Twin Paradox: Einstein's Explanation and Alternative Interpretations

[cos]

It is my understanding that the twin paradox arose from the fully reciprocal nature of special theory which shows that if a clock is moving past me in outer space that clock is ticking over at a slower rate than my clock but that from the point of view of a person accompanying that clock it is my clock that is ticking over at a slower rate than his clock; the paradox, apparently, being that both clocks cannot be ticking over at a slower rate than the other one (the original ‘clock’ paradox).

In his 1918 Naturwissenschaften article Einstein attempted to negate this paradox insisting that it is only the clock that has been made to move to the other clock’s location that incurs time dilation on the basis that it experiences forces of acceleration however in chapter 4 of his 1905 article ‘On the Electrodynamics of Moving Bodies’ Einstein wrote:-

“If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by ·5tv²/c²...”

My reason for posting this message is that, having been made to move from A to B clock A (although Einstein does not refer to this fact) must have accelerated.

The alternative is that clock A incurred instantaneous velocity which, I assume, is a concept that Einstein would not have tolerated ergo his chapter 4 depiction effectively provides a similar explanation for the eventual discrepancy between clocks A and B as did his 1918 article.

On the basis that Einstein’s chapter 4 STR clock A accelerated, moved toward B at v then decelerated this is analogous to an astronaut’s return journey following turn-around.

As a result of his outward-bound journey the astronaut’s clock will lag behind his twin’s clock by ·5tv²/c².. As a result of his inward-bound trip the astronaut’s clock will lag behind the twin’s clock by an additional ·5tv²/c².

I have read several interpretations of the twin paradox one of which insists that the traveler’s clock does not (as Einstein expressed it in chapter 4) ‘go more slowly’ than the Earth clock but that the Earth clock, from the traveler’s point of view, ticks over at a faster rate than his own clock but only during the astronaut’s period of acceleration following turn-around however it is my understanding that the concept of time contraction was, for Einstein, an anathema.

Although I have included Einstein’s chapter 4 equation it would very much be appreciated if responses did not incorporate mathematical ‘proofs’ or explanations.

I am, as was Faraday, one of those annoying self-taught persons who has no comprehension of mathematics and, like Faraday, prefers simple, every-day language interpretations.

Einstein insisted that as far as the propositions of mathematics are certain, they do not refer to reality and I tend to agree.

 

[granpa]

all of the above

 

[granpa]

seriously, one of them is from the point of view of the stationary twin and the other is from the point of view of the traveling twin. there is no contradiction.

 

[atyy]

I prefer not to think of time dilation, although that is valid. The accumulated proper time of a person is simply the "length" of his trajectory in spacetime.

In normal geometry, a straight line between two points has the shortest length. Still in normal geometry, the edge of a square has a shorter length than the diagonal.

The difference in spacetime geometry is that a straight line has the longest length. If you draw the situations you described in spacetime, and apply this principle, you will reproduce the standard time dilation results.

The reason for defining spacetime length in this somewhat strange way is that it ensures that the speed of light is the same for "normal" observers moving at constant velocity relative to each other, which is an experimental observation.

Although you prefer not to have the equations, here is a link, just in case:
http://www.math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html

 

[Fredrik]

...it would very much be appreciated if responses did not incorporate mathematical ‘proofs’ or explanations.

I am, as was Faraday, one of those annoying self-taught persons who has no comprehension of mathematics and, like Faraday, prefers simple, every-day language interpretations.

Einstein insisted that as far as the propositions of mathematics are certain, they do not refer to reality and I tend to agree.

This is one place where you're going wrong. Not the last part. I agree with that. A theory is a mathematical abstraction. You can think of a theory as an approximate description of our universe or as an exact description of a fictional universe that resembles our own, but not as an exact description of our universe, so you got that part right. Your mistake is to think that there is a non-mathematical answer to this problem.

The reason why there isn't, is that the only way that anyone can believe that the paradox exists in the first place is to use the theory (i.e. the mathematics) incorrectly. No one claims that the twin paradox is something that happens in the real world. The claim is that there's a paradox in the theory. But the "paradox" is just a mistake in a calculation, so there's no way to resolve it without examining the calculation and showing what the mistake is.

