Understanding Time Dilation in Einstein's Special Theory of Relativity

[cos]

I have come to this thread long after it began and have not gone through every posting from the beginning to the end, so some one may have mentioned this already, if so I apologize for the duplication. Special relativistic effects on time measurements and general relativistic effects on time measurements are reaffirmed continuously everyday in the Global Positioning Satellite system. If these effects were not taken into account, then the GPS system would fail within about 30 minutes. Those who were discussing math vs reality might want to read this paper: http://relativity.livingreviews.org/Articles/lrr-2003-1/

I suspect that a great deal of discussion could be dispensed with by examining the details discussed in the above paper and realizing that, yes, relativity works.

Your contribution to the discussion may have been unneccesary if you had gone through previous postings wherein I made no suggestion that relativity does not work.

 

[phyti]

cos post;

In my previous thread ‘Time dilation’ dated Mar22-09 I wrote -
In section 4 STR Einstein wrote -

"If one of two synchronous clocks at A is moved in a closed curve
with constant velocity until it returns to A, the journey lasting
t seconds, then by the clock which has remained at rest the
travelled clock on its arrival at A will be a .5tv^2/c^2 second
slow. 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 under otherwise identical
conditions."

What do people think he meant by the phrase "...must go more
slowly..."?
Does anyone agree that he meant that the moving clock will tick
over at a slower rate than (i.e. incur time dilation relatively
to) the other clock?

********************

On the (probably erroneous) basis that some people may agree that
he did I follow that up with the question - On the basis of his
depiction of a clock that is made to move in a closed curve around
another clock is it correct for me to assume that Einstein meant
that the clock that is moving in a closed curve will “go more
lowly”(i.e. tick over at a slower rate) than the clock “which has
remained at rest.”?

The A clock must accelerate to leave the B clock, move at a
constant speed for most of the path, then decelerate to reunite
with the B clock. Since A initiated the trip, and traveled a
greater distance than B, in the same amount of elapsed B time, A
would have an average speed greater than B. Time dilation is a
functon of speed, therefore the A clock experiences more time
dilation than B. When reunited, A clock is lagging behind B clock.
This is not about perception of clocks, but the physics of clock
function according to light propagation. Each will
observe/perceive the others clock to be running faster or slower,
depending on direction of motion.

This is a repeat of my reply for the your first thread.

Here is a quote from the Max Born book, page 257, which you have (A and B swapped).

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

Because A & B are synchronized initially, the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

I can only add, time dilation is a real factor affecting particle accelerators and gps systems.

 

[cos]

The whole point is that in relativity there is no "physical" truth about the rate a clock is ticking...

So when Einstein wrote that clock A 'must go more slowly' than clock B he was not describing a 'physical' fact (or truth)?

But if you're defining "physical truth" in this way, then you'd have to agree that in the frame where A and B were initially in motion and then A came to rest after accelerating, it's a physical truth that A ticked more rapidly after accelerating.

By the comment "... A ticked more rapidly after accelerating." I take it that you mean that A ticked more rapidly whilst it was moving with uniform velocity (i.e. after accelerating)?

By "ticked more rapidly" did you mean that A ticks over at a faster rate than it did before it accelerated (or whilst it is accelerating) or that because A, applying the Lorentz transformations, 'determines' that B is ticking over at a slower rate than itself?

The idea of time contraction was, I believe, an unacceptable concept as far as Einstein was concerned.

You wrote, below, "...I agree that in this frame A was ticking more slowly after accelerating." It was 'this frame' to which I have consistently been referring.

Of course, an "observer" is just a shorthand for talking about a certain coordinate system; in reality, an intelligent observer is perfectly capable of determining the coordinates of events in a system other than his own rest frame. However, if you are only talking about what's true in the frame where A and B were initially at rest, I agree that in this frame A was ticking more slowly after accelerating.

"After accelerating"? I am of the opinion that the v in Einstein's equation .5tv^2/c^2 can be his instantaneous velocity whilst accelerating.

The slower rate of operation of clock A is not affected by the fact that he takes his foot off the gas pedal at any given instant.

But yes, relative to a particular choice of coordinate system there can be definite truths about which clock was ticking slower...

So A, having arrived at B's location, is, apparently, entitled to be of the opinion that his clock lags behind B due to the fact that whilst he was moving his clock was 'going more slowly' (i.e. ticking over at a slower rate) than B?

 

[cos]

In the 26 years that I have been researching Einstein's special theory I have read at least 100 popularization books and possibly thousands of articles on the subject however none of those authors have referred to a 'fictitious force'.

That is probably because all of those authors assumed that you understood Newtonian mechanics. Fictitious forces are a product of Newton, not Einstein.

They apparently not only assumed that I understood Newtonian mechanics but that the potentially millions of other people who were to read those books and articles also understood Newtonian mechanics!

Somewhat presumptuous of them wouldn't you agree?

You state that fictitious forces are not 'Einstein'. Do you mean that they are not special theory?

If they are incorporated in STR perhaps you would be so kind as to point out where in STR Einstein refers, or alludes to, fictitious forces?

That's my point, the scenario to which I was referring was not an out and return trip but was Einstein's initial (section 4) depiction of one clock that is made to travel to another clock's location.

In an out-and-return trip Einstein's depiction could be applied to a twin's return journey whereupon, according to Einstein, his clock will 'go more slowly' than it did before he started moving.

Ah, ok. So A and B are initially synchronized in B's rest frame and at different locations. Then A is moved with a velocity v (in B's frame) to B and is found to lag B. The calculation in B's frame shows that A and B started synchronized, A "went more slowly", and thus A was found to lag. C is an inertial observer in a frame where A is at rest after beginning to move. The calculation in C's frame shows that B started out ahead (relativity of simultaneity, see section 2), B "went more slowly", but A didn't catch up, and thus A was found to lag. In both cases the calculations show that A lags B by the same amount so there is no conflict between either calculation or the measured outcome.

There is NO observer C in Einstein's section 4 STR depiction and the introduction of same is nothing more than a deliberate obfuscation!

Determinations made by a hypothetical observer C can have no effect whatsoever on what is taking place as far as A or B are concerned nor on the rates of operation of their clocks!

 

[cos]

In the 26 years that I have been researching Einstein's special theory I have read at least 100 popularization books and possibly thousands of articles on the subject however none of those authors have referred to a 'fictitious force'.

That is probably because all of those authors assumed that you understood Newtonian mechanics. Fictitious forces are a product of Newton, not Einstein.

They apparently not only assumed that I understood Newtonian mechanics but that the potentially millions of other people who were to read those books and articles also understood Newtonian mechanics!

