Twin Paradox: Einstein's Explanation and Alternative Interpretations

[JesseM]

JesseM, just curious what you'd recommend as a good introduction to SR nowadays? I put down French and Rosser, but I don't even know if those are in print any more!

I know the A.P. French book is still out, that was what we used in my intro college course and I think it was fine, but for an introduction that has a little more of a conceptual focus (but doesn't skip the actual equations) I recommend Spacetime Physics by Taylor and Wheeler.

 

[cos]

You can certainly analyze the scenario Einstein describes in section 4 of the 1905 paper from the perspective of a frame that's different from the one where A and B are initially at rest (the one that Einstein chooses to label as the 'stationary' frame, although from the context it's clear that this is just for reference, and does not suggest the frame is meant to be 'stationary' in any absolute sense). For example, suppose that in their own initial rest frame, the clocks at A and B are 20 light-seconds apart, and initially synchronized. Now suppose that clock A is instantly accelerated to 0.8c relative to clock B, so that it takes 20/0.8 = 25 seconds to reach the position of B in B's frame. While it moves at 0.8c, in this frame its rate of ticking is slowed down by a factor of sqrt(1 - 0.8^2) = 0.6, so that in those 25 seconds it only advances forward by 0.6*25=15 seconds, meaning it will be 10 seconds behind clock B when it reaches the position of clock B. This is not quite the same as what's predicted by Einstein's formula of (1/2)*t*v^2/c^2, but that's because he earlier approximated (1 - sqrt(1 - v^2/c^2)) as (1/2)*v^2/c^2, "neglecting magnitudes of fourth and higher order". The non-approximate formula would be t*(1 - sqrt(1 - v^2/c^2)).

Now, the point is that there is no obligation to analyze this situation from the perspective of the frame where A and B are initially at rest. You could analyze this same situation described by Einstein from the perspective of a situation where A and B are initially in motion at speed v (which is 0.8c in my example),

One could analyse this situation from another perspective however other than overly complicating the discussion I see no reason whatsoever for doing so!

Simple question - in your opinion is the claim that the traveler is incapable of realizing that his is the clock that slows down, that he believes that the Earth clock physically (as opposed to seemingly) incurs time contraction, a valid claim?

 

[granpa]

Simple question - in your opinion is the claim that the traveler is incapable of realizing that his is the clock that slows down, that he believes that the Earth clock physically (as opposed to seemingly) incurs time contraction, a valid claim

he does not believe either. he does not believe that his clock slowed down. he does not believe that the Earth clock slowed down.

he believes that everything is relative. he believes that in his frame the coordinates of certain events is (x,y). he believes that in the Earth frame the coordinates of those same events is (x',y'). he does not believe that either is more real than the other.

 

[cos]

yes. but whether you conseder his clock to have been ticking faster or slower during the journey depends on whether you consider him to have been moving faster or slower than the Earth which depends on whether you consider the Earth to be stationary or not. you just seem to be confused like so many others by relativity of simultaneity. its where all beginners get stuck.

As previously pointed out - the astronaut's journey can be shown to be in complete accord with Einstein's chapter 4 depiction.

In that depiction Einstein pointed out that, initially, clocks A and B are stationary; he also points out that A is made to move which suggests that in Einstein's opinion, B remains stationary.

Prior to take off the astronaut the astronaut is of the opinion that he, and the planet, are - for all intents and purposes - stationary. He presumably knows that the Earth is moving through space however he is fully entitled to apply the reference 'stationary' to the planet.

He moves out into space and comes to a stop whereupon he is once again in the same reference frame as the Earth and it makes no difference whatsoever whether or not he considers that reference frame to be moving or stationary on the basis that the laws of physics apply equally to all inertial reference frames.

The relativity of simultaneity has no application whatsoever to this situation and I can only conclude that you are confused.

The relativity of simultaneity only comes into effect from the point of view of an observer in another reference frame relatively to which the Earth and the astronaut are moving however his opinion has nothing whatsoever to do with what either of the twin's determine.

 

[cos]

he does not believe either. he does not believe that his clock slowed down. he does not believe that the Earth clock slowed down.

In message no. 28 you wrote:-

"the traveler realizes that his clock is ticking over at a slower rate compared to the coordinate time of a particular inertial reference frame whose velocity we more or less arbitrarily set equal to zero than it was before he started moving regardless of the fact that it is ticking over at its normal rate."

Correspondence terminated.

 

[atyy]

Absolute things in special relativity
-spacetime metric
-worldlines and their intersections
-accumulated proper time of a worldline
-existence of global inertial reference frames in which the laws of physics are "simple"
-existence of accelerated reference frames in which the laws of physics are "complex"

Relative things in special relativity:
-the laws of physics are equally "simple" in all inertial reference frames.

Relativity allows us to know all of the above, to know which are absolute, which are relative, and the relationship between all of them. (This is not quite true, I'll leave the caveats to someone else Please don't ask me what "simple" means :rofl:).

 

[JesseM]

One could analyse this situation from another perspective however other than overly complicating the discussion I see no reason whatsoever for doing so!

The point is simply to demonstrate what I said earlier, that although you can say one clock's average rate of ticking is objectively slower, there is no basis for saying that one clock is ticking slower than the other at any given moment during the trip. Do you agree with that statement?

Simple question - in your opinion is the claim that the traveler is incapable of realizing that his is the clock that slows down, that he believes that the Earth clock physically (as opposed to seemingly) incurs time contraction, a valid claim?

He can certainly see that his clock ticked slower on average, just by comparing the time on his clock with the time on the Earth clock when they reunite.

 

[phyti]

On the basis that the objects exchange light signals it is assumed that they are sources of those signals so I fail to understand why you say that the objects are moving away from the source. They are moving away from the point in space where the source was located at the instant of emission not away from the source..

True, the source is part of the light clock, the emission point is fixed in space,
and becomes one end of an invariant interval. Most people will accept the invariant
interval of SR, yet reject the implication of fixed locations, even though the
event does not move!

The pilot, having accelerated away from the planet or following turn-around is fully justified in being of the opinion that he is moving thus that his is the clock A to which Einstein referred in chapter 4 thus that it is his clock which ‘goes more slowly’ than the Earth clock’; that the Earth clock does not incur time contraction.

Popular interpretations of SR leave the reader with the impression they have no
choice of frame. They will also cite the 1st postulate 'the rules of physics are
the same in all frames', yet state 'space contracts' for the space traveler. In
keeping things in perspective, the space traveler is the only one who perceives
earth time changing, the rest of the world does not. Like a person on drugs who
experiences hallucinations, they are in his mind and not shared by the rest of the
world, i.e., it's altered perception.

It is a primary tenet of physics that whilst a theory, such as STR’s concept of time dilation, can appear to have been experimentally verified on numerous occasions it only requires one experiment to invalidate any theory.

Although it is accepted that time dilation has been experimentally verified it’s absolutely essential counterpart - length contraction - has not!

If the theoretical concept of length contraction does not physically take place (as distinct from ‘mathematically’ or ‘seemingly’) then the concept of the constancy of the speed of light cannot be maintained.

Time dilation is a function of the speed of the traveler to the speed of light. It is the only physical effect of motion. Length contraction is a result of time dilation, the pseudo rest frame of the traveler, and the resulting axis of simultaneity, whereby he measures the ends of a length at different times.
The equations of SR can be formulated using only the constant speed of light. The measured constancy you refer to is a result of the geometry of the pseudo frame of rest.

I recommend "Einstein's Theory of Relativity" by Max Born, it's not too heavy on math, and the author is very thorough.

 

[cos]

The point is simply to demonstrate what I said earlier, that although you can say one clock's average rate of ticking is objectively slower, there is no basis for saying that one clock is ticking slower than the other at any given moment during the trip. Do you agree with that statement?

No; the Hafele-Keating experiment was based on Einstein's chapter 4 reference to "one of two synchronous clocks at A moved in a closed curve with constant velocity until it returns to A then by the clock which has remained at rest [the laboratory clocks] the traveled clock on its arrival at A will be a .5tv²/c² slow."

Einstein then referred to a balance-clock at the equator which, in his words "must go more slowly" than a clock at one of the poles. I read his comment 'go more slowly' as 'tick over at a slower rate than' or 'incur time dilation relatively to' hence his clock traveling in a closed curve will 'go more slowly than' (i.e. 'tick over at a slower rate than' or 'incur time dilation relatively to') the clock that has remained at rest.

It is my belief that Hafele and Keating (et al) accepted that during the first flight the clocks aboard the aircraft would 'go more slowly than' (incur time dilation relatively to) the laboratory clocks so during that flight they would have been fully justified in realizing that although their clocks appeared to be ticking over at the same rate as they were before their departure their clocks were, "at any given moment during the trip", physically ticking over at a slower rate than previously.

Einstein's 'closed curve' depiction was an extension of clock A moving in any polygonal line i.e. an astronaut's out-and-return journey.

The nonsensical claim - that from the astronaut's point of view the eventual difference between the clocks was not because his clock was going more slowly than the Earth clock but because the Earth clock was ticking over at a faster rate than his clock - would have Hafele and Keating insisting that their clocks were not 'going more slowly than' (incurring time dilation relatively to) the laboratory clocks but that the laboratory clocks were incurring time contraction and that the Earth's axial spin and orbit of the sun had physically increased!

I am of the opinion that if Hafele or Keating or anyone else had expressed such an opinion either before, during, or after that first flight they would have been ridiculed.

He can certainly see that his clock ticked slower on average, just by comparing the time on his clock with the time on the Earth clock when they reunite.

On the basis that he can see (i.e. realize or determine) that his clock "ticked slower on average" he is, presumably, not of the opinion that whilst he was traveling the Earth clock ticked faster than it did before he left.

Upon returning to the planet and concluding that "his clock ticked slower on average" during that initial out-and-return journey the astronaut, upon making an identical out-and-return journey should also be capable of realizing that although his clock appears to be ticking over at the same rate as it was before he commenced his trip that it is "on average" (apart from during turn-around) ticking over at a slower rate than it was before he left the planet.

