's what I consider to be the interesting observation. We can do the same thing for the other two IRFs and we get the same answer even though the IRF-dependent values are different. Let's look again at the second IRF diagram:

We see here that the turn-around event occurs at a coordinate time of 5 years but the Proper Time on the SAH twin's clock is about 3.9 years and the traveling twin is closer than before, only about 3.2 light-years away. (You have to count the red dots to determine what the Proper Time is on the SAH twin's clock.) But it doesn't take just 3.2 years for the image of the turn-around event to reach the SAH twin because he is moving away from it. We have to follow the path of light along a 45 degree angle to see where it intersects with the SAH twin. (Unfortunately, I didn't draw these diagram with the two axes having exactly the same scale so you have to pay attention to the grid lines when you define what 45 degrees means.) And here is the diagram showing the blue signal path for the second IRF. Again, you have to count the red dots to see that the SAH twin sees the turn-around event when his own clock reaches 10.6 years:

Now let's look again at the third IRF diagram:

Now the turn-around event occurs at a coordinate time of 11.8 years and the Proper Time on the SAH twin's clock is at about 9.1 years. And just like in the second IRF, the distance between the SAH twin and the traveling twin is only about 3.2 light-years away but it doesn't take 3.2 years for the SAH twin to see the traveling twin turn around because he is traveling towards him. In fact, it takes only about 1.5 years and once again, in this IRF, the SAH twin sees the turn-around event when his own clock reaches about 10.6 years. Here's a diagram showing the blue signal path for this IRF:

Don't you agree that this is an interesting observation? No matter what IRF we use, it doesn't change the observations that the observers make.

 

[ghwellsjr]

Lest you think that these issues come about because of the non-inertial nature of the traveling twin, I want to focus our attention on the two red guys who remain inertial throughout the scenario. In their common rest frame, they remain 8 light-years apart.

So let's ask ourselves the same question as before: how long does it take for the image of event A to reach the Stay-At-Home (SAH) twin? Well, in their common Inertial Reference Frame (IRF), event A occurs at a Coordinate Time of 4 years and since it is 8 light-years away, then it will take 8 more years to reach the SAF twin at which point his clock will read 12 years. Here's the IRF diagram with the red path of the signal going from event A to the SAH twin:

Please note that this signal happens to pass through the traveling twin at his Proper Time of 8 years (count the blue dots).

Now let's look at the same situation in the second and third IRF's:

As we can see, even though the Coordinate Distances are different than in the first IRF and even though the Coordinate Times are all different, still the signal going from event A passes right through the traveling twin at his Proper Time of 8 years (count the blue dots) and arrives at the SAH twin at his Proper Time of 12 years (count the red dots).

So if you want to consider any type of non-inertial frame or any frame that is a combination of IRF's, you need to be able to show that each observer continues to observe exactly what he observes in any IRF, plus you have to show the paths of the light signals remain consistent. And you have to do this for the entire scenario including all observers and all signals.

My question to those of you who are enamored by taking on this challenge: why does this appeal to you? What do you hope to learn? What do you think these other frames will show you that you cannot also see from any IRF?

 

[ghwellsjr]

If someone tells you that it takes 8 minutes and 20 seconds for light to get from the sun to the earth, you should realize that they are assuming (whether they know it or not) the common sun-earth inertial rest frame to be able to make that statement. Furthermore, that statement relies on the definition of a frame in Special Relativity such that it takes the same length of time for light to get from the Earth to the sun as it does for the light to get from the sun to the earth. Unless we make an assumption like this, we cannot and should not think that there is intrinsic in nature a meaning to the idea of simultaneous space that stretches between the Earth and the sun or between any other locations.

When we see an event on the sun such as a solar flare and note the time on our clock, we know that any definition of a frame or any theory that attempts to explain how light propagates will affirm that we did see that flare at that time but any claim that the solar flare actually happened 8 minutes and 20 seconds earlier is nothing more than a concept of simultaneity that we put into nature, not one that we take out of nature.

 

[zonde]

... but any claim that the solar flare actually happened 8 minutes and 20 seconds earlier is nothing more than a concept of simultaneity that we put into nature, not one that we take out of nature.

We assume that cosmological principle is attributable to nature i.e. we take it from the nature not the other way around. And simultaneity is a way how we can arrange our observations in a model that conforms with cosmological principle.

