Understand Special Relativity and Time paradox

[Dale]

There is no PeterDonis post no. 190.

Oops, my apologies. The excellent post I was referring to was PAllen's.

 

[PeterDonis]

I described the experiences an observer could only have by experiencing these things in a Minkowski space-time frame. For example he measures the speed of light to be c, as do all other observers moving relative to him. And he experiences the laws of physics, as do all other observers living in Minkowski space-time frames.

But that has nothing to do with whether a particular simultaneous space is "special". All of these experiences are local; all they show is that whatever "reality" is, it looks locally like Minkowski spacetime. But a simultaneous space isn't local; that's the whole point.

 

[Dale]

We will just have to agree to disagree.

This isn't a matter of opinion. Your position is mathematically wrong. Try to write it down mathematically and you will see that you violate the one-to-one requirement.

You concept of non-inertial simply does not apply to the two separate blue and red inertial frames in the above sketch.

Neither of the separate individual frames violates the mathematical requirements.

I never said that either the blue or the red frames are non-inertial. I said that the "simultaneous spaces" which you keep talking about define a non-inertial frame. If you restrict your comments to inertial frames and stop discussing the sequence of "simultaneous spaces" then it is clear that A comes before B in every possible inertial frame, including the blue and red ones.

If I had been talking about a single non-inertial coordinates, I might have tried using Rindler coordinates or something, but then I would have to explain the Rindler horizen, etc. But, we are confronted with no such situation here.

Yes, we are talking about non-inertial coordinates. Every time you bring in your sequence of "simultaneous spaces" idea for the traveling twin you are defining a simultaneity convention for a non-inertial frame. It is a perfectly valid simultaneity convention, but it does not cover the entire spacetime. Your problem is that you continue to try to apply it in a region of the spacetime that it cannot cover because it violates the mathematical requirements in that region.

In that case, your point is also clear--it is just wrong.

I can back mine up with math and references if you wish. Can you do the same?

Just look at the space-time diagram. The intersections of the blue and red simultaneous spaces with the 2nd black worldline are there to see. There can be no mistaken about where the intersections are.

And here you go from talking about inertial frames to talking about simultaneous spaces, thereby forming a non-inertial frame which is invalid in the region of the 2nd black worldline.

 

[PAllen]

This makes no sense to me. By that logic you must dismiss the use of Minkowski space-time diagrams entirely, since by that definition there is probably no object in the universe that is inertial. All objects have accelerated at one time or another along the history of its worldline.

Nope. Firstly, anyone can use any inertial frame for any SR analysis of a given scenario, involving any number of objects, distances, times. This is the the most practical approach. Choose the inertial frame for convenience (e.g. COM frame for many kinematic problems). There is no requirement I ever use a frame in which I am at rest.

A different question is what is 'experienced' as a simultaneity surface. You can approach this mathematically or physically.

I prefer physically, and note that there is a well defined 'frame' for an observer when/where different physically reasonable simultaneity definitions agree (to some desired precision). Thus, agreement to some precision between radar simultaneity and Born rigid ruler simultaneity defines the size of a physically meaningful frame for an observer. The longer since your last significant (to desired precision) deviation from inertial, the larger the spatial extent of your physically meaningful simultaneity slice.

Mathematically, along a world line, you can use whatever spacelike surfaces you want as simultaneity slice, for a region of spacetime in which they don't intersect. If you want to cover a region where one choice has intersections, choose a different set of surfaces that don't intersect there.

 

[bobc2]

Bear with me a little more here. I'm trying to get my head into your concept and use of Lorentz frames and simultaneous spaces.

Focus just on the stay-at-home twin. Is he in an inertial frame? Does he exist in a continuous sequence of simultaneous spaces, each one parallel to his X1 axis?

 

[Dale]

Focus just on the stay-at-home twin. Is he in an inertial frame?

He is in all inertial frames. Additionally, he is at rest in an inertial frame.

