Twin Paradox (thorough explanation needed)

[Eli Botkin]

Eli,

My understanding is that twin B can see the twin A clock ticking faster than his own "while B himself is non-inertial". He has to, for otherwise he could not agree with twin A on their relative aging when they are again co-located. The faster ticking is not the result or proper time speeding up, but only a change in the twin B POV due to frame transitioning, which causes the A-clock to advance faster along its worldline (per B).

GrayGhost

GrayGhost:
The phrase “twin B can see the twin A clock ticking faster than his own” can be misunderstood. I expect, though, that in this instance your SR expertise is not meaning to interpret that as “seeing through a telescope.” Rather I assume you mean the time that B assigns to events on A’s worldline based on B’s knowledge of the Lorentz transformation equations. If so, then I maintain that an observer will always assign fewer ticks to a moving clock than he experiences on his own clock. This keeps the dilation ratio < 1 even in the instantaneous inertial frames within a period of acceleration. The ratio would rise to 1 as the relative velocity approaches 0, and then fall again.

This assigned time and its resulting dilation rate are not measurable quantities. To see why there is agreement when they are again co-located, I suggest that the more usual meaning of “see” be used. If either observer views the other clock through a telescope, they will first see a slower clock than their own followed by a faster clock (Doppler). [Both of course see the same Doppler shifts.] B (who does the accelerating) will see the two Doppler-shifted periods of A’s clock-images for equal periods on his own (B’s) clock. However, A will see the slower image of B’s clock for a longer time (on A’s clock) than he sees the faster portion. Therefore, at get-together, B’s clock reads less than A’s. And they’ve both been witness to why this has occurred.

Thanks for “listening.”

 

[Mentz114]

If so, then I maintain that an observer will always assign fewer ticks to a moving clock than he experiences on his own clock.

All observers agree on the number of ticks. A tick is an event and events cannot be 'transformed away'. The interval between the ticks is observer dependent. I don't think it affects your argument if you amend the phrase in question.

 

[GrayGhost]

GrayGhost: The phrase “twin B can see the twin A clock ticking faster than his own” can be misunderstood. I expect, though, that in this instance your SR expertise is not meaning to interpret that as “seeing through a telescope.”

Yes, you're right. This is the 2nd time I've stated this poorly in this forum, as it's a bad habit when posting too quickly. Thanx. The faster ticking A-clock (per B) is ascertained from looking at the date/time, track, and navigation data after the fact. The faster ticking clock is "determined" (not seen) by analysis of the data while ignoring the relativistic effects during analysis ... which seems to be the way most folks look it unfortunately (I believe that an incorrect approach).

You're right, the light signals received from twin A will be doppler shifted, and so "the images" of the moving clock can show a clock ticking faster or slower than yor own "due to doppler effects". To know the real relative rate of 2 clocks, the spacetime dilation needs to be determined (and accounted for) from its corresponding relativistic doppler effects for twin B to know the correct A-velocity as it applies to relativity. If NOT DONE, then twin B will determine that twin A moved superluminally and the A-clock ticked (at times) faster than his own per himself. However IF DONE, then the A-velocity will always be luminal (or less) as it should be, and the A-clock will never tick faster than Bs per B.

At any rate, thanks for the correction there Eli.

GrayGhost

 

[Gulli]

@JesseM

I think I understand now.

The Earth and the colony share the same frame of reference and so agree with each other on when an event took place. The colony does not have to agree on simultaneity with the spaceman but they have confirmation from Earth (because Earth and the spaceman were at the same position during the event of the spaceship speeding by) that the spaceship passed them in 600.000. So when the signal from the spaceship speeding past Earth reaches the colony in 600.020 the signal will read "it is now 600.000 and a spaceship just passed us", so no head start. The spaceman does not have to agree on simultaneity with the Earth: the spaceman and Earth agree that the spaceship passed Earth in what they both consider to be 600.000, but the spaceman does not have to agree this event was simultaneous to the year 600.000 on the colony and can therefore infer a head start. The value of the head start depends on the relative velocity between the Earth/colony frame and that of the spaceman, and on the distance between Earth and the colony.

Thanks for this piece of enlightenment!

 

[JesseM]

@JesseM

I think I understand now.

