Visiting Princeton, the American home to Albert Einstein, I’m reminded of one of my favorite “paradoxes” of special relativity. And, even more so, one of my favorite versions of this paradox which, when I first heard it, it blew my mind. What paradox is this of which I speak? The twin paradox of course! Really just the plain old twin paradox? No. Much better than that: the twin paradox in donut space!
The twin “paradox” of special relativity is really one of the classics of undergraduate physics. Two twins, born minutes apart, over the years drift apart. One becomes an astronaut and the other a surf bum who enjoys hanging with his homies. Astronaut twin hops on a rocket ship which accelerates up to relativistic speeds, visits nearby star, says hi to the empty space around the star, and returns home to earth. Time dilation in special relativity tells us that the clock in the astronaut’s ship will tick slower than the one with the beach bum, and thus, when the astronaut returns home, he will be younger than his sibling. The “paradox” comes from thinking: but isn’t the whole experiment symmetric. Can’t I think about the whole experiment from the astronauts point of view, and won’t he see a similar symmetric path for his beach bum brother?
The resolution of the twin paradox comes from realizing that the whole setup isn’t really symmetric: one of those twins undergoes acceleration in his path (the astronaut) while the other twin just sits on his lazy bum surfside. Cool. Standard stuff from undergraduate special relativity.
But here is a fun twist to think about. Suppose that the universe we lived in was a donut (sit down Homer J, calm yourself.) A donut? Well suppose that if you traveled in one direction for a long time you would return to where you started. Just as if you travel on the surface of a donut (a two dimensional surface) you can walk “around” the donut (imagine yourself and ant on the donut) and end up where you started, you can imagine that the topology of the universe is such that if you walk in the right direction you will eventually end up back where you started. Such nontrivial topologies of the universe surprisingly aren’t ruled out by simple things like General Relativity, although experimental searches for such crazy topologies haven’t turned up any evidence that our universe has such a crazy topology.
But back to the twins. Now suppose that you run the twin experiment, but with the twins entirely in inertial frames. In other words, the beach bum sits in an inertial frame (we probably need to put him in space to get a good approximation of this, but ignore this for now, this is a thought experiment, damnit!), and his astronaut buddy also is in an inertial frame, but one which is moving with a constant velocity with respect to his surfing brother. So now, since we are in a donut topology for our universe, it is possible to set up this experiment such that the astronaut brother eventually goes “all the way around the universe” and ends up beside his surfing bro. Now, the relationship between the twins really is symmetric: neither of them ever shifts reference frames or accelerates or such. And thus there really does seem to be a paradox. Which of the two twins is younger? And why?
O brave new quantum world!
Shouldn’t the distance also seem to be shorter for the astronaut? So this would make the system asymmetric again.
This is a classic from my old Usenet days.
“This is a classic from my old Usenet days.”
When we Caltech students owned E. S. Nesnon [spell it backwards] — the vending machine cartel of cigarette machines, candy machines, soda machines, hot food machines, pinball machines, and the beta-test for Nolan Bushnell of “Pong” and “Computer Space” [which was re-released as “Asteroids”]. I owned 11% of E. S. Nesnon, and was the world champion of both games. “Computer Space” (and thus “Asteroids”) had its spaceships and flying saucers zooming through toroidal space.
We debated the Toroidal Twin Paradox then. But this was the late 1960s and early 1970s, so, by the nature of the times, I don’t remember the outcome of the debate.
if its symmetric, it’s symmetric
Wouldn’t the “join” of the two edges, a global property, determines “when” the traveling twin returns? You’re essentially matching up coordinate times across this cut, and you can’t always match up t with t in different frames, so it must pick out some preferred rest frame.
“I think that there aren’t non-trivial topologies with Lorentz signature, so this paradox doesn’t exist.”
I don’t think this is the problem. What do you mean by “aren’t.” Certainly I can formulate special relativity on M x R with M a three dimensional torus and put the Lorentz metric on this space-time….
I think that there aren’t non-trivial topologies with Lorentz signature, so this paradox doesn’t exist.
Wait a sec. How do they get to be in different inertial frames? We can consider their velocity relative to the inertial frame of their birth (which presumably is the same for each twin). One must have accelerated more than the other with respect to this frame (assuming they are following the same path, but at different velocities). Hence, no paradox. If they are born in different frames, then they weren’t the same to begin with. In this case the mothers acceleration between births resolves the paradox.
Clearly a ‘trick’ question. The twin who is born second is always “the younger twin”.. even if he ends up being older. Physicists forget that twins aren’t born simultaneously. Is it April 1st yet?
