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The symmetrical clock paradox

Some people get lost in the twin paradox presented here (with accelerations and all), so maybe a simpler, symmetrical paradox is in order. So, here is that "gedanken" experiment:

Let's have two identical rockets, carrying two identical high precision (in today's technology that would be atomic) clocks, with digital displays. The displays show the number of ticks (cycles) that each clock recorded (this could be for instance number of nanoseconds - a simple integer) since last reset. The rockets and clocks include a mechanism that fires the thrusters on detection of blue light and resets the clocks on detection of red light.

These two rockets are placed in space, away from any gravitational fields, so that they are pointed to one another, on a straight line. The rockets are far away from each other and we shall call the left one A and the right one B. Midway between rockets A and B we place a source of light we shall call S, capable of emitting blue or red light, each simultaneously to left and right, toward rockets A and B.

Rockets A and B, together with S form a single inertial frame of reference S for now (in other words, they are all stationary to one another).

Now we send blue light simultaneously from our source S toward rockets A and B. This starts the thrusters and the rockets burn their fuel, therefore being accelerated toward one another, on a collision course. The rockets burn all their fuel and each reaches constant velocity v in relation to our stationary frame of reference S, but in opposite direction. The speed at which the rockets are travelling is "normal" speed (i.e. not nearly close to the speed of light, but something a normal rocket could do).

After the burn is completed, we send red light simultaneously from our source S in the direction of rockets, therefore resetting the counters of our clocks. Now we have a system of three inertial reference frames, A - for rocket A, B - for rocket B and S - for our source of light S, in linear motion in relation to one another. And we have our clocks on rockets A and B synchronised, as observed from S.

Just before the collision occurs (i.e. when rockets A and B are in immediate vicinity of light source S) and if the trip of rockets is sufficiently long, we shall have the following situation in relation to clocks, as observed from different frames of reference, due to time dilation as per special theory of relativity:

• clock A displays N as observed from reference frame A
• clock A displays M (different from N) as observed from reference frame B
• clock A displays L (different from N and M) as observed from reference frame S

And due to symmetry of motion and identical construction of clocks:

• clock B displays N as observed from reference frame B
• clock B displays M (different from N) as observed from reference frame A
• clock B displays L (different from N and M) as observed from reference frame S

In comparison, such a system would show N on both clocks as observed from any frame of reference using only Newton's mechanics. --Jimbo2x (talk) 20:19, 3 January 2008 (UTC)

Since this is entirely original research we can't of course put it in an article, and it doesn't really belong on this talk page either. You might try Usenet. As a matter of fact it just so happens that someone (or you?) just asked exactly the same question on Usenet in sci.physics.relativity.

Anyway, I'll give you two hints here. (1) You write: "... in linear motion in relation to one another. And we have our clocks on rockets A and B synchronised." When they are in relative motion, they are not synchronized. They both see the other one's clock "run slower". (2) You created a symmetric situation, so you get symmetric results. Good luck on Usenet :-) - DVdm (talk) 21:44, 3 January 2008 (UTC)

Point taken about original research. I thought someone must have done this before (it seemed unlikely that this would be original). And it wasn't me on Usenet - so obviously not all that original ;-)

In terms of "synchronised" - they are synced in the sense that they are both running equally different as observed from S. About symmetric - yeah, that was my point. I think it would be useful to show that observers in different reference frames would read the same clock differently in immediate vicinity of each other according to SR.

--Jimbo2x (talk) 21:58, 3 January 2008 (UTC)

The idea of symmetry is of course not new, but I had never seen this kind of setup with the central light.

Little note on "... synchronised in the sense that...". Sure, but that's another kind of synchronization. From the three clocks in the setup, no pair of clocks is synchronized in any way. You could perhaps say that the clocks A and B are sort of equally desynchronized according to S, but as soon as you forget S, A and B are just two systems in relative motion - nothing special about them. In fact, one can construct a "central S" for every pair of systems in relative motion. That can't possibly make the pair physically special.

