Twin Paradox: Einstein's Explanation and Alternative Interpretations

[JesseM]

Some people may be able to "find inertial frames where the Earth twin's clock ticks slower than the traveling twin's clock during the trip away from the Earth" however this does not comply with Einstein's chapter 4 depiction and it is that depiction to which my posting specifically applies!

It makes no difference if clock A in Einstein's chapter 4 depiction travels a distance to clock B as Einstein shows or if clock A is initially at rest alongside clock B and moves away from B for the same length of time (t) at the same velocity (v).

In accordance with his equation (.5tv²/c²) clock A will lag behind B by the same amount. Clock B (the Earth clock) will not, according to Einstein, lag behind (having, as you suggested, ticked slower than) the traveling twin's clock (Einstein's clock A).

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

Now, the point is that there is no obligation to analyze this situation from the perspective of the frame where A and B are initially at rest. You could analyze this same situation described by Einstein from the perspective of a situation where A and B are initially in motion at speed v (which is 0.8c in my example), and then when A accelerates it comes to rest and B continues to move towards it at v. In terms of my example, if A and B were initially 20 light seconds apart in their rest frame before A accelerated, then in a frame where they were initially moving at 0.8c, the distance between them would be shrunk to 20*0.6 = 12 light-seconds due to Lorentz contraction. Also, is A and B were initially synchronized by the Einstein synchronization convention (which Einstein describes in section 2 of the paper) in their own rest frame, then in the frame where they are moving at 0.8c they will not be synchronized, thanks to the relativity of simultaneity--in general if two clocks are a distance L apart in their own rest frame and synchronized in that frame, then in a frame where they are moving at speed v along the axis between them, they will be out-of-sync by a factor of vL/c^2, with the time on the trailing clock being ahead of the time on the leading clock by this amount. So, in the frame where A and B are initially moving at 0.8c, they will be out-of-sync by (0.8c)*(20 light-seconds)/c^2 = 16 seconds. Since we are picking a frame where B is moving in the direction of A, B is the trailing clock here, so its time is the one that's ahead by 16 seconds. 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. Since the initial distance between them is 12 light-seconds in this frame, it will take 12/0.8c = 15 seconds for B to catch up with A. During this time A will advance forward by 15 seconds but B will only advance forward by 15*0.6 = 9 seconds. Since A started out reading 0 seconds at the moment it came to rest, and B started out reading 16 seconds "at the same moment" in this frame, then when B catches up with A, A will read 0 + 15 = 15 seconds, while B will read 16 + 9 = 25 seconds. So, in this frame we get the exact same prediction that A is behind B by 10 seconds when they meet, in spite of the fact that in this frame A was ticking faster than B after A accelerated, not slower. There is nothing about Einstein's thought-experiment that requires us to analyze it from the perspective of a particular inertial frame, we'll get the same final answer to what the clocks read when they meet regardless of what frame we use.

So according to your previous comment "you can find inertial frames where the Earth twin's clock ticks slower than the traveling twin's clock during the trip away from the Earth" the Earth clock will lag behind the traveler's clock presumably by .5tv²/c² "then the traveling twin's clock ticks slower than the Earth twin's on the return journey after the turnaround" also presumably by .5tv²/c² aren't the Earth clock and the traveler's clock going to read the same time?

Again, the 0.5tv^2/c^2 formula is just an approximation. And in this situation as in the previous one, you'll get the same answer to what the clocks read when they meet regardless of what frame you choose. You also have to take into account that if you pick a frame where the Earth is in motion, the time for the traveler to get a certain distance away from the Earth is not the same as the time for the traveler to return back to the Earth from that distance, as it would be in the Earth's rest frame.

For instance, suppose that in the Earth's rest frame, the ship moves away at 0.8c until it reaches a star 20 light-years from Earth (in the Earth's frame), then turns around and returns to Earth at 0.8c. In the Earth's frame, both the inbound leg and the outbound leg will last for 20/0.8 = 25 years, and the traveler's clock will be slowed down by a factor of 0.6 on each leg, so the traveler will only age by 25*0.6 = 15 years on the outbound leg, and will age another 25*0.6 = 15 years on the inbound leg, so when they reunite the Earth twin is 25 + 25 = 50 years older while the traveling twin is only 15 + 15 = 30 years older.

Now let's look at this from the perspective of a frame where the Earth is moving at 0.8c, and the traveler is at rest during the outbound leg, letting the distant star come to him, then when the star reaches him he accelerates in the direction of the Earth, moving towards the Earth at (0.8c + 0.8c)/(1 + 0.8*0.8) = 1.6c/1.64 = 0.975609756c (using the formula for relativistic velocity addition). If the star was 20 light years from Earth in the Earth rest frame, in this frame the distance between Earth and the star is only 20*0.6 = 12 light years due to Lorentz contraction. So when the traveler first comes to rest near the Earth, and the Earth moves away at 0.8c while the star moves towards him at 0.8c, it will take a time of 12/0.8 = 15 years for the star to catch up with him in this frame. At the moment the star catches up to him, the Earth is now 12 light years away, and then the traveler accelerates to 0.975609756c in the direction of the Earth, while the Earth continues to move away at 0.8c. So, the distance between them is only shrinking at a rate of 0.975609756c - 0.8c = 0.175609756c, which means it will take a time of 12/0.175609756 = 68.33333 years for the traveler to catch up with the Earth again, in this frame. During this return phase, the traveler's aging is slowed down by a factor of sqrt(1 - 0.975609756^2) = 0.2195122, so he'll only age by 68.33333*0.2195122 = 15 years during this phase. Meanwhile, during both the first phase and the second phase the Earth was always moving at 0.8c in this frame, and so the Earth's clock was always slowed down by a factor of 0.6, so during the first phase the Earth-twin aged 15*0.6 = 9 years, and during the second phase the Earth-twin aged 68.33333*0.6 = 41 years, and so the Earth twin aged a total of 9 + 41 = 50 years from the time the traveling twin departed to the time the traveling twin caught up with Earth again. So you see, this frame ends up predicting exactly the same thing about their ages when they reunite as was predicted in the Earth rest frame--this frame predicts that when they reunite, the Earth twin has aged 50 years while the traveling twin has only aged 15 + 15 = 30 years. That's despite the fact that in this frame, during the first phase of the the trip the Earth-twin aged less than the traveling twin, with the Earth-twin aging only 9 years from the time the traveling twin departed to the time the traveling twin turned around (again, relative to this frame's definition of simultaneity), and the traveling twin aging 15 years between departure and turnaround.

