Twin Paradox: Einstein's Explanation and Alternative Interpretations

[Dale]

edit: I suppose you might have meant something like "longer path through spacetime from the moment each clock was set to zero to the moment they met", in which case your statement would make sense since they'd been synchronized in B's rest frame before A accelerated.

Yes, exactly. From the point on each worldline where they were synchronized to the reunion event there is a shorter spacetime interval for one clock than the other.

This is no more surprising than the fact that someone driving from Boston to New York shows a smaller odometer reading than someone driving from Atlanta to New York even if they both reset their odometers at the beginning of their respective trips. You could even find two starting cities with the same lattitude or longitude to make the analogy more exact, but I am too lazy.

A doesn't "slow down". Both A and B do what they're supposed to, which is to measure a property of their respective world lines. It's the world lines that are different, not the clocks.

Well said.

 

[phyti]

... In Minkowski spacetime clocks don't slow down in any physical sense, they simply take a different path through spacetime and the different interval is a property of that path rather than a property of the clock that measures that path.

The slowdown is built in, aka, the calibration curve.

 

[JesseM]

The slowdown is built in, aka, the calibration curve.

What do you mean by "calibration curve"? The definition here seems to have nothing to do with relativity, but maybe you're using the term differently.

 

[matheinste]

Hello phyti

Are you referring to the calibration of axes using the invariant hyperbolae.

Matheinste

 

[atyy]

Let us imagine that the astronaut's outward journey is directly away from the South Pole and, having come to a stop, he is now looking back at a very large clock at that location which is mounted on a platform that allows it to remain stationary from the traveler's point of view (i.e. it is not spinning around with the planet).

According to my interpretation of Einstein's chapter 4, paragraph 1, his clock (Einstein's clock A) will then lag behind the Earth clock (Einstein's clock B) by .5tv²/c2. When he determines a lag created by the time that it takes that light to reach him as well as any gravitational time dilation created by the Earth's mass he can calculate the exact amount of that lag however his clock will then be ticking over at the same rate as the Earth clock on the basis that they are Einstein's paragraph 1, chapter 4, 'points A and B of K.

He then adjusts his clock so that it reads the same time as the Earth clock so yes, the astronaut's clock is (temporarily) synchronous with the Earth clock and whilst the Earth clock's rate of operation is affected by it's location in a gravitational tidal area the astronaut attains Einstein's chapter 4 (purely hypothetical) instantaneous velocity (of near-light speed for the astronaut) ergo his clock is then 'going more slowly' than the Earth clock by a factor of .5tv²/c2 and he will arrive back on the planet with his clock lagging behind the Earth clock by the same amount as it did at the end of his outward-bound trip.

Alternatively, if the astronaut (in a suitably equipped ship), accelerates at perhaps 100g his clock will very soon be 'going more slowly' than the gravitationally affected Earth clock.

A version of your depiction is that Einstein's paragraph 1, chapter 4, clocks A and B are twin astronaut's each in identical ships that, unlike Einstein's clocks A and B, are initially stationary alongside, and synchronous with, each other whereupon A moves in Einstein's paragraph 2 polygonal path (i.e. away from then back to B's location).

At the end of his 'outward-bound' trip A's clock, although lagging behind, is ticking over at the same rate as B's clock so A adjusts his clock in accordance with the calculated lag factor and they are once again synchronous.

At the end of his outward-bound trip the astronaut's clock lags behind his Earth-bound twin's clock and when he returns to the planet his clock lags even further behind his twin's clock in accordance with Einstein's paragraph 1, chapter 4 depiction ergo, according to that depiction, the astronaut will have aged at a slower rate than his twin thus the Earth-bound twin will be the elder.

My interpretation of ‘reality’ is in relation to an event that takes place in an observer’s reference frame.

It seems to me that although "In Minkowski spacetime clocks don't slow down in any physical sense..." equatorial clocks do, according to Einstein "go more slowly" than polar clocks.

I believe these statements are correct.

The equatorial clock is a clock in Minkowski space - that is just the technical name for the spacetime that Einstein discusses in Chap 4. So both are correct descriptions of the same physical situation.

The initial clock synchronization involves equalizing the zero-settings of the clocks and the rate of the clocks.

Once a reference frame is stated to be "real", the traveling clock is the one that "really" slows - in the sense that it lags.

This is exactly equivalent to saying that the Minkowski rate of the clock is unchanged. To resynchronize the clock, one only has to readjust its zero-setting (as you said in the above quotes), and not its rate (just like resetting an odometer).

 

[Fredrik]

The slowdown is built in, aka, the calibration curve.

I don't know what you mean by calibration curve, but for a clock to even have a ticking rate, we have to be talking about a specific point on its world line, and a specific coordinate system. This means that you can only compare the ticking rates of two different clocks if you specify the points on their world lines where you will compare, and the two coordinate systems you will use for the comparison. Since you can get any result you want to just by choosing appropriate coordinate systems, this comparison doesn't make much sense unless there's a situation where there's a "natural" choice of an event and a coordinate system to associate with the second clock once you have chose an event and a coordinate system to associate with the first clock.

The only one I can think of is an event where the two world lines intersect. We use one coordinate system: the co-moving inertial frame of one of the clocks. The ticking rates can be defined as "change of proper time"/"change of coordinate time". (This gives one of the clocks a ticking rate of 1, but the other can have any value).

 

[Fredrik]

Well said.

Of course it is. It's almost exactly the same thing you said earlier. (I didn't see that until now).

 

[phyti]

JesseM;
from your post 31, with [corrections]

A now ticks at the normal rate in this frame since it's at rest. Since the initial distance between them is 12 light-seconds [20 ls] in this frame, it will take 12/0.8c = 15 [20/.8 = 25] seconds for B to catch up with A. During this time A will advance forward by 15 [25] seconds but B will only advance forward by 15*0.6 = 9 [25*.6 = 15] seconds.

The example cited by cos would be trivial if it just meant one clock lagging behind the other, i.e., you could set either ahead and prove any scenario. Einstein intended to demonstrate the connection between time dilation and motion. Your example just mirrors the original setup, with A and B swapped. As the diagram shows, the longer leg has less proper time.

 

[cos]

One particular section of one particular work is a rather narrow focus. I am trying to provide you with a much more general conceptual tool.

'One particular work' - Einstein's special theory - is my specific interest and whilst I appreciate that you are, perhaps, providing a more general conceptual tool in relation to the twin paradox it is not the 'twin paradox' per se in which I am particularly interested but my argument relates to the claim that the astronaut is of the opinion that his clock does not 'go more slowly' than it did before he started moving but that the Earth clock physically 'goes faster' than it did when he had come to a stop at the end of his outward-bound trip.

One particular section (chapter 4) of one particular work (special theory) is the crux of the argument from my point of view.

