Greater than the “real time” between ticks of A’s clock? Yes.
Before I go into details I again point out that my presentation is
specifically in relation to Albert Einstein’s chapter 4 of his 1905 article ‘On the Electrodynamics of Moving Bodies’ depiction.
I have been accused of arguing the validity of the mainstream understanding of relativity however I fail to see why my referring to chapter 4 of relativity could be arguing that mainstream understanding of relativity.
It seems to me that some people are attempting to pretend that chapter 4 of special theory does not exist. If chapter 4 of relativity argues the validity of the mainstream understanding of relativity then I suggest that people should look at chapter 4. It is part and parcel of relativity thus should not be ignored even it
is uncomfortable or inconvenient.
In that chapter (paragraph 1) 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 which has remained at B by .5
tv2/c2 ... t being the time occupied in the journey from A to B.”
In paragraph 3 Einstein refers to a clock that has remained at rest compared with an identical clock that has moved in a closed curve around that clock. I am of the opinion that he implied that clock B in paragraph 1 remains at rest ergo, in that chapter, Einstein does not allow that clock B “moves through spacetime”.
If someone were to draw a diagram of clocks A and B moving through spacetime they would be presenting that phenomenon from the point of view of another reference frame however Einstein specifically pointed out that the event is “viewed in the stationary system” (i.e. clock B’s reference frame).
Although Einstein made no mention of the fact - it is obvious that clock A in paragraph 1 must accelerate in order to move to B’s location. On the basis that the attainment of an instantaneous velocity can be mathematically ‘ratified’ I can only repeat Einstein’s comment that as far as the propositions of mathematics are certain, they do not refer to reality and as far as I am concerned physics should be a study of physical reality thus Einstein’s paragraph 1, chapter 4, clock A must accelerate and as Einstein showed in his 1918 Naturwissenschaften article, it is only the clock that experiences forces of acceleration (clock A) that incurs variations in it’s rates of operation not the clock referred to in his 1918 article which continues to move with uniform velocity (clock B).
In paragraph 3, chapter 4, Einstein wrote:- “Thence we conclude that a balance clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles...”
My interpretation of the phase ‘must go more slowly’ is that the equatorial clock ticks over at a slower rate than (i.e. that it incurs time dilation relatively to) the polar clock.
On the basis that the equatorial clock ‘must go more slowly’ than the polar clock then it must progressively lag further and further behind the polar clock.
I interpret Einstein’s paragraph 1 conclusion that A will lag behind B as being in accordance with his later comment that A (his paragraph 3 equatorial clock) ‘goes more slowly’ (i.e. ticks over at a slower rate) than B (his paragraph 3 polar clock).
I am of the opinion that if Einstein’s paragraph 1 clock A was not located some distance away from B but was initially stationary alongside and synchronous with B then traveled away from B at the same velocity (v) and for the same length of time (t) as does Einstein’s clock A then comes to a stop it will then lag behind B by the same amount as Einstein’s clock A lagged behind B in accordance with his equation .5tv2/c2.
This is analogous to an astronaut’s outward bound trip however, having come to a stop and adjusted his clock so that it then indicates the same time as the Earth clock (and, allowing for the fact that the Earth clock’s rate of operation is affected by it’s location in a gravitational tidal area is ticking over at the same rate as the Earth clock), the situation is precisely analogous to Einstein’s paragraph 1, chapter 4 depiction of synchronous clocks located at points A and B of K.
Having synchronized his clock with the Earth clock (having allowed for the time that it takes for light to traverse the intervening distance) the astronaut immediately accelerates and soon attains an instantaneous velocity whereby his clock is ticking over at the same rate as the (gravitationally affected) Earth clock after which his clock starts to progressively ‘go more slowly’ than the Earth clock.
Upon his arrival back at the planet he finds that his clock lags behind the Earth clock whereupon he can either conclude that his clock ‘went more slowly’ than the Earth clock during that trip or that the Earth clock ticked over at a faster rate than his own clock (i.e. the Earth clock incurred time ‘contraction’) yet, in accordance with Einstein’s 1918 article, he knows that his was the reference frame that accelerated.
As as I detailed in another posting - when Galileo wrote his book ‘Two New Sciences’ he was already in trouble with authorities so he presented it as a purely hypothetical discussion between a teacher and two of his students. Einstein also presented his 1918 article as a purely hypothetical discussion - this time between a relativist and a critic.
Having already been criticized for having stated, in his 1916 general theory, that the special theory ‘law’ of the constancy of the velocity of light required modification and, in that same year in his book ‘Relativity’ that this law was not fully valid, it was, perhaps, an attempt on his behalf to avoid further criticisms that he wrote his 1918 article in that format.
In his self-published book ‘Fiction Stranger Than Truth’ Nikolai Rudakov wrote :-
“Very few relativists have actually adopted Einstein’s explanation [of the twin paradox]. Not many authors mention the 1918 dialogue, and some who do imply that Einstein was wrong.”
The simple fact is that that when we are discussing Einstein’s concepts regarding relativity per se (not just his 1905 article) we should, I believe, refer to all of his relevant materiel including his 1918 article as well as his chapter 4 depictions and if any of it ‘argues against the validity of the mainstream understanding of relativity’ then it is Einstein who should be criticized not me! I’m just the messenger.
In my opinion Einstein’s chapter 4 can be shown to describe an out-and-return trip by an astronaut and that the ludicrous claim that the astronaut does not accept that his clock incurs time dilation but insists that the Earth clock incurs time contraction and only during his period of acceleration following turn around does not comply with (argues against) Einstein’s ‘understanding’ of relativity as presented in that chapter.
In the normal space analogy, Y and Z start at the same point in space, then
both of them move through space in different paths, eventually meeting at another point in space. At that point, they find that they have accumulated different amounts of "real distance".
Does this mean that the "real distance" between ticks of Y's rulers were greater?
On the assumption that Z is the astronaut - yes.
(In the first scenario you depicted A and B with B being the astronaut so I assume that when you refer to Y and Z above you are classifying Z as the astronaut.)
On the basis that the astronaut’s clock incurs time dilation (i.e. the “real time” between the ticks of Z’s clock are greater) his rule ‘must’ accordingly incur length contraction (i.e. the “real distance” between the ticks on his rule will be shorter).
Given the analogy, I suggest that if the answer is "no" for the second scenario, it must also be "no" for the first scenario. If so, then we can ask if it makes any sense to say that "time" goes more slowly for B. If it is to make sense, then "time" in that statement cannot be "real time".
My answer to both scenarios is yes.
I confused myself totally too.