On the basis that a majority of those items were popularization works this attitude would be somewhat presumptuous of those authors wouldn't you agree?

You state that fictitious forces are not 'Einstein'. Do you mean that they are not special theory?

If they are incorporated in STR perhaps you would be so kind as to point out where in STR Einstein refers, or alludes to, fictitious forces?

That's my point, the scenario to which I was referring was not an out and return trip but was Einstein's initial (section 4) depiction of one clock that is made to travel to another clock's location.

In an out-and-return trip Einstein's depiction could be applied to a twin's return journey whereupon, according to Einstein, his clock will 'go more slowly' than it did before he started moving.

Ah, ok. So A and B are initially synchronized in B's rest frame and at different locations. Then A is moved with a velocity v (in B's frame) to B and is found to lag B. The calculation in B's frame shows that A and B started synchronized, A "went more slowly", and thus A was found to lag. C is an inertial observer in a frame where A is at rest after beginning to move. The calculation in C's frame shows that B started out ahead (relativity of simultaneity, see section 2), B "went more slowly", but A didn't catch up, and thus A was found to lag. In both cases the calculations show that A lags B by the same amount so there is no conflict between either calculation or the measured outcome.

There is NO observer C in Einstein's section 4 STR depiction and the introduction of same is nothing more than a deliberate obfuscation!

Determinations made by a hypothetical observer C can have no effect whatsoever on what is taking place as far as A or B are concerned nor on the rates of operation of their clocks!

 

[cos]

No one has suggested that that the unaccelerated "inertial reference frame clock" experiences a variation in its rate of ticking, at least not in any inertial reference frame (if we consider non-inertial coordinate systems, virtually anything can be true about the rate of ticking of any clock). The point is just that although the accelerated clock does change its rate of ticking in almost every inertial frame (except the frame where its direction changes but its speed stays the same), there are some frames which say it ticks slower after the acceleration than it was ticking before the acceleration, and other frames which say it ticked slower before the acceleration than it did after. Do you disagree with this?

Some people insist that after A accelerates (and is then moving with uniform velocity toward B) observer A 'determines', in accordance with the Lorentz transformations, that B is ticking over at a slower rate than it was before A started moving.

I wholeheartedly agree that "...there are some frames which say it ticks slower after the acceleration than it was ticking before the acceleration, and other frames which say it ticked slower before the acceleration than it did after." The point that I'm trying to make is that irrespective of what those frames 'say' or 'determine' their observations have absolutely no affect whatsoever on clocks A or B!

As I have also pointed out, the systems that Einstein depicted could be contained in an otherwise empty universe in which there are obviously no other frames!

 

[cos]

cos, Einstein never used the word "physical" to refer to time dilation and he never used the word "real" or "illusion" at all. So the question remains, what do you mean by those words? I have suggested what I think you mean (although you are not consistent in your usage), but you haven't even had the courtesy to say yes or no to it. I don't care how you use those words, but just define them and use them consistently so that we can communicate.

It is not important what I think about the words 'physical' or 'real' but what Einstein meant by the words 'must go more slowly'!

I am of the opinion that he meant that clock A 'physically' or 'really' or 'actually' goes more slowly (i.e. ticks over at a slower rate) than a clock at one of the poles or, analogously, than a clock around which it has moved in a closed curve or, analogously, relative to which it has moved in any polygonal line including it's original trip to B's location.

What do you believe he meant by the words 'must go more slowly'?

 

[JesseM]

So when Einstein wrote that clock A 'must go more slowly' than clock B he was not describing a 'physical' fact (or truth)?

Not if "physical truth" is defined to mean something that is true regardless of your choice of conventions about coordinate systems, and as I said before, this is how most physicists nowadays use the word "physical". If you want to use a different definition that's fine, it's just semantics, but I would ask that you spell out what you mean by physical. That's what DaleSpam asked you in post #90 above, and I also asked you about this in post #85 which you were responding to here:

The whole point is that in relativity there is no "physical" truth about the rate a clock is ticking, if "physical" is taken to mean something objective that doesn't depend on an arbitrary choice of coordinate system (which is how physicists usually use the word 'physical'). Similarly, there is no "physical" truth about which of two objects has a greater x-coordinate; it depends on what coordinate system you use, where you place the origin and how you orient the x-axis of that system. Perhaps you are just using a different definition of "physical"? Would you say that a "physical" truth need not be frame-invariant, but can be relative to one's choice of coordinate system? If so, I would certainly agree that in the frame where A and B were initially at rest, it is a physical truth that A ticked more slowly after accelerating. But if you're defining "physical truth" in this way, then you'd have to agree that in the frame where A and B were initially in motion and then A came to rest after accelerating, it's a physical truth that A ticked more rapidly after accelerating.

Could you please answer these questions? Specifically, do you want to define "physical" differently from how most physicists define it, so that it no longer implies coordinate-invariance? If so, do you acknowledge that under such a nonstandard definition, there could also be a "physical truth" about which of two objects has a greater x-coordinate, even though this truth can obviously only be decided relative to a particular (arbitrary) choice of how to orient our coordinate axes?

By the comment "... A ticked more rapidly after accelerating." I take it that you mean that A ticked more rapidly whilst it was moving with uniform velocity (i.e. after accelerating)?

Yes, in the frame where it came to rest after accelerating.

By "ticked more rapidly" did you mean that A ticks over at a faster rate than it did before it accelerated (or whilst it is accelerating) or that because A, applying the Lorentz transformations, 'determines' that B is ticking over at a slower rate than itself?

Since A accelerated, A does not have a single rest frame, so the meaning of "applying the Lorentz transformations" is ambiguous--what frame would you have A use, the frame where A was at rest before the acceleration, or the frame where A was at rest after the acceleration? I was thinking of the inertial frame where A was at rest after acceleration, and in this frame both of your above statements are true; in this frame A ticks faster after acceleration than before acceleration, and in this frame B ticks more slowly than A after A has accelerated.

The idea of time contraction was, I believe, an unacceptable concept as far as Einstein was concerned.

I don't understand what you mean by "time contraction", can you explain this? No clocks rate of ticking is faster than the rate coordinate time is passing in any inertial frame, if that's what you mean; it can only be ticking at the same rate as coordinate time (if it's at rest in the chosen frame), or slower than coordinate time (if it's moving in this frame).

"After accelerating"? I am of the opinion that the v in Einstein's equation .5tv^2/c^2 can be his instantaneous velocity whilst accelerating.