The Earth and the colony share the same frame of reference and so agree with each other on when an event took place. The colony does not have to agree on simultaneity with the spaceman but they have confirmation from Earth (because Earth and the spaceman were at the same position during the event of the spaceship speeding by) that the spaceship passed them in 600.000. So when the signal from the spaceship speeding past Earth reaches the colony in 600.020 the signal will read "it is now 600.000 and a spaceship just passed us", so no head start. The spaceman does not have to agree on simultaneity with the Earth: the spaceman and Earth agree that the spaceship passed Earth in what they both consider to be 600.000, but the spaceman does not have to agree this event was simultaneous to the year 600.000 on the colony and can therefore infer a head start. The value of the head start depends on the relative velocity between the Earth/colony frame and that of the spaceman, and on the distance between Earth and the colony.

Thanks for this piece of enlightenment!

Yup, sounds like you've got it! Glad I could help.

 

[Eli Botkin]

All observers agree on the number of ticks. A tick is an event and events cannot be 'transformed away'. The interval between the ticks is observer dependent. I don't think it affects your argument if you amend the phrase in question.

Mentz114:
You are, of course, correct to say ” All observers agree on the number of ticks” on a worldline.

I should not carelessly assume and then omit clarifying phrases.
I should have written:
If so, then I maintain that an observer will always assign fewer ticks to a moving clock during any sequence of ticks on his own clock. “Fewer ticks” is what is implied by the term “time dilation.

Said another way: Time on a moving clock is “slower “ when compared to the “non-moving” clock. And again: dt/dT < 1 if t is time measured on the moving clock and T is time measured on the “non-moving” clock. Of course, the two clocks must be equivalent in every way (except for where they’re placed).

 

[Gulli]

@JesseM

So then am I correct in assuming the twin paradox has the exact same solution as my spaceman/colony problem because in the twin paradox the traveler returns and so has to turn around somewhere and thus passes a point where his velocity (well, at least the radial component) in relation to Earth is zero and that this point is analogous to the colony?

 

[DTThom]

Everyone is quick to teach and slow to learn. The tens of thousands of documents purporting to explain the Twins Paradox illustrates this as well as anything.

In the process of being so quick to teach, the purveyors of these documents have overlooked the simple and obvious truth about clock rates.

Why do people make things so difficult for themselves? Did it never occur to them that when two reunited clocks (meaning they are now once again at the same place-moment) show an ACTUAL disparity in their recorded time, that there must necessarily have been an ACTUAL difference in clock rates involved while they were in relative motion with each other?

One should never suggest (as they so often do) that there was some sort of "jump in time" involved with the change of inertial frame (meaning at the turn-around point). The simple act of starting a clock as an inbound astronaut passes an outbound astronaut cannot possibly create a "jump in time". (Remember, the outbound astronaut hands off his clock reading to the inbound astronaut.)

The time contraction formula [t' = t * sqr rt of (1 - v^2)] is not linear. That is why the party who changes frames to bring the two parties back together will register the least amount of time on his clock with the symmetry of the situation preserved.

The actual distances and speeds relative to the universe will vary depending on which party changes frames, but the parties involved cannot possibly detect that. That is in keeping with the postulates and deductions of special relativity.

Time-keeping, distance and speed are interminably bound in one equation. Therefore, actual differences in clock rates implies actual length contraction dependent on actual speed relative to the universe. Actual length contraction works in combination with actual time-keeping contraction to preserve the symmetry of measures across inertial frames.

There is clock functioning at every level, dependent on actual light speed, at even the atomic level. Our observations and measuring paradigms of every nature are constrained by the speed of light, as is our "synchronizing" of clocks.

Special relativity can be charted out in actual terms (absolute terms), where light speed is constant in an actual sense. All the results of special relativity, including the consistent measured speed of light, fall naturally into place when charting these actualities against the (experimentally undetectable) rest state of the universe.

Actual time-keeping and length contraction arise naturally from the fact that all phenomena are dependent on the speed of light, which is itself invariant in actuality, being massless.

Consider that A.P. French writes on page 150 of Special Relativity: "Note, though, that we are appealing to the reality of A's acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twin paradox exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large."

And I feel very sorry for any physicist who doesn't understand that.

Michio Kaku states on page 80 of Einstein's Cosmos that bringing the twins together "determines which twin was "really" moving."