 

[Dale]

We assume that cosmological principle is attributable to nature i.e. we take it from the nature not the other way around. And simultaneity is a way how we can arrange our observations in a model that conforms with cosmological principle.

How does simultaneity come from the cosmological principle?

 

[zonde]

How does simultaneity come from the cosmological principle?

Cosmological principle means certain requirement of objectivity. Simultaneity is a way how to implement this objectivity into our models. We can view simultaneity as correspondence between different observers that does not single out any observer as special.

If you use single observer centered model you can't really demonstrate that your model is objective.

 

[Dale]

Cosmological principle means certain requirement of objectivity. Simultaneity is a way how to implement this objectivity into our models. We can view simultaneity as correspondence between different observers that does not single out any observer as special.

If you use single observer centered model you can't really demonstrate that your model is objective.

That doesn't answer my question. What I am asking is how you can define an actual simultaneity convention that way.

You have two events A and B, no (or maybe two) observers, and the cosmological principle. How do you determine if A and B are simultaneous or not? I may be missing the obvious, but I see no way of doing that.

 

[zonde]

That doesn't answer my question. What I am asking is how you can define an actual simultaneity convention that way.

You have two events A and B, no (or maybe two) observers, and the cosmological principle. How do you determine if A and B are simultaneous or not? I may be missing the obvious, but I see no way of doing that.

You expect too much from cosmological principle. It gives you criterion how to judge different models but it doesn't say how to come up with these models.

 

[Dale]

You expect too much from cosmological principle.

It isn't my expectation at all. But if you cannot derive a simultaneity convention from the cosmological principle then YOUR earlier claim "simultaneity is a way how we can arrange our observations in a model that conforms with cosmological principle" seems like a dubious and unsubstantiated claim.

 

[zonde]

It isn't my expectation at all. But if you cannot derive a simultaneity convention from the cosmological principle then YOUR earlier claim "simultaneity is a way how we can arrange our observations in a model that conforms with cosmological principle" seems like a dubious and unsubstantiated claim.

Well, English is not my native language so maybe I somehow misstated what I meant.
"model conforms with cosmological principle" means:
a) that model fulfils certain criterion (cosmological principle)
b) that model can be derived from certain criterion (cosmological principle)

I believe that what I said means a)

 

[Dale]

Even if you cannot derive a simultaneity convention from the cosmological principle, can you even produce one which is demonstrably compatible with it? If not, then your claim is still dubious and unsubstantiated.

 

[bobc2]

If someone tells you that it takes 8 minutes and 20 seconds for light to get from the sun to the earth, you should realize that they are assuming (whether they know it or not) the common sun-earth inertial rest frame to be able to make that statement. Furthermore, that statement relies on the definition of a frame in Special Relativity such that it takes the same length of time for light to get from the Earth to the sun as it does for the light to get from the sun to the earth. Unless we make an assumption like this, we cannot and should not think that there is intrinsic in nature a meaning to the idea of simultaneous space that stretches between the Earth and the sun or between any other locations.

When we see an event on the sun such as a solar flare and note the time on our clock, we know that any definition of a frame or any theory that attempts to explain how light propagates will affirm that we did see that flare at that time but any claim that the solar flare actually happened 8 minutes and 20 seconds earlier is nothing more than a concept of simultaneity that we put into nature, not one that we take out of nature.

ghwellsjr, 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). Nature gave us the worldlines to follow through space-time along with the simultaneous space in which to experience nature. These simultaneous spaces, for each different observer, are unique. Further, nature manifests the laws of nature through the continuous sequence of simultaneous spaces we experience as we move along our worldines. If you were one of the ficticious hyperdimensional observers looking at the block universe (pedagogically speaking--refer to earlier post with the hyperdimensional observers), these Lorentz simultaneous spaces would not have the same significance as for one of us 3-dimensional creatures. However, even the hyperdimensional creature could make out patterns of 4-dimensional objects that can be identified as unique patterns, from which laws of physics could be derived. And those laws would be recognized as associated with the Poincare' group of transformations.

Thanks again for the latest posts with the graphics--a good job as usual of summarizing the way we’ve been describing these inertial frames and coordinates. Here’s my summary that I was preparing just before your last post was presented (I was having some trouble with precision with one of the diagrams, so just hijacked yours). Again, my pictures are messy as compared to yours.