I am of the opinion that having read and accepted special theory including chapter 4 as well as Einstein's 1918 article - an astronaut should be capable of realizing that although his clock appears to be ticking over at its standard rate as he returns to Earth it is, according to Einstein 'going more slowly' than it was before he started moving.

As I have pointed out, a person who moves to the top of a mountain could be of the opinion that a clock at that location is ticking over at the same rate as a clock at sea-level because they are both ticking over at the same rate as his own clock which is, in his opinion, ticking over at its normal rate when he is at both locations.

Alternatively he could (as Confucius suggested) apply his (assumed) knowledge of the Wallops Island experiment and realize that his clock is ticking over at different rates depending on his location in a gravitational field.

As the astronaut starts to accelerate for the return trip he will see his clock continuously appearing to be ticking over at a constant rate however, due to the fact that his velocity (v) is increasing his clock will, in accordance with Einstein's chapter 4 equation (.5tv²/c²), be 'going more slowly' than it was at a lesser instantaneous velocity.

Although Einstein's chapter 4 is 'one particular section of one particular work' I fail to see any reason why it should be ignored or not taken into consideration.

If Einstein's chapter 4 depiction is 'wrong' or if it argues against mainstream physics community interpretations of relativity then perhaps it should be expunged or publicly denounced as members of the, then, scientific community (representatives of The Church) would have liked to do with Galileo's 'Two New Sciences' or as members of the, then, physics community would have liked to do with Einstein's general theory comment that the law of the constancy of the velocity of light required modification and his comment in 'Relativity' that the same law is not fully valid - that it can only be applied in conditions that do not, to the best of extant scientific knowledge, exist (i.e. zero gravity).

Ignoring, or in our discussions not taking into account, 'one particular section of one particular work' will not make it go away.

If other sections of a particular work (special theory) contradict that particular section then it seems that special theory must contain an inconsistency but that's not what I'm saying!

What I'm saying is that if an observer is located in an inertial reference frame there is no internal dynamic experiment that he can carry out to determine if he is moving with uniform velocity or if he is 'at rest' and that the same thing applies to all inertial reference frames thus that no reference frame takes precedence over any other reference frame however, in chapter 4 (as well as in his 1918 article), Einstein points out that if a clock is made to move it will 'go more slowly than' an identical clock that remained at rest in the original reference frame. On the basis that a clock is made to move it must undergo acceleration.

According to Nikolai Rudakov in his book 'Fiction Stranger Than Truth' - "Very few relativists have actually adopted Einstein's [1918] explanation [of the twin paradox]. Not many authors mention the 1918 dialogue, and some who do imply that Einstein may have been wrong."

It is my opinion that Einstein's chapter 4 depictions are analogous (albeit sans reference to acceleration) to his 1918 article thus that some people may similarly insist that chapter 4 was wrong thus should not be referred to on the (assumed) basis that it argues against mainstream physics community interpretations of relativity however it is Einstein's relativity - not interpretations of same - to which I refer and it does contain chapter 4.

 

[JesseM]

JesseM;
from your post 31, with [corrections]

A now ticks at the normal rate in this frame since it's at rest. Since the initial distance between them is 12 light-seconds [20 ls] in this frame, it will take 12/0.8c = 15 [20/.8 = 25] seconds for B to catch up with A. During this time A will advance forward by 15 [25] seconds but B will only advance forward by 15*0.6 = 9 [25*.6 = 15] seconds.

What is the basis for your "corrections"? I specified that the initial distance between A and B was 20 light-seconds in the rest frame of B. Do you disagree that, according to relativity, this means that if we analyze the same situation from the perspective of the frame where A is at rest after accelerating (a frame moving at 0.6c relative to B's rest frame), the initial distance between A and B must have been 12 light-seconds in this frame, not 20 light-seconds?

The example cited by cos would be trivial if it just meant one clock lagging behind the other, i.e., you could set either ahead and prove any scenario.

No, you couldn't "set either ahead" by any amount you like, because Einstein specified that the clocks were initially synchronized in the rest frame of B. According to relativity, this uniquely determines the amount that A and B were initially out-of-sync in any other frame moving with some specified velocity relative to the rest frame of B. If A and B were initially 20 light-seconds apart and synchronized in the frame of B, then in a frame moving at 0.6c relative to this frame, moving in the direction from A to B, B must have been initially ahead of A by precisely 12 seconds. Do you disagree? Would you like me to demonstrate this using the Lorentz transformation?

Einstein intended to demonstrate the connection between time dilation and motion. Your example just mirrors the original setup, with A and B swapped.

It's not a different example, it's the same scenario viewed from the perspective of a different frame (you can see that I didn't just swap A and B by noting that in both Einstein's description and my description, A was the clock that accelerated, and in my description A and B were both originally in motion rather than both originally at rest as in Einstein's description). Do you understand that the Lorentz transformation shows us how to describe the same physical situation from the perspective of different reference frames?

As the diagram shows, the longer leg has less proper time.

The time and distance intervals in your diagram are incorrect--see my question about your "corrections" above--but the shape of the two worldlines is correct. I don't know what "tB" is supposed to represent, and I don't know what segment of each worldline you're talking about when you talk about the amount of proper time on each (obviously the endpoint is when the two clocks meet, but what's the starting point on each worldline? Is it the event of each clock reading t=0? If so, note that the starting point along the B worldline should be lower on the page than the starting point on the A worldline, since the B clock is ahead in this frame).

 

[Dale]

whilst I appreciate that you are, perhaps, providing a more general conceptual tool in relation to the twin paradox it is not the 'twin paradox' per se in which I am particularly interested

As I already mentioned previously the utility of Minkowski geometry is not in any way limited to the twin paradox. It is generally applicable to all scenarios, including the section 4 that has captivated your attention, the astronaut scenario, the mountain scenario, and any other scenario you will invent or encounter. It is a complete mathematical framework for attacking any problem in special relativity with immediate application to general relativity also.

In many fields of study one examines special case solutions because the more general approach is overly cumbersome or otherwise intractable. You take appropriate limits and examine the results in the hope of gaining some insight, but you recognize that your insights are inherently approximations of the intractable general approach and can fail. Such is not the case for Minkowski geometry, it is both completely general and more straightforward than other approaches to special relativity. It is one of the rare instances where you do not need to sacrifice generality for convenience.

Although Einstein's chapter 4 is 'one particular section of one particular work' I fail to see any reason why it should be ignored or not taken into consideration.
...
Ignoring, or in our discussions not taking into account, 'one particular section of one particular work' will not make it go away.

I never suggested you ignore it nor did I suggest it was wrong. I simply provided a better conceptual framework for understanding it.

Do you think it is reasonable to ignore such a useful tool simply because it was not explicitly presented in the seminal manuscript?

 

[cos]

Once a reference frame is stated to be "real", the traveling clock is the one that "really" slows - in the sense that it lags.