Please note that Einstein's equation above is an amount of time, not an instantaneous rate of ticking, and certainly not an instantaneous velocity. It's meant to tell you how much a moving clock will lag behind a non-moving clock in a given frame after some time t has passed in that frame. Also, it is only an approximation; the non-approximate equation would be t*(1 - \sqrt{1 - v^2/c^2}). For example, if clock A is moving at 0.6c and clock B is at rest in a certain frame, and they start off showing the same time, then after t=10 seconds of coordinate time have passed in that frame, clock A will lag behind clock B by 10*(1 - \sqrt{1 - 0.6^2}) = 10*(1 - 0.8) = 2 seconds. If you use Einstein's approximation you predict that clock A lags behind clock B by 0.5*10*0.6^2 = 1.8 seconds, which is close to the correct value of 2 seconds although a little off.

The slower rate of operation of clock A is not affected by the fact that he takes his foot off the gas pedal at any given instant.

It's true that if you accept the notion of "instantaneous rate of ticking" (which you objected to earlier), then the instantaneous rate of ticking in a given frame depends solely on the instantaneous velocity in that frame, it doesn't depend on whether the clock is accelerating or moving at constant speed. In any case, in most thought-experiments in SR we just assume the accelerations are instantaneous.

But yes, relative to a particular choice of coordinate system there can be definite truths about which clock was ticking slower...

So A, having arrived at B's location, is, apparently, entitled to be of the opinion that his clock lags behind B due to the fact that whilst he was moving his clock was 'going more slowly' (i.e. ticking over at a slower rate) than B?

It is meaningless to state an "opinion" about clock rates without specifying what coordinate system you want to use, and of course A is "entitled" to use absolutely any frame he wants, even one where neither he nor B have been at rest at any point during the experiment. If A wishes to calculate things relative to the inertial frame where B is at rest, then in this frame it is certainly true that A lagged behind because A ticked more slowly after accelerating. But again, you always have to specify a choice of frame you use when talking about rates of ticking, to do otherwise would be like saying "it's my opinion that Earth has a greater x-coordinate than the Sun" without specifying where you want the origin of your coordinate system to be and which direction the x-axis is pointing relative to this origin.

 

[JesseM]

I am of the opinion that he meant that clock A 'physically' or 'really' or 'actually' goes more slowly (i.e. ticks over at a slower rate) than a clock at one of the poles

But you're still haven't defined what you mean by words like "physically", "really", or "actually"! It is true that relative to a particular choice of coordinate system A ticks more slowly, but we understand that this choice is an arbitrary one and the universe doesn't care which coordinate system we use, nothing in the laws of physics justifies the idea that one coordinate system's point of view is somehow more correct than another's. Similarly, if you pick a coordinate system where the origin is at the center of the Sun and the positive x-direction is pointed towards the Earth, it's true in this coordinate system that the Earth has a greater x-coordinate than the Sun...but would you say that the Earth "physically" or "really" or "actually" has a greater x-coordinate than the Sun, in spite of the fact that we are obviously free to use a coordinate system where the origin is at a different position? Please answer this question about whether you would use words like "physical" to describe statements about which of two objects has a greater x-coordinate, and it will help me to understand what you mean by the word "physical" when you talk about clock rates.

 

[cos]

In my previous thread ‘Time dilation’ dated Mar22-09 I wrote -
In section 4 STR Einstein wrote -

"If one of two synchronous clocks at A is moved in a closed curve
with constant velocity until it returns to A, the journey lasting
t seconds, then by the clock which has remained at rest the
travelled clock on its arrival at A will be a .5tv^2/c^2 second
slow. 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 under otherwise identical
conditions."

What do people think he meant by the phrase "...must go more
slowly..."?
Does anyone agree that he meant that the moving clock will tick
over at a slower rate than (i.e. incur time dilation relatively
to) the other clock?

********************

On the (probably erroneous) basis that some people may agree that
he did - I follow that up with the question - On the basis of his
depiction of a clock that is made to move in a closed curve around
another clock is it correct for me to assume that Einstein meant
that the clock that is moving in a closed curve will “go more
slowly” (i.e. tick over at a slower rate) than the clock “which has
remained at rest.”?

The A clock must accelerate to leave the B clock, move at a
constant speed for most of the path, then decelerate to reunite
with the B clock. Since A initiated the trip, and traveled a
greater distance than B, in the same amount of elapsed B time, A
would have an average speed greater than B. Time dilation is a
function of speed, therefore the A clock experiences more time
dilation than B. When reunited, A clock is lagging behind B clock.
This is not about perception of clocks, but the physics of clock
function according to light propagation. Each will
observe/perceive the others clock to be running faster or slower,
depending on direction of motion.

In Einstein's section 4 depiction of an equatorial clock which 'must go more slowly' than a polar clock there is no relationship to "...the physics of clock function according to light propagation."

"The physics of clock function according to light propagation." is solely in relation to Doppler shift and light travel time which, I believe, has absolutely no physical effect whatsoever on any clocks physical rate of operation.

As you point out, above, "...Time dilation is a function of speed." It is not a function of Doppler shift or light travel time which, although bought about by relative speed, only create an illusion of time variations.

You wrote "...Since A initiated the trip, and traveled a greater distance than B..." Clock B travels no distance! It has, according to Einstein, remained at rest!

You wrote "..therefore the A clock experiences more time dilation than B." Clock B, in Einstein's depiction "...has remained at rest..." ergo it incurs no time dilation!

This is a repeat of my reply for the your first thread.

Here is a quote from the Max Born book, page 257, which you have (A and B swapped).

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

My interpretation of that comment is that every internal process in A's system (including the rate of operation of his clocks) must take place more slowly (i.e. his clocks must 'go more slowly') than the same process (i.e. the rate of operation of the clocks) in system B.

Because A & B are synchronized initially, the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

I wholeheartedly agree - the time effect (i.e. the slower rate of operation of A's clock compared to it's rate of operation before he started moving) is, according to Einstein, caused by the motion and as you point out above "Time dilation is a function of speed, therefore the A clock experiences [sic. more] time dilation."

I can only add, time dilation is a real factor affecting particle accelerators and gps systems.

Irrelevant, I made no suggestion whatsoever that time dilation (as depicted by Einstein's section 4 STR comments) is not 'a real factor'!

 

[Dale]

You state that fictitious forces are not 'Einstein'. Do you mean that they are not special theory?

Correct, the concept of fictitious forces predates Einstein by more than 100 years.

There is NO observer C in Einstein's section 4 STR depiction and the introduction of same is nothing more than a deliberate obfuscation!

Determinations made by a hypothetical observer C can have no effect whatsoever on what is taking place as far as A or B are concerned nor on the rates of operation of their clocks!