Now, see if I can summarize our differences in the consideration of implications arising from our understanding of the frame coordinates. I think a chief problem you and the others have with my understanding can be seen with the sketches a), b), c) and d) below. I began earlier in this PF thread by providing a representation of the turnaround region that discretized the otherwise continuously accelerating turnaround motion. I analyzed the continuous turnaround as a sequence of inertial straight line increments as shown in sketch a). Particularly objectionable to some was the sequence of momentary simultaneous spaces shown.

Sketch b) zoomed in on the turnaround, showing discrete events for which you could assign momentary velocities. This of course means that momentary simultaneous spaces would be assigned in accordance with the requirement that a photon worldline must bisect the angle between X4 and X1 at any moment (this assures photon speed c for all inertial frames).

This procedure then led to my taking note of the interesting feature of sequential X1 lines (corresponding to the simultaneous spaces) intersecting the worldline of a 2nd Red guy displaced to the right in the Red rest frame (see earlier posts) with a negative time sequence along the 2nd Red guy worldline. This of course in no way implied that time was going in reverse for the 2nd Red guy at rest in his own rest frame.

But, now I think one of the most objectionable aspects of my analysis of the accelerating twin is shown in sketches c) and d). Here, I am presenting the case for the twin in constant deceleration-acceleration. For in this case it is clear that no signals can be received by the twin from the region identified in sketch c). And no signals can be sent by the twin to any place located in the region shown in sketch d).

So I think our disagreement comes down to whether or not there can be any physical meaning attached to the twin’s momentary spaces that extend into regions for which no experimental signals can be exchanged. For the logical positivist the case is closed. No meaning should be attached. For the hard realist the reality is there and described by what events are presented to the simultaneous momentary spaces. For the soft realist, external reality exists independent of the observer, but a line is drawn for regions like this, where no signals can be exchanged.

[edit: Expanded on the initial response to latest post by ghwellsjr]

 

[PAllen]

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

I want to focus on this, as the way it bundles things gets at our disagreements.

1) To me 'relativity of simultaneity' means exactly that if two inertial observers in relative motion follow the same convention for synchronizing separated clocks at rest with respect to them, they will come to different conclusions about which events are simultaneous. Nothing more, nothing less. It does not mean, even for inertial observers, that there is some absolute nature to simultaneity. (I believe that Einstein used relativity of simultaneity strictly in the sense I describe, though that is only an argument by authority). You want to interpret relativity of simultaneity to mean each observer, at each moment, has a particular absolute simultaneity; rather than there is no such thing as an absolute sense of simultaneity.

2) Little need to discuss constancy of speed of light for inertial frames.

3) " Further, nature manifests the laws of nature through the continuous sequence of simultaneous spaces we experience as we move along our worldines." This I don't think follows from (1) or (2), nor do I see how it can be justified except as an article of faith. We do not experience simultaneous spaces nor are relativistic laws of nature expressed in terms of simultaneous spaces. For SR, they may take a simplest for in any global inertial frame. (Of course, in GR, global inertial frames don't exist, and global preferred simultaneity is a non-starter). No real observer's experience exactly matches a global inertial frame, but any observer can pick any such frame to make their analysis simpler. [Note: global, for spacetime, means global in space and time, obviously; a global inertial frame covers all of spacetime.]

 

[PeterDonis]

This procedure then led to my taking note of the interesting feature of sequential X1 lines (corresponding to the simultaneous spaces) intersecting the worldline of a 2nd Red guy displaced to the right in the Red rest frame (see earlier posts) with a negative time sequence along the 2nd Red guy worldline. This of course in no way implied that time was going in reverse for the 2nd Red guy at rest in his own rest frame.

Here you are saying that the simultaneous spaces have no real physical meaning; the "interesting feature" is no more than that. But here...

For the hard realist the reality is there and described by what events are presented to the simultaneous momentary spaces.

...you are saying that the simultaneous spaces *do* have physical meaning. But the "interesting feature" is that the order in which some events are "presented to the simultaneous momentary spaces" is not well-defined; that's the whole point.

You can't have it both ways. If the simultaneous spaces don't have meaning, then the "interesting feature" doesn't lead to any contradictions, but you can't use simultaneous spaces to argue for your view of "reality". If you want to use simultaneous spaces to argue for your view of "reality", then the "interesting feature" is more than that: it's a genuine contradiction.