I don't understand what you mean by 'Once a reference frame is stated to be "real'" but does your comment that 'the traveling clock is the one that "really" slows - in the sense that it lags.' mean not just that the traveling clock (Einstein's clock A) is found to lag behind the 'stationary' clock (Einstein's clock B) when A arrives at B's location or does it mean that the traveling clock physically 'goes more slowly than' the stationary clock whilst it is moving? In other words - that the traveling clock ticks over at a slower rate than it did before it started moving - that the traveling clock incurs time dilation whilst it is moving?

If so, then an observer accompanying clock A is fully entitled to realize that although his clock appears to be ticking over at the same rate as it was before they started moving that it is, in reality, ticking over at a slower rate than it was before they started moving.

If this is correct then the claim in relation to which I made the original posting - that the astronaut insists that his clock is not ticking over at a slower rate than it did before he started moving back to the planet but that it is the Earth clock (Einstein's clock B) that is physically ticking over at a faster rate than it was before he started moving and only during his period of acceleration is, as I suspected, in contradiction of special theory - specifically chapter 4.

The astronaut's concept of 'reality' is not limited to what he sees nor even to what he, applying the Lorentz transformations, predicts as some people insist but should also take into account what he knows in the same way that whilst a person sees a steel rod appear to bend at the surface of a body of water this is not taking place in reality- the rod does not bend.

 

[cos]

I never suggested you ignore it nor did I suggest it was wrong. I simply provided a better conceptual framework for understanding it.

By 'understanding it' are you referring to the twin paradox per se or to Einstein's reference to clocks A and B of K and the subsequent relocation of A?

Although it may be a better conceptual framework I (as one of those annoying self-taught people including Faraday who was similarly ignorant of mathematical processes) find the Minkowski concepts far more complicated than Einstein's depiction.

Do you think it is reasonable to ignore such a useful tool simply because it was not explicitly presented in the seminal manuscript?

On the basis that I, due to my ignorance, prefer to keep things as simple as possible - yes.

 

[atyy]

I don't understand what you mean by 'Once a reference frame is stated to be "real'" but does your comment that 'the traveling clock is the one that "really" slows - in the sense that it lags.' mean not just that the traveling clock (Einstein's clock A) is found to lag behind the 'stationary' clock (Einstein's clock B) when A arrives at B's location or does it mean that the traveling clock physically 'goes more slowly than' the stationary clock whilst it is moving? In other words - that the traveling clock ticks over at a slower rate than it did before it started moving - that the traveling clock incurs time dilation whilst it is moving?

If so, then an observer accompanying clock A is fully entitled to realize that although his clock appears to be ticking over at the same rate as it was before they started moving that it is, in reality, ticking over at a slower rate than it was before they started moving.

If this is correct then the claim in relation to which I made the original posting - that the astronaut insists that his clock is not ticking over at a slower rate than it did before he started moving back to the planet but that it is the Earth clock (Einstein's clock B) that is physically ticking over at a faster rate than it was before he started moving and only during his period of acceleration is, as I suspected, in contradiction of special theory - specifically chapter 4.

The astronaut's concept of 'reality' is not limited to what he sees nor even to what he, applying the Lorentz transformations, predicts as some people insist but should also take into account what he knows in the same way that whilst a person sees a steel rod appear to bend at the surface of a body of water this is not taking place in reality- the rod does not bend.

The statement of a reference frame is crucial for any of the statements in your first two paragraphs to make sense, let alone be true or false. They are true in a particular reference frame. That is why I said, "Once a reference frame is stated to be 'real'".

An ideal clock does not incur time dilation only at the instant of acceleration.

Your example of a rod that appears bent illustrates that it is crucial to state the reference frame. Just because a person sees a rod bend at the surface of the water does not mean that the rod does not bend. He needs to know the angle at which he is looking relative to the surface of the water and the angle at which the rod enters the water. Similarly, one should always state the reference frame when talking about time dilation as real, for without it, the astronaut will never be able to know reality.

 

[cos]

The statement of a reference frame is crucial for any of the statements in your first two paragraphs to make sense, let alone be true or false. They are true in a particular reference frame. That is why I said, "Once a reference frame is stated to be 'real'".

You didn’t answer my question:-

“I don't understand what you mean by 'Once a reference frame is stated to be "real'" but does your comment that 'the traveling clock is the one that "really" slows - in the sense that it lags' mean not just that the traveling clock (Einstein's clock A) is found to lag behind the 'stationary' clock (Einstein's clock B) when A arrives at B's location or does it mean that the traveling clock physically 'goes more slowly than' the stationary clock whilst it is moving? In other words - that the traveling clock ticks over at a slower rate than it did before it started moving - that the traveling clock incurs time dilation whilst it is moving?”

I try to the very best of my ability to respond directly to questions asked by (most) people and I think it is only courteous for them to reciprocate.

What do you mean by “The statement of a reference frame...”?

In that paragraph I intimated that if the traveling clock physically ticks over at a slower rate in its own reference frame than it did before it started moving then it physically ticks over at a slower rate than it did before it started moving regardless of the possibility that somebody viewing it from another reference frame may, or may not, determine otherwise!

The point of view of, or determinations made from, another reference frame has absolutely nothing whatsoever to do with what the clock does only with what they think the clock is doing!

I have absolutely no interest whatsoever in what an observer in another reference frame determines or does not determine on the basis that his determinations (observations; measurements; predictions) have absolutely NO physical affect on that clock!

My second paragraph to which you referred was:-

“If so, then an observer accompanying clock A is fully entitled to realize that although his clock appears to be ticking over at the same rate as it was before they started moving that it is, in reality, ticking over at a slower rate than it was before they started moving.”

Are you unable to see that when I referred to what an observer accompanying clock A realizes that I am talking about what takes place in THAT observer’s reference frame?

An ideal clock does not incur time dilation only at the instant of acceleration.

I did NOT say that it DOES!

A clock starts to incur time dilation at the instant of acceleration!

Your example of a rod that appears bent illustrates that it is crucial to state the reference frame. Just because a person sees a rod bend at the surface of the water does not mean that the rod does not bend. He needs to know the angle at which he is looking relative to the surface of the water and the angle at which the rod enters the water.

I hate to disillusion you but a steel rod does NOT bend when it is placed into a tank of water!

The observer KNOWS the angle at which he is looking relative to the surface of the water AND the angle at which the rod enters the water because he is LOOKING AT the event!

Similarly, one should always state the reference frame when talking about time dilation as real, for without it, the astronaut will never be able to know reality.

In my first paragraph where I talked about the moving clock ‘going more slowly than’ the stationary clock the same phenomenon applies to observers in both reference frames. Clock A ‘goes more slowly than’ clock B as far as both observers are concerned!

In my second paragraph I specifically referred to what an observer accompanying clock A determines and if that’s not ‘stating the reference frame’ (i.e. identifying same) I fail to see what else you would call it!

 

[JesseM]

In that paragraph I intimated that if the traveling clock physically ticks over at a slower rate in its own reference frame than it did before it started moving then it physically ticks over at a slower rate than it did before it started moving regardless of the possibility that somebody viewing it from another reference frame may, or may not, determine otherwise!

