Time is the denominator of existence, and bits come to be in it by Daryl Janzen

snp:

This is a very simple experiment, which you are overcomplicating. There is no need to 'break your head' on it, because it's purposely very clear and very simple. And there have been many tests that have confirmed the validity of SR in its applicable domain, although this particular experiment is so simple that it hardly needs to be carried out.

Eckard:

thanks very much for your post! I'll respond to it over there.

Daryl

To anyone who's interested:

I just added two posts at Eckard's site that I thought I'd put here as well, in case anyone's interested in this discussion we're having and might miss it there. Please feel free to comment.

First of all, here's his reply:

In order to test my understanding of the velocity of light and of simultaneity, Daryl Janzen introduced two gunslingers (this word is not in my dictionary, I just assume receivers of the same signal) who are located on a train with equal distance from the common a source of that signal located in the middle of the train.

Yes, according to the endnotes of my essay, they will see the signal at the same moment. It is reasonable and possible to choose only one co-ordinate system that refers to the train.

An observer on the ground may sees the train moving to the right. This motion does not matter.

Eckard

Here's my first response:

Eckard,

Thanks for your answer. I'm sorry you felt that I mean to 'test' you with this question. I just thought it would be a nice concrete place to start a discussion about relativity and check on what we can agree, as you previously suggested.

And sorry for the confusion over the definition of a 'gunslinger'. From Wikipedia: Gunfighter and gunslinger /ˈɡʌnslɪŋər/, are 20th-century words, used in cinema or literature, referring to men in the American Old West who had gained a reputation as being dangerous with a gun.

In the scenario I posted, which I adapted from Greene's 'Fabric of the Cosmos', the two men are dueling with laser pistols, so their 'bullets' travel at c. And someone observes it all from outside the train. I asked: "Do the gunslingers see the signal at the same time, or not? According to which perspective? If no, is it still a fair fight (assuming each stands his ground)? How do you reconcile this with there being one common time?"

We don't agree on the answer to the first question, so let's consider your suggestion 3: "The velocity of light c equals to the distance d between the position of emitter at the moment of emission and the position of receiver at the moment of detection divided by the time of flight t: c=d/t."

In the frame of the observer outside the train, the signal propagates at c towards both gunslingers, from the place of emission. While the signal is propagating, the guy on the left is approaching that point of emission and the guy to the right is moving away from it. The distance that the light eventually travels in order to reach the guy on the left should therefore be less, in the outside observer's frame of reference, than the distance that the light eventually travels in order to reach the guy at the right. With c constant, this means, by your suggestion 3, that the signal reaches the guy at the left in less time than it takes to reach the guy to the right.

On the other hand, in the gunslingers' proper frame of reference on the train, they never move relative to the place where the signal is emitted, so the distance that the light travels is the same in either direction, takes the same amount of time to get to both gunslingers, and is therefore observed by each of them at the same time.

Do you disagree that the signal will be observed synchronously in the gunslingers' frame, but the gunslinger on the left will see the signal before the one on the right in the frame of the observer standing outside the train? If we can agree on this basic picture, which doesn't say anything about what's *really* going on, but only demonstrates the issue that Einstein and others realised, then we can move on to discuss how we would interpret it. The key, in my opinion, has to do with what Paul brought up in his first post above, on Jun. 12, 2013 @ 18:47 GMT. As I keep saying, synchronicity and simultaneity are different things.

Regards,

Daryl

And here's the second:

Oh I can't help myself. Can I say already what I think is the problem with Einstein's proposal that synchronous events are simultaneous? It's perfectly exemplified in the following quotation from Greene (next three paragraphs):

"So: *if you buy the notion that reality consists of the things in your freeze-frame mental image of right now, and if you agree that your* now *is no more valid than the* now *of someone located far away in space who can move freely, then reality encompasses all of the events of spacetime*. The total loaf exists [he's been chopping up space-time like a loaf of bread]. Just as we envision all of space as *really* being out there, as *really* existing, we should also envision all of time as *really* being out there, as *really* existing, too. Past, present, and future certainly appear to be distinct entities. But, as Einstein once said, "For we convinced physicists, the distinction between past, present, and future is only an illusion, however persistent." The only thing that's real is the whole of spacetime.