 

[zonde]

Even if you cannot derive a simultaneity convention from the cosmological principle, can you even produce one which is demonstrably compatible with it? If not, then your claim is still dubious and unsubstantiated.

We have Einstein's simultaneity convention. I see no reason to look further. And of course we have to use another convention - we have to pick some inertial frame. And we can pick some frame that is close to COM rest frame of some region of universe (say Local group).

 

[Austin0]

Quote by bobc2 View Post

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

I want to focus on this, as the way it bundles things gets at our disagreements.

1) To me 'relativity of simultaneity' means exactly that if two inertial observers in relative motion follow the same convention for synchronizing separated clocks at rest with respect to them, they will come to different conclusions about which events are simultaneous. Nothing more, nothing less. It does not mean, even for inertial observers, that there is some absolute nature to simultaneity. (I believe that Einstein used relativity of simultaneity strictly in the sense I describe, though that is only an argument by authority). You want to interpret relativity of simultaneity to mean each observer, at each moment, has a particular absolute simultaneity; rather than there is no such thing as an absolute sense of simultaneity.

2) Little need to discuss constancy of speed of light for inertial frames.

I agree completely with #1

But I think #2 might bear some discussion.

The constancy of the speed of light for inertial frames can have two interpretations.
1) That the speed of light is actually isotropically constant relative to all frames through some unknown mechanism.
or
2)It is only measured to be constant by conventionally synchronized clocks.
I.e. It is made to be isotropically invariant by that very convention.

Judging by his quote "Nature gave us a speed of light that is the same for all inertial frames. " I suspect that bobc2 favors the first interpretation.
Which of course makes sense because the concept of actual simultaneity defined by light signals and the concept of actual constant light speed are integrally related. Perhaps even circularly ;-)

 

[ghwellsjr]

ghwellsjr, I tend to feel that nature has put the relativity of simultaneity into our physics and into our reality.

Tendencies and feelings don't count in our physics and our reality. You have to come up with some hard evidence. I won't ask you for any because I already know it doesn't exist.

Nature gave us a speed of light that is the same for all inertial frames.

Nature gave us a value for the speed of light in all inertial frames which can be measured only by round-trip techniques.

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).

Prior to Einstein, Lorentz and Poincare' had a perfectly good explanation for how light propagated at c only in a single inertial reference frame which was at rest with respect to a presumed ether. There is no experiment that could be performed to indicate that they were wrong. I won't ask you for one because I already know none exists.

Nature gave us the worldlines to follow through space-time along with the simultaneous space in which to experience nature. These simultaneous spaces, for each different observer, are unique. Further, nature manifests the laws of nature through the continuous sequence of simultaneous spaces we experience as we move along our worldines. If you were one of the ficticious hyperdimensional observers looking at the block universe (pedagogically speaking--refer to earlier post with the hyperdimensional observers), these Lorentz simultaneous spaces would not have the same significance as for one of us 3-dimensional creatures. However, even the hyperdimensional creature could make out patterns of 4-dimensional objects that can be identified as unique patterns, from which laws of physics could be derived. And those laws would be recognized as associated with the Poincare' group of transformations.

You are claiming that it is possible to track the propagation of light. Please read the wikipedia article on The One Way Speed of Light.

Thanks again for the latest posts with the graphics--a good job as usual of summarizing the way we’ve been describing these inertial frames and coordinates. Here’s my summary that I was preparing just before your last post was presented (I was having some trouble with precision with one of the diagrams, so just hijacked yours). Again, my pictures are messy as compared to yours.

.

Aside from the precision of my diagrams, do you feel compelled to mark them up because they are inadequate on their own? I have asked you why you are enamored to seek out a more complicated way to understand relativity than simply using a single Inertial Reference Frame (IRF) and then using the Lorentz Transformation process to create any other single IRF. Are you ever going to answer?

Now, see if I can summarize our differences in the consideration of implications arising from our understanding of the frame coordinates. I think a chief problem you and the others have with my understanding can be seen with the sketches a), b), c) and d) below. I began earlier in this PF thread by providing a representation of the turnaround region that discretized the otherwise continuously accelerating turnaround motion. I analyzed the continuous turnaround as a sequence of inertial straight line increments as shown in sketch a). Particularly objectionable to some was the sequence of momentary simultaneous spaces shown.