Of course, "its own reference frame" needs clarification. In SR one typically looks only at inertial reference frames (because these are the only frames where SR laws such as 'moving clocks run slow' are guaranteed to apply), and there is no inertial frame where the clock is at rest both before and after the acceleration. You can look at the frame of an inertial observer who was at rest relative to the clock before it accelerated, and in this frame the clock ticks slower after it accelerates; or, you can look at the frame of an inertial observer who is at rest relative to the clock after it accelerated, and in this frame the clock was ticking slower before the acceleration. You may choose to ignore my posts, but hopefully others can see that your statement above is unclear without clarification about which of these two frames is being called the clock's "own reference frame" (unless you are talking about a non-inertial frame, in which case as I said the standard laws about moving clocks running slow can no longer be expected to apply in this frame).

 

[Fredrik]

The clock's "own reference frame" can be interpreted in more than one way. Two options occur to me:

1. The co-moving inertial frame.

2. The local non-inertial frame constructed by taking the world line to be the time axis, with proper time as the time coordinate, and letting the spatial coordinates be defined by the "radar" definition of simultaneity in the largest neighborhood of the world line where this is possible.

The funny thing about these two options is that both of them say the same thing about the ticking rate of any clock in its own reference frame: It's always the same. No clock can have a higher ticking rate in its own frame than any other. (How would you define "ticking rate" if not as "proper time"/"coordinate time"? In both 1 and 2 above, coordinate time is proper time, so the ticking rate is always 1).

 

[Dale]

By 'understanding it' are you referring to the twin paradox per se or to Einstein's reference to clocks A and B of K and the subsequent relocation of A?

I thought I was very clear that I am referring to understanding all of SR including Einstein's reference to clocks A and B, the twin paradox, and everything else. This is what I meant by "generally applicable" above. I have repeated this statement at least 3 times already. The geometric approach applies to your question as well as to everything else in SR.

Although it may be a better conceptual framework I (as one of those annoying self-taught people including Faraday who was similarly ignorant of mathematical processes) find the Minkowski concepts far more complicated than Einstein's depiction.

On the basis that I, due to my ignorance, prefer to keep things as simple as possible - yes

That is, frankly, nothing more than an absurd excuse. Einstein's depiction in section 4 used much more math than I have here, so I don't think your excuse is even valid. I have provided you a very non-mathematical and intuitive introduction to Minkowski geometry as it applies to relativity with the analogy to drivers and odometers. This is a simple and powerful analogy that will allow you to intuitively grasp any SR scenario involving a single spatial dimension. If you cannot even make the minimal mental effort required to understand that simple analogy then you really are not that interested in learning SR.

Again, I am willing to answer questions you might have about the analogy and how to apply it. I am sure there are many open points for confusion and I am willing to work with you to clarify them, but so far it seems that you have dismissed it entirely without making any effort.

 

[paw]

As the astronaut starts to accelerate for the return trip he will see his clock continuously appearing to be ticking over at a constant rate however, due to the fact that his velocity (v) is increasing his clock will, in accordance with Einstein's chapter 4 equation (.5tv²/c²), be 'going more slowly' than it was at a lesser instantaneous velocity.

Whether you realize it or not, this statement makes it appear you believe there is some sort of absolute motion. I think that misunderstanding is at the heart of your inability to accept the many correct interpertations you have been given by JesseM, DS et al.

To demonstrate how this statement of yours is meaningless in any physical way, imagine an astronaut in a completely enclosed, inertial spaceship. The astronaut is in free fall within his ship. He cannot in any physical way determine whether he is moving or at rest. Do you agree so far?

Now suppose he ignites the engine and applies 1g accelleration for 1sec. Does he now (do you) think he is going faster than before the acceleration? How would he know?

He had no way of knowing how fast he was moving, or even IF he was moving, to begin with. He also had no way of knowing his orientation wrt any possible motion. All he knows is that he underwent an acceleration. To an outside observer he may have slowed down, speeded up or even stayed at the same speed but changed direction.

So please tell me how could he possibly say his clock 'will be going more slowly' than it was?

I hope you can see from this there is no way to make any statement about clocks ticking more slowly in anything but a relative sense. You MUST specify a reference frame to make any sense at all. And if you accept SR you must also accept that all inertial frames are equally valid so for any frame where clock A is seen to tick more slowly than clock B you can find a frame where B ticks more slowly than A.

This is what JesseM and others have been saying all along and it's why their various analyses from various different frames are valid and correct.

 

[cos]

The clock's "own reference frame" can be interpreted in more than one way. Two options occur to me:

1. The co-moving inertial frame.

This is what I mean by 'in its own reference frame'. In other words, as far as an observer accompanying that clock is concerned.

The funny thing about these two options is that both of them say the same thing about the ticking rate of any clock in its own reference frame: It's always the same. No clock can have a higher ticking rate in its own frame than any other. (How would you define "ticking rate" if not as "proper time"/"coordinate time"? In both 1 and 2 above, coordinate time is proper time, so the ticking rate is always 1).

I refer you to my analogy of an observer at the top of a mountain who moves to sea-level.

In both locations he could be of the opinion that the ticking rate of clocks at those locations is the same as his own clock at those locations thus that the sea-level clock and the mountain top clock are ticking over at the same rate as each other or he could apply his knowledge of gravitational time dilation and realize that what appears to be taking place is not reality. That the sea-level clock, as well as his own clock at that location, are physically ticking over at a slower rate than the mountain top clocks.

I know, full well, that this is applicable to general theory whereas my argument is in relation to special theory however the analogy is in respect to what the observer applying his knowledge determines is taking place.

Similarly, an observer accompanying Einstein's paragraph 1, chapter 4, clock A although of the opinion that the clock appears to be ticking over at the same rate as it was before they started moving it is physically 'going more slowly' than it was before they started moving.

In the same way, when Hafele and Keating conducted the first leg of their experiment they would have been fully justified in realizing that although their clocks appeared to be ticking over at their normal rate their clocks were, in fact, ticking over at a slower rate than they were before the flight commenced - that the eventual difference between their clocks and the laboratory clocks is due to the fact that their clocks physically ticked over at a slower rate than the laboratory clocks NOT that the laboratory clocks ticked over at a faster rate than they did prior to the commencement of the flight.

Having determined that their clocks lag behind the laboratory clocks what other conclusion could they have arrived at other than that their clocks ticked over at a slower rate during the flight than they did before the flight commenced?

Having arrived at B's location and finding that his clock lags behind clock B what conclusion can the observer accompanying clock A in Einstein's paragraph 1, chapter 4, depiction (knowing that prior to his relocation A and B were synchronized) arrive at other than that his clock must have ticked over at a slower rate than clock B?