"In this way of thinking, events, regardless of when they happen from any particular perspective, just *are*. They all exist. They eternally occupy their particular point in spacetime. There is no flow. If you were having a great time at the stroke of midnight on New Year's Eve, 1999, you still are, since that is just one immutable location in spacetime. It is tough to accept this description, since our worldview so forcefully distinguishes between past, present, and future. But if we stare intently at this familiar temporal scheme and confront it with the cold hard facts of modern physics, its only place of refuge seems to lie within the human mind.

"Undeniably, our conscious experience seems to sweep through the slices. It is as though our minds provide the projector light referred to earlier, so that moments of time come to life when they are illuminated by the power of consciousness. The flowing sensation from one moment to the next arises from our conscious recognition of change in our thoughts, feelings, and perceptions. And the sequence of change seems to have a continuous motion; it seems to unfold into a coherent story... The intuitive image of a projector light that brings each new *now* to life just doesn't hold up to careful examination. Instead, every moment is illuminated, and every moment remains illuminated. Every moment *is*. Under close scrutiny, the flowing river of time more closely resembles a giant block of ice with every moment forever frozen into place."

People do think of space-time as existing, but not always just as such a frozen block. In the general relativistic picture, objects are more often thought to move around, warping space-time as they go. How often have you heard that when something falls into a black hole, it has to keep falling towards the singularity at r=0 because r is the timelike direction within the event horizon, so even light can't escape it? It can move in any spatial direction it likes, but even light has to keep going towards r=0. Let me ask you: if one of these gunslingers we're talking about jumped into a black hole, could he shoot a laser bullet towards r=2m and one towards r=0 (say he's got two guns and fires them simultaneously in either 'direction') so that, although they'd both fall towards the singularity out of necessity, the latter bullet would actually get there 'first'? Should that be any more possible to do than for you to take a gun and point it towards the past and another and point it towards the future and have the latter make it to 2014 before the former? The whole concept is so completely inconsistent and blatantly wrong!--and it's truly remarkable that it's persisted as long as it has.

So, the first point I addressed in my essay--which I couldn't avoid having to address because nothing else I could say would make any sense from the point of view of the current incorrect paradigm in physics--is the blatant inconsistency in this common way of thinking of space-time as something that exists: due to the "relativity of simultaneity", people *do* think of space-time as existing, as the Greene quotation illustrates, *but the idea smuggles in an extra dimension that's not formally part of the theory*! They think of a block universe--all of space-time--as existing, which sneaks in the same sense of temporality as we think of when we think of a block of wood as existing. Just as a 3D block of wood sitting somewhere as time passes is a 4D concept, described by 4D physics with three spatial and one temporal dimensions, a 4D block universe existing as Greene has described it is a *5D* concept, described by four space-time dimensions and one temporal dimension. There's more unobservable (and completely unjustifiable) structure in this view than there is when we just assume absolute simultaneity and a true rest frame, which is what Einstein rejected from the point of view of parsimony; i.e., he was so parsimonious that his theory led to a conception of reality with *more* added junk than if he'd just accepted what's *obvious* from the beginning.

But the 5D idea that Greene describes really is a misrepresentation of what Einstein's SR is actually supposed to imply. So: what does Einstein's proposal that simultaneity is relative *really* mean? The block universe that's a logical consequence of the proposal is *just* a 4D slice of that 5D reality. The block universe doesn't exist; it's just a temporally singular thing that pops in and out of that 'existence' in an instant.