Do try the Usenet thread - it might become interesting :-)

Cheers, DVdm (talk) 22:25, 3 January 2008 (UTC)

There are only two clocks in the setup - A and B. If two equidistant stationary clocks can be synced by passing a beam of light simultaneously from the midway to them to "reset" them, so can the same two equidistant clocks in uniform opposing motion be synced as observed from S, unless the beam of light somehow travels to the left at different speed than to the right :-) Sure, from A, B is observed to be different (due to relative motion) and vice versa, but for an observer in S, they will be in sync (i.e. explicitly, both clocks display L when rockets meet, as observed from S). I never thought nor do I think that S would make A and B special. S is just another inertial frame of reference and an observer, just like you said.

--Jimbo2x (talk) 23:07, 3 January 2008 (UTC)

Well, you really do have a third clock in there. The minute you mention "reference frame S", you have it. No reference frame without a clock :-)

Do try Usenet for this - it's better suited for this kind of thing. This page is for discussions about the format and content of the article, not the subject. Cheers, DVdm (talk) 08:59, 4 January 2008 (UTC)

No worries. --Jimbo2x (talk) 09:41, 4 January 2008 (UTC)

Actually, all observers in the above example would see one and the same value on both clocks. That is to say, my example does not reach correct conclusions. --Jimbo2x (talk) 23:19, 14 February 2008 (UTC)

The problem is: whenever somebody tries to make a setup, which clearifies, what one is talking about, this "problem" is solved with the axe original research. The twin paradox has to be a mystery. There are so many ways to misunderstand what is argued. Simple example: what is a clock. It could be so simple: Two identical laboratories are connected by a loaded spring. This mechanism is released and according to conservation of momentum both labs gain the same speed, relative to each other. Now the have the following instruments: a source of light, that is a resonator, amplifying a certain spectral line of a certain gas of atoms and a prisma or better, grating. This light is directed to a semi reflecting mirror, twisted 45°, so 50% of the light is directed to the grating, 50% is send in direction of the second "twin" lab. So, the grating deflects two different beams: one from the local source and one from the distant source. It should be possible, by elementary calculation to determine the ratio of the detected deflection angles. And it should be possible to have a shutter which modulates the outgoing beam with this determined ratio on and of. And again it should be possible to compare the on/off ratio of the received and send signals. I have to point out: this is no OR, it is just a try to find a common language of what is a clock, how to measure a frequency..., how to measure distance and time in an environment, where nothing is absolute, except "c". ErNa (talk) 06:32, 9 May 2008 (UTC)

The common language as regards editing the article is in the literature. Check (=read) the references, and you'll find definitions of "clock", "distance", "time" and lots of other terms. If you go and invent your own definitions, chances are extremely high that you will either get it wrong and somebody has it done before, or you get it right and somebody has it done before. If you're really lucky, you might get it wrong in a spectacular new way. If you want to learn about relativity, you might consider enrolling in Wikiversity. ;) Paradoctor (talk) 00:52, 1 May 2009 (UTC)

Accelerated rocket calculation

IMO the "Accelerated rocket calculation" doesn't add anything valuable to the article, and certainly doesn't add enough to justify its length.--76.93.42.50 (talk) 15:36, 17 March 2008 (UTC)

Seconded. Oddity- (talk) 09:14, 7 May 2008 (UTC)

Making the article easier to understand

The twin paradox is much easier to understand if one imagines a long line of stationary, evenly spaced, and synchronized clocks extending from the stationary twin to the point where the other twin turns around. Imagine that as these clocks pass the moving twins window a strobe flashes so he can read off the elapsed time. Even though the non-moving twins clock seems to the moving twin to be ticking at half the rate of his own, the elapsed time, as told by the clock immediately outside his window, is passing at twice the rate of his own. More importantly, just before he stops, in order to turn around, the line of clocks are, from his perspective, out of synch but the moment he stops the line of clocks will be perfectly synchronized again which means that the nonmoving twins clock now reads the same as the clock he is next to. That means that his calculation of what the nonmoving twins clock said jumps suddenly while he decelerates (which leads to general relativity). Of course, this doesnt add anything to the article that wasnt there before but it does make it easier to understand. Em3ryguy (talk) 08:17, 20 May 2008 (UTC)