On the assumption that he realizes that, all appearances to the contrary, his clock did tick over at a slower rate than it did before they started moving if he repeats the experiment he could, during that transit period, realize that whilst the rate of operation of his clock (A) appears to be normal it is, as indicated by his previous trip, physically ticking over at a slower rate than it was before they started moving.

 

[JesseM]

This is what I mean by 'in its own reference frame'. In other words, as far as an observer accompanying that clock is concerned.

Fredrik's option 1 was not really specific enough--it's still not clear whether you're talking about a non-inertial observer who was accompanying clock A both before and after it accelerated, or about an inertial observer who is accompanying it after the acceleration, but saw A in motion relative to himself prior to that. Again, even if you refuse to answer requests for clarification from me, perhaps someone else can press you on this question.

I refer you to my analogy of an observer at the top of a mountain who moves to sea-level.

In both locations he could be of the opinion that the ticking rate of clocks at those locations is the same as his own clock at those locations thus that the sea-level clock and the mountain top clock are ticking over at the same rate as each other or he could apply his knowledge of gravitational time dilation and realize that what appears to be taking place is not reality. That the sea-level clock, as well as his own clock at that location, are physically ticking over at a slower rate than the mountain top clocks.

I know, full well, that this is applicable to general theory whereas my argument is in relation to special theory however the analogy is in respect to what the observer applying his knowledge determines is taking place.

In the general theory there is the principle of "diffeomorphism invariance" (see this article) which allows you to construct your coordinate systems in absolutely any way you like and still find that the Einstein field equations give correct predictions. Although it probably would not be very practical, you certainly could construct a coordinate system where at some particular instant the mountain-top clock is ticking slower than the sea-level clock, although this coordinate system will agree with all other coordinate systems in predictions about local events like how fast observers positioned at each clock will see the other clock ticking (they both agree the sea-level clock appears to be ticking slower) or what the two clocks will read when brought together (if the clocks were initially synchronized at sea level before the second clock was placed on the mountain, the sea-level clock will have elapsed less time when they're brought together again). In the general theory just as in the special theory, there is no single absolute truth about which of two clocks is ticking faster at a particular instant.

Similarly, an observer accompanying Einstein's paragraph 1, chapter 4, clock A although of the opinion that the clock appears to be ticking over at the same rate as it was before they started moving it is physically 'going more slowly' than it was before they started moving.

If you're talking about an inertial observer, this is of course wrong. This observer will find that clock B was ahead of clock A before A accelerated (they were not initially synchronized in his frame), and that B was ticking slower than A after A accelerated, but because of B's "head start" it is still ahead when A catches up to it.

Having arrived at B's location and finding that his clock lags behind clock B what conclusion can the observer accompanying clock A in Einstein's paragraph 1, chapter 4, depiction (knowing that prior to his relocation A and B were synchronized) arrive at other than that his clock must have ticked over at a slower rate than clock B?

Your comment "knowing that prior to his relocation A and B were synchronized" shows that you still do not understand the relativity of simultaneity (there's a good introduction here)--assuming A and B were synchronized in B's rest frame as Einstein suggested, then in the frame of an inertial observer who sees A at rest after the acceleration, A and B were not synchronized in his own frame prior to the acceleration--from his perspective the procedure used to "synchronize" A and B was simply incorrect, because it assumed that light moves at the same speed in all directions in B's rest frame, whereas this observer assumes that light moves at the same speed in all directions in his own frame (and 'synchronization' of clocks at different locations involves sending light-signals from one to the other in SR). You can see from Einstein's own words in Chapter VIII and Chapter IX of one of his books that simultaneity is relative to one's choice of frame, there is no single absolute truth about whether two clocks are synchronized.

Since a failure to appreciate the relativity of simultaneity seems to be the key to your misunderstanding of Einstein's section 4 thought-experiment, I recommend that anyone else discussing the issue with you emphasizes this point, if you are unwilling to engage in discussion with me.

 

[cos]

I thought I was very clear that I am referring to understanding all of SR including Einstein's reference to clocks A and B, the twin paradox, and everything else. This is what I meant by "generally applicable" above. I have repeated this statement at least 3 times already. The geometric approach applies to your question as well as to everything else in SR.

I was confused by your reference in one sentence to 'generally applicable' and, in another sentence 'understanding it', excuse me for attempting to unravel my confusion.

That is, frankly, nothing more than an absurd excuse. Einstein's depiction in section 4 used much more math than I have here, so I don't think your excuse is even valid.

Einstein's depictions in paragraphs 1 through 3 of section 4 to which my postings specifically apply contain one single mathematical equation which I find to be superfluous to his comment that A lags behind B and that the same result is arrived at when A is made to move in any polygonal line and if A is made to move in a a closed curve NON of which requires any mathematics to be understood. I am of the opinion that the single mathematical equation that he provides is not in relation to whether or not clock A 'goes more slowly' than B but is in relation to how much A lags behind B!

I have provided you a very non-mathematical and intuitive introduction to Minkowski geometry as it applies to relativity with the analogy to drivers and odometers. This is a simple and powerful analogy that will allow you to intuitively grasp any SR scenario involving a single spatial dimension. If you cannot even make the minimal mental effort required to understand that simple analogy then you really are not that interested in learning SR.

I am not interested in learning SR! I believe that I have never made any comment in that respect!

I have pointed out on numerous occasions (much more than 3 times already) that my specific interest is in relation to paragraphs 1 through 3 of chapter 4!

Again, I am willing to answer questions you might have about the analogy and how to apply it. I am sure there are many open points for confusion and I am willing to work with you to clarify them, but so far it seems that you have dismissed it entirely without making any effort.

I realize that I hereby leave myself open to criticism again however to what do you refer by the word 'analogy'?

To what do you refer by the word 'it' in your assumption that I have dismissed 'it' entirely?

Contrary to your assumption I assure you that I have made every effort to understand, and respond to, messages from people of reasonable attitude.

Let's get back to basics.

In paragraph 1, chapter 4, OEMB Albert Einstein wrote:-

"If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other one..."

There is, as far as I can tell, only two explanations as to why A is found to lag behind B:

1. During that trip A ticks over at a slower rate than it did before it started moving OR -

2. The rate of operation of clock B increases whilst A is moving.

My original posting was in relation to the claim by some relativists that A (the astronaut's clock) does not tick over at a slower rate than it did before he started his return journey but that clock B (the Earth-bound twin's clock) ticks over at a faster rate than it did before the astronaut started his return trip.

My response to that claim back in the mid-90s was that in my opinion it contradicted Einstein's chapter 4 depictions and in an attempt to attain the opinion of members of an obviously more prestigious group than the one in which that claim was made I posted the concepts in this group.

I am of the (presumably correct) opinion that the Hafele-Keating experiment was based on Einstein's paragraph 3, chapter 4, depiction of a clock being made to move in a curved path around an 'at rest' identical clock. In his book 'Was Einstein Right' Clifford Will pointed out that both legs of the HKX should be looked at from the point of view of all of those clocks (the traveled clocks and the laboratory clocks) moving around a hypothetical master clock at the center of the planet.