My point is that when one finally understands, and makes this clear distinction, and denies the temporality that our thoughts always want to sneak into the idea, then it should be very clear that the Einsteinian view, that synchronous events should be simultaneous, *must* be wrong. The reason is obvious: *something* exists; there is *some* sense in which time passes, because right now is earlier than right now is earlier than right now, etc.--or at least it's not all on par as we perceive it. That much is true, even if it's because all of eternity *exists* in the 5D sense described by Greene, and our consciousnesses simply flow through our worldtubes like a river that flows everywhere and never runs dry. For that consciousness to flow, and the block to exist, that fifth dimension is required. The pure 4D block universe, unadulterated by our thoughts, is impossible to reconcile with any realistic sense of the world, and those who argue for it always do fall back on the 5D concept at one time or other, if not always so overtly as Greene does.

So, what I propose is that only the three-dimensional world around us exists, and there is only one true sense of simultaneity. In the gunslingers example, the signal either reaches them simultaneously or it doesn't, regardless of whether that is described as synchronous in the chosen frame of reference or not. This bit of structure that's necessary to form a coherent theory of existence that's consistent with the apparent flow of time, etc., precludes any informational bits that might come to be. Above all else, without *existence*, bits can't exist--for bits that exist can't be the cause of their own existence.

So how do we reconcile the results of the gunslingers example with the notion of absolute simultaneity? Take the outside observer to be perfectly at rest in the cosmic rest-frame. Now consider the perspective of the two gunslingers. Is it so difficult to see that from their perspective, if they'd just lift the blinds so they can see the world around them, then they too would realise that the guy to the left is going to see the signal first, because he meets it part-way between his position at the time of emission and the signal's position at the time of emission?

Of course it's not difficult to see that that's going to be their perception. Just because everything can also be described as if the train were at rest and the Universe were zipping past--just because he can bounce a ball on the floor, or toss it in the air, and have it come right back to his hand--doesn't mean the gunslingers are unable to come to grips with the fact that they're actually moving, and the sense that the guy to the left is going to see the signal first.

But this is the rock that the whole relativity church was built upon: Mach's failed argument that even if there is a cosmic rest frame we could never observe it; Einstein's wrong argument that it's just superfluous structure and the theory's just as good without it. WE HAVE A VERY PRECISE OBSERVATION OF A COSMIC REST-FRAME, and all the relative motion between galaxies, which is very small compared to the speed of light, is full well understood to be motion through the Universe.

So let's go back to Greene's statement: *if you buy the notion that reality consists of the things in your freeze-frame mental image of right now, and if you agree that your* now *is no more valid than the* now *of someone located far away in space who can move freely*. This statement has been fed to us for a hundred years, and it's just plain wrong. For which freeze-frame mental image of right now are we supposed to say is the valid one for the gunslingers to hold: the one with the blinds shut or the one with them open? If the former is no more valid a mental image to them than the latter, and acceptance of the latter in light of all the cosmological evidence we've found over the past century is also consistent with the apparent fact that time does flow, then why the **** should we hold the former up as the crown jewel of objective thought, which proves to us without a doubt that there's no such thing as the passage of time, and all eternity 'exists'? If the freeze-frame mental image of right now that's held by the gunslingers when they've blocked out the evidence from the world around them leads to an unrealistic description of physical reality when we assume that it's a true representation of "right now", then we should instead assume that the true representation of "right now" is the freeze-frame mental image of right now that's held by the gunslingers when they've opened the blinds!

Cheers,

Daryl

Thanks, Ed.

I skimmed through his essay earlier and couldn't see how he was proposing something different from teleparallelism, but I'll have another look. Also, I don't see why he's setting the cosmological constant to zero. It doesn't have to (and I think it shouldn't) be interpreted as dark energy, but can be treated as a geometrical constant. The full vacuum Einstein equation is R_ab=Lambda*g_ab.

Daryl

Hi Israel,

It's nice to see you here as well! Thanks for reading my essay and commenting. I see much to clarify based on your comments, so I'm glad you've given me a chance to do that. I'll address things point-by-point.

"...based on the mathematical formulation of special relativity (SR) one can conclude that the world doesn't evolve."

Yes, I'm arguing that one can, and often does--but really doesn't have to--conclude this.