So I'm afraid that a resolution is always going to look like e.g. posts #3 and #142 in this thread. They require that you know some mathematics, or that you at least understand simultaneity in the context of inertial frames in special relativity.

 

[phyti]

...
My reason for posting this message is that, having been made to move from A to B clock A (although Einstein does not refer to this fact) must have accelerated.

The alternative is that clock A incurred instantaneous velocity which, I assume, is a concept that Einstein would not have tolerated ergo his chapter 4 depiction effectively provides a similar explanation for the eventual discrepancy between clocks A and B as did his 1918 article.

I have read several interpretations of the twin paradox one of which insists that the traveler’s clock does not (as Einstein expressed it in chapter 4) ‘go more slowly’ than the Earth clock but that the Earth clock, from the traveler’s point of view, ticks over at a faster rate than his own clock but only during the astronaut’s period of acceleration following turn-around however it is my understanding that the concept of time contraction was, for Einstein, an anathema.

Although I have included Einstein’s chapter 4 equation it would very much be appreciated if responses did not incorporate mathematical ‘proofs’ or explanations.

I am, as was Faraday, one of those annoying self-taught persons who has no comprehension of mathematics and, like Faraday, prefers simple, every-day language interpretations.

Einstein insisted that as far as the propositions of mathematics are certain, they do not refer to reality and I tend to agree.

Aw Freddie, we can at least try.

The fundamental causes of time dilation are the properties of light. [1] It's propagation speed in (matter free) space is constant and independent of its source. Light is the messenger between objects, big and small. Imagine two objects separated vertically or horizontally by a space, and not moving relative to the Earth lab. The objects exchange light signals periodically, once per second (1 tick). Then a force is applied to move both to the right. Because of [1], the speed of the objects does not change the speed of light, the objects are moving away from the source, and it takes longer to exchange the signals. Observers in the lab see the tick rate decrease.
Copy these objects, assemble as a clock, put the clock in a capsule with a pilot and launch it into space. As it moves past earth, the lab sees the clock rate slower than the lab clock.
In the capsule, the clock and the pilot are moving, therefore the rate of signal exchange is the same for both, i.e., slower. The pilot is thus not aware of the slower rate and sees his clock as 'normal'. The speed of the capsule alters the perception of the pilot (or device).
Because the capsule is moving at a constant speed (no acceleration), SR allows the pilot to assume he is not moving. If he chooses this option, he will calculate the lab clock rate to be slow. His other option is to accept his motion, adjust his time, and the strange things (anomalies) disappear.
Acceleration definitely makes the twin scenario asymmetrical, but does not explain the time difference, since it's a constant part of the travel time. As the duration of the trip increases, so does the time difference.
Time dilation has been experimentally verified, so it is real and not a mathematical contrivance.
Hope this helps.

 

[cos]

seriously, one of them is from the point of view of the stationary twin and the other is from the point of view of the traveling twin. there is no contradiction.

The contradiction, as I see it, is that some people insist that from the point of view of the traveling twin the Earth clock is incurring time contraction as he accelerates toward the planet following turn-around yet it is my understanding that Einstein refused to accept this idea.

It is a ‘contradiction’ of the laws of physics that the astronaut, having accelerated to an instantaneous velocity of close to the speed of light (thereupon generating a gamma factor of 40,000), would be of the opinion that the planet is spinning on its axis at around 64 million k-h.

 

[cos]

Although you prefer not to have the equations, here is a link, just in case:
http://www.math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html

In your opinion, does that link show that the traveler truly believes that his clock is not ticking over at a slower rate than it was before he commenced acceleration following turn-around but that it is the Earth clock that is physically ticking over at a faster rate?

In chapter 4 STR, as well as in his 1918, article Einstein effectively wrote that clock A ‘goes more slowly’ than B not that B ‘goes faster’ than A.