Sketch b) zoomed in on the turnaround, showing discrete events for which you could assign momentary velocities. This of course means that momentary simultaneous spaces would be assigned in accordance with the requirement that a photon worldline must bisect the angle between X4 and X1 at any moment (this assures photon speed c for all inertial frames).

This procedure then led to my taking note of the interesting feature of sequential X1 lines (corresponding to the simultaneous spaces) intersecting the worldline of a 2nd Red guy displaced to the right in the Red rest frame (see earlier posts) with a negative time sequence along the 2nd Red guy worldline. This of course in no way implied that time was going in reverse for the 2nd Red guy at rest in his own rest frame.

But, now I think one of the most objectionable aspects of my analysis of the accelerating twin is shown in sketches c) and d). Here, I am presenting the case for the twin in constant deceleration-acceleration. For in this case it is clear that no signals can be received by the twin from the region identified in sketch c). And no signals can be sent by the twin to any place located in the region shown in sketch d).

So I think our disagreement comes down to whether or not there can be any physical meaning attached to the twin’s momentary spaces that extend into regions for which no experimental signals can be exchanged. For the logical positivist the case is closed. No meaning should be attached. For the hard realist the reality is there and described by what events are presented to the simultaneous momentary spaces. For the soft realist, external reality exists independent of the observer, but a line is drawn for regions like this, where no signals can be exchanged.

[edit: Expanded on the initial response to latest post by ghwellsjr]

The only way in Special Relativity that there are regions that cannot be reached by signals traveling at the speed of light is if the twin can travel faster than the speed of light. (Or if he doesn't exist at some points in time.) So I'd have to say your discovery is bogus.

I summarized my disagreement with you in post #217:

So if you want to consider any type of non-inertial frame or any frame that is a combination of IRF's, you need to be able to show that each observer continues to observe exactly what he observes in any IRF, plus you have to show the paths of the light signals remain consistent. And you have to do this for the entire scenario including all observers and all signals.

My question to those of you who are enamored by taking on this challenge: why does this appeal to you? What do you hope to learn? What do you think these other frames will show you that you cannot also see from any IRF?

Unless you can meet my challenge, I'm not going to share your enthusiasm for simultaneous spaces.

 

[BruceW]

I am kind of jumping into this thread late on. But I wanted to say that the idea of the simultaneous hypersurface is a perfectly valid thing. Take some frame of reference, then all the events at t=0 according to that frame lie on the simultaneous hypersurface of that frame. On the other had, the concept has the potential to be wrongly interpreted (as all concepts do).

Another question is whether the concept is useful. Most often, I would say it is not useful, but it is sometimes useful if you want to relate relativity to something that we humans can get our heads around. Analogously, in general relativity, it can be useful to choose a particular (non-rigid) reference frame, defined such that adjacent clocks in space (which are all attached to the non-rigid reference frame) have vanishingly small differences in the time they show. (I think Einstein called this a reference-mollusc).

 

[PeterDonis]

The only way in Special Relativity that there are regions that cannot be reached by signals traveling at the speed of light is if the twin can travel faster than the speed of light. (Or if he doesn't exist at some points in time.) So I'd have to say your discovery is bogus.

No, his "discovery" is correct; it's just a long-winded way of observing that any accelerated observer has a Rindler horizon. He basically thinks that you are claiming the region of spacetime behind the Rindler horizon of the accelerated observer doesn't exist. Which, of course, you aren't.

 

[ghwellsjr]

The only way in Special Relativity that there are regions that cannot be reached by signals traveling at the speed of light is if the twin can travel faster than the speed of light. (Or if he doesn't exist at some points in time.) So I'd have to say your discovery is bogus.

No, his "discovery" is correct; it's just a long-winded way of observing that any accelerated observer has a Rindler horizon. He basically thinks that you are claiming the region of spacetime behind the Rindler horizon of the accelerated observer doesn't exist. Which, of course, you aren't.

Ok, thanks, I learned something. I had overlooked that infinite time can never be reached. So is the point that an Inertial Reference Frame is inadequate to deal with infinite time and so we need to use a non-inertial reference frame thereby proving that non-inertial reference frames are superior to IRF's?

 

[PeterDonis]

I had overlooked that infinite time can never be reached.

That's not the point he was making, or the point of the Rindler horizon. Given a worldline that has a constant proper acceleration for all time (i.e., it looks like a hyperbola x^2 - t^2 = constant in some inertial reference frame), there will be a region of spacetime that can't send light signals to any event on that worldline (the region bounded by the future Rindler horizon), and a region of spacetime that no event on that worldline can send light signals to (the region bounded by the past Rindler horizon). These regions are at finite coordinates; they aren't at "infinite time".