According to Einstein's chapter 4 depiction there is an "objective truth about whose clock is ticking slower at any given moment". According to Einstein it is clock A that is ticking slower than B at any given moment.

Nope, Einstein would disagree--you're just failing to understand the relativity of simultaneity (which Einstein briefly discusses at the end of section 2 of the 1905 paper, and which he also explains in more detail in sections VIII and IX of this book), which means that if A and B were synchronized in their own rest frame, they are out-of-sync in other frames. So, in a frame where they're out-of-sync, the time on B may be significantly ahead of the time on A at the moment that A changes speed, meaning that even though B is ticking more slowly as it approaches A in such a frame, it will still be true that B is ahead when they meet.

 

[atyy]

cos, although these notes by 't Hooft are probably too mathematical as an introduction, they do contain a statement you may find congenial: "It is of importance to realize what this implies: although we have the well-known postulate that an experimenter on a moving platform, when doing some experiment, will find the same outcomes as a colleague at rest, we must rearrange the results before comparing them. http://www.phys.uu.nl/~thooft/lectures/gr.html

The books I actually learned from were by Anthony French and WGV Rosser. Rindler has a book that is good for the logical subtleties.

 

[JesseM]

According to the principle of relativity - if I am located in a windowless room I am unable to carry out any experiment to determine if the room is stationary or if it is moving with uniform velocity. On that basis i am fully justified in believing that the room is stationary.

Sure.

If I then move to another room that has a window I am fully justified in assuming that it is the various bits of the universe that are moving - that I am stationary in an absolute sense.

You can use a coordinate system where you're stationary and all the other bits of the universe are moving, but that's not what I meant by stationary in an "absolute" sense. An "absolute" definition of stationary would be one which excludes all other frame's definitions of stationary--although it would be perfectly valid for you to use a coordinate system where you were stationary, it would also be perfectly valid for you to use a coordinate system where some other bit of the universe was stationary and you were in motion, so you cannot say you are "stationary in an absolute sense" because you acknowledge that both frames are equally valid.

My response, above, was in relation to the comment that the Earth could be moving at relativistic speed however, as I tried to point out, "the laws of physics work exactly the same in different inertial reference frames" hence it makes no difference to the conclusion at which Einstein arrived in chapter 4 if the Earth is moving with a uniform relativistic velocity or if it was 'at rest'.

That's right, you can use a frame where the Earth is moving with a uniform relativistic velocity or one where it's at rest, either way you'll get the same prediction about the reading on the Earth's clock and the reading on a space traveler's clock when they reunite after the traveler had departed Earth earlier (I gave an example of different frames giving the same prediction about clock readings in my previous post).

 

[granpa]

Having come to a stop at the end of his outward bound trip the astronaut, regardless of the speed at which he moved or the distance traveled, is then of the opinion that he and the planet are contained in the same reference frame!

The astronaut, along with the planet, is also moving around, and traveling along with, the galaxy at the same velocity as the planet but he is neither moving toward, nor traveling away from, the planet. They are at rest with respect to each other.

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

you will of course get the same result when the traveler gets back to Earth but you will get different results at the point where he turns around.

http://en.wikipedia.org/wiki/Relativity_of_simultaneity

 

[atyy]

That's right, you can use a frame where the Earth is moving with a uniform relativistic velocity or one where it's at rest, either way you'll get the same prediction about the reading on the Earth's clock and the reading on a space traveler's clock when they reunite after the traveler had departed Earth earlier (I gave an example of different frames giving the same prediction about clock readings in my previous post).

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

 

[JesseM]

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

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

 

[cos]

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

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

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

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

 

[granpa]

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

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

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

 

[cos]

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

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

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

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

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

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

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

 

[cos]

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

In message no. 28 you wrote:-

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

Correspondence terminated.

 

[atyy]

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

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

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

 

[JesseM]

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

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

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

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

 

[phyti]

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

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

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

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

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

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

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

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

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

 

[cos]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[cos]

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

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

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

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

 

[granpa]

motion is relative. acceleration is not

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[granpa]

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

.

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

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

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

 

[yogi]

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

 

[yogi]

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

 

[cos]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Again, it is NOT!

Gravitational time dilation WAS TAKEN INTO ACCOUNT!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

That is precisely what I am doing.

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

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

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

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

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

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

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

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

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

 

[cos]

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

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

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

 

[phyti]

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

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

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

Every book can't be perfect!

 

[atyy]

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

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

 

[cos]

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

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

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

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

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

 

[cos]

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

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

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

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

 

[atyy]

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

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

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

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

 

[JesseM]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Again, it is NOT!

Gravitational time dilation WAS TAKEN INTO ACCOUNT!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[JesseM]

(continued from previous post)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[granpa]

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