That is a bit of an extreme reaction. The whole point of relativity is that you can use any inertial reference frame you choose. The results will always be the same, as I explained.

Anyway, I only introduced C because you already refused to use non-inertial reference frames like A's. If you only admit B's reference frame (because A's is non-inertial and because no other inertial frame is explicitly mentioned) then it is hard to see what you are objecting to.

 

[cos]

Not if "physical truth" is defined to mean something that is true regardless of your choice of conventions about coordinate systems, and as I said before, this is how most physicists nowadays use the word "physical". If you want to use a different definition that's fine, it's just semantics, but I would ask that you spell out what you mean by physical. That's what DaleSpam asked you in post #90 above, and I also asked you about this in post #85 which you were responding to here:

You have allowed this thread to deteriorate into a totally in appropriate philosophical discussion.

When Einstein wrote that the equatorial clock 'must go more slowly' than a clock at one of the poles did he mean that the equatorial clock goes more slowly than a polar clock?

 

[Dale]

I made no suggestion that relativity does not work.

Then what is the discussion about?

 

[JesseM]

You have allowed this thread to deteriorate into a totally in appropriate philosophical discussion.

When Einstein wrote that the equatorial clock 'must go more slowly' than a clock at one of the poles did he mean that the equatorial clock goes more slowly than a polar clock?

As I said, I think he meant the average over an entire orbit, and I believe it would be true in all inertial frames that over a complete orbit an equatorial clock would tick less than a clock at the pole. Can we focus on the other situation Einstein discusses in section 4 where clock A and clock B are initially some distance apart, then A is briefly accelerated and afterwards moves inertially towards B? Do you assert that in this example A is "physically", "really", or "actually" ticking slower than B between the time it's accelerated and the time it reaches B, in spite of the fact that there are perfectly valid inertial frames where it is B that's ticking slower during this period of time? I just want to understand if you use words like "physically", "really" and "actually" to mean something that there is a single correct answer about, or if you just use these words to refer to the perspective of particular frames, so that you would be equally fine with saying that it is B that "physically", "really", and "actually" ticks slower than A in certain choices of frames.

 

[Dale]

It is not important what I think about the words 'physical' or 'real' but what Einstein meant by the words 'must go more slowly'!

I am of the opinion that he meant that clock A 'physically' or 'really' or 'actually' goes more slowly

The problem is that because you refuse to define "physical", "real", etc. I still don't know what you mean by that last. I cannot tell if we agree or disagree, and I don't know what words to use to clearly communicate my position back to you. It is, in fact, important what you think about those words because you are the one I am trying to communicate with. I even made it easy for you and suggested some definitions, all you have to do is say yes or no.

I think it is rather hypocritical that you accused me of "deliberate obfuscation" above.

 

[matheinste]

This doesn't answer the question but may be of some interest for those like me who have not seen it before.

http://www.sigmapisigma.org/radiations/2005/electrodynamics_fall05.pdf

"--------If there are two synchronously running clocks at A, and one of them is moved along a closed curve with constant velocity until it has returned to A, which takes, say, t sec, then,on its arrival at A, this clock will lag ½t(v R /c)2 sec [to lowest order in v R /c] behind the clock that has not been moved. From this we conclude that a balance-wheel clock located at the Earth’s equator must, under otherwise identical conditions, run more slowly by a very small amount than an absolutely identical clock located at one of the Earth’s poles.---------”

There are legion experimental demonstrations of time dilation, such as the ubiquitous muons-in-cosmic-rays example that appears in all the textbooks. When time could be measured to nanosecond precision fifty years after Einstein wrote these preceding lines, an experiment was done that recalled Einstein’s prediction explicitly:

Hay’s experiment as described J. Bronowski, The Ascent of Man, Little & Brown (1973), p. 255.

The experiment was done by a young man called H.J. Hay at Harwell. He imagined the Earth squashed flat into a plate, so that the North Pole is at the centre and the equator runs round the rim. He put a radio-active clock on the rim and another at the center of the plate and let it turn. The clocks measure time statistically by counting the number of radio-active atoms that decay. And sure enough, the clock at the rim of Hay’s plate keeps time more slowly than the clock at the centre. This goes on in every spinning plate, on every turntable. At this moment, in every revolving gramophone disc, the centre is ageing faster than the rim with every turn

Matheinste.

 

[DrGreg]

cos,

you have repeatedly been asked to explain what you mean by "physically", "really", or "actually". You might think these words are obvious and require no further explanation, but in relativity things that seemed obvious in Newtonian theory are no longer so. If you describe something as "physically slower" (for example), you need to explain what measurements or calculations you would perform to decide whether something is "physically slower" or not.

You often refer to a clock "ticking more slowly" but you fail to say slower than what and as measured by whom. In relativity these are not optional extras: different observers get different answers. It makes sense to assert "A ticks more slowly than B as measured by C". If we shorten this to "A ticks more slowly than B" this only makes sense (for instantaneous tick rates) in a context where the "as measured by C" is understood -- often the context is "as measured by B". To say "A ticks more slowly" makes no sense at all unless everyone implicitly understands what B and C are.

Note that in relativity it is possible for

"A ticks (instantaneously) more slowly than B as measured by B"

"B ticks (instantaneously) more slowly than A as measured by A"

to be simultaneously true. It's not a contradiction because A & B use different measurement procedures.

But "A actually ticks slower" (without mention of a B or C) means nothing. Can you give an unambiguous operational definition (what numbers you would measure or calculate) of what you think it means?

When Einstein wrote that the equatorial clock 'must go more slowly' than a clock at one of the poles did he mean that the equatorial clock goes more slowly than a polar clock?

He meant the equatorial clock ticks more slowly than a polar clock as measured by a polar clock.



============
Note: everything above applies to "instantaneous" clock rates. If you are talking about average clock rates where clocks A and B are initially together, separate and come back together again, everyone will agree which clock ticked fewer ticks than the other over the whole round-trip journey, but then at least one of the clocks must have accelerated (I'm assuming Special Relativity in the absence of gravity), so simple inertial frame analysis is not sufficient.

 

[phyti]

Jesse, post 81

Well, if you're just talking about average rate of ticking for a non-inertial clock between the times it departs from and returns to an inertial clock, then you aren't saying anything controversial if you say that the non-inertial clock has a slower average rate of ticking between these events, since this is true in all frames. I thought you were saying something more, Well, if you're just talking about average rate of ticking for a non-inertial clock between the times it departs from and returns to an inertial clock, then you aren't saying anything controversial if you say that the non-inertial clock has a slower average rate of ticking between these events, since this is true in all frames. I thought you were saying something more, that the clock A in his example in section 4 was objectively ticking slower than clock B during the after it was accelerated to the time it met clock B; that would be incorrect, but if you didn't mean to suggest this, please clarify. that would be incorrect, but if you didn't mean to suggest this, please clarify.

that the clock A in his example in section 4 was objectively ticking slower than clock B during the after it was accelerated to the time it met clock B

If it wasn't, when or where in the trip does the difference in time occur?