He should be capable of realizing that the Earth clock is not, on average - or at any time during his voyage, physically ticking over at a faster rate than it was prior to his departure.

He should be capable of realizing that the Earth's axial spin and orbit of the sun has not increased! (if Earth seconds contract so, too, would Earth days and years).

"Knowledge is one-dimensional; the proper application of knowledge is multi-dimensional. Only the extremely wise, and the exceptionally foolish, are not prepared to change." (Confucius).

The claim, by some people, that the astronaut would not be aware that his clock is incurring time dilation during his trips but that it is the Earth clocks that are ticking over at a faster rate than they were before he left home does not, in my opinion, comply with chapter 4 of that theory!

If chapters 1 through 3 of special theory (or perhaps more to the point - interpretations of those chapters) ratify that claim then I can only conclude that there's something wrong somewhere because it seems to me that neither chapter 4 nor Einstein's 1918 article support such a claim but appear to contradict same.

 

[cos]

Popular interpretations of SR leave the reader with the impression they have no choice of frame. They will also cite the 1st postulate 'the rules of physics are
the same in all frames', yet state 'space contracts' for the space traveler. In
keeping things in perspective, the space traveler is the only one who perceives
earth time changing, the rest of the world does not. Like a person on drugs who
experiences hallucinations, they are in his mind and not shared by the rest of the
world, i.e., it's altered perception.

It is imperative to my argument that whilst the traveler "perceives Earth time changing" he should be capable of realizing, in accordance with Einstein's chapter 4 depiction and 1918 article, that his, being the accelerated and moving clock, is the one that physically incurs time dilation - that the Earth clock is NOT changing!

I recommend "Einstein's Theory of Relativity" by Max Born, it's not too heavy on math, and the author is very thorough.

Thank you but I have a copy; I find his analogy of length contraction to that of a cucumber sliced at different angles to be nonsensical.

 

[granpa]

motion is relative. acceleration is not

 

[JesseM]

No; the Hafele-Keating experiment was based on Einstein's chapter 4 reference to "one of two synchronous clocks at A moved in a closed curve with constant velocity until it returns to A then by the clock which has remained at rest [the laboratory clocks] the traveled clock on its arrival at A will be a .5tv²/c² slow."

The Hafele-Keating experiment was more complicated in that it involved gravitational time dilation as well as velocity-based time dilation; also, unlike in Einstein's thought-experiment, it did not involve one clock being moved in a straight line at constant velocity to the other clock. See more details on this experiment here.

Einstein then referred to a balance-clock at the equator which, in his words "must go more slowly" than a clock at one of the poles. I read his comment 'go more slowly' as 'tick over at a slower rate than' or 'incur time dilation relatively to' hence his clock traveling in a closed curve will 'go more slowly than' (i.e. 'tick over at a slower rate than' or 'incur time dilation relatively to') the clock that has remained at rest.

Since Einstein was writing in 1905 before the discovery of gravitational time dilation, presumably we can assume that the mass of the sphere he discusses in section 4 can be treated as negligible so that there is no gravitational time dilation (a hollow sphere rotating in flat spacetime, say). And when he says the clock at the equator is ticking slower, from the context I think it can be understood that he is talking about the total elapsed time over the course of one full rotation of the sphere, not saying that there is any objective sense in which the clock at the equator is ticking slower at every instant during the course of one rotation. Certainly it is true that regardless of what inertial frame we choose, a clock at the equator of a rotating sphere will tick less over the course of a full rotation than a clock at the pole; but it is not true that the clock at the equator is ticking slower than the clock at the pole at every single instant, because in a frame where the sphere's center is in motion, there can be moments when the clock at the pole actually has a higher velocity than the clock at the equator, so in such a frame the clock at the pole will be ticking slower at that instant. Do you deny that there are valid inertial frames where this is true? If not, do you think Einstein failed to realize this, or that he denied that all inertial frames are equally valid?

It is my belief that Hafele and Keating (et al) accepted that during the first flight the clocks aboard the aircraft would 'go more slowly than' (incur time dilation relatively to) the laboratory clocks so during that flight they would have been fully justified in realizing that although their clocks appeared to be ticking over at the same rate as they were before their departure their clocks were, "at any given moment during the trip", physically ticking over at a slower rate than previously.

Again, the Hafele-Keating experiment is complicated by gravitational time dilation, so we can't analyze the path of the aircraft from the perspective of the type of inertial frame seen in SR. But if we were talking about aircrafts flying around a massless sphere in flat spacetime, I am sure Hafele and Keating would agree that there is no objective truth about which of the two clocks is ticking faster at any given instant, since different inertial frames disagree on this, although it's true that over the course of the whole trip one clock elapses more time in total.

Einstein's 'closed curve' depiction was an extension of clock A moving in any polygonal line i.e. an astronaut's out-and-return journey.

Sure, but of course the velocity of the ship at each point on the curve is different in different frames, and in every frame the rate his clock is ticking at any given instant depends only on his velocity at that instant.

The nonsensical claim - that from the astronaut's point of view the eventual difference between the clocks was not because his clock was going more slowly than the Earth clock but because the Earth clock was ticking over at a faster rate than his clock

I don't know what you mean by this distinction--in any given frame, if clock A is ticking slower than clock B, then how is that different from saying that clock B is ticking faster than clock A in this frame? Of course it is true that in any given frame, clocks can never move forward faster than the rate that the frame's time coordinate is moving forward, only slower than the time coordinate--is that what you mean?

would have Hafele and Keating insisting that their clocks were not 'going more slowly than' (incurring time dilation relatively to) the laboratory clocks but that the laboratory clocks were incurring time contraction and that the Earth's axial spin and orbit of the sun had physically increased!

Again, if we talk merely about the relative rate of one clock as compared to another, I don't see the distinction from saying "A is ticking slower than B" vs. "B is ticking faster than A". On the other hand, if we talk about the rate that either clock is ticking relative to coordinate time in any given inertial frame, it is true that clocks can only tick slower than the time coordinate, never faster. And the rate a clock is slowed down at a given instant in a given inertial frame depends only on its velocity at that instant in that frame--if a clock is moving at speed v at some instant, at that instant it is always slowed by a factor of $$\sqrt{1 - v^2/c^2}$$. So of course if two clocks A and B are moving relative to one another, then at any given instant it is always possible to find an inertial frame #1 where A has a higher v than B, and thus A is ticking more slowly than B in frame #1 at that instant, as well as another inertial frame #2 where B has a higher v than A, and thus B is ticking more slowly than A in frame #2 at that instant. Nevertheless, if A and B start out at the same position with their times synchronized, then they move apart and at some later time come together again, if we analyze the entire problem from beginning to end in each frame, both frames will make the same prediction about which clock has elapsed less time when they reunite, even though they disagreed about which was ticking slower at one particular instant.

On the basis that he can see (i.e. realize or determine) that his clock "ticked slower on average" he is, presumably, not of the opinion that whilst he was traveling the Earth clock ticked faster than it did before he left.

Slower than what? Faster than what? The Earth clock was ticking faster than the astronaut's clock on average, but if we pick some inertial frame, it must be true that on average the astronaut's clock was ticking slower than the frame's coordinate time by a greater amount than the Earth's clock was ticking slower than the frame's coordinate time...but only on average, not at any given instant.

The claim, by some people, that the astronaut would not be aware that his clock is incurring time dilation during his trips but that it is the Earth clocks that are ticking over at a faster rate than they were before he left home does not, in my opinion, comply with chapter 4 of that theory!

Again, you need to be clear about whether you are comparing the two clocks to each other, or comparing both of them to the coordinate time of some coordinate system. If the first, I see no distinction between A ticking slower than B vs. B ticking faster than A; if the latter, I agree both can only tick slow relative to coordinate time, never faster, but I'd like to know who the "some people" are who have claimed otherwise, I think perhaps you misunderstood someone's comments there.

But aside from this issue, you started this post by denying this claim of mine: "although you can say one clock's average rate of ticking is objectively slower, there is no basis for saying that one clock is ticking slower than the other at any given moment during the trip." Are you saying there is a basis for saying that, at a single moment during the trip, one clock is objectively ticking slower than the other? Do you deny that if you have two clocks A and B moving relative to one another in flat spacetime, then at any given moment, it is possible to find a frame #1 where A is ticking more slowly than B (because A has a higher instantaneous velocity in frame #1 at that moment), and also possible to find a frame #2 where B is ticking more slowly than A (because B has a higher instantaneous velocity in frame #2 at that moment)? Do you deny that all inertial frames are equally valid in SR, and that they'll all make the same predictions about questions like what two clocks read when they meet each other?

 

[granpa]

It is imperative to my argument that whilst the traveler "perceives Earth time changing" he should be capable of realizing, in accordance with Einstein's chapter 4 depiction and 1918 article, that his, being the accelerated and moving clock, is the one that physically incurs time dilation - that the Earth clock is NOT changing!

.

he knows that the Earth clock isn't changing and that he is accelerating, but it doesn't follow that he must conclude that his clock is now ticking slower. in a frame where the Earth is moving, acceleration could cause the travelers clock to tick faster. not over the whole trip including the return of course, but over one leg of it.

since all frames are equally valid he has no way of knowing who is actually moving. he only knows the relative velocity. obviously for convenience we choose to arbitrarily set the Earth's velocity to zero.

and yes of course he is capable of determining how someone in the Earth frame would calculate his spacetime coordinates at any point.

 

[yogi]

COS - your post #16 ...You are interpreting Einsteins Interpretation in a way that leads directly to the paradox - the round trip can be broken down into two one way trips - there is a spacetime path followed by the A clock that is different than the spacetime path followed by the Earth clock E and the B clock (my example). The total age difference is the amount A is behind B when A arrives at B plus the time A is behind E when A returns to E...it makes no difference which was put in motion (the E-B frame or the frame containing the A clock). This is where Einstein created a false asymmetry by synchronizing A and B in the same frame and then putting A into motion - but if A and E are already in motion - it should be obvious that if A passes E and continues on until reaching B, A will read less than all clocks in the EB frame upon arriving at B...to get the total double the result of the one way difference

 

[yogi]

In terms of STR parlance, the distance between E and B is a proper distance vt where t is the time measured by the E and B clocks during transient. And the temporal distance measured in the EB frame is ct (a proper time.) These two factors determine Gamma and the time dilation. We can't say that clocks run slower or faster - age difference is simply a result of a particular experiment - in the case of the one way trip it is due to the invariance of the interval - the combination of the space distance and temporal distance in EB frame must total the temporal distance and space distance in the A frame

 

[cos]

The Hafele-Keating experiment was more complicated in that it involved gravitational time dilation as well as velocity-based time dilation; also, unlike in Einstein's thought-experiment, it did not involve one clock being moved in a straight line at constant velocity to the other clock.