Does he exist in a continuous sequence of simultaneous spaces, each one parallel to his X1 axis?

Simultaneity is a convention. You can certainly choose that convention if you like, but you don't have to.

Also, there is no empirically discernable sense in which he exists in one simultaneity convention and not in another. If he exists, then he exists regardless of which simultaneity convention you adopt.

 

[ghwellsjr]

Very good graphics, ghwellsjr. Your graphics make it very clear. Good job.

Thanks, I'm glad you like them.
...

The sketch below gets messy, but it illustrates a couple of more details that one may or may not find interesting. Notice that the event A on the 2nd Red stay-at-home guy (displaced from the first red twin) is presented to the returning twin’s trip simultaneous space before it is presented to the outgoing twin’s simultaneous space. Notice that this does not in any way imply that the 2nd Red guy's time is flowing backwards for that Red guy sitting at rest in his own black inertail frame. It's just a feature of special relativity and is no more mysterious than the two twins having different ages after they reunite.

By the way, the blue dots on the traveling twin's worldline are placed with same proper time increments as the black worldline dots (one year intervals of proper time on both worldlines, in accordance with your preference). The hyperbolic calibration curves show the five year lapses.

I have attempted to replicate your drawing without the extra lines:

Now I transform to the IRF in which the traveling twin is stationary during the outbound portion of his trip:

And the transform to the IRF in which the traveling twin is stationary during the inbound portion of his trip:

Now how do you get information from these last two IRF's to draw the extra lines in the first IRF and to come to the conclusions that you do?

 

[bobc2]

Thanks, I'm glad you like them.
...

I have attempted to replicate your drawing without the extra lines:

Now how do you get information from these last two IRF's to draw the extra lines in the first IRF and to come to the conclusions that you do?

Once again a very good job. You have presented all of the information in an easily understood picture that accounts very well for the difference in the twins' ages at the reunion. And this of course is exactly what was originally requested at the beginning of this thread. No additonal comments were really needed at that point.

I had originally simply tried to give the discussion a larger context by expanding the picture by adding frame coordinates for the outgoing and return trip frames. Of course subsequent comments (beginning with an observation by Vandam) led to the addition of a second Red guy in the rest frame with events A and B and the intersections of the blue X1 and X1' axes with that 2nd Red guy's worldline. My picture of the blue guy jumping frames (to use a phrase from Rindler's textbook) then generated a series of push-backs from others.

But, to answer your question I've displayed my construction, sketch b) below, to show how my additional features would be added to your sketch a). I just wanted to show the coordinates associated with your frames (Rindler makes a distinction between "frames" and "frame coordinates"). So, I began by establishing the X4 and X4' coordinates along the direction of the inertial worldlines. Then, I used 45-degree green lines to represent photon worldlines. The simultaneous spaces for the two coordinate systems are then established by adding in the X1 and X1' axes. These X1 and X1' axes are of course placed such that the photon worldlines bisect the angles between the X4-X1 pairs.

The 2nd Red guy worldline was just added into illustrate the interesting feature mentioned by Vandam.

[edit: Note the slight discrepancy in my sketch b). The green photon worldline is not exactly 45-degrees. I hope this does not distract from illustrating the basic concepts.

 

[ghwellsjr]

Once again a very good job. You have presented all of the information in an easily understood picture that accounts very well for the difference in the twins' ages at the reunion. And this of course is exactly what was originally requested at the beginning of this thread. No additonal comments were really needed at that point.

And once again, thanks.

I had originally simply tried to give the discussion a larger context by expanding the picture by adding frame coordinates for the outgoing and return trip frames. Of course subsequent comments (beginning with an observation by Vandam) led to the addition of a second Red guy in the rest frame with events A and B and the intersections of the blue X1 and X1' axes with that 2nd Red guy's worldline. My picture of the blue guy jumping frames (to use a phrase from Rindler's textbook) then generated a series of push-backs from others.

Maybe you're thinking of a different thread since Vandam got himself banned before this thread was started.