Please don't use the 1st postulate as a defense, it get's old fast!

 

[JesseM]

If it wasn't, when or where in the trip does the difference in time occur?

In a frame where the clock A that was accelerated was not ticking slower than B (because A and B initially had a nonzero speed and A's speed decreased after accelerating, from the perspective of this frame), the reason A was behind B when they met was because they were not synchronized in the first place (always remember the relativity of simultaneity!). In this frame B's time was always ahead of A's time by a constant amount prior to A accelerating, and after A accelerated its reading was "gaining on" B's reading, but not fast enough for it to surpass B's time by the time they met. If A was allowed to continue on at constant velocity past B after they crossed paths, then in this frame A's time would eventually surpass B's time.

Note that I gave a numerical example that worked like this back in post #64:

Suppose for example A and B are a distance of 60 light-seconds apart in the "stationary" frame K, and both are synchronized in this frame. Then if A is moved at 0.6c towards B at the moment when both clocks read a time of t=0, it will take 100 seconds in this frame for A to reach B, during which time A will only tick 80 seconds due to time dilation (the Lorentz factor being 1.25), so when A meets B, B will read t=100 seconds while A reads t=80 seconds.

Now consider things from the perspective of the inertial frame where A and B were initially moving at 0.6c and then A was accelerated to come to rest in this frame while B continued to move towards it at 0.6c. In this frame the clocks were not synchronized initially, so when A read t=0, B already read t=36 seconds according to this frame's definition of simultaneity. Then it takes 80 seconds in this frame for B to reach A (because the initial distance between them was 48 light-seconds in this frame due to length contraction, and 48 light-seconds/0.6c = 80 seconds), during which time B only ticks forward by 80/1.25 = 64 seconds due to time dilation, meaning B reads t=36 + 64 = 100 seconds when they meet, while A reads t=80 seconds when they meet. So you see that both frames make the same prediction about their respective times, even though in the first frame A was ticking slower while in the second frame B was ticking slower.

 

[cos]

When Einstein wrote that the equatorial clock 'must go more slowly' than a clock at one of the poles did he mean that the equatorial clock goes more slowly than a polar clock?

As I said, I think he meant the average over an entire orbit, and I believe it would be true in all inertial frames that over a complete orbit an equatorial clock would tick less than a clock at the pole.

On the basis that the equatorial clock does, on average over an entire orbit, 'go more slowly' than the polar clock I am of the impression that during this orbit the equatorial clock also 'goes more slowly' (i.e. ticks over at a slower rate) than the polar clock.

On the basis that, after one orbit, the equatorial clock is slower than (i.e lags behind) the polar clock then in my opinion it must, during that orbit, have ticked over at a slower rate than it would had it been located alongside the polar clock.

I think there is a distinction between the idea that the equatorial clock is slower than (i.e. lags behind) a polar clock after an entire orbit and that it 'goes more slowly' (i.e. ticks over at a slower rate) than the polar clock.

Can we focus on the other situation Einstein discusses in section 4 where clock A and clock B are initially some distance apart, then A is briefly accelerated and afterwards moves inertially towards B? Do you assert that in this example A is "physically", "really", or "actually" ticking slower than B between the time it's accelerated and the time it reaches B, in spite of the fact that there are perfectly valid inertial frames where it is B that's ticking slower during this period of time?

I agree that "...in this example A is "physically", "really", or "actually" ticking slower than B between the time it's accelerated and the time it reaches B." however I do not accept that "...there are perfectly valid inertial frames where it is B that's ticking slower during this period of time."

In my opinion B's rate of operation can in no way be affected by A's acceleration or deceleration or rate of uniform travel toward (or away from) B!

So when you say that it is B that's ticking slower during this period of time this is nothing more than a comparison of the calculated rate of operation of B to that of the rate of operation of clock A thus clock A 'is' ticking over at a faster rate than it was before it started accelerating however I am of the understanding that the concept of time contraction was, for Einstein, unacceptable.

Ergo, on that basis, A is not ticking over at a faster rate (time contraction) than it was before it started accelerating and on the basis that A's actions of any kind have no affect whatsoever on B's rate of operation A cannot be of the opinion that B is ticking over at a slower rate than it was before he started accelerating but can only conclude that B appears (according to his calculations) to be ticking over at a faster rate than it was previously.

In my opinion, which is probably controversial, clock A's instantaneous velocity can be substituted for v in the Lorentz transformations.

A is accelerating toward B and has attained a instantaneous velocity of s. He switches his rockets off and at that very instant is moving at the same (albeit, now) uniform velocity of s (i.e he is moving toward B at the same speed as he was at the very instant the rocket shut down).

I fail to see why the calculated rate of operation of clock A from B's point of view (or the calculated rate of clock B from A's point of view) will be any different if A has attained an instantaneous velocity of s or is moving toward B with a uniform velocity of s.

I just want to understand if you use words like "physically", "really" and "actually" to mean something that there is a single correct answer about, or if you just use these words to refer to the perspective of particular frames, so that you would be equally fine with saying that it is B that "physically", "really", and "actually" ticks slower than A in certain choices of frames.

Saying that B 'is' ticking slower (or faster) than A in certain choices of frames is NOT the same as saying that B ticks over at slower (or faster) rates than it did before A started moving (or accelerating). I reiterate - I am of the opinion that B's rate of operation is not affected by A's actions of any kind, it merely appears, from A's point of view (in accordance with his calculations), to change!

An astronaut has come to a stop at the end of his outward bound trip. He accelerates and attains an instantaneous velocity whereby a gamma factor of 400 000 is attained. He 'sees' (or 'determines' or 'calculates') the Earth clock ticking over at a fsater rate than his own clock by a factor of one second Earth time for each 400 000 seconds of his own time.

He flicks a switch extinguishing his rocket and at that very instant 'sees' (or 'determines' or 'calculates') that the Earth clock is no longer ticking over at that faster rate than his own clock but is immediately ticking over at a slower rate than his own clock by a factor of 400 000 seconds for each of his own seconds.

Is he not likely to ask himself what affect this would have on the clock (and on all of the planet's inhabitants) which instantaneously reverts from being 400 000 times faster than his clock to being 400 000 times slower?