In his book ‘Was Einstein Right?” (54, Oxford University Press, 1990) Clifford M Will shows that the differences between the traveled clocks and the laboratory clocks were determined after the gravitational time dilation effect was taken into account!

I did not suggest that the Hafele-Keating was the same as Einstein’s paragraph 1, chapter 4 depiction of a clock “being moved in a straight line at constant velocity to the other clock.” but that it was analogous to his chapter 4, paragraph 3 depiction of one clock being moved in a closed curve with constant velocity until it returns to the other clock precisely as took place in the Hafele-Keating experiment.

Since Einstein was writing in 1905 before the discovery of gravitational time dilation, presumably we can assume that the mass of the sphere he discusses in section 4 can be treated as negligible so that there is no gravitational time dilation (a hollow sphere rotating in flat spacetime, say).

As pointed out, above, any gravitational time dilation created by the mass of a sphere (such as the Earth) is taken into account!

And when he says the clock at the equator is ticking slower, from the context I think it can be understood that he is talking about the total elapsed time over the course of one full rotation of the sphere, not saying that there is any objective sense in which the clock at the equator is ticking slower at every instant during the course of one rotation.

In paragraph 1, chapter 4, Einstein wrote that clock A lags behind clock B hence he is, in that paragraph, talking about the total elapsed time over the course of that trip however in paragraph 3 his comment is that “a balance clock at the equator must go more slowly than a clock at one of the poles.” (see below)

Certainly it is true that regardless of what inertial frame we choose, a clock at the equator of a rotating sphere will tick less over the course of a full rotation than a clock at the pole; but it is not true that the clock at the equator is ticking slower than the clock at the pole at every single instant, because in a frame where the sphere's center is in motion, there can be moments when the clock at the pole actually has a higher velocity than the clock at the equator, so in such a frame the clock at the pole will be ticking slower at that instant.

What, precisely, do you mean by “a frame where the sphere's center is in motion.”? Are you depicting a sphere that is mounted on a rod through its center and the sphere is stationary but the rod is in motion (i.e. is spinning)?

If so, I can see no relationship whatsoever to a sphere that is spinning! Is it not possible that you could stick to the subject under discussion (i.e. specifically Einstein’s chapter 4 depictions) and not resort to inappropriate fanciful concepts?

Do you deny that there are valid inertial frames where this is true? If not, do you think Einstein failed to realize this, or that he denied that all inertial frames are equally valid?

If you are referring to a totally inapplicable sphere mounted on a spinning rod or any other fanciful ‘valid’ inertial frames - no.

Although an out-and-return trip by an astronaut could also come under the heading of ‘fanciful’ I am of the opinion that there is no difference between such a concept and that of the Hafele-Keating experiment.

Again, the Hafele-Keating experiment is complicated by gravitational time dilation,

Again, it is NOT!

Gravitational time dilation WAS TAKEN INTO ACCOUNT!

so we can't analyze the path of the aircraft from the perspective of the type of inertial frame seen in SR.

No, but we can “analyze the path of the aircraft from the perspective of” Einstein’s chapter 4, paragraph 3 in SR!

Every single experiment that has been conducted here on the surface of this planet that has been cited as providing proof of SR similarly does not comply with “the type of inertial frame seen in SR”. Do you dismiss all of them for that reason?

But if we were talking about aircrafts flying around a massless sphere in flat spacetime, I am sure Hafele and Keating would agree that there is no objective truth about which of the two clocks is ticking faster at any given instant, since different inertial frames disagree on this, although it's true that over the course of the whole trip one clock elapses more time in total.

Imagine that the Earth is a massless transparent sphere with a clock at the ‘equator’ (A) and another clock at one of the ‘poles’ (B). An observer standing alongside clock B would continuously see clock A ticking over at a slower rate than his own clock. At any given instant he would see that the time indicated by that clock lapses even further behind his own time than the time indicated by that same clock at a previous instant indicating to him that clock A has continuously ticked over at slower rate than his own clock between those instances (observations) and, on that basis, it is (irrespective of the fact that he may be consciously unable to discern same) physically ticking over at a slower rate than his own clock in the one-tenth of a second that it takes for his cerebral processes to inform him that he is looking at that clock.

An observer accompanying clock A would be of the opinion that clock B continuously ticks over at a faster rate than his own clock but on the basis that he has read and fully accepts Einstein’s chapter 4, paragraph 3 - pointing out that his (equatorial) clock ‘goes more slowly’ than the (polar) clock B - he takes Einstein’s word for it and realizes that clock B is NOT incurring time contraction which (as I have previously stated was apparently, for Einstein, an anathema) but that it is his clock that is ticking over at a slower rate than B (see below).

Sure, but of course the velocity of the ship at each point on the curve is different in different frames, and in every frame the rate his clock is ticking at any given instant depends only on his velocity at that instant.

It’s velocity is different but its speed remains constant! It is a clock’s rate of travel (i.e. its speed) that dictates its SR rate of time dilation not its direction of travel.

I don't know what you mean by this distinction--in any given frame, if clock A is ticking slower than clock B, then how is that different from saying that clock B is ticking faster than clock A in this frame?

As previously pointed out - the claim is that according to the astronaut the Earth clock is physically ticking over at a faster rate than it was before he commenced his trip and for the astronaut to be of the opinion that this is physically taking place he must also believe (predict, determine) that the Earth’s axial spin and orbit of the sun have physically increased.

Again, if we talk merely about the relative rate of one clock as compared to another, I don't see the distinction from saying "A is ticking slower than B" vs. "B is ticking faster than A".

I quite agree however, as pointed out above, ‘we’ (that is, my side of the discussion) are not simply talking about “the relative rate of one clock as compared to another” (which is effectively out of context) but ‘we’ are saying that if the astronaut considers that the Earth clock is physically ticking over at a faster rate than his own clock (which he considers to be ticking over at an unchanged rate i.e. that his clock is ticking over at the same rate as it was before he started moving) then he must also believe that the Earth’s axial spin and orbit of the sun has physically increased.

Let us assume that our intrepid astronaut has accelerated to a velocity of close to the speed of light thereby generating the particle acceleration attained gamma factor of 40,000 as a result of which the Earth clock is, according to his calculations, ticking over at a rate of 40,000 seconds for each of his own seconds. It is not only every Earth second that has been compressed by that factor but also every Earth minute; hour; day and year.

On the basis that Earth days are compressed (dilated) by a factor of 40,000 the planet must, according to his calculations, be spinning on its axis at 64 million kilometers an hour.

Furthermore, on the basis that Earth years are compressed by that same amount, the planet would, according to his calculations, be orbiting the sun at the (SR forbidden) velocity of 4c!

(His trip takes him directly along the solar system’s axis and, having come to a stop and turned his ship around, he is now looking at the Earth orbiting the sun analogous to the tip of a second-hand moving around a clock face).

Assuming that the astronaut possesses a smidgin of intelligence he must be able to conclude that, regardless of what his calculations indicate, the Earth is not spinning on its axis at 64 million kilometres a second otherwise, presumably, this would have some affect on the population as well as everything else that’s not tied down.

Similarly on the basis that the Earth 'cannot' be orbiting the sun at 4c he must come to the conclusion that what his calculations indicate (or predict) is taking place - is not!

If he is able to come to the conclusion that Earth years, days, hours and minutes are not compressed by a factor of 40,000 he must also be able to come to the conclusion that Earth seconds are similarly not compressed by a factor of 40,000 yet this is precisely what particle acceleration experiments show will take place.

Again, you need to be clear about whether you are comparing the two clocks to each other...

That is precisely what I am doing.

I'd like to know who the "some people" are who have claimed otherwise, I think perhaps you misunderstood someone's comments there.

It was sometime in the mid 90s but I will not provide that author’s name as I have no intention of possibly besmirching an innocent party. There was no misunderstanding on my behalf irrespective of the fact that your baseless comment implies otherwise. The claim to which I refer was that the traveler determines that the eventual discrepancy between the two clocks was not because his clock ‘went more slowly than’ the Earth clock but because the Earth clock incurred time contraction and only during his period of acceleration following turn-around.

But aside from this issue, you started this post by denying this claim of mine: "although you can say one clock's average rate of ticking is objectively slower, there is no basis for saying that one clock is ticking slower than the other at any given moment during the trip." Are you saying there is a basis for saying that, at a single moment during the trip, one clock is objectively ticking slower than the other?

On the basis that there is no such thing as an instantaneous moment - that time flows continuously - yes, I am saying that.

Do you deny that if you have two clocks A and B moving relative to one another in flat spacetime, then at any given moment, it is possible to find a frame #1 where A is ticking more slowly than B (because A has a higher instantaneous velocity in frame #1 at that moment), and also possible to find a frame #2 where B is ticking more slowly than A (because B has a higher instantaneous velocity in frame #2 at that moment)?

It would be very much appreciated if you would stick to the subject on hand and not introduce flights of fantasy.

Do you deny that all inertial frames are equally valid in SR, and that they'll all make the same predictions about questions like what two clocks read when they meet each other?

No I do not deny that but what I’m talking about is specifically what the astronaut believes is taking place i.e. the predictions or determinations generated in his reference frame.

Furthermore, I’m not talking about “what two clocks read when they meet each other” but what it is claimed the astronaut ‘sees’ (or ‘predicts’ or ‘determines’) whilst he is moving toward the planet!