Martin Gardner writes on page 114 of Relativity Simply Explained: "There is one all important difference between the relative motion of the astronaut and the relative motion of the stay-at-home. The stay-at-home does not move relative to the universe."

Both Kaku and Gardner were using the simplest of twins paradox scenarios, in which one party is assumed to be at rest with the cosmos. But that need not be the case. There can be any number of "in between" situations, leading to a lesser time differential. It is also not necessary for the twins to reunite to determine which one was "really moving". The noted asymmetry (noted by both parties) in the time-keeping difference builds incrementally, beginning at the moment of inertial change for one party, when radio or light signals are regularly sent forth and back to check on current clock status.

One should do a search on Einstein's clock synchronization, and its bearing on spacetime diagrams. He or she will find that the notorious "jump in time" is built into that clock synchronization, because it is a one way synchronization, which gets instantly replaced with a different synchronization when a new inertial frame is adopted.

There is all the difference of night and day between predicting and explaining. We can use Einstein's clock synchronization and spacetime to predict a time differential, but we must look at relativity in the universal frame of reference to explain not only that time differential, but also all the mutually symmetrical measures made across inertial frames.

The preceding remarks were copied off my copyrighted web document.

By the way, "observations" and "measures" are strictly synonomous, and constrained by light speed. You might think that a "visual observation" is something different than measuring, but the combination of eye to brain processing is precisely a form of measuring. Identically, biological aging is precisely synonomous with clock functioning of all types, right down to the atomic level and things such as the Doppler effect.

Also, the following phrase is a meaningless muddling of terminology: ".. the distance between them really is shorter in the spaceship's own rest frame.

"Really" means "really", a reality independent of anyone's inertial frame.

You may call it the "God's eye view" or the "view from a higher dimension". Such view is the instantaneous view, free from the constraint of light speed, which is finite.


---------------------------------------------------------------
"First they tell you you're wrong and they can prove it;
then they tell you you're right but it isn't important;
then they tell you it's important but they knew it all along."

Charles Kettering, former head of General Motors
---------------------------------------------------------------

 

[DTThom]

I need to clarify one paragraph in the mid portion of my previous post. Here is the first half of that paragraph, better stated:

Both Kaku and Gardner were using the simplest of twins paradox scenarios, in which one party is assumed to be at rest with the cosmos, and the other party both has motion relative to the universe and changes frames. But that need not be the case. There can be any number of "in between" situations -- such as both parties having motion relative to the universe, with still only one party changing frames (same time differential); or both parties changing frames, leading to a lesser time differential.

 

[Dale]

Hi DTThom, welcome to PF!

Why do people make things so difficult for themselves? Did it never occur to them that when two reunited clocks (meaning they are now once again at the same place-moment) show an ACTUAL disparity in their recorded time, that there must necessarily have been an ACTUAL difference in clock rates involved while they were in relative motion with each other?

Consider two cars that travel from Miami to New York, one travels via Washington DC and the other travels via Denver. When the two cars are reunited in New York they show an ACTUAL disparity in their recorded mileage. Does that necessarily imply that there must necessarily have been an ACTUAL difference in their odometer rates? No, both odometers could still have accurately measured one mile per mile and yet obtained different odometer readings.

 

[JesseM]

@JesseM

So then am I correct in assuming the twin paradox has the exact same solution as my spaceman/colony problem because in the twin paradox the traveler returns and so has to turn around somewhere and thus passes a point where his velocity (well, at least the radial component) in relation to Earth is zero and that this point is analogous to the colony?

If you choose to analyze the twin paradox from the perspective of the Earth frame this could be one way of looking at it, but the answer to which twin ages less is frame-indpendent, you could equally well analyze everything in the frame of an inertial ship which was traveling at the same velocity as the traveling twin during his outward journey, but which kept traveling inertially in the same direction when the traveling twin turned around.

 

[bobc2]

Hi DTThom, welcome to PF!Consider two cars that travel from Miami to New York, one travels via Washington DC and the other travels via Denver. When the two cars are reunited in New York they show an ACTUAL disparity in their recorded mileage. Does that necessarily imply that there must necessarily have been an ACTUAL difference in their odometer rates? No, both odometers could still have accurately measured one mile per mile and yet obtained different odometer readings.

Exactly. Right on the money, DaleSpam. (Could it be that each twin travels his own world line at light speed--but their world lines have different path lengths, just like your two different paths from Miami to New York?)