There are language and reference problems in your essay. Also sounds like WP:OR because most of the time dilation issues in the article are discussion points on old/accepted papers and books on relativity, and you quote none. Jok2000 (talk) 02:55, 21 May 2008 (UTC)

First, its a suggestion not as essay. Second, it can be rewritten by anyone to any form they like. Third, since it doest change anything or add anything to the article that wasn't there before and moreover since every statement made is trivial I dont see why a citation would be necessary. I am simply suggesting it as a way of illustrating what is being talked about. Em3ryguy (talk) 03:57, 21 May 2008 (UTC)

Where do you draw the line between stating the obvious and publishing original research? If one can do nothing but repeat what has already been published and one cant even make a trivial observation then I have to question the usefulness of WP. People can read the published articles themselves. They look to wikipedia to illustrate and explain these complex ideas in simple terms. At least, I do. Em3ryguy (talk) 08:45, 24 May 2008 (UTC)

Terracentric universe, redux???

I'm a bit confused.

Surely the homebody is subject to several sources of rotational accelaration: the Earth's axial rotation, the Earth's orbit around the sun and the sun's orbit around the galactic core, as well as whatever gravitation forces are affecting the Milky Way's celestial transit.

There's one bit that may address this, but I've no way of knowing whether it actually does or not: The standard textbook approach treats the twin paradox as a straightforward application of special relativity. Here the Earth and the ship are not in a symmetrical relationship: the ship has a "turnaround" in which it undergoes non-inertial motion, while the Earth has no such turnaround. (my emphasis)

So does this "non-inertial" bit mean that relativity doesn't apply to gravitational acceleration? Or is it that the acceleration experienced by the Earth is negligible in relativistic terms (the speed of the Earth's rotation being a sixth of a millionth of c, and orbit a ten thousandth)?

I think we should be told!

Prof Wrong (talk) 14:16, 4 July 2008 (UTC)

The twin paradox is a Gedankenexperiment, and as such assumes a simplified set-up. In this context "Earth" is meant to represent an inertial observer, one that doesn't undergo accelerations. To avoid confusing readers like Prof Wrong, the article should use something else, like another spaceship, or at least point out that the of effect of accelerations on Earth-based observers is negligible compared to what the traveler suffers. Paradoctor (talk) 00:24, 1 May 2009 (UTC)

Capitalisation

A quick check in a couple of well known text books ('Introducing Einstein's Relativity' by Ray d'Inverno and 'Special Relativity and its Experimental Foundations' by Yuan Zhong Zhang) confirmed that 'special relativity is not normally capitalised. Martin Hogbin (talk) 10:09, 31 December 2008 (UTC)

Indeed, Downwards' edit was on the mark, Eeekster's revert was off. DVdm (talk) 12:18, 31 December 2008 (UTC)

Article needs massive rewrite

The twin's paradox is poorly stated here. The issue of acceleration is a red herring. The real paradox is this: If Twin T travels in an inertial frame near the speed of light, he doesn't age according to Twin S in the stationary frame. This is an experimental fact: See muon lifetime. Therefore, Twin T outlives Twin S, according to Twin S.

All inertial frames are equivalent. Therefore, according to Twin T, it is Twin S that is moving near the speed of light. Therefore, according to Twin T, Twin S dies first.

The paradox is that we have two apparently contradictory histories: in one history Twin T outlives Twin S, and in the other history things are reversed. Paradox occurs because we believe that only one of these two histories can be true.

I'd guess that the resolution of this paradox is not in any acceleration, because we need not speculate upon a return trip to establish matters. The resolution may be that simultaneity of events is not universally agreed upon. So various observers in various frames all will have varying views of which twin died first.