 

[cos]

COS - your post #16 ...You are interpreting Einsteins Interpretation in a way that leads directly to the paradox - the round trip can be broken down into two one way trips - there is a spacetime path followed by the A clock that is different than the spacetime path followed by the Earth clock E and the B clock (my example). The total age difference is the amount A is behind B when A arrives at B plus the time A is behind E when A returns to E...it makes no difference which was put in motion (the E-B frame or the frame containing the A clock). This is where Einstein created a false asymmetry by synchronizing A and B in the same frame and then putting A into motion - but if A and E are already in motion - it should be obvious that if A passes E and continues on until reaching B, A will read less than all clocks in the EB frame upon arriving at B...to get the total double the result of the one way difference

The point that I'm trying to make is that as far as I am concerned Einstein's chapter 4 of special theory contradicts the claim that from the traveler's point of view his clock does not incur time dilation but that the Earth clocks physically tick over at a faster rate than they did prior to his departure.

Thanks for your response but I prefer to deal with that situation rather than introducing extraneous concepts.

 

[phyti]

It is imperative to my argument that whilst the traveler "perceives Earth time changing" he should be capable of realizing, in accordance with Einstein's chapter 4 depiction and 1918 article, that his, being the accelerated and moving clock, is the one that physically incurs time dilation - that the Earth clock is NOT changing!.

The drawing shows A and B both accelerating, they both are effected by time dilation to equal degrees, so your argument is not correct.

Thank you but I have a copy; I find his analogy of length contraction to that of a cucumber sliced at different angles to be nonsensical.

Every book can't be perfect!

 

[atyy]

In paragraph 1, chapter 4, Einstein wrote that clock A lags behind clock B hence he is, in that paragraph, talking about the total elapsed time over the course of that trip however in paragraph 3 his comment is that “a balance clock at the equator must go more slowly than a clock at one of the poles.”

Einstein's prediction applied to the real spinning Earth with gravity is wrong.
http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_58/iss_9/12_1.shtml

 

[cos]

The drawing shows A and B both accelerating, they both are effected by time dilation to equal degrees, so your argument is not correct.

On the basis that “The drawing shows A and B both accelerating” it follows that ‘the drawing’ does not comply with Einstein’s chapter 4 depiction wherein he pointed out that it is clock A - and clock A alone - that is made to move not clock B and it is his comments to which my arguments apply NOT ‘the drawing’.

Einstein does not suggest that A and B are both made to move and in the twin paradox it is only the traveler who experiences a force of acceleration not the Earth so your comment that my argument is not correct, based on ‘the [inapplicable] drawing’ has no validity.

Why don’t you at least try to stick to the subject on hand rather than introduce red herrings in an attempt to obfuscate same using inappropriate materiel?

Picasso provided a drawing of a clock that was severely distorted thus could not possibly tick over let alone incur time dilation however I see no reason whatsoever for accepting the reality of his depiction.

 

[cos]

Einstein's prediction applied to the real spinning Earth with gravity is wrong.

Einstein's prediction that “a balance clock at the equator must go more slowly than a clock at one of the poles.” corresponds directly with the Hafele-Keating experiment in respect to which, as Clifford M Will points out in his book 'Was Einstein Right?", any effects of gravitational time dilation were taken into account.

As Will's points out, a clock at one of the poles could be substituted with a hypothetical master clock at the center of the planet thereby becoming the 'at rest' clock referred to in Einstein's paragraph 3 relatively to which the other clock moves in a closed curve.

The clock at the equator effectively becomes the clock that, according to Einstein, moves in a closed curve as did the Hafele-Keating clocks.

 

[atyy]

Einstein's prediction that “a balance clock at the equator must go more slowly than a clock at one of the poles.” corresponds directly with the Hafele-Keating experiment in respect to which, as Clifford M Will points out in his book 'Was Einstein Right?", any effects of gravitational time dilation were taken into account.

As Will's points out, a clock at one of the poles could be substituted with a hypothetical master clock at the center of the planet thereby becoming the 'at rest' clock referred to in Einstein's paragraph 3 relatively to which the other clock moves in a closed curve.

The clock at the equator effectively becomes the clock that, according to Einstein, moves in a closed curve as did the Hafele-Keating clocks.

Einstein's prediction did not take gravity into account and is wrong. The Hafele-Keating experiement and analysis did take gravity into account and is correct.

 

[JesseM]

In his book ‘Was Einstein Right?” (54, Oxford University Press, 1990) Clifford M Will shows that the differences between the traveled clocks and the laboratory clocks were determined after the gravitational time dilation effect was taken into account!

That's what I said, the experiment was more complicated than Einstein's thought experiment because they had to take gravitational time dilation into account.

I did not suggest that the Hafele-Keating was the same as Einstein’s paragraph 1, chapter 4 depiction of a clock “being moved in a straight line at constant velocity to the other clock.” but that it was analogous to his chapter 4, paragraph 3 depiction of one clock being moved in a closed curve with constant velocity until it returns to the other clock precisely as took place in the Hafele-Keating experiment.

It depends what you mean by "analogous". Certainly the real Hafele-Keating experiment involved gravitational time dilation, while Einstein's thought experiment in chapter 4 did not involve gravitational time dilation (which hadn't even been discovered yet), only velocity-based time dilation.

Since Einstein was writing in 1905 before the discovery of gravitational time dilation, presumably we can assume that the mass of the sphere he discusses in section 4 can be treated as negligible so that there is no gravitational time dilation (a hollow sphere rotating in flat spacetime, say).

As pointed out, above, any gravitational time dilation created by the mass of a sphere (such as the Earth) is taken into account!

By who? Not by Einstein when he was writing the 1905 paper, though of course it was taken into account by Hafele and Keating.

And when he says the clock at the equator is ticking slower, from the context I think it can be understood that he is talking about the total elapsed time over the course of one full rotation of the sphere, not saying that there is any objective sense in which the clock at the equator is ticking slower at every instant during the course of one rotation.

In paragraph 1, chapter 4, Einstein wrote that clock A lags behind clock B hence he is, in that paragraph, talking about the total elapsed time over the course of that trip however in paragraph 3 his comment is that “a balance clock at the equator must go more slowly than a clock at one of the poles.” (see below)

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere", not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

What, precisely, do you mean by “a frame where the sphere's center is in motion.”? Are you depicting a sphere that is mounted on a rod through its center and the sphere is stationary but the rod is in motion (i.e. is spinning)?

Er, why would you imagine I meant that? Of course I am talking about Einstein's thought experiment where the sphere is spinning. The point is that the sphere has a center, we can either pick an inertial frame where the position of the center of the sphere remains constant over time, or we can pick a frame where the center of the sphere is moving at some nonzero constant velocity.

If you are referring to a totally inapplicable sphere mounted on a spinning rod or any other fanciful ‘valid’ inertial frames - no.

I have no idea why you would imagine that the sphere must be "mounted" on anything in order for its center to be in motion. Imagine a sphere in space, its center not accelerating, and the sphere spinning on its axis. Is it not obvious that there will be one frame where the center is at rest, and other frames where the center is moving at constant velocity? The basic notion of different inertial frames is that they assign different velocities to the same object.

Again, the Hafele-Keating experiment is complicated by gravitational time dilation,

Again, it is NOT!

Gravitational time dilation WAS TAKEN INTO ACCOUNT!

Why do you think I was saying otherwise? "Complicated by" does not mean it was not taken into account, it just means that the analysis is more complex than the analysis of Einstein's thought experiment in section 4 of his 1905 paper, where we know he was just talking about special relativity rather than general relativity (since general relativity had not yet been invented).

so we can't analyze the path of the aircraft from the perspective of the type of inertial frame seen in SR.

No, but we can “analyze the path of the aircraft from the perspective of” Einstein’s chapter 4, paragraph 3 in SR!

No, you can't. You must use GR to analyze the path of an aircraft moving around the Earth. On the other hand, you could use SR to analyze the path of an aircraft moving around a massless moving sphere, because in that case spacetime would not be curved so GR would not be necessary.

Every single experiment that has been conducted here on the surface of this planet that has been cited as providing proof of SR similarly does not comply with “the type of inertial frame seen in SR”. Do you dismiss all of them for that reason?

GR reduces to SR locally, so for any experiment conducted in a small region of space, the curvature of spacetime due to the Earth's mass will be negligible and the experiment can be adequately analyzed using SR only. But the Hafele-Keating experiment covers a very large region where the curvature of spacetime cannot be treated as negligible--you said yourself that they had to take into account gravitational time dilation, which only occurs in the curved spacetime of GR, not the flat spacetime of SR.

But if we were talking about aircrafts flying around a massless sphere in flat spacetime, I am sure Hafele and Keating would agree that there is no objective truth about which of the two clocks is ticking faster at any given instant, since different inertial frames disagree on this, although it's true that over the course of the whole trip one clock elapses more time in total.

Imagine that the Earth is a massless transparent sphere with a clock at the ‘equator’ (A) and another clock at one of the ‘poles’ (B). An observer standing alongside clock B would continuously see clock A ticking over at a slower rate than his own clock.

But time dilation in SR is not a matter of what any observer sees visually, something that's influenced by the Doppler effect. Time dilation is based on the coordinate times assigned to successive clock-ticks in an inertial coordinate system. For example, if you are moving towards me at 0.6c, and in my frame both my clock and your clock read "0 seconds" at coordinate time t=0 seconds, then at coordinate time t=10 seconds in my frame, my clock will read "10 seconds" but your clock will read only "8 seconds", in accordance with the time dilation formula which says an object moving at 0.6c in some frame should be slowed down by a factor of sqrt(1 - 0.6^2) = 0.8. However, if I actually watch your clock as you approach me, it won't look like it's ticking slower than mine visually, in fact it will appear to be ticking twice as fast as mine because of the relativistic Doppler effect. On the other hand, if you were moving away from me at 0.6c, then if I watch your clock it will appear to be ticking twice as slow as mine, an apparent visual slowdown greater than the "actual" slowdown of 0.8 predicted by the time dilation formula (which is what I'd calculate if I factored out the light transit time for light from each successive tick of your clock).