Does he truly believe that this takes place because he flicked that switch?

I won't bother detailing the following but on the basis of the identical nature of time contraction and length contraction he would 'see' the planet instantaneously change from being shaped like a pancake of around 32 meters thick at the center (tapering to almost zero at its perimeter) to being more than 5 billion kilometers deep (in his direction of travel).

Would he not be of the opinion that this instantaneous and somewhat considerable change in that dimension could have a detrimental affect on the planet and a devastating effect on its inhabitants.

In fact, as his instantaneous velocity increases, so too 'does' the distance from him to the planet (as 'does' the rate of operation of the Earth clock) but when he flicks that switch the (seemingly increasing) distance to the planet instantaneously reverts to a close proximity to the plant - simply because he flicked a switch?

 

[cos]

It is not important what I think about the words 'physical' or 'real' but what Einstein meant by the words 'must go more slowly'!

I am of the opinion that he meant that clock A 'physically' or 'really' or 'actually' goes more slowly.

The problem is that because you refuse to define "physical", "real", etc. I still don't know what you mean by that last.

OK; what do you think Einstein meant by the phrase "...must go more slowly..."?

My opinion is that he was saying that a clock at the equator must tick over at a slower rate than it would if it were located at one of the poles or that it ticks over at a slower rate than a polar clock.

I cannot tell if we agree or disagree, and I don't know what words to use to clearly communicate my position back to you. It is, in fact, important what you think about those words because you are the one I am trying to communicate with. I even made it easy for you and suggested some definitions, all you have to do is say yes or no.

Your response to the question above might indicate whether or not we agree or disagree.

I think it is rather hypocritical that you accused me of "deliberate obfuscation" above.

The term 'hypocritical' implies that you are of the opinion that I have introduced 'deliberate obfuscation'.

Whilst my responses, or lack thereof, may have created obfuscation this was not my intention!

 

[cos]

cos,

you have repeatedly been asked to explain what you mean by "physically", "really", or "actually". You might think these words are obvious and require no further explanation, but in relativity things that seemed obvious in Newtonian theory are no longer so. If you describe something as "physically slower" (for example), you need to explain what measurements or calculations you would perform to decide whether something is "physically slower" or not.

When Einstein wrote in section 4 STR that an equatorial clock 'must go more slowly' he, I believe, related this to his equation .5tv^2/c^2 . having referred to that same equation in my postings I assumed that readers would automatically apply that same equation.

As to what measurements or calculations I would perform (as distinct from applying) I am of the opinion that Einstein did not explain what measurements or claculations he would perform (as distinct from applying) so perhaps you point should be directed to his depiction.

You often refer to a clock "ticking more slowly" but you fail to say slower than what and as measured by whom. In relativity these are not optional extras: different observers get different answers. It makes sense to assert "A ticks more slowly than B as measured by C". If we shorten this to "A ticks more slowly than B" this only makes sense (for instantaneous tick rates) in a context where the "as measured by C" is understood -- often the context is "as measured by B". To say "A ticks more slowly" makes no sense at all unless everyone implicitly understands what B and C are.

Perhaps you could provide a reference as to where I referred to a clock that is ticking more slowly but where I failed to say to what it is ticking more slowly than?

I do not care who makes the measurement. In my opinion a clock's intrinsic rate of operation will remain unchanged regardless of the point of view, or the determinations of, an observer.

Note that in relativity it is possible for

"A ticks (instantaneously) more slowly than B as measured by B"

"B ticks (instantaneously) more slowly than A as measured by A"

This may well be true of the previous sections of relativity however in section 4 Einstein points out that a clock at the equator must go more slowly than a clock at one of the poles and it is my understanding that from the point of view of observer B (at one of the poles) A does not 'tick more slowly' than his own clock as per you statement above ("A ticks (instantaneously) more slowly than B as measured by B") but faster!

to be simultaneously true. It's not a contradiction because A & B use different measurement procedures.

It matters not that A and B "...use different measurement procedures." Nothing that any of them 'measure' or 'calculate' or 'determine' or 'predict' will have any affect whatsoever on a clock's intrinsic rate of operation.

But "A actually ticks slower" (without mention of a B or C) means nothing. Can you give an unambiguous operational definition (what numbers you would measure or calculate) of what you think it means?
He meant the equatorial clock ticks more slowly than a polar clock as measured by a polar clock.

And as measured by the equatorial observer!

He determines that his clock is 'going more slowly' than the polar clock; he realizes that his having moved to the equator has had no affect whatsoever on the rate of operation of the polar clock which is still ticking over at the same rate as it was when he was at the same location.

In order to qualify that suggestion - neither of the observers can actually see what the other clock is doing but on the basis that the equatorial observer has read, and agrees with, Einstein's section 4 depiction he could agree with Einstein that his clock is 'going more slowly' than the polar clock and at a slower rate than it was before he moved to the equator.

Note: everything above applies to "instantaneous" clock rates. If you are talking about average clock rates where clocks A and B are initially together, separate and come back together again, everyone will agree which clock ticked fewer ticks than the other over the whole round-trip journey, but then at least one of the clocks must have accelerated (I'm assuming Special Relativity in the absence of gravity), so simple inertial frame analysis is not sufficient.

Most of the above applies to Einstein's section 4 STR depiction of equatorial and polar clocks where the the clocks are not separated and come back together again however observer A, as i pointed out in other message, could initially have been located at one of the poles where his clock was ideally synchronous with an identical clock. He moves to the equator then back to the pole (i.e. travels in a closed curve relative to the polar clock) whereupon he finds, as Einstein depicted in his description of a clock that moves in a closed curve relative to another clock, that his clock now lags behind that clock ergo he should be able to conclude, as Einstein pointed out, that his clock (progressively) went more slowly (i.e.ticked over at a slower rate) than the polar clock.

 

[Dale]

OK; what do you think Einstein meant by the phrase "...must go more slowly..."?

I think that he meant that A's proper time is slower than the coordinate time \left(\frac{dt}{d\tau}>1\right) in system K, the reference frame where A and B were initially at rest and synchronized. I believe that he understood that this is a frame-variant statement, which was the reason why he clearly identified frame K.

The term 'hypocritical' implies that you are of the opinion that I have introduced 'deliberate obfuscation'.

Whilst my responses, or lack thereof, may have created obfuscation this was not my intention!

I will accept this statement at face value and not impugn your motives. I would ask that you show me the courtesy of doing the same.

By the way, I would appreciate it if you would try not to underline so many words. I often go back through a thread looking for a link, and they are very hard to find with so many non-links in underlined font. Italics and bold are much preferable (as you have done in this quoted post).