 

[DTThom]

Hi DTThom, welcome to PF!Consider two cars that travel from Miami to New York, one travels via Washington DC and the other travels via Denver. When the two cars are reunited in New York they show an ACTUAL disparity in their recorded mileage. Does that necessarily imply that there must necessarily have been an ACTUAL difference in their odometer rates? No, both odometers could still have accurately measured one mile per mile and yet obtained different odometer readings.

I can easily deduce that the two odometers in your example traveled actual different distances, due to the actual difference showing up in the number of odometer ticks at the same place-moment in New York. I can identically deduce that the two reunited clocks I spoke of ticked an actual different number of times while they were apart. I don't care what distance the clocks traveled. You provided a distance problem. An odometer is bound to read true to the distance of the road. I provided a clock ticking problem, and the number of ticks are dependent on the combination of speed and distance covered relative to the universe.

* I care only about the number of clock ticks involved, compared to the number of clock ticks registered on the clock at rest with the universe. *

You can do relativity the easy way, or you can make it hard for yourself, and never explain the missing time of the twins paradox.

Also, you cannot travel on a world line. A world line is a spacetime construct, not something physical.

 

[JesseM]

I can easily deduce that the two odometers in your example traveled actual different distances, due to the actual difference showing up in the number of odometer ticks at the same place-moment in New York. I can identically deduce that the two reunited clocks I spoke of ticked an actual different number of times while they were apart. I don't care what distance the clocks traveled. You provided a distance problem. An odometer is bound to read true to the distance of the road. I provided a clock ticking problem, and the number of ticks are dependent on the combination of speed and distance covered relative to the universe.

There is a rather precise analogy between problems involving paths through space and odometers, and problems involving paths through spacetime and clocks--see [post=2972720]this post[/post].

* I care only about the number of clock ticks involved, compared to the number of clock ticks registered on the clock at rest with the universe. *

The phrase "at rest with the universe" has no meaning in relativity. You can choose any inertial frame you like, each of which with a different definition of which objects are "at rest", and they will all make the same prediction about how much time elapses on a given clock between two events on its world line (like the event of leaving a space station and the event of returning to it).

Also, you cannot travel on a world line. A world line is a spacetime construct, not something physical.

"Travel on a world line" is just a common shorthand for taking a path through space and time that matches the world line (i.e the coordinates are the same). And how are you defining "something physical"? Would you say a path through space is also not "something physical" and that therefore you cannot travel on a path through space?

 

[DTThom]

Exactly. Right on the money, DaleSpam. (Could it be that each twin travels his own world line at light speed--but their world lines have different path lengths, just like your two different paths from Miami to New York?)

You can travel through space (a distance dimension) at light speed (a speed dimension).

You cannot travel through spacetime at any speed. No meaning can be attached to such a statement.

Distance = speed * time (relativity or not)

"At rest with the universe" has a clear meaning, relativity or not. Relativity can be fully developed in absolute (universal) terms, and in doing so, it is seen just why it is that we cannot determine our true state of motion relative to the universe.

For the benefit of the recent posters, I'll present these few more comments:


One can hold two reunited clocks in ones hand and see that one registered more clicks than the other WHILE they were apart. "While" sounds like a "time" word to me. We must say that the two clocks ticked at different rates, i.e., ticks per unit "time". That "time" can only be some "time" by which to distinguish the "time" recorded by the two clocks. The only way to avoid circular reasoning is to acknowledge a "time" as kept by a clock at rest with the universe.

Light has a finite and constant speed relative to the universe. It is the speed by which we define all lesser speeds. That is your clue as to the functioning of a photon clock, which must necessarily complete fewer cycles as it (the clock itself) increases its translatory speed through the universe.

Without absolutes (actualities) we are left with only circular definitions (strictly relative definitions). But the time differential showing up at the same place-moment does not fit with circular definitions. The disparity in the clock readings is real. Therefore the two clocks did actually tick a different number of times while they were apart.

====================

I have better things to do. Goodbye.

 

[Dale]

I can easily deduce that the two odometers in your example traveled actual different distances, due to the actual difference showing up in the number of odometer ticks at the same place-moment in New York. I can identically deduce that the two reunited clocks I spoke of ticked an actual different number of times while they were apart.