In your opinion, does that link show that the traveler truly believes, having accelerated to an instantaneous velocity of (or moving with uniform velocity at) close to the speed of light generating a gamma factor of 40,000, that the Earth is physically spinning on its axis at 64 million k-h?

 

[cos]

This is one place where you're going wrong. Not the last part. I agree with that. A theory is a mathematical abstraction. You can think of a theory as an approximate description of our universe or as an exact description of a fictional universe that resembles our own, but not as an exact description of our universe, so you got that part right. Your mistake is to think that there is a non-mathematical answer to this problem.

There is a non-mathematical solution to this problem (by which I take it you refer to the paradox) and Einstein provided same in chapter 4 as well as in his 1918 article. Observers accompanying both clocks know that they have incurred acceleration thus both of them know that their’s is the moving clock.

The equation provided by Einstein in chapter 4 was not a mathematical solution but a method of determining the amount of lag incurred by clock A.

The reason why there isn't, is that the only way that anyone can believe that the paradox exists in the first place is to use the theory (i.e. the mathematics) incorrectly. No one claims that the twin paradox is something that happens in the real world. The claim is that there's a paradox in the theory. But the "paradox" is just a mistake in a calculation, so there's no way to resolve it without examining the calculation and showing what the mistake is.

Doesn’t ‘the theory’ (i.e. the mathematics of chapters 1 through 3) show that the determinations are fully reciprocal? That from A’s point of view B’s clock slows down and from B’s point of view A’s clock slows down?

So I'm afraid that a resolution is always going to look like e.g. posts #3 and #142 in this thread. They require that you know some mathematics, or that you at least understand simultaneity in the context of inertial frames in special relativity.

As far as I’m concerned, Einstein provided a resolution of the twin paradox without requiring any knowledge of mathematics or understanding of simultaneity on my behalf.

My specific interest is in relation to the claim that the traveling twin is not allowed to determine that he is moving (thus that his is the clock that ‘goes more slowly’ than the Earth clock) but determines that the Earth clock incurs time contraction.

 

[cos]

The fundamental causes of time dilation are the properties of light. [1] It's propagation speed in (matter free) space is constant and independent of its source. Light is the messenger between objects, big and small. Imagine two objects separated vertically or horizontally by a space, and not moving relative to the Earth lab. The objects exchange light signals periodically, once per second (1 tick). Then a force is applied to move both to the right. Because of [1], the speed of the objects does not change the speed of light, the objects are moving away from the source, and it takes longer to exchange the signals. Observers in the lab see the tick rate decrease.

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.

Your depiction is, of course, a slightly more complicated version of the textbook light clock gedanken.

Copy these objects, assemble as a clock, put the clock in a capsule with a pilot and launch it into space. As it moves past Earth, the lab sees the clock rate slower than the lab clock.

In the capsule, the clock and the pilot are moving, therefore the rate of signal exchange is the same for both, i.e., slower. The pilot is thus not aware of the slower rate and sees his clock as 'normal'. The speed of the capsule alters the perception of the pilot (or device).

Because the capsule is moving at a constant speed (no acceleration), SR allows the pilot to assume he is not moving. If he chooses this option, he will calculate the lab clock rate to be slow. His other option is to accept his motion, adjust his time, and the strange things (anomalies) disappear.

My specific interest is in relation to the claim that the pilot, having accelerated following turn-around, is of the opinion that it is the Earth clock that is incurring time contraction which is, of course, on the basis that, as you point out, he assumes that he is not moving however, as you also point out, he can accept his motion whereupon the anomaly (that the Earth clock ‘is’ physically ticking over at a faster rate than it was before he started his return trip and that the planet is spinning faster on its axis) disappears. That’s the very point I’m trying to get across.

You have, perhaps albeit unintentionally, ratified my argument.

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.

Acceleration definitely makes the twin scenario asymmetrical, but does not explain the time difference, since it's a constant part of the travel time. As the duration of the trip increases, so does the time difference.

(As Einstein pointed out in chapter 4.)

Time dilation has been experimentally verified, so it is real and not a mathematical contrivance.
Hope this helps.