What say you? Brews ohare (talk) 22:09, 1 January 2009 (UTC)

Muon lifetime is not directly related to the twin paradox. The essence of the TP is that both twins are together at one event, get separated, and then reunite in another event, arranged in such a way that, between those two events, one twin remains in one inertial frame, whereas the other twin does not. The latter either undergoes accelerations, or jumps from one inertial frame to another. Failing to understand this difference is the origin of the paradox. You seem to confuse the TP with so-called mutual time dilation. Check The Twin Paradox in the Usenet Physics FAQ. DVdm (talk) 22:25, 1 January 2009 (UTC)

I was just about to say much the same. Martin Hogbin (talk) 22:28, 1 January 2009 (UTC)

Maybe you can help me out here. I feel that the usual explanation of the twin paradox provided in the article based upon acceleration really is an easy cop-out that amounts to saying: "Hey, there is no paradox because you broke the rules; you have to stay in an inertial frame." It's true that the rules were broken, but "no fair" is neither a very interesting nor a very persuasive explanation. However, to me there is a more interesting paradox, which does not break the rules. You refer to it as the mutual time dilation paradox, but there seems to be no discussion of same on Misplaced Pages. Is that so?

This paradox is that, with no tricks about acceleration, only different inertial frames, Twin T says that he aged more than Twin S, but Twin T says the opposite.

As an example, suppose Twin S and Twin T move toward each other at speed c/(2 + ε) according to observer O. They both die on his doorstep at the same time. All observers agree on simultaneity of events occurring at the same location. Observer O says that Twin S and Twin T aged the same amount.

Now let O move toward Twin T at constant speed. Then Twin T appears to move rapidly toward rendezvous with a slow clock, while Twin S moves slowly toward rendezvous with a fast clock. We arrange that O arrives at rendezvous exactly as Twins S & T meet and expire. For things to work out, we require that Twin S travel a short distance and Twin T a long distance so both have the same number of clicks on their clocks upon meeting. In other words the rendezvous according to O occurs just enough nearer Twin S's house, and just enough farther from Twin T's house. I suppose this what relativity predicts?

I find this version more interesting because it draws upon relativity of simultaneity, and not a cop-out. Brews ohare (talk) 23:10, 1 January 2009 (UTC)

I do not quite follow your example, but it is not what is generally known as the twins paradox, see the history section of the article. Martin Hogbin (talk) 23:53, 1 January 2009 (UTC)

There are plenty of paradoxes in SR based on the relativity of simultaneity, for example the Ladder paradox. Martin Hogbin (talk) 23:57, 1 January 2009 (UTC)

'Brews ohare', you're talking about time dilation or the relativity of simultaneity, more than the twin paradox, which is a way of using those to show the counterintuitive nature of relativity. You're right that acceleration isn't required for time dilation; you can even do a variation of the twin paradox without acceleration, by having three ships resetting their clocks as they coast past each other.

I think saying someone is 'in an inertial frame' is problematic. A frame of reference is a set of coordinates overlaid on spacetime — all events of interest can be located in any frame. It's just that some frames are convenient to use, because some events occur at the same time or in the same place.

You might be interested in the diagram in http://en.wikipedia.org/Talk:Twin_paradox/Archive_12#Stationary_in_relation_to_which_frame.3F showing the events of the twin paradox in the three rest frames, side by side.

—WWoods (talk) 23:10, 2 January 2009 (UTC)

I had not looked at the archives; thanks. Brews ohare (talk) 06:11, 3 January 2009 (UTC)

Rotational motion

Most particle accelerators use rings, and particles move in circles. Circular motion is non-inertial. Thus, it would seem that muon lifetime experiments based upon collider experiments are outside special relativity. See Muons in accelerators & Muons in flight. Is it just fortuitous that lifetime dilation in circles is the same as in straight paths? Do you know of some discussion of same?