Exactly. Note that this does not imply that the rate of either clock has changed, simply that they are measuring the length of different paths through spacetime (aka worldlines).

I don't care what distance the clocks traveled. You provided a distance problem. An odometer is bound to read true to the distance of the road.

Similarly, a clock is bound to read true to the spacetime interval of its worldline.

You can do relativity the easy way, or you can make it hard for yourself

I agree completely. I find the geometric approach by far the easiest and most intuitive.

A world line is a spacetime construct, not something physical.

A worldline is simply a path through spacetime. It is every bit as physical as a path through space.

 

[JesseM]

"At rest with the universe" has a clear meaning, relativity or not. Relativity can be fully developed in absolute (universal) terms, and in doing so, it is seen just why it is that we cannot determine our true state of motion relative to the universe.

Why do you believe there is such a thing as "true" rest relative to the universe? Given that relativity has no need for such a concept and it does just fine at predicting the results of all measurements, this seems like a kind of religious faith.

One can hold two reunited clocks in ones hand and see that one registered more clicks than the other WHILE they were apart. "While" sounds like a "time" word to me. We must say that the two clocks ticked at different rates, i.e., ticks per unit "time". That "time" can only be some "time" by which to distinguish the "time" recorded by the two clocks.

Yes, it's coordinate time in different inertial frames. But different inertial frames disagree about the relative rates the two clocks were ticking at different phases of the trip--for example one frame may say the traveling clock was ticking slower than the inertial clock for both the inbound and outbound leg of its journey (this would be true in the rest frame of the inertial clock), another frame may say the traveling clock was ticking faster than the inertial clock during the outbound leg but slower than the inertial clock during the inbound leg (this would be true in the inertial frame where the traveling clock was at rest during the outbound leg), and a third may say the traveling clock was ticking slower than the inertial clock during the outbound leg and faster during the inbound leg (this would be true in the inertial frame where the traveling clock was at rest during the inbound leg). All these frames would nevertheless agree that the total elapsed time of the traveling clock was less than the inertial clock, so it had a slower rate on average over the whole trip, even if they disagree about the relative rates during particular phases of the trip.

Again this is analogous to odometers, as you can see if you read my [post=2972720]linked post[/post] on the geometric analogy. Instead of talking about the rate that a clock is ticking relative to coordinate time t in some inertial frame, we can talk about the rate a car's odometer reading is increasing relative to the car's coordinate position x along the x-axis in some Cartesian spatial coordinate system. Different Cartesian coordinate systems with their axes oriented at different angles will disagree about (change in odometer/change in x-coordinate) during different phases of the trip, but they will all be able to calculate the total change in odometer reading as a function of how (change in odometer/change in x-coordinate) varies along the path (and the rate at each point is just a function of the path's slope at that point), and will all agree that the car that traveled in a straight line had a smaller total change in odometer reading than the one that didn't. This is just like how different inertial frames disagree about (change in clock reading/change in t-coordinate) during different phases of the trip, but they can all calculate the total change in clock reading as a function of how (change in clock reading/change in t-coordinate) varies along the path (and the rate at each point is just a function of the clock's speed at that point), and will all agree that the clock that moved inertially had a greater total change in clock reading than the one that didn't.

The only way to avoid circular reasoning is to acknowledge a "time" as kept by a clock at rest with the universe.

No, you can just talk about coordinate time in different inertial frames, without the need to single out one frame as the one that's "at rest with the universe". In the odometer example, all Cartesian coordinate systems agree on the average rate of (change in odometer reading/change in x-coordinate) for both cars, and agree that the car with the straight path had a smaller average rate than the one with the non-straight path, but presumably you wouldn't say here that we must somehow conclude that we must single out one Cartesian coordinate system as the one with the "true" x-axis direction, so that its value of (change in odometer reading/change in x-coordinate) at any point on a car's path is the only "true" value.

Light has a finite and constant speed relative to the universe. It is the speed by which we define all lesser speeds.

No, light has a constant speed relative to all inertial frames, not "relative to the universe". If two events on the worldline of a light ray have (difference in position/difference in time) = c in the coordinates of one inertial frame, then these same events also have (difference in position/difference in time) = c in the coordinates of any other inertial frame, even though different frames may disagree about the specific values of (difference in position) and (difference in time) for a given pair of events. You can use the Lorentz transformation to verify that this is the case.