"So, to solve this problem you're suggesting to add an additional absolute time dimension, is this correct? As I understood, the idea that you expose in your essay is that the four dimensional space-time of relativity should be embedded in an absolute time dimension?"

No, that's the point of confusion. I'm saying that very often people *do* think of space-time as existing, and that's a five-dimensional concept that's not acknowledged. I quoted Weyl and Ehlers in the essay; above, I've quoted a few paragraphs from Brian Greene's Life of the Cosmos, which really illustrates this. See the big long post that's two above here, on Jun. 29, 2013 @ 23:30 GMT.

To understand what I mean by this, think of a block of wood that's just sitting somewhere. That's a four-dimensional concept. It occupies a three-dimensional place as time passes. The space-time concept that's often been derived erroneously from relativity is a similar five-dimensional concept, extending throughout all of space-time as some extraneous time passes. There are two important differences, though: there is some flow of thought through the timelike dimension of space-time, like a river where the map stays the same but there's a constant flux through it; and, due to general relativity, space-time usually isn't thought to be just a frozen block, but something that changes. Think about it: space-time has to be thought to exist, in an extra dimension of time, in order to be able to change.

Consider the following from Einstein's autobiography: "It is a wide-spread error that the theory of relativity is supposed to have, to a certain extent, first discovered, or at any rate, newly introduced, the four-dimensionality of the physical continuum. This, of course, is not the case. Classical mechanics, too, is based on the four-dimensional continuum of space and time. But in the four-dimensional continuum of classical physics the subspaces with constant time value have an absolute reality, independent of the choice of the reference system. Because of this the four-dimensional continuum falls naturally into a three-dimensional and a one-dimensional (time), so that the four-dimensional point of view does not force itself upon one as *necessary*."

In a completely analogous way, I'm arguing that because of relativity and a really sloppy way of thinking of things which has a lot to do with the complete inconsistency of our languages with the whole block universe concept, people do very commonly think of space-time five-dimensionally, which is a very wrong way of thinking. They think of space-time existing, as a four-dimensional view of things, in much the same way that people used to incorrectly think of classical mechanics as describing a three-dimensional view of things.

Does that make sense? I'm just arguing that the whole concept is completely muddled and wrong. That's what I meant in my abstract when I said the concept of time in physics is a mess. I then offered just the existence of the three-dimensional Universe, as a way out of the mess that's motivated by cosmology.

You then asked how I would define time. I'd define time in terms of existence, and I'd say that the three-dimensional Universe exists; it evolves from one time to the next, continuously. All that exists is the three-dimensions of reality--and that is a four-dimensional concept, described by four-dimensional physics with time as one dimension. Do you see what I mean?

"You discuss the issue of simultaneity and the twin paradox"

I don't think I wrote anything at all about the twins paradox. I searched the PDF to be certain, and the word "twin" isn't there. Perhaps you're referring to the discussion of time dilation and the relativity of simultaneity. That was meant to illustrate that Einstein's interpretation of the relativity of simultaneity is wrong.

"...I just wonder if you are aware of this."

I'm well aware of the literature on the relativity of simultaneity and its implications.

With regard to the rest, I prefer to keep a chin up. Reason based on sound logic and empirical evidence will prevail. And in that regard, thank you for recommending my essay to Daniel and others. I appreciate that.

And I very much look forward to reading your essay. Thanks very much for commenting, and if you would like further clarification on any points in my essay, please don't hesitate to ask.

Best wishes,

Daryl

Hi Israel,

Thanks for your reply. Again, since you've written a lot I've got a lot to say in response. I really want to be clear about the whole five-dimensionality thing before I can properly respond to a lot of what you've said/asked, so I'll have to submit multiple posts.

First of all, you say that my point that "'existence' in itself constitutes a dimension of temporality" is not easy to grasp at first. I don't deny that it is, but then my question is: isn't that remarkable? I mean, nothing should be more obvious, from a geometrical perspective, than to say that "a point exists" presents a one-dimensional concept; the entire history of the zero-dimensional point is described by a line. This doesn't mean that the line itself exists, and the point moves along it--in fact, can you now begin to see that THAT is a two-dimensional concept? I mean, that "the existence of a line" is already a two-dimensional concept, with ITS existence described by another dimension?

