Time paradox

DaleSpam

Not just the red guys frame, but all inertial frames.

And here you go with your non inertial frame.

A sequence of simultaneous spaces is a simultaneity convention. And I have already told you that the naive simultaneity convention used here cannot cover the red worldline because it violates the few mathematical requirements of a coordinate system.

Your statement is mathematically invalid, as I pointed out well over 100 posts ago. I don't know why you persist in it.

zonde

I would like to bring up this earlier comment. I am trying to understand bobc2's position.
Paradoxes do not represent reality. Paradox reveals contradiction and we do not allow idea that reality could be self contradictory. Only models of reality can be self contradictory (and therefore flawed).

So I would ask in what model Andromeda Paradox is supposed to appear.

PAllen

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.

Austin0

Quote by Austin0

yes of course ,instant turnaround is a convenient idealization,
it was understood that this depicted inertial phases before and after turnaround.
When you attribute the x ₁ axis to the traveler you are implicitly applying it to an accelerated frame. I.e. there is no turnaround without acceleration. So whether the traveler frame is accelerating at that time is not relevant , In the context of the overall trip it is non-inertial.
Quote by Austin0

Well you have completely avoided answering my question again.
Whether or not you are talking about a single accelerated system I am asking your thought regarding the x₁ axis as it would apply to such a system (with the edit above).

If there was such a co-moving system at that time after turnaround (inertial) would it correspond (be congruent) to the x₁ axis in your chart?
Would there, or not, be a traveler at A with a clock reading of t'=5+(a hair)??

zonde

When we speak about reality, do we mean only single moment of space or do we include all past and all future?
I believe that with reality we mean single slice of spacetime i.e. we do not include all past and all future.

PAllen

I value my past and future possibilities perhaps more than you More seriously what slice? Through a given event (e.g. me hitting submit for this post), there are uncountably infinite spacelike slices.

bobc2

You folks still seem to be troubled by the approximaty of the simultaneous spaces to the accelerated turnaround point in my sketches. So, here is a sketch where we consider the simultaneous spaces far far removed (years and millions of miles) from the turnaround neighborhood. We still have the same interesting feature about the order of events. Event A occurs before event B in the 2nd Black's rest frame. However, for the travelling twin moving along his worldline, event B is presented to his return trip simultaneous space before it is presented to his outgoing simultaneous space.

To emphasize the distinction between the outgoing frame and the return trip frame (not a single acceleration frame), I've colored the outgoing frame blue and the return frame red. The stay-at-home twin has the worldline along the black X4 axis.

DaleSpam

Again, as we have already covered over and over again and again, your comments about simultaneous spaces are defining a simultaneity convention, and that convention is non-inertial. All of my previous comments hold. This approach violates the mathematical requirements in the region of the 2nd (now) black observer, so it is not a valid simultaneity convention for that region. If my many posts on this topic were not sufficiently clear, then please read PeterDonis' post 190, which is very well written.

You seem to think that I am having difficulty understanding your point. I understand your point quite clearly. Your point is not unclear, it is wrong.

In neither the blue frame nor in the red frame does B come before A.

EDIT: oops, it is PAllen's post 190

PeterDonis

Bobc2, a while back you did say that this "reversal of time" was just an interesting feature, nothing more. But now you appear to be saying that it *is* more; that there is some genuine physical meaning to the "interesting feature". And the pushback you are getting is because of the obvious paradoxical consequences of such a claim. So which is it?

PAllen

The question is the occurrence of this 'interesting feature' is part of the definition of the applicability of this simultaneity convention to the non-inertial observer. That is, the maximal spacetime region in which this simultaneity convention is applicable is the region in which there are no intersections of simultaneity surfaces. I could describe additional, physical plausibility criteria as well, but it is at least mathematically valid to use such a convention for the non-inertial observer as long as you don't try to cover a region of spacetime including such intersections.

Of course, I remain convinced that, even where applicable, making such statements as 'this is where the distant clock really runs faster than mine' are physically meaningless and conceptually grossly misleading.

bobc2

We will just have to agree to disagree. 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. 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.

There is no PeterDonis post no. 190.

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

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.

bobc2

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.

bobc2

Yes, an interesting feature. However, I did say that there is a sense in which one could consider his simultaneous space (as described by a Minkowski space-time diagram) as special. 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. If he did not "live" in a Lorentz-Minkowski-Einstein frame he would not have those experiences, so that is the sense of which I referred.

However you misinterpreted my use of the word "experienced." Webster's dictionary gives two or three definitions for the use of that term. You chose the wrong definition where I thought the context made the definition I was applying very clear.

I'm not insisting anyone embrace that sense of specialness. Of course forum members can consider that observation or dismiss it--whatever their preference.

DaleSpam

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

PeterDonis

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.

DaleSpam

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.

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.

Yes, we are talking about non-inertial coordinates. Every time you bring in your sequence of "simultaneous spaces" idea for the travelling 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.

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

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

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.