Without absolutes (actualities) we are left with only circular definitions (strictly relative definitions).

"Relative" is not the same as "circular". Do you have a problem with defining the position of various points on a 2D plane relative to a Cartesian x-y coordinate system, even though you know you have a choice of different Cartesian coordinate systems with their axes pointing at different angles? And note that you can use any given coordinate system to calculate absolute quantities that are the same from one coordinate system to another, like the straight-line distance between two points--if coordinate system #1 assigns points A and B coordinates (x_A, y_A) and (x_B, y_B), then it will calculate the straight line distance between them as \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} (Pythagorean theorem), while coordinate system #2 might assign the same points A and B coordinates (x'_A, y'_A) and (x'_B, y'_B), and therefore compute the distance as \sqrt{(x&#039;_B - x&#039;_A)^2 + (y&#039;_B - y&#039;_A)^2}. But despite these different coordinates and calculations they will both end up with the same value for the straight-line distance between A and B, i.e. it will be true that \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(x&#039;_B - x&#039;_A)^2 + (y&#039;_B - y&#039;_A)^2}. It's exactly the same in relativity, where there are plenty of absolutes that different coordinate systems agree on, like the spacetime interval between two events which is computed with the formula \sqrt{(x_B - x_A)^2 - c^2 (t_B - t_A)^2}.

 

[bobc2]

bobc2 comment after DaleSpam's: Exactly. Right on the money, DaleSpam. (Could it be that each twin travels his own world line at light speed--but their world lines have different path lengths, just like your two different paths from Miami to New York?)

You can travel through space (a distance dimension) at light speed (a speed dimension).

You cannot travel through spacetime at any speed. No meaning can be attached to such a statement.

I think we understand the context when we refer to "moving along the world line at light speed." It's not fruitful at this point to rehash that point (Herman Weyl: "...an observer crawls along his world line.", etc.). But, in any case, the absolute universe you imply would necessarily be characterized as a "block time" model. You've got a big can of worms on your hands if you pursue that concept very far.

"At rest with the universe" has a clear meaning, relativity or not. Relativity can be fully developed in absolute (universal) terms, and in doing so, it is seen just why it is that we cannot determine our true state of motion relative to the universe.

No further comment is needed. DaleSpam's and JesseM's last posts sum it up pretty well.

 

[GrayGhost]

You can travel through space (a distance dimension) at light speed (a speed dimension). You cannot travel through spacetime at any speed. No meaning can be attached to such a statement.

Hello DTThorn,

Try this on for size maybe ... You and I are inertial, and of a luminal relative velocity. We each hold ourself stationary passing only thru time, and the other passing thru space over time. Which is it? Are we each moving thru space w/o realizing it, or are neither of us moving? If the later, then relative motion is an illusion of sort. If the former, then why do we not realize we pass thru space while stationary?

My opinion is that the former is true. We each move thru a spacetime continuum, and I'd personally agree with bobc2 that we all travel at c thru the continuum. That our relative motion depends only on "the extent of parallelness" of our velocity vectors thru 4d spacetime, otherwise known as our arrows-of-time. When parallel, then zero relative motion. The Minkowski model revealed this notion.

As to why we do not perceive ourselves passing thru space while inertial, is yet another matter. And an interesting one at that, IMO. We perceive space and time differently, yet we know that the inertial fellow over yonder is indeed passing thru 3-space while he knows himself only to transcend time. It seems clear IMO that "space exists", although we measure it differently. Clearly, there is more to time than what is presently known. At this point, we know there is "a spacetime" that exists, because we measure it and we exist in it. It's not just that space and time are interwoven, but also matter and energy ... so all 4 working in concert in unison, all a part of the same single mechanism. That said, it's time for a party!

GrayGhost

 

[yogi]

"Cause" is a vague word, but in SR the fact that one twin accelerates between meetings while the other does not is a necessary and sufficient condition for the twin that accelerated to be younger when they reunite.

It's irrelevant which frame you take to be "at rest", all frames calculate the same answer for the ages of the twins when they reunite.