But, to answer your question I've displayed my construction, sketch b) below, to show how my additional features would be added to your sketch a). I just wanted to show the coordinates associated with your frames (Rindler makes a distinction between "frames" and "frame coordinates"). So, I began by establishing the X4 and X4' coordinates along the direction of the inertial worldlines. Then, I used 45-degree green lines to represent photon worldlines. The simultaneous spaces for the two coordinate systems are then established by adding in the X1 and X1' axes. These X1 and X1' axes are of course placed such that the photon worldlines bisect the angles between the X4-X1 pairs.

The 2nd Red guy worldline was just added into illustrate the interesting feature mentioned by Vandam.

I take it that you are applying angles to create your lines rather than having a computer program do it for you. That would explain why my diagram didn't line up with the axes as you had intended. But now that I understand what your intent was, I can redraw my diagrams to portray what you want. However, I need to move the second red guy out a little further because he is too close to the intersection of the X₁ and X'₁ axes to show what you want. So here is another set of drawings starting with the original and final rest frame of all the participants:

Now the rest frame for the traveler during the outbound:

You should know that I adjusted the coordinate time of event B in the first frame so that the event appears simultaneous with the common origin of all the frames. That is another way of forcing event B to be simultaneous with your X₁ axis.

And the rest frame for the traveler during the inbound:

Again, I adjusted the coordinate time of event A in the first frame so that the event appears simultaneous with the traveling twin's turn-around event. That is another way to force event A to be simultaneous with your X'₁ axis.

Now I think your interesting observation was that in the traveler's simultaneous space, event A occurs after event B. However, I think you can see that following your definition of simultaneous space, you really should say that the interesting observation is that in the traveler's simultaneous space, event A occurs both before and after event B. And now that I've pointed that out, you can easily go back to your original sketch or mark up mine to show this interesting observation.

 

[Dale]

I think you can see that following your definition of simultaneous space, you really should say that the interesting observation is that in the traveler's simultaneous space, event A occurs both before and after event B.

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

 

[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).

 

[PAllen]

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).

Actually, it is more like a small (generally) chart centered on a world line; like to world tube: it covers the whole world line, however long its history (in proper time); but may be very limited in spatial extent. It is also completely different from a momentary comoving local inertial frame at a single event in the sense that connection components do not vanish - they encode inertial forces in the 'simplest possible way'.

 

[PeterDonis]

Actually, it is more like a small (generally) chart centered on a world line; like to world tube: it covers the whole world line, however long its history (in proper time); but may be very limited in spatial extent. It is also completely different from a momentary comoving local inertial frame at a single event in the sense that connection components do not vanish - they encode inertial forces in the 'simplest possible way'.

Yes, good point; I was really thinking of something more like an MCIF, but if one is willing to let the connection coefficients be nonzero, one can construct a "world-tube chart" as you describe that is not limited in extent along the worldline of interest.

 

[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.

 

[Dale]

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.

I don't think that this is the source of the disagreement. While it is true that for the constant proper acceleration case the use of the naive simultaneity convention only leads to problems in a region behind the Rindler horizon, the same is not true in the case of non-constant proper acceleration. In those cases there is still a region where the simultaneity convention fails even though those regions can easily exchange signals with the traveling twin.

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.

Thanks for not bringing in solipsism! I don't think any of the philosophies mentioned are relevant since the problem is a mathematical one, but at least it isn't as absurd as talking about solipsism.

 

[PAllen]

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.

You're doing nothing wrong. I never use any other approach to compute anything in SR, nor would I use any other approach to explain it to someone learning it or having confusion. Invariance means you can use any coordinates; why not pick the simplest?

The rest is just to answer "what if someone in a spinning, thrusting, rocket really wants to set up coordinates it which the rocket is at rest and not spinning; the SR analog of merry go round coordinates". The case where this isn't just stubbornness is to introduce GR techniques and actually bridge to GR via the principle of equivalence.

 

[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.