An observer standing next to clock B doesn't have their own inertial rest frame because they're not moving inertially. If we choose the inertial frame in which the center of the sphere is at rest, then in this frame B will be moving at constant speed so it's true that B will be ticking at a constant slowed-down rate, while A will be ticking at a normal rate. On the other hand, if we choose a different inertial frame in which the center of the sphere is moving inertially at some constant velocity, then in this frame B's speed will be different at different moments so its rate of ticking will be variable as well, while A will be ticking at some constant slowed-down rate, so there may be particular moments when A's rate of ticking is slower than B's in this frame.

At any given instant he would see that the time indicated by that clock lapses even further behind his own time than the time indicated by that same clock at a previous instant indicating to him that clock A has continuously ticked over at slower rate than his own clock between those instances (observations) and, on that basis, it is (irrespective of the fact that he may be consciously unable to discern same) physically ticking over at a slower rate than his own clock in the one-tenth of a second that it takes for his cerebral processes to inform him that he is looking at that clock.

Again, the formulas of special relativity are not concerned with visual appearances, but with the coordinates of events in inertial reference frames. As I said, a clock moving towards you would actually appear to be ticking faster than your own clock visually, but in your inertial rest frame it would still take a longer coordinate time between ticks than the coordinate time between ticks of your own clock, by an amount given by the time dilation formula.

he takes Einstein’s word for it and realizes that clock B is NOT incurring time contraction which (as I have previously stated was apparently, for Einstein, an anathema) but that it is his clock that is ticking over at a slower rate than B (see below).

I still have no idea what you mean by "time contraction". Do you understand that in relativity there is no frame-independent truth about whether a clock is ticking slow or not, that we can only talk about its rate of ticking relative to some inertial coordinate system? Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster, but the clock is not ticking slow in any "objective" sense, and different inertial frames will disagree about which of two clocks is ticking slower (relative to their own coordinate time) at any given instant. And Einstein made clear that all inertial frames are equally valid, there is no reason to consider one frame's perspective to be more "true" than any other's.

Sure, but of course the velocity of the ship at each point on the curve is different in different frames, and in every frame the rate his clock is ticking at any given instant depends only on his velocity at that instant.

It’s velocity is different but its speed remains constant! It is a clock’s rate of travel (i.e. its speed) that dictates its SR rate of time dilation not its direction of travel.

Only in one particular inertial frame. If an object is moving in a circle at constant speed in the inertial rest frame of the center of the circle, then in a different inertial frame where the center of the circle is moving at constant velocity, the speed of the object will be variable (and in this frame the path of the object will look like some type of cycloid rather than a circle). Again, in SR all inertial frames are equally valid.

I don't know what you mean by this distinction--in any given frame, if clock A is ticking slower than clock B, then how is that different from saying that clock B is ticking faster than clock A in this frame?

As previously pointed out - the claim is that according to the astronaut the Earth clock is physically ticking over at a faster rate than it was before he commenced his trip and for the astronaut to be of the opinion that this is physically taking place he must also believe (predict, determine) that the Earth’s axial spin and orbit of the sun have physically increased.

If the astronaut understands relativity at all, he knows that to talk about the rate a clock is ticking in any objective "physical" sense is totally meaningless, you can only talk about the rate a clock is ticking in one inertial coordinate system or another, and different coordinate systems give different (equally valid) answers. If you don't understand this, you really have missed one of the most basic ideas about relativity!

 

[JesseM]

(continued from previous post)

I quite agree however, as pointed out above, ‘we’ (that is, my side of the discussion) are not simply talking about “the relative rate of one clock as compared to another” (which is effectively out of context) but ‘we’ are saying that if the astronaut considers that the Earth clock is physically ticking over at a faster rate than his own clock (which he considers to be ticking over at an unchanged rate i.e. that his clock is ticking over at the same rate as it was before he started moving) then he must also believe that the Earth’s axial spin and orbit of the sun has physically increased.

Again, in relativity it is quite meaningless to talk about how fast any clock is ticking "physically" in a frame-independent sense. No matter what clock you are dealing with, different frames assign it different rates of ticking, and for any pair of clocks, different frames will disagree about whose rate of ticking is slower (since different frames will disagree about which clock's speed is greater). Einstein makes it quite clear that there is no reason to prefer one inertial frame's perspective over any other, and any relativity textbook you might care to look at should make this clear as well.

Let us assume that our intrepid astronaut has accelerated to a velocity of close to the speed of light

"a velocity of close to the speed of light" relative to what? If the astronaut is moving at close to the speed of light in the rest frame of the Earth, then in the astronaut's own inertial rest frame the astronaut is at rest and the Earth has a velocity close to the speed of light. There is no objective physical truth about which is "really" at rest and which is "really" moving at close to light speed, that's why they call it relativity, because quantities like speed and the time dilation factor can only be defined relative to some frame of reference or another.

thereby generating the particle acceleration attained gamma factor of 40,000 as a result of which the Earth clock is, according to his calculations, ticking over at a rate of 40,000 seconds for each of his own seconds. It is not only every Earth second that has been compressed by that factor but also every Earth minute; hour; day and year.

"According to his calculations"? If the astronaut is moving inertially, then in his own rest frame, it is the Earth that has the large velocity while he is at rest, and the time dilation formula must work the same way in every inertial frame according to the first postulate of relativity, so he must calculate that the Earth's clock is ticking 40,000 times slower than his own if he does the calculations relative to his own inertial rest frame, not 40,000 times faster.

Assuming that the astronaut possesses a smidgin of intelligence he must be able to conclude that, regardless of what his calculations indicate, the Earth is not spinning on its axis at 64 million kilometres a second otherwise, presumably, this would have some affect on the population as well as everything else that’s not tied down.

You appear to have badly misunderstood the principle that the laws of physics work the same in all inertial reference frames, which was one of the two basic postulates of SR that Einstein put forward in his 1905 paper. Every frame must predict that clocks moving in that frame slow down, not speed up. Two observers moving inertially relative to one another will each calculate that the other one's clock is running slower than their own. And despite this seemingly counterintuitive result, all frames will nevertheless get identical predictions about all local events like what two clocks read at the moment they pass next to one another (you could take a look at this thread where I diagrammed an example of two rows of clocks moving at constant speed next to one another, where in each row's rest frame it was the clocks of the other row that were running slow, yet both frames predict the same thing about what any given pair of clocks read at the moment they pass next to one another).

But aside from this issue, you started this post by denying this claim of mine: "although you can say one clock's average rate of ticking is objectively slower, there is no basis for saying that one clock is ticking slower than the other at any given moment during the trip." Are you saying there is a basis for saying that, at a single moment during the trip, one clock is objectively ticking slower than the other?

On the basis that there is no such thing as an instantaneous moment - that time flows continuously - yes, I am saying that.

Relativity deals with plenty of instantaneous quantities such as instantaneous velocity, as do all dynamical theories of physics expressed using calculus. Do you know the basics of calculus? Do you understand, for example, if we have some curve y(x) graphed on the x-y plane, then the value of dy/dx at a particular value of x represents the instantaneous slope at the point on the function with that x-value? Do you understand that in physics, dx/dt at a particular value of t represents the instantaneous velocity at that exact value of the t-coordinate? Do you understand that the time dilation formula gives you a clock's instantaneous rate of ticking as a function of the clock's instantaneous velocity (relative to whatever frame you're using)? If you know the clock's velocity as a function of time v(t) in your frame, then to find the total elapsed time on the clock between two coordinate times t0 and t1, you'd do an integral over the instantaneous rate of ticking at every value of t between t0 and t1, i.e $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$

Do you deny that if you have two clocks A and B moving relative to one another in flat spacetime, then at any given moment, it is possible to find a frame #1 where A is ticking more slowly than B (because A has a higher instantaneous velocity in frame #1 at that moment), and also possible to find a frame #2 where B is ticking more slowly than A (because B has a higher instantaneous velocity in frame #2 at that moment)?

It would be very much appreciated if you would stick to the subject on hand and not introduce flights of fantasy.

"Flights of fantasy"? All of special relativity revolves around the idea that you can analyze a problem from the perspective of any inertial reference frame, and that the laws of physics will work exactly the same in every inertial frame, so no frame should be physically preferred over any other. Again, the very name "relativity" refers to the fact that certain quantities, such as the rate a clock is ticking, can only be measured relative to different (equally valid) inertial frames.

Do you deny that all inertial frames are equally valid in SR, and that they'll all make the same predictions about questions like what two clocks read when they meet each other?

No I do not deny that but what I’m talking about is specifically what the astronaut believes is taking place i.e. the predictions or determinations generated in his reference frame.

Furthermore, I’m not talking about “what two clocks read when they meet each other” but what it is claimed the astronaut ‘sees’ (or ‘predicts’ or ‘determines’) whilst he is moving toward the planet!

So do you agree that if the astronaut is moving inertially, then in his inertial rest frame he is at rest while the planet is moving towards him at high speed, therefore in this frame his own clock is ticking at the normal rate while the planet's clock is ticking slower?

 

[granpa]

time on Earth does seem to the traveler to speed up while he is accelerating. you consider this absurd yet you don't think it absurd that the travelers clock appears to slow down. why can a clock, in your opinion, slow down but not speed up?

 

[JesseM]

time on Earth does seem to the traveler to speed up while he is accelerating.

Unlike with inertial frames, there is no preferred way to define the coordinate system of an accelerating observer, so this really depends on a totally arbitrary choice of coordinate systems. You can find coordinate systems where the accelerating observer is at rest and time runs faster on the Earth clock, but you can also find coordinate systems where the accelerating observer is at rest and time runs slower on the Earth clock, or runs at exactly the same rate, or alternates running fast and slow, etc.

 

[granpa]

I'm just using the instantaneous frame of the traveler at each instant. I didnt think there was anything controversial about it.

 

[JesseM]

I'm just using the instantaneous frame of the traveler at each instant. I didnt think there was anything controversial about it.

In the instantaneous inertial frame of the traveler at each instant, the Earth-clock is always ticking slower at that instant in that frame, not faster. Only if you construct a non-inertial coordinate system which has the property that its definition of simultaneity at each point on the traveler's worldline matches the definition of simultaneity in the traveler's instantaneous inertial rest frame at that point, and the coordinate time along the traveler's worldline matches the traveler's proper time, can you say that the Earth's clock will be ticking faster in this non-inertial coordinate system.