When Hafale and Keating arrived back at the laboratory and found that their clocks lagged behind those clocks they could either have assumed that, all appearances to the contrary, their clocks are physically 'going more slowly' than they did before the flight commenced OR that the laboratory clocks as well as every other clock on the planet are physically 'going faster' than they did before the flight commenced.

I am of the opinion that Hafele and Keating (et al) would have preferred the former explanation thus had they repeated that first leg of the experiment they could have realized that although their clocks appeared to have remained unchanged they were, in reality, physically ticking over at a slower rate than they were before the flight commenced thus at a slower rate than the laboratory clocks as determined by the results of their first trip.

Please answer this question - in your opinion, would Hafele or Keating have been able to realize, during that repeat performance, that their clocks, all appearances to the contrary, were physically ticking over at a slower rate than they were before the flight commenced?

I am of the opinion that an observer accompanying clock A in Einstein's paragraph 1, chapter 4, OEMB depiction could, if he repeats that experiment, be of the opinion that although his clock appears to be ticking over at the same rate as it was before he started moving that it is, in reality, 'going more slowly' than it was before he started moving AND that an astronaut returning to Earth could also arrive at that conclusion.

 

[cos]

Whether you realize it or not, this statement makes it appear you believe there is some sort of absolute motion. I think that misunderstanding is at the heart of your inability to accept the many correct interpertations you have been given by JesseM, DS et al.

Please explain the difference between my statement that the astronaut starts to accelerate (i.e. he starts to move) and Einstein's paragraph 1, chapter 4 comment that clock A is made to move?

Admittedly Einstein's depiction does not take into account that clock A accelerates however other than a mathematical proposition which Einstein described as not referring to reality the idea that clock A instantaneously attains a velocity without undergoing acceleration is not, as far as I am concerned, the physical realities to which the subject of physics should be applied.

To demonstrate how this statement of yours is meaningless in any physical way, imagine an astronaut in a completely enclosed, inertial spaceship. The astronaut is in free fall within his ship. He cannot in any physical way determine whether he is moving or at rest. Do you agree so far?

According to the principle of relativity - yes.

Now suppose he ignites the engine and applies 1g accelleration for 1sec. Does he now (do you) think he is going faster than before the acceleration? How would he know?

During that period of acceleration he knows that he is moving relatively to his situation before he started accelerating due to the fact that he feels a force of acceleration and he has put his foot on the gas pedal however on the basis that he was not moving before he accelerated he is not 'going faster than before the acceleration' (on the bais that he was not 'going fast' before he accelerated) but knows that he is moving relatively to his previous 'at rest' situation.

Having taken his foot off the gas pedal he knows that he is still moving away from his original location and because he knows that he accelerated at 1g for 1 second he knows his rate of travel away from that original location.

He had no way of knowing how fast he was moving, or even IF he was moving, to begin with. He also had no way of knowing his orientation wrt any possible motion. All he knows is that he underwent an acceleration. To an outside observer he may have slowed down, speeded up or even stayed at the same speed but changed direction.

The point of view of an outside observer has absolutely nothing whatsoever to do with what the person you described determines!

So please tell me how could he possibly say his clock 'will be going more slowly' than it was?

'He', in my depiction, is an observer accompanying Einstein's paragraph 1, chapter 4, OEMB presentation of clocks A and B or an astronaut who has returned to the planet following an out-and-return voyage both of whom, unlike your astronaut, are able to compare the time on their clock with those of Einsteins clock B or the astronaut's stay-at-home twin's clock.

Your depiction makes no allowance whatsoever for him to compare his clock with Einsteins clock B or the Earth-bound twin's clock but let's assume that, having previously looked out a window, he determines that he is analogous to Einstein's chapter 4 clock A located at one of the points A and B of K with his twin located in an identical ship at point B and his clock is, as Einstein depicted therein, synchronous with clock B's clock.

He moves away from the window and carries out several experiments in order to determine if he (and his twin) are stationary or are moving at the same uniform velocity with no success.

He starts moving toward his twin and, in accordance with Einstein's paragraph 1, chapter 4, depiction he will find, upon his arrival at B's location, that his clock lags behind his twin's clock.

He can either assume that the lag was caused by his clock ticking over at a slower rate than it was before he started moving or that his twin's clock was ticking over at a faster rate than it was before he (A) started moving.

I hope you can see from this there is no way to make any statement about clocks ticking more slowly in anything but a relative sense. You MUST specify a reference frame to make any sense at all. And if you accept SR you must also accept that all inertial frames are equally valid so for any frame where clock A is seen to tick more slowly than clock B you can find a frame where B ticks more slowly than A.

So on the basis that we can find a frame where B ticks more slowly than A where, upon those clock moving together, it will be found that B lags behind A the fact that this contradicts Einstein's suggestion that the complete opposite takes place - that A lags behind B - is, apparently, of no relevance.

This is what JesseM and others have been saying all along and it's why their various analyses from various different frames are valid and correct.

I repeat my comment above - "The point of view of an outside observer has absolutely nothing whatsoever to do with what the person you described determines!

Although 'various analyses from various different frames' may very well be 'valid and correct' they are only 'valid and correct' from their point of view and have absolutely nothing whatsoever to do with what an observer accompanying Einstein's clock A or an astronaut making an out-and-return journey determine!

The fact that an observer in a different frame may determine that Einstein's chapter 4 clock B ticks more slowly than A this has no affect whatsoever on what takes place in Einstein's depiction or during the astronaut's trip!

 

[matheinste]

Hello cos.

Quote:-

----There is, as far as I can tell, only two explanations as to why A is found to lag behind B:

1. During that trip A ticks over at a slower rate than it did before it started moving OR -

2. The rate of operation of clock B increases whilst A is moving.
---------------------------------------------------------------------------

Quote:-

I am not interested in learning SR! I believe that I have never made any comment in that respect!
------------------------------------------------------------

There is another interpretation open to an observer colocated with clock A. This would be:- I, unlike some people, who by their own admission have no wish to study SR, have done so and conclude that because I have traveled with clock A along a non inertial spacetime path, whereas the other clock, B, has remained on an inertial path, I will have have accumulated less proper time than clock B, as predicted by SR, and so the time on the clock colocated with me will show an earlier time than that on clock B. I.e. it will lag clock B.

Matheinste.

 

[Dale]

I am not interested in learning SR!

If that were true then you would not be reading Einstein's writings on the subject nor asking questions about it.

There is, as far as I can tell, only two explanations as to why A is found to lag behind B:

1. During that trip A ticks over at a slower rate than it did before it started moving OR -

2. The rate of operation of clock B increases whilst A is moving.

The whole reason I am bringing up the geometric approach is because these are not the only two explanations. Here is a driving analogy. Two drivers reset their odometers, they each drive due north on parallel roads and note that their odometers read the same. Then, driver A's road bends towards driver B's road. When they meet they notice that driver A's odometer now reads more than driver B's.