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.

 

[atyy]

In your opinion, does that link show that the traveler truly believes that his clock is not ticking over at a slower rate than it was before he commenced acceleration following turn-around but that it is the Earth clock that is physically ticking over at a faster rate?

No. The traveller believes that everyone has his own clock, which ticks according to his own proper time. His clock has accumulated less time, because he traveled a shorter path in spacetime.

In chapter 4 STR, as well as in his 1918, article Einstein effectively wrote that clock A ‘goes more slowly’ than B not that B ‘goes faster’ than A.

I haven't read the article. In my view, no one's clock ever goes faster or slower, it's just a question of the distance they cover distance in spacetime.

For this paragraph, consider just normal space, not spacetime. A friend and I fly from Boston to San Francisco. He flies directly across the United States. I fly from Boston to London to Singapore to Japan then to San Francisco. My route is obviously longer but that is not because my rulers contracted in length compared to my friend's rulers. That is just nonsensical (actually, it can make sense, but that's another story).

The time dilation comparison only makes sense in the usual, but very special case that one of the twins moves along a straight line in spacetime. In the general case, where both twins travel curly spacetime paths, and meet again, they will have aged by different amounts which is best explained by the different distances they covered in spacetime.

In your opinion, does that link show that the traveler truly believes, having accelerated to an instantaneous velocity of (or moving with uniform velocity at) close to the speed of light generating a gamma factor of 40,000, that the Earth is physically spinning on its axis at 64 million k-h?

I'm not sure your calculation is right, but I'll answer in the spirit of it - of course not. The twin will feel and be able to measure his acceleration, so he will know that he has changed reference frames, and understanding the theory of relativity he will always be able to correct his measurements to infer what the people on Earth are experiencing.

 

[yogi]

Some good observations COS - here is my take on the TP. First Einstein explained things in a way that seemed to make a difference as to which clock was put in motion - then in 1918 he shifted his argument to a pseudo G force - but a correct explanation should be able to resolve which clock logs the most time using an one way trip where there is no acceleration - for example have the A clock already in motion and start it when it passes Earth on its way to B clock. When A arrives at B it will have accumulated less time than the Earth clock and the B clock (they will read the same since they can by syced and are always in the same frame and not moving wrt to one another). So the whole paradox falls apart in that you are simply measuring one clock traveling between two fixed clocks and that will always lead to an actual difference in the time logged by the single clock when timed by the two clocks - moreover, it doesn't make any difference if the Earth and B are moving or if A is moving

 

[Fredrik]

Doesn’t ‘the theory’ (i.e. the mathematics of chapters 1 through 3) show that the determinations are fully reciprocal? That from A’s point of view B’s clock slows down and from B’s point of view A’s clock slows down?

Not quite. In order to be able to say something like that, we have to take "B's point of view" to always be the co-moving inertial frame. (B is the astronaut twin). If we do, then what you said is true at all points on B's world line except the turnaround event. That much is true, but this does not imply that A is younger when they meet again. To see that, you have to understand simultaneity in the context of inertial frames in special relativity. See my spacetime diagram for some of the details. (Use the link in my previous post).

 

[cos]

No. The traveller believes that everyone has his own clock, which ticks according to his own proper time. His clock has accumulated less time, because he traveled a shorter path in spacetime.

So you obviously agree with me that the traveler realizes that his clock is ticking over at a slower rate than it was before he started moving regardless of the fact that it appears to him to be ticking over at its normal rate.

 

[granpa]

So you obviously agree with me that the traveler realizes that his clock is ticking over at a slower rate than it was before he started moving regardless of the fact that it appears to him to be ticking over at its normal rate.

slower relative to what? to what it was before. suppose the Earth is moving at relativistic speed and the traveler is actually slowing down.