Now, to go back to the one-dimensional concept of "a point existing", this doesn't have to mean that there is ever anything real but the point itself. The point itself may *constitute all of reality at any instant*, even though its existence is one-dimensional. This is how the dimension of time is viewed in classical mechanics, and it's why, as Einstein indicated, there was widespread misunderstanding about the dimensionality of physical reality as described by classical mechanics; i.e. while the dimensionality of the physical world described by classical mechanics is four, the three dimensions of space constitute all of reality at any instant.

Therefore, while the dimensionality of the physical worlds described by classical mechanics and relativity are the same (four), the significant difference between the two lies in the implication, because of the relativity of simultaneity, that the *four*-dimensions of space-time constitute all of reality at any instant, as that may be arbitrarily defined.

But do you see that in this explanation of the difference between the two theories I have tactfully avoided saying anything like "relativity implies that all of space-time is real"? That's because it's wrong to do so. "Is" carries existential meaning, and as I've tried to explain (because for whatever reason it IS a difficult concept to grasp), the concept of anything's existence constitutes an extra dimension of temporality, above and beyond the number of dimensions of the thing itself. That's why the conception of space-time as "being" all real is five-dimensional--i.e. because the "being" already constitutes another dimension.

The reason why this is important is that people do often think of space-time as existing--that the reality of all space-time that's thought to be indicated by relativity HAS led many to think of space-time as something that exists. And the problem that's arisen because of that, is that, as a thing that exists, people then go on to think of it as something that EVOLVES and CHANGES. And while it may be subtle and hard to grasp the point that "all of space-time is real" IS a five-dimensional concept, there's nothing at all subtle about the five-dimensionality of the concept that "all of space-time is real and constantly changing".

I guess the thing is, that it's a lot easier to grasp the fact that the classical mechanics description of a three-dimensional Universe that exists is really a four-dimensional concept, described by four-dimensional physics, than it is to grasp the reality of all four-dimensions of relativistic space-time WITHOUT sneaking a fifth dimension into the concept. Would you agree with that? Because that's where all the inconsistency that comes with thinking of space-time as warping and changing and evolving as bodies move around in it, etc., enters.

That's really enough to say in one post, and I've hardly begun to respond to all the points you've brought up, so I'll break here and get back to your comments in another post.

(CONTINUED FROM PREVIOUS POST)

Now, based on everything I said in the last post, I want to answer to your point that "In Newton's mechanics time is seen as a linear flow with the same rate for observers", and your comment/question a little further down, that "In this case I was expecting something more like your notion of time. I mean, what do you understand by the word "time"? A flow, change, a substance, etc.?"

You weren't satisfied with my definition of time in terms of existence, and were looking for a word like "flow". I intentionally avoided using the word "flow" even though that's roughly what I mean by "exist" anyway. I mean that I would define time just as Newton defined absolute time, except that I'd add a note that by "flow" I DO NOT mean flow through a substantive dimension, which the definition can easily be taken to mean. Rather than flow along a substantive dimension, I suppose I'd say more properly I mean flow as an absolute dimension, so that, as with classical mechanics, I mean that at any instant the three-dimensional Universe, and only the three-dimensional Universe, constitutes all of reality.

If you understood my previous post, you'll know what I mean by this.

Now, in between the two statements I just answered to, you noted that relativity complicates such a view and asked if I have a solution to the problem. Actually, more specifically you asked, "Since the mathematical formulation is the one that matter for practical purposes. So mathematically speaking, do you have any solution?"

I have a suspicion that you've got something else entirely in mind, and therefore may not consider my answer too carefully, but I'll give it to you anyway: Actually, the statement that the mathematical formalism is all that matters for practical purposes is incorrect. In practice, in order to correctly make use of the mathematical formalism, we need to take observation into account as well--and the cosmological data clearly indicate that there is an ultimate cosmic rest-frame. I discussed this in my last essay, referenced it in the current one, and tried to explain it to you further during the previous contest. I discussed why this matters, practically, in both of my essays; but I'll briefly explain that here.