 

[JesseM]

On the basis that the equatorial clock does, on average over an entire orbit, 'go more slowly' than the polar clock I am of the impression that during this orbit the equatorial clock also 'goes more slowly' (i.e. ticks over at a slower rate) than the polar clock.

Only on average over the whole orbit, not necessarily at every moment, depending what frame you choose. In certain frames there will be periods of time where the equatorial clock has a smaller speed than the polar clock, so during these periods of time the polar clock must be ticking slower in such a frame.

I agree that "...in this example A is "physically", "really", or "actually" ticking slower than B between the time it's accelerated and the time it reaches B."

You "agree"? I did not say that was what I thought.

however I do not accept that "...there are perfectly valid inertial frames where it is B that's ticking slower during this period of time."

So do you disagree with the math in the second paragraph in my example from post #64 below?

Suppose for example A and B are a distance of 60 light-seconds apart in the "stationary" frame K, and both are synchronized in this frame. Then if A is moved at 0.6c towards B at the moment when both clocks read a time of t=0, it will take 100 seconds in this frame for A to reach B, during which time A will only tick 80 seconds due to time dilation (the Lorentz factor being 1.25), so when A meets B, B will read t=100 seconds while A reads t=80 seconds.

Now consider things from the perspective of the inertial frame where A and B were initially moving at 0.6c and then A was accelerated to come to rest in this frame while B continued to move towards it at 0.6c. In this frame the clocks were not synchronized initially, so when A read t=0, B already read t=36 seconds according to this frame's definition of simultaneity. Then it takes 80 seconds in this frame for B to reach A (because the initial distance between them was 48 light-seconds in this frame due to length contraction, and 48 light-seconds/0.6c = 80 seconds), during which time B only ticks forward by 80/1.25 = 64 seconds due to time dilation, meaning B reads t=36 + 64 = 100 seconds when they meet, while A reads t=80 seconds when they meet. So you see that both frames make the same prediction about their respective times, even though in the first frame A was ticking slower while in the second frame B was ticking slower.

Unless you think the math is wrong in the second paragraph, the second paragraph is saying that in the frame where A and B were initially moving at 0.6c, it must be true that at the moment A accelerates (and we can assume the acceleration is instantaneously brief), A reads t=0 seconds and B reads t=36 seconds, and at the moment B and A meet, A reads 80 seconds and B reads 100 seconds. So B ticked forward by 64 seconds, while A ticked forward by 80 seconds, therefore B was ticking slower during this period, from the perspective of this frame. If you don't disagree with the math--and I can show you that these numbers follow directly from applying the Lorentz transformation to the scenario described in the first paragraph from the perspective of B's rest frame--then how can you say you disagree with the statement that "there are perfectly valid inertial frames where it is B that's ticking slower during this period of time"?

In my opinion B's rate of operation can in no way be affected by A's acceleration or deceleration or rate of uniform travel toward (or away from) B!

No one said anything about B's rate of operation being accelerated by A's acceleration! In the frame described by the second paragraph, B is always ticking at a rate of 0.8 ticks per second of coordinate time, both before and after A accelerates. Before A accelerates A is also ticking at 0.8 ticks per second of coordinate time, but then after accelerating A comes to rest in this frame, and is now ticking at 1 tick per second of coordinate time. Here are a few numbers to make this clear:

At coordinate time of t=-30 seconds in this frame, A reads -24 seconds, B reads 12 seconds

At coordinate time of t=-20 seconds in this frame, A reads -16 seconds, B reads 20 seconds

At coordinate time of t=-10 seconds in this frame, A reads -8 seconds, B reads 28 seconds

At coordinate time of t=0 seconds in this frame (the moment that A accelerates), A reads 0 seconds, B reads 36 seconds

At coordinate time of t=10 seconds in this frame, A reads 10 seconds, B reads 44 seconds

At coordinate time of t=20 seconds in this frame, A reads 20 seconds, B reads 52 seconds

At coordinate time of t=30 seconds in this frame, A reads 30 seconds, B reads 60 seconds

...So, you can see that for each interval of 10 seconds of coordinate time prior to A's acceleration, B advances forward by 8 seconds (from 20 to 28 seconds between t=-20 and t=-10 seconds, for example), and A also advances forward by 8 seconds (from -16 to -8 seconds between t=-20 and t=-10). On the other hand, for each interval of 10 seconds of coordinate time after A's acceleration, B still advances forward by the same amount of 8 seconds (from 44 to 52 seconds between t=10 and t=20, for example), while A now advances forward at the faster rate of 10 seconds (from 10 to 20 seconds between t=10 and t=20). So B's rate of ticking never changes in this frame, only A's rate of ticking changes, from ticking at the same rate as B before it accelerates to ticking faster than B afterwards. Again, do you disagree that these are the numbers we get if we apply the Lorentz transformation to the scenario as it was described in B's rest frame, or do you agree with the math but think that the description in B's rest frame describes what "really", "actually" happens while the description in this second frame is some sort of illusion?

So when you say that it is B that's ticking slower during this period of time this is nothing more than a comparison of the calculated rate of operation of B to that of the rate of operation of clock A thus clock A 'is' ticking over at a faster rate than it was before it started accelerating however I am of the understanding that the concept of time contraction was, for Einstein, unacceptable.

You still have never explained what "time contraction" means. You can see in the above scenario that even though A speeds up after accelerating, its rate of ticking can never be faster than the rate that coordinate time is passing--is that what you mean by time contraction? Or do you just think relativity forbids a clock's rate from ever speeding up at all after it changes velocities? If the latter, you're incorrect, if a clock changes velocities in such a way that its speed becomes smaller in a given frame, then its rate of ticking will get faster than what it was before changing velocities, from the perspective of that frame.

In my opinion, which is probably controversial, clock A's instantaneous velocity can be substituted for v in the Lorentz transformations.

Sure, if you want to figure out what things look like in the frame where A is instantaneously at rest at that instant (but in this frame A was not always at rest if it is accelerating; by definition an inertial frame must travel at the same constant velocity forever).

A is accelerating toward B and has attained a instantaneous velocity of s. He switches his rockets off and at that very instant is moving at the same (albeit, now) uniform velocity of s (i.e he is moving toward B at the same speed as he was at the very instant the rocket shut down).

Of course, did you think I had suggested otherwise?

I fail to see why the calculated rate of operation of clock A from B's point of view (or the calculated rate of clock B from A's point of view) will be any different if A has attained an instantaneous velocity of s or is moving toward B with a uniform velocity of s.