 

[cos]

Some good observations COS - here is my take on the TP. First Einstein explained things in a way that seemed to make a difference as to which clock was put in motion - then in 1918 he shifted his argument to a pseudo G force - but a correct explanation should be able to resolve which clock logs the most time using an one way trip where there is no acceleration - for example have the A clock already in motion and start it when it passes Earth on its way to B clock. When A arrives at B it will have accumulated less time than the Earth clock and the B clock (they will read the same since they can by syced and are always in the same frame and not moving wrt to one another). So the whole paradox falls apart in that you are simply measuring one clock traveling between two fixed clocks and that will always lead to an actual difference in the time logged by the single clock when timed by the two clocks - moreover, it doesn't make any difference if the Earth and B are moving or if A is moving

It is imperative that my discussion applies solely to Einstein's chapter 4 depiction as well as an out-and-return journey and that the traveler, or an observer accompanying clock A, be permitted to realize that his clock does incur time dilation (i.e. tick over at a slower rate than it did before he started moving) regardless of the fact that it appears to be ticking over at an unchanged rate.

It does make a difference 'if the Earth and B are moving or if A is moving' on the basis that, according to Einstein's chapter 4 depiction as well as his 1918 article, it is the accelerated clock that incurs time dilation not the unaccelerated clock (i.e. the Earth clock or Einstein's chapter 4 clock B.

 

[granpa]

does or doesn't make a difference?

 

[JesseM]

It is imperative that my discussion applies solely to Einstein's chapter 4 depiction as well as an out-and-return journey and that the traveler, or an observer accompanying clock A, be permitted to realize that his clock does incur time dilation (i.e. tick over at a slower rate than it did before he started moving) regardless of the fact that it appears to be ticking over at an unchanged rate.

It does make a difference 'if the Earth and B are moving or if A is moving' on the basis that, according to Einstein's chapter 4 depiction as well as his 1918 article, it is the accelerated clock that incurs time dilation not the unaccelerated clock (i.e. the Earth clock or Einstein's chapter 4 clock B.

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

 

[cos]

slower relative to what? to what it was before.

You answered your own question.

suppose the Earth is moving at relativistic speed and the traveler is actually slowing down.

Suppose the two clocks Einstein referred to in chapter 4 are, initially, moving at relativistic speed and, as Einstein pointed out, clock A moves to B's location; will that have any affect on Einstein's conclusion?

Are you suggesting that Einstein's chapter 4 depiction only applies if the reference frame in which clocks A and B are initially located is stationary - to which I respond - stationary relatively to what?

You wrote "suppose the Earth is moving at relativistic speed" to which I apply your question - relative to what?

Are you of the opinion that the Earth could be moving at relativistic speed? Is there any evidence to support such an idea? Is there any evidence which indicates that this could be a valid point of view? Is there any evidence to prove that the tooth fairy does not exist?

In my opinion physics should be a study of reality.

 

[granpa]

You answered your own question.



Suppose the two clocks Einstein referred to in chapter 4 are, initially, moving at relativistic speed and, as Einstein pointed out, clock A moves to B's location; will that have any affect on Einstein's conclusion?

Are you suggesting that Einstein's chapter 4 depiction only applies if the reference frame in which clocks A and B are initially located is stationary - to which I respond - stationary relatively to what?

You wrote "suppose the Earth is moving at relativistic speed" to which I apply your question - relative to what?

Are you of the opinion that the Earth could be moving at relativistic speed? Is there any evidence to support such an idea? Is there any evidence which indicates that this could be a valid point of view? Is there any evidence to prove that the tooth fairy does not exist?

In my opinion physics should be a study of reality.

it was a hypothetical question.
i believe jesse put it very well in his post. i will leave it at that.

 

[JesseM]

Are you suggesting that Einstein's chapter 4 depiction only applies if the reference frame in which clocks A and B are initially located is stationary - to which I respond - stationary relatively to what?

Einstein always made clear that the coordinates of inertial reference frames are to be defined relative to inertial rulers and clocks, and that there is no absolute meaning to the word "stationary"--you can only talk about an object being stationary relative to a particular ruler/clock system, not "stationary" in any absolute sense. The first postulate of relativity is that the laws of physics work exactly the same in different inertial reference frames (different ruler/clock systems moving relative to one another), so there can be no basis for thinking one system's coordinates are more "physical" than any other's. See Einstein's original 1905 paper on special relativity for some more on this stuff.