Since the predictions made through the mathematical formalism have been verified, it's reasonable to assume for now that there's nothing the matter with the math. The issue, then, becomes one of consistently reconciling the mathematical formalism with ALL the observations. And in that sense, yes, I do think I have a solution. Put far too simply to be entirely convincing to everyone, the solution is that, according to the cosmological observations, there IS an ultimate cosmic rest-frame, and an associated "true"--i.e. absolute--simultaneity-relation. Mathematically speaking, in any inertial frame BUT the cosmic frame, the hypersurfaces of absolute simultaneity won't be orthogonal to the proper time axis--i.e. they'll be tilted--BUT that doesn't mean that they won't still "flow equably along that axis" (please recall, that by flowing equably along an axis, I don't mean a substantive dimension).

I don't like using the words "privileged" or "preferred" reference frame to describe the cosmic reference frame, because these are the words of relativists who historically wanted to claim the objectivity of their stance, and argued that each observer's proper frame should be considered to be as good a reference frame through which to describe "true reality" as anyone else's, and therefore liked to claim that there's no such thing. I call bulls**t.

The point that I tried to make in my essay with the Albert and Henri example is that this is actually wrongheaded. Sure, Henri is free to frame things in such a way that the clock across the train car "remains at a fixed distance from him, and both of them remain motionless". But if he opens his eyes to the world around him, he should see that he's actually moving--i.e. he's not "truly" motionless. Then, he should realise that a photon that travels from the clock to him DOESN'T truly make it all the way across that fixed distance, because from point of emission to point of observation Henri ACTUALLY moved forward and met it part way. Mathematically speaking, this all works perfectly well according to what I said above about absolute simultaneity being tilted in Henri's frame of reference.

Now, the Machian argument against this is "How can Henri or Albert really *know* that Albert's frame is the correct one to use?" In actuality, it isn't: the Earth orbits the Sun, which orbits the Galaxy, etc. But what matters is that there IS one objective cosmic frame, which we have been able to observe to an unprecedented degree of confidence. We "know", with as great a degree of scientific knowledge that we "know" anything, that we are actually moving through the Universe at 370 km/s--and that's really the bottom line. The relativist argument is pure speculation based on the relativity of inertia; science says otherwise, and it's perfectly consistent with the relativity of inertia.

The last thing you talked about was the twins paradox, and it seems you want to perpetuate a much too common error about acceleration being the solution. I'll address that in one more post.

Dear Eckard,

Thank you for the gentle nudge. I apologise. I did see your post there only late last night when I had to go to bed, and I woke up with this post from Israel that bothered me for a couple of reasons, so that I wanted to respond right away. Unfortunately, I am leaving shortly to go camping for my daughter's 8th birthday this weekend, so I fear I won't get to it before then. I hope it's not too much to ask if I can respond then? I see now that Paul has posted another reply there, too, and I'd like to keep up the discussion.

Peter, If you manage to see this, I did notice your last post above and haven't had a chance to read through and give it the consideration it deserves. As I said, I'll be gone for a couple of days but will respond when I get back.

My apologies to both of you, and best regards,

Daryl

(CONTINUED ON FROM PREVIOUS POST)

First of all, from your statement,

"So, if you use this... to demonstrate that SR is plagued with paradoxes, our colleagues will reply with the same argument, i.e., that Henry changed from an inertial frame to a non-inertial frame and therefore there is no paradox. This is what I meant when I said: "I just wonder if you are aware of this." But from your comments I have the impression that you were not."