If the twins were placed on contrary rotating concentric carousels, they could compare lifetimes periodically when they happened to ::meet. If we sit in a stationary frame, and the carousels rotate in opposite directions at the same rate, the twins age the same way, and have the same age each time they meet. Suppose for example, the two twins die simultaneously on their third meeting. Then any observer must agree that this event occurred on the third meeting, because everybody agrees on simultaneity of events occurring at the same location.

If, on the other hand, we rotate with angular rate Ω, one twin ages faster than the other, according to us. That means one will die earlier, and just how much earlier depends on how fast we rotate, and not upon the twins at all!! Nonetheless, on their third meeting, both must be observed by us to die, regardless of our Ω.

Perhaps the circumference of one carousel shrinks and the other enlarges so the twins always meet at the same number of clicks on their differently paced clocks?? That would require the radii of the carousels to shrink or enlarge, even though the radii are everywhere perpendicular to the rotational velocity, and would lead to increased radial separation of the perimeters of the carousels as our Ω increases.

So a more likely explanation seems to be that our notion of the angular rate of the carousels changes with our Ω in a way that reconciles the clocks of the twins? The faster clock must travel less far than the slower clock before rendezvous, so rendezvous occurs at the same number of clicks on each clock? Brews ohare (talk) 01:23, 2 January 2009 (UTC)

Well, exploration of the literature shows that this subject is complicated by gravitational time dilation. I haven't found a thorough discussion, though I added some "Further Reading" to the article. Brews ohare (talk) 04:40, 2 January 2009 (UTC)

I removed the totally ridiculous vanity-press book you added, but I question the "further reading" section entirely. There are already (way too many) external sources in the special relativity article. This section adds nothing to the article and just invites people to add or spam their personal favorites, or use internet searches to book-mine potentially crappy sources they have never read or do not understand. Tim Shuba (talk) 06:45, 2 January 2009 (UTC)

Brews, referring to your "...particles move in circles. Circular motion is non-inertial. Thus, it would seem that muon lifetime experiments based upon collider experiments are outside special relativity.". Please note that circular motion and even non-inertial motion is completely and 100% within the realm of special relativity. The only thing that is outside SR, is gravitation. DVdm (talk) 16:58, 2 January 2009 (UTC)

DVdm: It appears that to the stationary observer the rotating clock is slowed by SR but for a rotating observer the clock is slowed by the gravitational time dilation due to centrifugal force. So the effect on clocks seems to be outside SR for the rotating observer (e.g. the muon). Do you agree? See Grøn Øyvind Brews ohare (talk) 20:50, 2 January 2009 (UTC)

Brews, I had not (nor have now) read your above comments beyond your first paragraph that I quoted and commented upon. Whatever you had in mind, I had to cut it short early. That's also the reason why I had reverted (without paying sufficient attention,) your section on the rotational version. DVdm (talk) 21:32, 2 January 2009 (UTC)

Bear in mind that, if we are going to delve onto this subject, we need to make the distinction between an object that is held in circular motion by gravity (for example the moon) which is in inertial motion, and one that is held in circular motion by other forces (for example a particle in a collider) which is not in inertial motion. I would suggest most of this discussion is outside the scope of this article.Martin Hogbin (talk) 11:18, 4 January 2009 (UTC)

Yes, by all means. That's why I still think we don't need that little section. I vote for removal. DVdm (talk) 11:21, 4 January 2009 (UTC)

I have made a correction. I suspected something was wrong, so I checked the reference. See exercise 9.25 in book on page 227. Since R > 3 mu, we have Delta(Tau_B)/Delta(Tau_A) > 1, so Alice is younger than Bob. She must be, as can be easily seen from an inertial frame attached to the center of the planet. Alice orbits and Bob does not, so her proper time integral is smaller than his. Afaiac, the section can stay now. I have added the page and a direct scholar search link to the reference. DVdm (talk) 13:07, 4 January 2009 (UTC)