You wrote "suppose the Earth is moving at relativistic speed" to which I apply your question - relative to what?

Relative to a particular ruler/clock system. Again, there is no absolute notion of speed in relativity, because the laws of physics are identical in all inertial frames.

Are you of the opinion that the Earth could be moving at relativistic speed? Is there any evidence to support such an idea? Is there any evidence which indicates that this could be a valid point of view? Is there any evidence to prove that the tooth fairy does not exist?

There is no evidence whatsoever to support the notion that any inertial reference frame is physically preferred over any other, and therefore there is no evidence to support the idea that objects have any "true" speed. And of course, no matter what the object, you can find some frame where the object is at rest, some frame where it's moving at 0.4c, some frame where it's moving at 0.999999c, etc.

 

[cos]

It is true that, no matter which frame you choose, the average rate of ticking on the clock of the traveling twin must be slower than the average rate on the clock of the Earth twin. But you can find inertial frames where the Earth twin's clock ticks slower than the traveling twin's clock during the trip away from the Earth,

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).

then the traveling twin's clock ticks slower than the Earth twin's on the return journey after the turnaround; you can also find frames where the opposite is true, and the traveling twin's clock is slower on the outbound trip but faster on the inbound leg. So, there is no objective truth about whose clock is ticking slower at any given moment, even if the average of the traveling twin's clock is always slower than the Earth twin's clock over the course of the whole trip.

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?

The traveler moves identical distances at identical velocities so where does the eventual discrepancy come into existence?

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.

I repeat, it is specifically Einstein's effective depiction of the twin paradox with which I am concerned not other "frames where the opposite is true."

 

[atyy]

So you obviously agree with me that the traveler realizes that his clock is ticking over at a slower rate than it was before he started moving regardless of the fact that it appears to him to be ticking over at its normal rate.

Essentially. But let me edit your statement a bit to agree with my biases: So you obviously agree with me that the traveler realizes that his clock is ticking over at a slower rate compared to the coordinate time of a particular inertial reference frame than it was before he started moving regardless of the fact that it is ticking over at its normal rate.

I haven't read the two articles you mentioned, so I don't know if they are right or wrong (and anyway, I may be wrong). But I should caution you that Einstein also published a paper saying that the General Theory of Relativity could not possibly be a correct theory.

 

[granpa]

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.

 

[granpa]

but suppose the traveler doesn't move at such high speed. suppose he only moves at a few hundred km/s. why would he presume that the Earth is stationary? the Earth is moving around the galaxy which is itself moving. we can try to measure some of this velocity by the cmb but we may never know exactly how fast the Earth is really moving. if indeed it can be said to have a 'real' velocity.

 

[cos]

Einstein always made clear that the coordinates of inertial reference frames are to be defined relative to inertial rulers and clocks, and that there is no absolute meaning to the word "stationary"--you can only talk about an object being stationary relative to a particular ruler/clock system, not "stationary" in any absolute sense. The first postulate of relativity is that the laws of physics work exactly the same in different inertial reference frames (different ruler/clock systems moving relative to one another), so there can be no basis for thinking one system's coordinates are more "physical" than any other's.

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.

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.

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'.

 

[cos]

it was a hypothetical question.

It was a question which attempted to disparage my argument - hypothetical or otherwise.

 

[cos]

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.

That's what I said!

 

[granpa]

That's what I said!

any observer can always calculate what any other observer in any other frame will observe or calculate. the traveler knows what a person in the Earth frame would say his velocity and clock rate were.

 

[cos]

but suppose the traveler doesn't move at such high speed. suppose he only moves at a few hundred km/s. why would he presume that the Earth is stationary? the Earth is moving around the galaxy which is itself moving. we can try to measure some of this velocity by the cmb but we may never know exactly how fast the Earth is really moving. if indeed it can be said to have a 'real' velocity.

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.

 

[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!