Wrong - the frame selected as "at rest" defines the proper distance which must be traveled between the two clocks that measure time in the "at rest" frame. This distance will be longer than the distance measured between the two points by the moving twin. If the A frame is selected as the at rest frame and two clocks are separated by a distance L in the A frame both the traveler in the B frame and the fixed twin in the A frame will agree that clocks in the A frame accumulate more time than the moving twins clock during the one way experiment

Acceleration is not a necessary condition to resolve the twin's time difference - two one way trips with clocks synchronized on the fly (close pass by) at the start and stopped when the traveler passes the second clock on the fly can simply be doubled - no acceleration is involved

If the B frame is taken as at rest - then the experiment is different since the proper length will be laid out in the B frame as the distance between two clocks in that frame and at the end of the one way trip both twins willl agree that it is the twin in the B frame who has aged most .,

A round trip is not necessary - to solve problems such as this it is helpful to reduce the situation to one which can eliminate all acceleration and show there is no ambiguity by substituting two one way trips for a round trip

For a true round trip, GPS satellites verify the time dilation due to SR every time they complete an orbit - the satellite is in an inertial frame during the entire experiment (one orbit)and the Earth - while nonetheless in a constant G field, still provides a valid platform for synchronizing ground clocks

 

[JesseM]

Wrong - the frame selected as "at rest" defines the proper distance which must be traveled between the two clocks that measure time in the "at rest" frame.

You're misusing the phrase "proper distance", proper distance is measured between a pair of events (or along a spacelike worldline), and it's frame-invariant.

This distance will be longer than the distance measured between the two points by the moving twin. If the A frame is selected as the at rest frame and two clocks are separated by a distance L in the A frame both the traveler in the B frame and the fixed twin in the A frame will agree that clocks in the A frame accumulate more time than the moving twins clock during the one way experiment

I wasn't talking about a one-way experiment, I was talking about the twin paradox. However, you are wrong that in the one way experiment both frames will agree that the A twin accumulates more time, you are failing to take into account the relativity of simultaneity here. In the rest frame of twin B, the event of him arriving at the distant destination is simultaneous with an event on A's worldline where A is younger. For example, if B travels at 0.6c to a destination which is 15 light-years away in A's frame, then in A's frame it takes 15/0.6 = 25 years to get there, but B is only 25*0.8 = 20 years older on arrival. But in B's frame, the moment he arrives at the destination with his clock showing an elapsed time of 20 years, this is simultaneous with the event of A's clock showing an elapsed time of 20*0.8=16 years.

Acceleration is not a necessary condition to resolve the twin's time difference - two one way trips with clocks synchronized on the fly (close pass by) at the start and stopped when the traveler passes the second clock on the fly can simply be doubled - no acceleration is involved

In this case you aren't dealing with the elapsed time on anyone clock, but rather the sum of two intervals on different clock worldlines. The total path that you are adding the times along is still a bent one in spacetime, no single clock undergoes acceleration but the fact remains that a "straight" path between two points in spacetime will always show a greater proper time than a non-straight path between the same two points, the only relevance of "acceleration" is that it is the cause of a single object's path through spacetime being non-straight.

Again think of the geometric analogy--if you have two roads between points 1 and 2 on a flat surface, and one is straight while the other is not, a car driving along the straight road will always show a smaller increase in its odometer reading than a car on the non-straight road. You're free to imagine two cars traveling along a non-straight V-shaped road, which "synchronize" their odometers at the moment they pass at the tip of the V, so that neither car actually has to deviate from a straight path, but it will still be true that the total odometer increase along the V-shaped road is greater than the odometer increase along the straight road between points 1 and 2.

If the B frame is taken as at rest - then the experiment is different since the proper length will be laid out in the B frame

Why do you say that? You are free to repeat exactly the same experiment while considering the B frame to be at rest, in my above example the distance between A and the destination (both of which are moving in the B frame) could be 0.8*15=12 light years in the B frame, matching the original statement that the distance would be 15 light years in the A frame. The statements about aging in each frame would also be identical, in the A frame A would have aged 25 years while B had aged 20 years at the moment B passed the destination, while in the B frame A would have aged 16 years while B had aged 20 years at the moment B passed the destination.

For a true round trip, GPS satellites

Not SR but GR, so there's no single inertial frame that can cover the entire trip, and the twin paradox is specifically assuming one twin traveled inertially so that the SR time dilation formula applies.

 

[GrayGhost]

You're misusing the phrase "proper distance", proper distance is measured between a pair of events (or along a spacelike worldline), and it's frame-invariant.