Attached is a geometric figure showing the situation. You can print it out and perform the following little experiment. Take two identical rulers and with ruler A measure the length of path A between a0 and a1, with ruler B measure the length of path B betwen b0 and b1. Note that they are the same. Now, with ruler A measure the length of path A between a1 and a2, with ruler B measure the length of path B betwen b1 and b2. Note that this time they are not the same, the measurement for A is longer.

You could certainly explain this by either of the following two explanations:
1) ruler (or odometer) A shrinks during the second measurement
2) ruler (or odometer) B expands during the second measurement

But do you really think those are the only two explanations possible? Do you not think that a third explanation is at least concievable?
3) the distance between a1 and a2 is greater than the distance between b1 and b2

Now, finally, if we replace the y-axis with t (in units where c=1) and if we replace the distance measured with rulers or odometers (Euclidean metric) with the interval measured with clocks (Minkowski metric) then we have a geometric spacetime diagram of exactly the situation described in section 4. Can you not see the geometric third explanation?
3) the interval between a1 and a2 is smaller than the interval between b1 and b2

I hope this more detailed explanation clarifies the "odometer analogy". I hope it also clarifies the fact that this geometric approach is applicable to the section 4 scenario. It is very easy to derive the equation Einstein presents in section 4 from the Minkowski metric, but I will spare you the math.

I encourage you to ask further questions about Minkowski spacetime.

 

[RandallB]

There is, as far as I can tell, only two explanations as to why A is found to lag behind B:

1. During that trip A ticks over at a slower rate than it did before it started moving OR -

2. The rate of operation of clock B increases whilst A is moving.

You potential explanations here are incomplete and IMO get at the root of your confusion.
First option 2 should be rejected as unrealistic as no change can occur to B since it remains stationary in a single frame. SR certainly will not support the #2 option.

But #1 is much to incomplete an explanation;
it has three different possibilities you have not detailed (or considered) since in the travel of A it must use two different Frames; one outbound and one inbound;
this gives four possibilities for the rate of clock A wrt B:

a) both A outbound and A inbound run FAST wrt B
b) both A outbound and A inbound run SLOW wrt B
c) A outbound runs SLOW and A inbound run FAST wrt B
d) A outbound runs FAST and A inbound run SLOW wrt B

SR only rejects option “a”
but based on the given information of the problem “b” “c” or “d” could be true.
Options “c” & “d” work as long as the amount of time A spends at the slow rate is long enough when summed with the time built up by A at the fast rate nets to a total time less than experienced by B for the duration of the round trip.

One of the three conditions will be observed by any random observer C moving at any fixed speed wrt B and will always give the same net change from start to finish for both A & B (B always less than A by the same amount) no matter what speed you use for observer C.

 

[paw]

According to the principle of relativity - yes.

Good. We're making some progress. You agree the astronaut in a closed ship cannot know anything about his state of motion before the acceleration. Not his speed nor his direction. Keep this in mind.

Having taken his foot off the gas pedal he knows that he is still moving away from his original location and because he knows that he accelerated at 1g for 1 second he knows his rate of travel away from that original location.

He does not! He only knows his state of motion is different than it was before. It could be faster, slower or even the same depending on whether the acceleration was in the direction of his (unknown) motion, against it or at an angle. Only an outside observer could make the distinction.

The point of view of an outside observer has absolutely nothing whatsoever to do with what the person you described determines!

It most certainly does. Only an outside observer can make a meaningful statement about the astronauts state of motion before or after the acceleration.

'He', in my depiction, is an observer accompanying Einstein's paragraph 1, chapter 4, OEMB presentation of clocks A and B or an astronaut who has returned to the planet following an out-and-return voyage both of whom, unlike your astronaut, are able to compare the time on their clock with those of Einsteins clock B or the astronaut's stay-at-home twin's clock.

Your 'observer accompanying Einstein's paragraph 1, chapter 4, OEMB presentation' IS an outside observer. It is this outside observer, who you won't allow me or anyone else to reference, that is fooling you into believing the astronaut can make a meaningful statement of his inertial motion. If you remove this outside observe, as I have done above, you'll realize that the astronaut cannot make a prediction about how his clock is ticking in any absolute sense.

 

[phyti]

for those who inquired;

The calibration curve is referred to in the book by Max Born, and I've seen it mentioned on other forums, but don't know if it's a popular term. In essence it is the hyperbola t = sqrt(x^2+1), shown in the drawing. The point where the observers (green) path intersects it, represents his unit of time on the vertical time scale. His path angle is determined by his speed, thus the relation between clock rate and speed is still there, i.e., incorporated into the geometry of the space-time diagram. The 45° (red) line is the asymptote to the hyperbola, and never meets it. It is also the light path, so even if a clock could move at c, it would never tick.

 

[phyti]

from JesseM post 31;

So, if A suddenly decelerates and comes to rest in this frame when it reads 0 seconds, B will already read 16 seconds at the "same moment" in this frame. From then on B will be moving towards A at 0.8c, and hence slowed down by a factor of 0.6 in this frame while A now ticks at the normal rate in this frame since it's at rest.

You state A is at rest, therefore he measures 20 ls, only B who still moves at .8c would measure 12 ls.

 

[matheinste]

Hello phyti.

The clibration curve, hyperbolae, invariant hyperbolae, appears in many books on relativity. The one i have open at the moment is Schutz - a First Course in General Relativity, page 17. It is as you describe it. Rindler, Essential Relativity, page 39 has a more complicated drawing than that of Schutz. I have seen it in other books not at present to hand. These drawings make it clear that the scale of the original and the trnsformed axes is not the same.

Matheinste

 

[JesseM]

You state A is at rest, therefore he measures 20 ls, only B who still moves at .8c would measure 12 ls.

This argument suggests confusion about length contraction--the observer "at rest" does not always measure the longest length (of course, there is no absolute truth about who's at rest in relativity anyway, it's all relative to your choice of frame), rather the observer at rest relative to the thing being measured measures the longest length, if the thing being measured is moving in A's frame, then A will measure a shorter length. I stated that the initial distance between A and B was 20 ls in B's rest frame, meaning that if you had a rod 20 ls long in this frame with B attached to one end, then A would originally be positioned next to the other end. So naturally in the frame where B is moving at 0.8c--the frame where A is "at rest" after A accelerates (though in this frame A was also moving at 0.8c before it accelerated)--the rod is moving at 0.8c too, so the rod must be 12 light-seconds long in this frame. So naturally if B was at one end and A was at the other end before A accelerated, that means A and B were 12 light-seconds apart in this frame before A accelerated and came to rest.

 

[cos]

Hello cos.