 

[granpa]

its ticking slower but there is a change of simultaneity from frame to frame. I'm just taking that into account.

 

[phyti]

Hey cos;
I'm a salmon guy, not herring.
Anyway, I agree with you that acceleration implies a change in course, and in the simple twin problem there are only three parts to the trip. But as you add more accelerations for both, it's the total path that counts. At some point, words are not sufficient to explain these things.
Here is a geometrical explanation for the twin aging problem.
Time is not altered by motion, only the rate of activity and thus time measurement. When the twins separate and later rejoin, the same amount of time has elapsed for the universe, but the two clocks have sliced that time interval into different lengths.

The vertical line representing the height of the triangle, is the time for an object not moving, i.e., the minimum interval. Any motion to the right or left results in a longer/dilated interval. Divide the paths at the change of direction for A. For each segment, rotate the minimum interval to horizontal. Where the paths cross the horizontal,draw a vertical
line. Where the vertical intersects the arc, represents the amount of time for each twin for that part of the trip. The one who travels the greatest distance in a given time, will travel the fastest, and thus have more time dilation.

 

[cos]

That's what I said, the experiment was more complicated than Einstein's thought experiment because they had to take gravitational time dilation into account.

So when they took gravitational time dilation into account and eliminated that factor they ended up with Einstein’s prediction which didn't take it into account in the first place.

Was the eventual lag between the HKX traveled clocks after the gravitational time dilation factor was removed from the results any different from Einstein’s predicted lag, unaware of any gravitational time dilation factor?

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere", not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

You have your interpretation of what Einstein meant by “more slowly”; I have mine.

Every instant that an observer alongside a ‘polar’ clock looks at an ‘equatorial’ clock on a transparent massless sphere the size of the Earth he will see that compared with his observation made at a previous instant the equatorial clock will lag even further behind his own clock indicating to him that between those instances the equatorial clock has continuously ticked over at a slower rate than his own clock.

Er, why would you imagine I meant that? Of course I am talking about Einstein's thought experiment where the sphere is spinning. The point is that the sphere has a center, we can either pick an inertial frame where the position of the center of the sphere remains constant over time, or we can pick a frame where the center of the sphere is moving at some nonzero constant velocity.

Other than as an attempt to confuse the debate why do we even need to “pick a frame where the center of the sphere is moving at some nonzero constant velocity.”? The opinions expressed by an observer in that frame have no bearing whatsoever on the determinations made by an observer in the “inertial frame where the position of the center of the sphere remains constant over time.”

An observer standing next to clock B doesn't have their own inertial rest frame because they're not moving inertially. If we choose the inertial frame in which the center of the sphere is at rest, then in this frame B will be moving at constant speed so it's true that B will be ticking at a constant slowed-down rate, while A will be ticking at a normal rate.

On the basis that the observer at the pole determines that the clock at the equator (B) is “ticking at a constant slowed-down rate” isn’t he of the opinion that clock B is incurring time dilation?

Isn’t an observer accompanying clock B of the opinion that clock A is (or at least appears to be) ticking over at a faster rate than his own clock?

If so, he can either conclude that clock A is incurring time contraction OR that his clock is incurring time dilation (i.e. is “ticking at a constant slowed-down rate”)!

Assuming that he is aware of, and accepts, Einstein’s paragraph 3, chapter 4 comment - that the clock at the equator “goes more slowly” than (i.e. is “ticking at a constant slowed-down rate” compared to) the polar clock - might he not tend to take Einstein’s word for it thus determine that B is not ‘ticking over at a constant increased rate’ but realize that his clock is “ticking at a constant slowed-down rate”?

On the other hand, if we choose a different inertial frame in which the center of the sphere is moving inertially at some constant velocity...

On the other hand we could stick to the subject under discussion and not introduce extraneous materiel that obfuscates same.

Again, the formulas of special relativity are not concerned with visual appearances, but with the coordinates of events in inertial reference frames. As I said, a clock moving towards you would actually appear to be ticking faster than your own clock visually, but in your inertial rest frame it would still take a longer coordinate time between ticks than the coordinate time between ticks of your own clock, by an amount given by the time dilation formula.

Which is precisely why I usually stick the word ‘see’ in quotation marks and often follow it with parenthesised (‘determine’ or ‘calculate’)

I still have no idea what you mean by "time contraction".

On the basis of your word ‘still’ I assume that you previously had no idea what I meant by “time contraction” however I am not aware of any earlier comments of yours to that effect. If I had seen one I would have responded - if one clock (A) is ticking over at over at a slower rate than another clock (B) some people state that clock A is incurring time dilation (i.e. that it’s seconds are ‘compressed’ or ‘shorter’) on the other hand some people insist that clock A is not ticking over at a slower rate than B but that B is ticking over at a faster rate than A thus that clock B’s seconds are extended i.e. contracted!

On the basis that, as I understand it, the idea of time contraction was an anathema for Einstein I am of the opinion that he would not have accepted this concept.

Do you understand that in relativity there is no frame-independent truth about whether a clock is ticking slow or not, that we can only talk about its rate of ticking relative to some inertial coordinate system?

And in chapter 4 where Einstein wrote that a clock at the equator goes more slowly than a clock at one of the poles he was talking about it’s rate of ticking relative to the polar observer’s inertial coordinate system.

Whilst it is quite possible that Einstein might have been aware of the fact that the Earth is moving through space he was not positing what some purely hypothetical observer contained in another (‘different’) imaginary inertial reference frame would determine but his comment was strictly in relation to what is taking place in the Earth’s reference frame.

On the basis that, in relativity, we can only talk about some other clock’s rate of ticking relative to some inertial system then the claim that the traveling twin can (whilst he is accelerating following turn-around) ‘talk about’ the Earth clock ticking over at a faster rate than it was before he started moving appears, to me, to contradict relativity.

That’s what I’ve been saying!

Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster,

So when the traveler is returning to Earth at uniform velocity he cannot, according to relativity, say or determine or calculate that the Earth clock is ticking faster than it was before he started moving ergo he can only conclude that his clock (as Einstein posited in paragraph 2, chapter 4 with respect to a clock moving in any polygonal line) is ticking over at a slower rate than it was before he started moving toward the Earth clock analogous to clock A in Einstein’s paragraph 1, chapter 4 STR.

but the clock is not ticking slow in any "objective" sense, and different inertial frames will disagree about which of two clocks is ticking slower (relative to their own coordinate time) at any given instant. And Einstein made clear that all inertial frames are equally valid, there is no reason to consider one frame's perspective to be more "true" than any other's.

Whilst “Einstein made clear that all inertial frames are equally valid..” in chapter 4 he referred to a single inertial reference frame in each paragraph of that presentation!

Although it is quite obvious that an observer (C) in a different inertial reference frame may have an entirely different perspective his opinion has nothing whatsoever to do with, and has no bearing on, determinations made by observers accompanying Einstein’s clocks A and B!

Only in one particular inertial frame. If an object is moving in a circle at constant speed in the inertial rest frame of the center of the circle, then in a different inertial frame where the center of the circle is moving at constant velocity, the speed of the object will be variable (and in this frame the path of the object will look like some type of cycloid rather than a circle). Again, in SR all inertial frames are equally valid.

Which, of course, has absolutely nothing whatsoever to do with the subject on hand!

In chapter 4 Einstein made no reference whatsoever to what an observer in a different inertial reference frame would conclude but referred to what takes place in the stationary system of the respective ‘at rest’ clocks.

The conclusion arrived at by a third observer moving past Einstein’s paragraph 1 “points A and B of K” or his paragraph 3 clock moving in a closed curve around another clock that has remained at rest or past a planet that has clocks at the equator and at one of the poles has absolutely nothing whatsoever to do with what the person accompanying the respective stationary clocks determines!

If the astronaut understands relativity at all, he knows that to talk about the rate a clock is ticking in any objective "physical" sense is totally meaningless, you can only talk about the rate a clock is ticking in one inertial coordinate system or another, and different coordinate systems give different (equally valid) answers. If you don't understand this, you really have missed one of the most basic ideas about relativity!

So if an astronaut ‘talks about’ or ‘determines’ or ‘predicts’ that the Earth clock is physically ticking over at a faster rate than it was before he started moving I presume that he has missed one of the most basic ideas about relativity?

You wrote, above, “Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster.” It seems that you agree with me that the claim - that from the astronaut’s point of view his clock is not ticking over at a slower rate than it was before he started moving following turn-around but that it was the Earth clock that started ticking over at faster rate than it was before he started moving - is inappropriate.

Do you also agree with me that an astronaut, having come to stop at the end of his outward-bound trip is equivalent to Einstein’s paragraph 1, chapter 4 clock A at “points A and B of K” with the planet represented by ‘B’?

If so, do you also agree that his return trip is equivalent to Einstein’s paragraph 1, chapter 4 depiction of clock A moving to B’s location?

I assume that you would agree with Einstein’s comment that clock A (the astronaut’s clock) lags (even further) behind clock B (the Earth clock) when A arrives at B’s location?

Regardless of the fact that it may not actually be aware of the phenomenon - A can either conclude that it incurred time dilation (physically ticking over at a slower rate than it was before it started moving) or that B incurred time contraction and on the basis that “a clock can only tick slower than the coordinate time of an inertial frame, never faster.” does it not follow that A cannot say that B ticked over at a faster rate than it did before he started moving?

He can, of course, declare that B appeared to be ticking over at a faster rate than it did before he started moving however is he justified in insisting that it did, physically, tick over at a faster rate than it was before he started moving?

It would be very much appreciated if you did not refer to a conclusion arrived at by an observer in a different reference frame as, in my view, their opinion has absolutely nothing whatsoever to do with what A or B determine and can only obfuscate the discussion.

 

[cos]

Hey cos;
I'm a salmon guy, not herring.

Regardless of personal taste - there's something fishy.

Here is a geometrical explanation for the twin aging problem.