Martin: an object that is held in circular motion by gravity (for example the moon) which is in inertial motion, and one that is held in circular motion by other forces (for example a particle in a collider) which is not in inertial motion. I believe this statement to be an error from the viewpoint of SR, which shares with Newtonian mechanics the view that any accelerated frame, whether the acceleration originates in gravity or a Lorentz force, is a non-inertial motion. Brews ohare (talk) 16:00, 4 January 2009 (UTC)

Yes but if we are concerned with gravity we need to use GR as SR cannot deal with gravitation. Martin Hogbin (talk) 16:48, 4 January 2009 (UTC)

Indeed, but in this particular case we don't really need gravity. After all, Alice's orbital speed can be expressed as a function of the R and mu (which are used in the expression of the proper times ratio). So in this particular case we can ignore gravity, and use her speed w.r.t the inertial frame attached to the center of the planet to calculate her proper time integral. Same conclusion. DVdm (talk) 17:04, 4 January 2009 (UTC)

"Rotational version" irrelevant

How does this section appear irrelevant? It is a twin paradox like that outlined in the intro, with the difference that the paths are circular. In fact, the author of the cited reference draws the parallel several times in the book. Brews ohare (talk) 20:53, 2 January 2009 (UTC)

Oops, sorry, you are right, I was confused. I thought you had gravitational time dilation in mind, where the concept of "mutual time dilation" is not present, and therefore cannot be the cause of wrongly claiming symmetry and thus generating the paradox. I slightly changed the text to make it silently implie that the twins remain at the same distance from the center of the gravitating body, for if they don't, the result can depend on whether Bob travels towards or away from the center and back. DVdm (talk) 21:23, 2 January 2009 (UTC)

History

Based on the description of the well known historian A.I. Miller (ref. 1), I rewrote the intro and the history section to include the contributions of Max von Laue. Already in 1913 he alluded to the fact, that one twin uses two frames for his journey, while the other remains in one, and this accounts for the different aging. Laue also was the first to illustrate those connections by using Minkowskian spacetime. German (p. 58): Von allen Weltlinien, welche zwei gegebene Weltpunkte 1 und 2 verbinden, hat die gerade Verbindung die längste Eigenzeit. (My translation: From all world lines, which connect two given world points 1 and 2, the straight connection has the maximal proper time.") --D.H (talk) 15:42, 4 January 2009 (UTC)

That is interesting history. The attribution to Lord Halsbury in several papers and texts goes back to a reference supposedly in Discovery in 1955, which appears in fact to be non-existent, as does Lord Halsbury himself as a contributor to relativity. Brews ohare (talk) 16:03, 4 January 2009 (UTC)

Discovery was a British science magazine published 1920-1966. They carried part of the second Dingle controversy, consisting of letters from Fisher, Mccrea, Dingle, Halsbury and a couple of others, and an article by Bondi. Also commenting on the debate is a rather amusing paper by Goodhart on biological time featuring cold-blooded physicists and astronautical frogs. If somebody wants my scans, drop a line on my talk page. Paradoctor (talk) 22:29, 30 April 2009 (UTC)

Separate page for discussion of Twin scenario itself

I have set up a separate page. Please post your reply there, as this is really the place for discussions on how to improve the article. Martin Hogbin (talk) 10:04, 17 January 2009 (UTC)

Quote from the boilerplate on top of this page: "This is not a forum for general discussion about the article's subject.", and "This" means all of Misplaced Pages. Please self-delete the page, or at least move it to your user space. The latter would be against Misplaced Pages policy, too, but my interest is limited to keeping the talk page on topic. Paradoctor (talk) 00:08, 1 May 2009 (UTC)

Apples and Oranges

Is it valid to say that, in the "Specific example" given in the article, the travelling twin effectively moved at 1.73c? I understand the length contraction but for practical purposes the aim of his trip was to travel 8.9 light years (from the home reference frame) and he accomplished it in 5.14 years (from his own reference frame). 83.146.14.12 (talk) 11:34, 14 April 2009 (UTC)

Speed is defined as distance divided by time. Normally one does not divide distance as measured by one person by time as measured by another. DVdm (talk) 16:26, 14 April 2009 (UTC)

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