I feel both misrepresented and slightly insulted. Regarding the misrepresentation, in describing a well-known "paradox" my intention was not to argue that SR, or certain of its assumptions, should be rejected because it "is plagued with paradoxes". In my opinion, that in itself is a pretty weak form of argument. What I did was move on from a derivation of the paradoxical result, to provide an intuitive and realistic resolution to it, that's in objective agreement with the empirical data (see my previous post), as opposed to simply going with Einstein's suggestion that we should just accept that "that's how it's got to be". And regarding the insult, you weren't being very clear about what you meant, which is I think why you felt the need to explain your meaning more clearly. It seems you perhaps meant to test my knowledge, so see if I'd infer your meaning from what you had written, and found that because I didn't respond as you'd expect someone with your knowledge to respond, you thought I must be unaware of something. Please: if you've not been overtly clear about your meaning, don't presume that I don't know my subject because I haven't answered unasked questions.

There are a couple of reasons why I couldn't possibly have inferred your meaning. First of all, the "clock paradox" that you refer to is not IDENTICAL to the "twins paradox", as you've suggested. The "clock paradox" is an important result from SR that's used in *constructing* the "twins paradox", which runs specifically as follows. Consider two twins, standing together: at some point in time, they separate via Lorentz boost and remain in constant relative motion awhile; then at some point in time another Lorentz boost causes them to approach each other with (for simplicity) the same relative velocity; when they come together at the same place, a final Lorentz boost keeps them together. According to the symmetry of relative time-dilation (meaning both, that clocks tick relatively slowly by the exact same amount whether they're approaching or receding with constant relative velocity AND from any perspective, a clock in uniform translatory motion will tick slowly--which you've called the "clock paradox") either twin should therefore expect the other to be younger when they meet again, according to a pure relativist perspective, for the following reason: since, according to pure relativists, there's no such thing as actual motion, as it's all just relative, throughout the entire scenario EITHER TWIN can claim to be "perfectly at rest" while his brother "went off on a journey"; therefore, BOTH of them should expect their brother to be younger when they return, but the two points mentioned in brackets above.

The "paradox" is already ill posed, because there is such a thing as REAL motion, as I've already discussed. Therefore, only one of the brothers can claim to have remained motionless the entire time; i.e., at some point, one of the two brothers HAS to have ACTUALLY hopped from one frame of reference to the other. They can both determine this by looking at the world around them. The solution then runs as in Schutz's Introduction to Relativity textbook, which you might be interested to look at because it has all the right details in it.

The resolution has nothing whatsoever to do with acceleration. It's a mathematically ill-posed problem in SR, which is derived without reference to acceleration, and it's by correctly posing it IN SR that it needs to be properly resolved.

And this is the other reason why I didn't pick up on what you were driving at before: I do know that there has been a common misconception that the twins paradox can be resolved by saying someone has to accelerate, but I'd like it if everyone would move beyond that, so I don't tend to think of it too much. It's wrong, and that's all there is to it.

This can be proven as follows (e.g., see Tim Maudlin's new space and time book; I think Fig 11? Although he's got the paradox itself wrong, this bit is great!): consider the scenario as described in the frame of the twin who remains at rest; his twin heads off into space and at some point turns around and heads back with the same constant velocity. We know it's the twin who actually went on the journey who aged less, and the reason can be stated geometrically: less proper time passes along the "longer" worldline, as drawn in Euclidean space. Now consider giving the twin who stays at home some short Lorentz boosts so that he moves for a short time, in the middle of his brother's absence, at the same "outward" velocity as his brother (no relative motion), then turns around at the same time (considered still in the same frame; so there's still no relative motion), and then comes back to rest and sits there waiting for his brother. He's been "accelerated" just as much, but still his worldline is shorter, and he will still have aged more than his brother. You can actually give him multiple of these boosts, so he's actually "accelerated" MORE, and he'll still be the older one when he and his twin come back together.

That's all I wanted to say in response to your detailed post. I hope I haven't said anything offensive, and I'm sorry that I took a bit of offense to your remark. I really appreciate your interest, and feel you've hit on some very good points here, and therefore also appreciate being given the opportunity to respond as I have. Please do post a reply if you see any further interesting points of discussion, and I'll gladly take them up with you.

Sincerely yours,

Daryl