Thank for your presentation however I am not looking for an explanation of same but am seeking comments in relation to the claim that from the astronaut's point of view, his clock does not tick over at a slower rate than it did before he started moving but that the Earth clock physically ticked over at a faster rate than it did prior to his starting to move.

When the twins separate and later rejoin, the same amount of time has elapsed for the universe...

In accordance with the above claim, it is (from that astronaut's point of view) not only the Earth-bound twin that ages at a faster rate than the astronaut but also the entire universe.

 

[JesseM]

So when they took gravitational time dilation into account and eliminated that factor they ended up with Einstein’s prediction which didn't take it into account in the first place.

Was the eventual lag between the HKX traveled clocks after the gravitational time dilation factor was removed from the results any different from Einstein’s predicted lag, unaware of any gravitational time dilation factor?

I don't think you can break down the total time dilation into a linear sum of SR velocity-based time dilation and GR gravitational time dilation in the way you're suggesting. I think the only way to calculate the total time elapsed on the clocks is to do an integral over the path of each clock in curved spacetime; paths through the same regions of space at different speeds would have different times elapsed, so in that sense you're taking into account velocity-based time dilation, but I'm pretty sure you can't calculate what the velocity-based time dilation would be in flat spacetime and add it to a pure gravitational-based time dilation that ignores velocity to get the total time dilation.

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere", not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

You have your interpretation of what Einstein meant by “more slowly”; I have mine.

So are you saying you do think he meant "more slowly at every instant in an objective frame-independent sense"? If so, what you're saying goes against the basics of relativity, and is objectively wrong, it's not just a matter of opinion. On the other hand, if you don't mean to imply that one clock is going more slowly than another at every moment in a frame-independent sense, then please spell out explicitly what you do mean.

Every instant that an observer alongside a ‘polar’ clock looks at an ‘equatorial’ clock on a transparent massless sphere the size of the Earth he will see that compared with his observation made at a previous instant the equatorial clock will lag even further behind his own clock indicating to him that between those instances the equatorial clock has continuously ticked over at a slower rate than his own clock.

But as I said, visual appearances are completely different than time dilation. If you talk about what would be happening in the inertial rest frame of the observer at the pole, in this case it's true that the clock at the equator is ticking slower than his own clock. However, there are other equally valid inertial frames where at certain times the clock at the pole is ticking slower than the clock at the equator (because the clock at the pole has a higher speed at those times in that frame).

Other than as an attempt to confuse the debate why do we even need to “pick a frame where the center of the sphere is moving at some nonzero constant velocity.”? The opinions expressed by an observer in that frame have no bearing whatsoever on the determinations made by an observer in the “inertial frame where the position of the center of the sphere remains constant over time.”

Because I am making the point that there is no objective frame-independent truth about which of two clocks is ticking slower at any given moment, since all inertial frames are equally valid and there are some frames where the clock at the pole is ticking slower at some moments. That has been my point all along--do you agree with it or disagree? If you agree, but just want to talk about the frame-dependent facts about what is happening in the rest frame of the observer at the pole (in which case I certainly agree that the clock is ticking slower at every moment in that frame), then just say so! But that's the only point I was making all along, if you never disagreed with it than you could have said so earlier and we would have saved a lot of time.

On the basis that the observer at the pole determines that the clock at the equator (B) is “ticking at a constant slowed-down rate” isn’t he of the opinion that clock B is incurring time dilation?

Yes, in his own rest frame, but not in an objective frame-independent sense.

Isn’t an observer accompanying clock B of the opinion that clock A is (or at least appears to be) ticking over at a faster rate than his own clock?

Again, visual appearances are a totally different matter than time dilation in SR, as evidenced by the fact that a clock moving towards you will appear to be ticking faster than yours even though in your rest frame it is actually ticking slower. If the observer at the equator considers what is happening in the inertial rest frame where he is at rest at a particular moment (i.e. the frame of an inertial observer moving in a straight line whose instantaneous velocity is the same as the instantaneous velocity of the observer at the equator at that instant), then in that frame the clock at the pole is ticking slower at that moment, since in that frame the clock at the pole has a nonzero velocity while his instantaneous velocity is zero.

Assuming that he is aware of, and accepts, Einstein’s paragraph 3, chapter 4 comment - that the clock at the equator “goes more slowly” than (i.e. is “ticking at a constant slowed-down rate” compared to) the polar clock - might he not tend to take Einstein’s word for it thus determine that B is not ‘ticking over at a constant increased rate’ but realize that his clock is “ticking at a constant slowed-down rate”?

If he is only interested in the rate each clock is ticking in the rest frame of the observer at the pole (or any inertial observer at rest relative to the center of the sphere), then in this frame he'll certainly realize that his clock is ticking at a constant slowed-down rate. But if he understands relativity he knows this is a frame-dependent fact, not an objective physical fact, since he could equally well look at the problem from the perspective of a different inertial frame and get a different answer, and all inertial frames have equal validity in SR.

On the other hand, if we choose a different inertial frame in which the center of the sphere is moving inertially at some constant velocity...

On the other hand we could stick to the subject under discussion and not introduce extraneous materiel that obfuscates same.

It certainly is not extraneous to the subject of whether there is any objective physical truth about which of two clocks is ticking slower at a given moment, which is the one I have been focused on all along. If you have no objection to the idea that there is no objective answer to this question, only different frame-dependent answers, then you shouldn't have objected to my statement in my first post on the thread (post #18) where I said:

It is true that, no matter which frame you choose, the average rate of ticking on the clock of the traveling twin must be slower than the average rate on the clock of the Earth twin. But you can find inertial frames where the Earth twin's clock ticks slower than the traveling twin's clock during the trip away from the Earth, then the traveling twin's clock ticks slower than the Earth twin's on the return journey after the turnaround; you can also find frames where the opposite is true, and the traveling twin's clock is slower on the outbound trip but faster on the inbound leg. So, there is no objective truth about whose clock is ticking slower at any given moment, even if the average of the traveling twin's clock is always slower than the Earth twin's clock over the course of the whole trip.

Do you, in fact, have any objection to this statement?

Again, the formulas of special relativity are not concerned with visual appearances, but with the coordinates of events in inertial reference frames. As I said, a clock moving towards you would actually appear to be ticking faster than your own clock visually, but in your inertial rest frame it would still take a longer coordinate time between ticks than the coordinate time between ticks of your own clock, by an amount given by the time dilation formula.

Which is precisely why I usually stick the word ‘see’ in quotation marks and often follow it with parenthesised (‘determine’ or ‘calculate’)

Well, you didn't put "see" in quotation marks in your statement about the astronaut, nor did you give any indication that you meant "see" to stand for some set of calculations:

At any given instant he would see that the time indicated by that clock lapses even further behind his own time than the time indicated by that same clock at a previous instant indicating to him that clock A has continuously ticked over at slower rate than his own clock between those instances (observations) and, on that basis, it is (irrespective of the fact that he may be consciously unable to discern same) physically ticking over at a slower rate than his own clock in the one-tenth of a second that it takes for his cerebral processes to inform him that he is looking at that clock.

If you are saying the astronaut calculates that clock A is ticking slower, can you spell out what these calculations are? Is he calculating the rate that A is ticking in some inertial frame?

On the basis of your word ‘still’ I assume that you previously had no idea what I meant by “time contraction” however I am not aware of any earlier comments of yours to that effect.

I didn't respond to the specific phrase "time contraction", but I said that I didn't understand the distinction you were making between the idea that his clock was ticking slower than the other clock vs. the idea that the other clock was ticking faster than his clock, and I assumed that your distinction between "time dilation" and "time contraction" was basically the same thing.

If I had seen one I would have responded - if one clock (A) is ticking over at over at a slower rate than another clock (B) some people state that clock A is incurring time dilation (i.e. that it’s seconds are ‘compressed’ or ‘shorter’) on the other hand some people insist that clock A is not ticking over at a slower rate than B but that B is ticking over at a faster rate than A thus that clock B’s seconds are extended i.e. contracted!

And it is exactly this distinction that I said I didn't understand, and still don't understand. For me, if A is ticking slower than B that is the same as saying that B is ticking faster than A, just like if a number N is larger than M that is the same as saying M is smaller than N. What we can say is that in inertial frames, clocks can only run slow relative to coordinate time in that frame, never faster, but I already brought this idea up and you didn't agree that this is what you were talking about.

On the basis that, as I understand it, the idea of time contraction was an anathema for Einstein I am of the opinion that he would not have accepted this concept.

Your use of the phrase "time contraction" still seems meaningless to me if it's supposed to be a comparison between two clocks. Again, it is true that in relativity a clock can only slow down relative to coordinate time in a particular inertial frame, never speed up relative to coordinate time, but apparently this isn't what you're talking about. Maybe what it comes down to is that you do think there is some objective frame-independent truth about whether a clock is running slow, and you're saying that a clock can never be running fast in this objective sense; but if so, that's the whole point I've been disputing, since I'm saying there is no objective answer to the question of how fast a clock is ticking, only an infinite number of different (and equally valid) frame-dependent answers.

And in chapter 4 where Einstein wrote that a clock at the equator goes more slowly than a clock at one of the poles he was talking about it’s rate of ticking relative to the polar observer’s inertial coordinate system.

I don't know if that's what he meant, but if you recall I did suggest that as at least one possibility:

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere" [which of course is the same as the rest frame of the polar observer], not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

So, I'd certainly have no objection to the idea that this might have been what I meant. The only point I have been making from the beginning is that there is no objective-frame dependent answer to the question of which of any two clocks is ticking slower at a given moment, but of course there is a single correct answer to the question of which clock is ticking slower at a given moment in a particular inertial frame. Again, do you have any disagreement with this point, or have you been arguing with me for no reason?

Whilst it is quite possible that Einstein might have been aware of the fact that the Earth is moving through space he was not positing what some purely hypothetical observer contained in another (‘different’) imaginary inertial reference frame would determine but his comment was strictly in relation to what is taking place in the Earth’s reference frame.