So is the point that an Inertial Reference Frame is inadequate to deal with infinite time and so we need to use a non-inertial reference frame thereby proving that non-inertial reference frames are superior to IRF's?

I can't speak for bobc2, but given an accelerated worldline, the regions of spacetime behind its Rindler horizons (future and past) are also the regions of spacetime where the "naive" definition of surfaces of simultaneity that he is proposing breaks down. To me (and apparently to most others in this thread), that's a reason not to use the "naive" definition of surfaces of simultaneity, or at least not to attribute "physical reality" to it. I'll leave it to him to clarify his position on that.

But none of that affects which regions of spacetime can or can't send light signals to or receive light signals from events on a particular worldline; the observation that bobc2 made about that was valid in itself, even if one doesn't agree with the use he is going to put it to.

 

[PAllen]

I can't speak for bobc2, but given an accelerated worldline, the regions of spacetime behind its Rindler horizons (future and past) are also the regions of spacetime where the "naive" definition of surfaces of simultaneity that he is proposing breaks down. To me (and apparently to most others in this thread), that's a reason not to use the "naive" definition of surfaces of simultaneity, or at least not to attribute "physical reality" to it. I'll leave it to him to clarify his position on that.

The naive simultaneity surfaces have problems beyond the Rindler horizon case. Such a horizon is a feature of a world line only if it always has and always will accelerate. For the proposed W shaped traveler twin path, there are no horizons for the traveling world line because it is inertial before some proper time t1, and inertial again after some proper time t2. Thus, it has no horizon. Nonetheless, the 'naive simultaneity surfaces' fail to provide a mathematically (or conceptually) valid simultaneity mapping between the home world line and the traveling world line (purely due to intersection of the surfaces leading to a multiple labeling).

To answer gwellsjr, I see no purpose to non-inertial coordinates in SR except to make analogies with or bridge to GR. That, however, is strictly a personal preference. There is no problem with non-inertial coordinates as long as you know the requirements for valid coordinates and don't over-interpret them; in particular, a global inertial frame is possible in SR, but global accelerated frame (rather than coordinates) is not possible at all in SR any more than GR.

 

[PeterDonis]

The naive simultaneity surfaces have problems beyond the Rindler horizon case. Such a horizon is a feature of a world line only if it always has and always will accelerate.

Yes, that's true; but you can still figure out where the horizon for an accelerating portion of a worldline *would* be if that same acceleration were extended through all of spacetime (by looking for the asymptotes of the hyperbola of which the accelerating portion of the worldline is a section), and that tells you where the "naive" simultaneity convention will start running into problems because multiple surfaces of simultaneity start intersecting. Those asymptotes don't define a global horizon, but they do define a boundary that's of interest; unfortunately there doesn't seem to be a single word for it.

a global inertial frame is possible in SR, but global accelerated frame (rather than coordinates) is not possible at all in SR any more than GR.

A minor point of terminology: I think you mean "frame field" here, rather than "frame"? A "frame" is defined at a single event; a "frame field" is a continuous mapping of frames to events over some region of spacetime.

 

[PAllen]

A minor point of terminology: I think you mean "frame field" here, rather than "frame"? A "frame" is defined at a single event; a "frame field" is a continuous mapping of frames to events over some region of spacetime.

This is a point of terminology varying by author. The specific thing I am thinking of is the construction described as a "proper reference frame of an accelerated observer" in section 13.6 of MTW. I think of this as the closest analog of a inertial frame for an accelerated observer in SR (or for any observer in GR). This construction becomes a global inertial frame the case of flat spacetime, zero acceleration and spin.

 

[PeterDonis]

The specific thing I am thinking of is the construction described as a "proper reference frame of an accelerated observer" in section 13.6 of MTW. I think of this as the closest analog of a inertial frame for an accelerated observer in SR (or for any observer in GR). This construction becomes a global inertial frame the case of flat spacetime, zero acceleration and spin.

That clarifies your usage, yes. A frame in this sense is still centered on a specific event (the origin of the frame), but it's more like a coordinate chart on a patch of spacetime centered on that event than it is like a set of four vectors at that event (which is the usage of "frame" I was thinking of).