In SR we're only calculating things from the "point of view" of particular inertial coordinate systems, I don't know what it means to calculate things "from A's point of view" since A changes velocity at a certain point. If you want to calculate things from the point of view of the inertial frame where A is at rest after accelerating, that is exactly what I was doing in the second paragraph of my numerical example above.

Saying that B 'is' ticking slower (or faster) than A in certain choices of frames is NOT the same as saying that B ticks over at slower (or faster) rates than it did before A started moving (or accelerating).

As I point out above, I have never claimed this, so I don't know why you're making this point. When I said "...there are perfectly valid inertial frames where it is B that's ticking slower during this period of time", I thought it was fairly clear from the context that I meant B is ticking slower than A during this period of time, not that B is ticking slower than B was ticking prior to A's acceleration. Perhaps you misunderstood my meaning? If so, now that I have clarified, do you disagree that "there are perfectly valid inertial frames where it is B that's ticking slower than A during this period of time" (i.e. the period of time between A accelerating and A and B meeting)? If you don't disagree, then would you say that B is "physically", "really", and "actually" ticking slower than A during this period of time, from the perspective of these frames?

 

[cos]

OK; what do you think Einstein meant by the phrase "...must go more slowly..."?

I think that he meant that A's proper time is slower than the coordinate time \left(\frac{dt}{d\tau}>1\right) in system K, the reference frame where A and B were initially at rest and synchronized. I believe that he understood that this is a frame-variant statement, which was the reason why he clearly identified frame K.

His comment "...must go more slowly..." was not in relation to "...in system K, the reference frame where A and B were initially at rest and synchronized." but to his later reference to a clock at the equator which, he insisted 'must go more slowly' than a clock at one of the poles.

For the purpose of edification I submit his section 4 comments -

"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 .5tv^2/c^2 (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B.

It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide.

If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be a .5tv^2/c^2 second slow. 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 under otherwise identical conditions."

I will accept this statement at face value and not impugn your motives. I would ask that you show me the courtesy of doing the same.

Noted - but I also trust that you will not impugn my motives in relation to any other of my postings and I will reciprocate

By the way, I would appreciate it if you would try not to underline so many words. I often go back through a thread looking for a link, and they are very hard to find with so many non-links in underlined font. Italics and bold are much preferable (as you have done in this quoted post).

"Sorry 'bout that Chief."

 

[cos]

On the basis that the equatorial clock does, on average over an entire orbit, 'go more slowly' than the polar clock I am of the impression that during this orbit the equatorial clock also 'goes more slowly' (i.e. ticks over at a slower rate) than the polar clock.

Only on average over the whole orbit, not necessarily at every moment, depending what frame you choose. In certain frames there will be periods of time where the equatorial clock has a smaller speed than the polar clock, so during these periods of time the polar clock must be ticking slower in such a frame.

On the basis that in my opinion this may well be the crux of the matter I make no apology for the fact that, although having read same, I have removed the remainder of your posting on the basis of my physical disabilities which severely restricts the time that I am able to devote to my responses.

"Depending what frame [is chosen]"?

There are, as I have previously stipulated, only two frames in this otherwise empty universe - that of the polar observer and that of the equatorial observer.

The planet could, as I have also previously stipulated, be replaced by a large rotating, transparent, massless sphere.

Apart from rotating, the sphere is not moving. From both points of view the equatorial clock is always moving at the same speed relative to the polar clock i.e. 1600K-h.

At no time does "...the equatorial clock [have] a smaller speed than the polar clock..."

 

[JesseM]

There are, as I have previously stipulated, only two frames in this otherwise empty universe - that of the polar observer and that of the equatorial observer.

First of all, frames are just coordinate systems. They don't depend on what objects happen to be present in the universe, you can certainly assign coordinates to events using a coordinate system where no object in the universe happens to be at rest. So, regardless of what objects happen to be present in the universe, you have an infinite number of different inertial frames you can use.

Second, in SR we usually stick to talking about inertial frames. The object at the equator does not remain at rest in any inertial frame, because it's moving non-inertially. You can think about non-inertial coordinate systems if you wish, but the usual rules of inertial frames, like light always moving at c or moving clocks running slow by a factor of \sqrt{1 - v^2/c^2}, no longer apply in non-inertial coordinate systems, so it's easier to just use inertial ones.

Finally, If you don't want to reply to my entire post that's fine, but could I request that you clarify what you meant when you said (speaking of the A-B thought experiment rather than the equatorial and polar clock thought experiment) that you do not agree with my statement that "...there are perfectly valid inertial frames where it is B that's ticking slower during this period of time"? Did you misunderstand what I meant by "it is B that's ticking slower", thinking I was saying that B's rate of ticking slowed down after A accelerated in some inertial frame? If so, please note that I didn't mean to imply any change in B's rate of ticking, I just meant that there are inertial frames where B is ticking slower than A after A accelerates, because in these frames A's rate of ticking speeds up after it accelerates (since its speed decreases in these frames) while B's rate of ticking remains unchanged. So with that clarification, would you still disagree that "there are perfectly valid inertial frames where it is B that's ticking slower than A during this period of time (the period after A accelerates)", or would you now agree with it? I'm not asking for a detailed answer here, just a simple "agree" or "disagree".

 

[cos]

So, regardless of what objects happen to be present in the universe, you have an infinite number of different inertial frames you can use.

On the basis of my presentation that the sphere is located in an otherwise empty universe thus a universe in which there is not an infinite number of frames and your continued insistence on an infinite number of frames - which, in my opinion, is a reprehensible attitude - this discussion is terminated.

 

[matheinste]

cos

Maybe for frames you mean rest frames.

In a universe empty except for ANY number of objects there are an infinite number of frames. This applies equally well to a universe containing only one object or our populated universe. However, there is at most only one frame in which any individual object can be at rest at any instant. So in the two clock scenario there are an infinite number of frames but for each clock there is only one frame in which it is at rest at any instant.

Matheinste

 

[Dale]

His comment "...must go more slowly..." was not in relation to "...in system K, the reference frame where A and B were initially at rest and synchronized." but to his later reference to a clock at the equator which, he insisted 'must go more slowly' than a clock at one of the poles.

Then I suggest that you misunderstood him. He postulated the equivalence of inertial reference frames and then he derived the relativity of simultaneity earlier in the paper. From then on he was repeatedly careful to identify the reference frame in which his analysis held. I don't know how you can read that work and come to any conclusion other than that he understood simultaneity, time dilation, and length contraction to be frame-variant effects.

Why do you think he repeatedly identified the reference frame in his later analysis if he believed that his results were frame-invariant?