I wasn't talking about a one-way experiment, I was talking about the twin paradox. However, you are wrong that in the one way experiment both frames will agree that the A twin accumulates more time, you are failing to take into account the relativity of simultaneity here.

I'm just curious ... Let's say twin B starts at rest with A and ends at rest on planet X (at rest with the A frame). Would you say B aged less than A, or would you say it cannot be said who aged less w/o being "co-located at rest with each other" twice?

GrayGhost

 

[JesseM]

I'm just curious ... Let's say twin B starts at rest with A and ends at rest on planet X (at rest with the A frame). Would you say B aged less than A, or would you say it cannot be said who aged less w/o being "co-located at rest with each other" twice?

GrayGhost

I'd say it can't be said who aged less in any objective frame-independent way, as you could always take the perspective of an inertial observer who was at rest relative to B during B's travel, but did not accelerate when B did and just continued past X inertially. In this observer's frame, A ages less than B between the time B departs A and the time B arrives at planet X, and from there on they both age at the same rate so A remains younger than B in this frame.

 

[MikeLizzi]

Hi DTThom,
I don’t mean to interrupt your dialog with others but I am trying to find a statement that you quoted from the A.P. French book on Special Relativity.

You posted,
Consider that A.P. French writes on page 150 of Special Relativity: "Note, though, that we are appealing to the reality of A's acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twin paradox exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large."

I was shocked when I read that and searched for it in my A.P. French book but couldn’t find it. Page 150 in my book is in Chapter 5 “Relativistic Kinematics” in the section entitled “Looking at Moving Clocks and Other Objects”. Could you identify the section in your book from which that quote was taken?

 

[JesseM]

I was shocked when I read that and searched for it in my A.P. French book but couldn’t find it. Page 150 in my book is in Chapter 5 “Relativistic Kinematics” in the section entitled “Looking at Moving Clocks and Other Objects”. Could you identify the section in your book from which that quote was taken?

according to google books that note is at the bottom of p. 150, maybe you have a different edition?

 

[Gulli]

"At rest with the universe" has a clear meaning, relativity or not. Relativity can be fully developed in absolute (universal) terms, and in doing so, it is seen just why it is that we cannot determine our true state of motion relative to the universe.

If we knew the locations and mass of all objects in the universe we could calculate the center of mass of the universe. Being stationary to that center of mass sort of means being stationary to the universe, but 1) we do not have sufficient data to calculate the center of mass of the universe, 2) most galaxies (most likely including our own) are not stationary to the center of mass of the universe, so it wouldn't apply to any of the usual traveler problems and 3) relativity works just fine without assuming a universal rest frame.

 

[George Jones]

according to google books that note is at the bottom of p. 150, maybe you have a different edition?

In my copy,

http://books.google.com/books?id=mi...ook_result&ct=result&resnum=1&ved=0CCcQ6AEwAA,

the quote is at the bottom of page 156.

 

[JesseM]

If we knew the locations and mass of all objects in the universe we could calculate the center of mass of the universe.

The universe is not imagined to have a center of mass in current cosmological models, the FLRW models assume that on large scales the distribution of matter is fairly homogeneous throughout all of space (regardless of whether space is finite or infinite). These models might be oversimplifications in the context of chaotic inflation though, I'm not sure.

 

[Gulli]

The universe is not imagined to have a center of mass in current cosmological models, the FLRW models assume that on large scales the distribution of matter is fairly homogeneous throughout all of space (regardless of whether space is finite or infinite). These models might be oversimplifications in the context of chaotic inflation though, I'm not sure.

The universe contains objects with mass, so there most definitely is a theoretical center of mass. Though it may be impossible to calculate because some mass may be outside the observable universe (there's no way for us to know), likewise I can imagine the fact that the gravitational force has a finite "speed" (the speed of light) would cause some problems if one were to try to calculate an actual universal center of mass.

 

[Dale]

The universe contains objects with mass, so there most definitely is a theoretical center of mass.

Not necessarily. If you have a uniform density then the center of mass corresponds to the geometric center, but many manifolds do not have a gemoetric center. For example, a simple 2-sphere is a manifold with no center. A 2-sphere only has a center if embedded in a flat 3-D manifold, and even then the center is not part of the 2-sphere.