There is another interpretation open to an observer colocated with clock A. This would be:- I, unlike some people, who by their own admission have no wish to study SR, have done so and conclude that because I have traveled with clock A along a non inertial spacetime path, whereas the other clock, B, has remained on an inertial path, I will have have accumulated less proper time than clock B, as predicted by SR, and so the time on the clock colocated with me will show an earlier time than that on clock B. I.e. it will lag clock B.

I have, over the past 25 years, studied SR however I make no claim that I have learned SR.

Your false accusation that I have no wish to study SR is totally unwarranted but like most critics honesty seems to be something that you conveniently ignore.

Would you please explain the difference between your determining that your clock has 'accumulated less proper time than clock B' and determining that your clock has ticked over at a slower rate than clock B?

Wouldn't your clock accumulate less proper time because it has ticked over at a slower rate than clock B?

Won't your clock 'lag behind clock B' due to the fact that whilst it was moving it was ticking over at a slower rate than it was before it started moving?

Other than having ticked over at a slower rate than it was before it started moving what other explanation is there for the fact that A lags behind B - that A accumulates less proper time than B - other than that, during that trip, A ticked over at a slower rate than B and at a slower rate than it was before it started moving?

 

[phyti]

If A perceives the separation of himself and B as 12 ls while moving at .8c, he won't perceive the separation the same when he decelerates to 0. He would be in the frame prior to their mutual acceleration. Do you understand your own posts, or did you post in haste without proof reading?

 

[JesseM]

If A perceives the separation of himself and B as 12 ls while moving at .8c, he won't perceive the separation the same when he decelerates to 0. He would be in the frame prior to their mutual acceleration. Do you understand your own posts, or did you post in haste without proof reading?

You are apparently jumping to incorrect conclusions about my posts, try asking for clarification before assuming I'm the one making a mistake. I'm just talking about what is happening in a particular inertial frame, not what the non-inertial clock A "perceives". In this inertial reference frame, both A and B are initially moving at 0.8c with a separation of 12 light-seconds, then A decelerates and comes to rest in this frame while B continues to move towards A at 0.8c. If you like, you can think of this frame as representing the viewpoint of a third object C that always moves inertially, with A coming to rest relative to C after (but not before) A accelerates.

What do you mean by the word "perceives", anyway? Do you understand that even for an inertial observer, what they see visually is differently from what is true in their own inertial rest frame? For example, if a clock is approaching an inertial observer at 0.6c, then in his own rest frame the clock will be slowed down by a factor of sqrt(1 - 0.6^2) = 0.8, but what he will see visually is that it appears to be sped up by a factor of 2 due to the relativistic Doppler effect. Likewise, although an object moving at 0.6c in his frame will be shrunk in length along the axis of motion by a factor of 0.8 due to Lorentz contraction, visually he will see the length as unchanged due to Penrose-Terrell effect. The length and time between ticks in his frame must be based on correcting for the fact that the light from successive ticks (or the light from different parts of the object) had different distances to travel to reach his eyes. For example, if in 2036 I see the image of a ship 26 light years away according to my ruler with its clock reading 50 years, and then 4 years later in 2040 I see an image of the same ship 20 light years away according to my ruler with its clock reading 58 years (meaning I visually saw the clock advance 8 years in 4 years of my time, so it looks like it's sped up by a factor of 2), then I can correct for the time the light from each event took to reach me and conclude that the first event happened at a time-coordinate of 2036 - 26 = 2010 in my frame, and the second happened at a time-coordinate of 2040 - 20 = 2020 in my frame, meaning that in my frame the clock actually took 10 years to advance 8 years from 50 to 58, so it was slowed down by a factor of 0.8 in my frame.

At first I thought maybe you understood this, and when you were talking about what A "perceives", you might have meant what would be true in a non-inertial reference frame where A is at rest both before and after the acceleration. But you said A was initially "moving at .8c" so this wouldn't make sense--obviously in a coordinate system where A is at rest, A's coordinate velocity is always zero. In any case, the problem with non-inertial coordinate systems is that unlike with inertial frames there's no single "correct" way to define the coordinate system of a non-inertial observer, you can design a coordinate system in which that observer is at rest in any of an infinite number of different ways, which will give different answers about things like the distance between objects and the rates clocks are ticking and which events are simultaneous.

 

[cos]

I am not interested in learning SR!

If that were true then you would not be reading Einstein's writings on the subject nor asking questions about it.

Studying SR is a far cry from learning it!

There is, as far as I can tell, only two explanations as to why A is found to lag behind B:

1. During that trip A ticks over at a slower rate than it did before it started moving OR -

2. The rate of operation of clock B increases whilst A is moving.

The whole reason I am bringing up the geometric approach is because these are not the only two explanations. Here is a driving analogy. Two drivers reset their odometers, they each drive due north on parallel roads and note that their odometers read the same. Then, driver A's road bends towards driver B's road. When they meet they notice that driver A's odometer now reads more than driver B's.

Attached is a geometric figure showing the situation. You can print it out and perform the following little experiment. Take two identical rulers and with ruler A measure the length of path A between a0 and a1, with ruler B measure the length of path B betwen b0 and b1. Note that they are the same. Now, with ruler A measure the length of path A between a1 and a2, with ruler B measure the length of path B betwen b1 and b2. Note that this time they are not the same, the measurement for A is longer.

You could certainly explain this by either of the following two explanations:
1) ruler (or odometer) A shrinks during the second measurement
2) ruler (or odometer) B expands during the second measurement

But do you really think those are the only two explanations possible? Do you not think that a third explanation is at least concievable?
3) the distance between a1 and a2 is greater than the distance between b1 and b2

Now, finally, if we replace the y-axis with t (in units where c=1) and if we replace the distance measured with rulers or odometers (Euclidean metric) with the interval measured with clocks (Minkowski metric) then we have a geometric spacetime diagram of exactly the situation described in section 4. Can you not see the geometric third explanation?
3) the interval between a1 and a2 is smaller than the interval between b1 and b2

I hope this more detailed explanation clarifies the "odometer analogy". I hope it also clarifies the fact that this geometric approach is applicable to the section 4 scenario. It is very easy to derive the equation Einstein presents in section 4 from the Minkowski metric, but I will spare you the math.

I encourage you to ask further questions about Minkowski spacetime.

I find your reference to lengths to be confusing. Are you suggesting that in Einstein's chapter 4 depiction an observer alongside clock B determines that the distance traveled by A is less than the distance determined by A on the basis that the distance A to B contracts from A's point of view?

On the basis of the fully reciprocal nature of SR - as far as B is concerned the distance A to B also contracts due to the fact that A is moving however I fail to see that your analogy has anything to do with Einstein's suggestion that the moving clock 'goes more slowly' than the stationary clock.

My extremely limited understanding of Minkowski spacetime is that it is based on mathematical propositions which, according to Einstein, do not refer to reality.

If a concept can only be explained by the application of mathematics and cannot be presented in simple everyday language it is not, in my opinion, reality.

I am not suggesting that mathematics is not of extreme importance but that it does not, by itself, provide proof of a concept or of a theory as some people insist.