Reference frames are just coordinate systems, you don't need to have an actual physical observer at rest in a particular coordinate system in order to calculate the time-coordinates of events in that system, any more than you'd need an observer physically present at the center of the Earth in order to place the origin of your spatial axes there. No frame is any more or less "imaginary" than any other. But again, it might be true that Einstein was talking about what would be true in the inertial frame where the center of the Earth was at rest, it's not relevant to my main point which is that the only objective physical truths are the ones which are agreed on by all inertial frames.

On the basis that, in relativity, we can only talk about some other clock’s rate of ticking relative to some inertial system then the claim that the traveling twin can (whilst he is accelerating following turn-around) ‘talk about’ the Earth clock ticking over at a faster rate than it was before he started moving appears, to me, to contradict relativity.

That's not what I said, I just said that at any given moment, you can find a valid inertial frame where the Earth clock is ticking slower than the traveling twin's clock at that moment. Of course, since we're assuming the Earth is moving inertially, no matter what inertial frame you pick, the Earth will have a constant speed in that frame, so its clock will be ticking at an unchanging rate in that frame.

Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster,

So when the traveler is returning to Earth at uniform velocity he cannot, according to relativity, say or determine or calculate that the Earth clock is ticking faster than it was before he started moving ergo he can only conclude that his clock (as Einstein posited in paragraph 2, chapter 4 with respect to a clock moving in any polygonal line) is ticking over at a slower rate than it was before he started moving toward the Earth clock analogous to clock A in Einstein’s paragraph 1, chapter 4 STR.

Regardless of what inertial frame you choose, the Earth's speed is constant in that frame, so the rate of ticking of the Earth clock doesn't change. However, in some frames there will be particular moments when the speed of the traveling twin is lower than that of the Earth, and therefore in such a frame the Earth-clock is ticking slower than his clock at those moments. Do you disagree with that? Note that in such a frame there will also be moments when the speed of the traveling twin is greater than the Earth's and his clock is therefore ticking slower than the Earth's, and it will always be true that when you look at the average rate his clock was ticking from the beginning to the end, it is less than the (constant) rate the Earth-clock was ticking, so he'll have aged less when he returns to Earth.

Whilst “Einstein made clear that all inertial frames are equally valid..” in chapter 4 he referred to a single inertial reference frame in each paragraph of that presentation!

He didn't explicitly refer to any particular inertial frame in the last paragraph of section 4, for example. If you believe he was implicitly talking about the inertial frame where the Earth was at rest, you could be right, I don't have any wish to argue this point. Again, my point all along has just been that there is never a frame-independent truth about which of two clocks is ticking slower at a particular moment.

Although it is quite obvious that an observer (C) in a different inertial reference frame may have an entirely different perspective his opinion has nothing whatsoever to do with, and has no bearing on, determinations made by observers accompanying Einstein’s clocks A and B!

And this doesn't conflict with my point. But again, I'll note that although it is a usual convention that an observer calculate things from the perspective of his own inertial rest frame, this is just a convention, any observer is free to use any coordinate system for the purpose of calculations, there isn't any physical reason he's forced to treat the frame where he's at rest as his own "perspective".

Only in one particular inertial frame. If an object is moving in a circle at constant speed in the inertial rest frame of the center of the circle, then in a different inertial frame where the center of the circle is moving at constant velocity, the speed of the object will be variable (and in this frame the path of the object will look like some type of cycloid rather than a circle). Again, in SR all inertial frames are equally valid

Which, of course, has absolutely nothing whatsoever to do with the subject on hand!

The reason we got into this long discussion was because you had some kind of objection to my original post on the thread, and the fact that different frames disagree on which of two clocks is ticking slower is certainly relevant to the point I made in that post. If you just didn't follow what I was arguing there, but now have no objection to the point that there is no objective frame-independent truth about which of two clocks is ticking slower at a given moment (regardless of whether you think this point is interesting or relevant to what you were talking about earlier), then we'll be in agreement and can drop the whole thing.

(continued in next post)

 

[JesseM]

(continued from previous post)

I don't know what you mean by this distinction--in any given frame, if clock A is ticking slower than clock B, then how is that different from saying that clock B is ticking faster than clock A in this frame?

As previously pointed out - the claim is that according to the astronaut the Earth clock is physically ticking over at a faster rate than it was before he commenced his trip and for the astronaut to be of the opinion that this is physically taking place he must also believe (predict, determine) that the Earth’s axial spin and orbit of the sun have physically increased.

If the astronaut understands relativity at all, he knows that to talk about the rate a clock is ticking in any objective "physical" sense is totally meaningless, you can only talk about the rate a clock is ticking in one inertial coordinate system or another, and different coordinate systems give different (equally valid) answers. If you don't understand this, you really have missed one of the most basic ideas about relativity!

So if an astronaut ‘talks about’ or ‘determines’ or ‘predicts’ that the Earth clock is physically ticking over at a faster rate than it was before he started moving I presume that he has missed one of the most basic ideas about relativity?

For a fact to be objectively "physical" it must be frame-independent. So yes, for the astronaut to say the Earth clock is "physically" ticking faster or slower than his is to miss a basic idea about relativity, although he can certainly say the Earth clock is ticking faster or slower than his clock in a particular frame, or that the Earth clock is ticking slow (never fast) relative to the coordinate time of a particular inertial frame.

You wrote, above, “Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster.” It seems that you agree with me that the claim - that from the astronaut’s point of view his clock is not ticking over at a slower rate than it was before he started moving following turn-around but that it was the Earth clock that started ticking over at faster rate than it was before he started moving - is inappropriate.

You need to specify what you mean by the astronaut's "point of view". I would agree that if the astronaut picks some (arbitrary) inertial frame to call his "point of view", then the Earth clock will be ticking at a constant rate in that inertial frame (since it's moving at a constant speed in that frame).

Do you also agree with me that an astronaut, having come to stop at the end of his outward-bound trip is equivalent to Einstein’s paragraph 1, chapter 4 clock A at “points A and B of K” with the planet represented by ‘B’?

To make the situation completely equivalent, you'd need to have a clock at the planet B that was synchronized with the astronaut's clock the moment before he left Earth, in the rest frame of the Earth and the planet--in this case I'd agree the situations are equivalent. (by the way, it's not the first paragraph of section 4 where he talks about 'points A and B of K', it's actually the third-to-last).

If so, do you also agree that his return trip is equivalent to Einstein’s paragraph 1, chapter 4 depiction of clock A moving to B’s location?

On the return trip, for it to be fully equivalent you'd need to label the Earth as B rather than the distant planet as in the outward trip, and here you'd need the clock on Earth to be synchronized with the astronaut's clock the moment before he left the distant planet, in the rest frame of the Earth and the planet.

I assume that you would agree with Einstein’s comment that clock A (the astronaut’s clock) lags (even further) behind clock B (the Earth clock) when A arrives at B’s location?

In each case, if the clocks are initially synchronized in the way I describe, then yes.

Regardless of the fact that it may not actually be aware of the phenomenon - A can either conclude that it incurred time dilation (physically ticking over at a slower rate than it was before it started moving) or that B incurred time contraction

You leave out a possibility here, which is that B experienced time dilation, but B's clock was ahead of A's clock at the moment before A began to move towards it (instead of A's clock being synchronized with B's at this moment, as would be true according to the definition of simultaneity in the rest frame of B), so that even though B's clock was ticking slower than A's throughout the journey, B's clock is still ahead of A's when they meet thanks to that "head start". This is exactly what would be true in the inertial rest frame where A is at rest during the trip and B is moving towards him. If you exclusively want to talk about how things would work in the rest frame of B, that's fine, but if you just want to talk about a particular frame you should always spell this out explicitly.

and on the basis that “a clock can only tick slower than the coordinate time of an inertial frame, never faster.” does it not follow that A cannot say that B ticked over at a faster rate than it did before he started moving?

If you want A to make calculations from the perspective of some inertial frame (even though he will not remain at rest in this frame), then it's true that in any inertial frame B is moving at constant speed, and therefore B's clock was ticking at a constant rate. But depending on which frame is chosen, it may be that A's clock was ticking even slower than B during the approach.

It would be very much appreciated if you did not refer to a conclusion arrived at by an observer in a different reference frame as, in my view, their opinion has absolutely nothing whatsoever to do with what A or B determine and can only obfuscate the discussion.

Again, it's just a convention that inertial observers use their inertial rest frame to represent their "perspective", there is no physical reason that it should be harder to "determine" the coordinates of events in a frame where you are moving than it is to "determine" the coordinates of these events in your own rest frame. And A is not an inertial observer, he has different rest frames at different times, so I don't know what you mean by not using "different reference frames" here. Again, if we use the inertial frame where A is at rest during the journey, in this frame it is B's clock that is running slow while his is running normally, and the only reason B's clock is ahead when he reaches it is because it was already ahead at the start of the journey.

 

[phyti]

cos;

from the astronaut's point of view, his clock does not tick over at a slower
rate than it did before he started moving but that the Earth clock physically
ticked over at a faster rate than it did prior to his starting to move.

The dilation only applies to the ship and its contents, thus the pilot cannot
detect the difference within his frame. If he looks out, he should see a
polarized universe, events forward happening faster and events rearward
happening slower, a scenario which is not normal.
It should be obvious from studying the light clock, the rate can only
physically decrease. Observers will perceive the others clock rate increase
when converging, and decrease when diverging, but these are doppler effects.

When the twins separate and later rejoin, the same amount of time has elapsed
for the universe...

In accordance with the above claim, it is (from that astronaut's point of
view) not only the Earth-bound twin that ages at a faster rate than the
astronaut but also the entire universe.

This is true if you change 'faster' to 'slower', where 'slower' is 'measured
as slower'. All other clocks do not change rates, because the astronaut initiated the
motion, and the amount of energy used to move the ship would not move the rest
of the universe in the opposite direction, i.e., the situation is not symmetrical by reason of the conservation of energy.
The ship moving cannot alter the rules of physics in distant parts of the
universe, especially instantaneously. That's the nonsense!
It violates postulate (1) that the theory is built on.
And then there's this:
SR theory transforms coordinates not objects.
Would you please pass the cucumbers.