@Joe Fitzsimons
Well, I don’t care about twins. Let’s assume you have two clocks, accelerate one and let it fly by the other one. The moment they’re at the same place (approximetly) you synchronise them. Then one clock takes it’s trip around the donut (for reasons of simplicity I think of a cylinder). When it`s back and they’re at the same spot again (if only for a very short period of time) you compare the passed time.
I believe the problem lies with the metric, or more specific that there are different geodesics connecting two events.
~G
Okay, you’ve suffered enough (or at least thought about it for a while.) Spoiler alerg! http://arxiv.org/abs/gr-qc/0101014
It depends which way Alice, the “stationary” twin, looks. At every point, Alice can look at Bob, the “moving” twin, either moving away from her or coming back at her, from the opposite direction. If she ever changes the direction she looks in, I suppose it might be said that there is effectively an acceleration.
Ultimately, the dilation effects can be cashed out in terms of Bob transmitting a time signal every second, which Alice can receive from both the Bobs she can see (if Bob always transmits, Alice will be overwhelmed by an infinite number of signals). If Alice refuses to recognize the incoming Bob signal as the same Bob (he went the other way, the other signal must be an imposter, an alter-Bob; unless she can confirm his identity), she will happily know from the signals she is receiving that Bob’s time is properly relativistically transformed, without any acceleration. If she recognizes both Bob signals as from the same Bob, she will see different Doppler effects for the two signals, and will have to decide how to reconcile them within the reference frames she decides to use.
Alternatively, since there is not a unique distance from Alice to Bob, so also there is not a unique velocity between them.
Someone who remembers a theorem or a reference should confirm or deny Jose. I think I remember that there are harmonic solutions, for example, on T^n, but it’s not obvious, except, I suppose, to a differential geometer.
Err…. “alert” that is. What the heck is “alerg”?
~g: I’m unsure what you mean. Which frame do you synchronise them in?
The clocks are very tiny (pointlike), and they’re really close at one moment. Synchronising means setting both clocks to zero simultanous. Since they’re at the same point in space, simultanity is meaningful.
Well, I admit there is a big deal of idealization here.
~g
~g: For any distance that is not exactly 0, I don’t see how this can work.
If the universe was a donut and if a twin was younger than the other and you are saying that the time frame goes a lot slower? Then what about their biological clock get messed up along with jet lag and other conditions?
If this is really symmetric, and if both are aging at the same rate in their own reference frames, then wouldn’t each see the other as younger? I mean, two sticks passing each other both see the other as shorter, right? Then if one entered the other’s reference frame, the one who changed frames would be younger. Then again, it’s been a while since I worked with special relativity…
Professor Philip Vos Fellman emailed me to
“You know if both brothers actually stay in intertial frames without acceleration, then you get Yaneer’s ZM theory – a first order theory of spacetime. Thought it was worth noting in passing. It’s not actually a relativity problem it’s a ZM theory problem.”
Is a first order space-time theory possible?
Yaneer Bar-Yam
New England Complex Systems Institute
Abstract:
Einstein’s general relativity relates the curvature of space time, a second order differential property, to the stress-energy-momentum tensor. In this paper we ask whether it is possible to develop a first order theory relating space-time angles to the energy-momentum vector. We suggest several concepts that would be relevant, including quantum mechanical concepts that are usually treated separately. Phenomenological Lorentz covariance arises from both field and coordinate transformation and the Dirac equation becomes a special case of the space-time field equation. We reinterpret Kaluza-Klein theory in this context by considering the compact fifth dimension as the quantum wavefunction phase. Further directions for development are suggested.
I think that ThePolynomial has solved the paradox. Each twin sees the other younger than himself, and there is no contradiction because we are comparing observations made in different reference frames.
Very good ThePolynomial!
I think the solution is that in a toroidal universe there is a rest-frame for the diameter of the universe. Entities moving wrt that frame will measure the diameter smaller than it really is. So the brother that sees the smallest diameter will be the younger brother.
I don’t think the situation is symmetric, because the travelling twin will have a winding number of 1 while the other will have winding number 0.
Joe said:
No, it’s the recoil of the birth that put the second twin in a cosmic orbit.
Wow. I just realized that given my solution there are very weird consequences to absolute velocity. The ‘you’ that is one diameter away is both not moving AND in a different inertial frame of reference. It would be as if winding had some kind of repulsive gravitational effect.
| The resolution relies on the selection of a preferred frame singled out by the topology of the space.
Hey, I was right?
No offense to the authors, but this is a published paper? Isn’t it more like an exercise in a GR class?
I’m not sure I’d agree with the statement that there is a preferred frame singled out by the topology. Instead, I’d say that the selection of a flat metric given the topology singles out a preferred frame. Or, maybe to put it another way, in a periodic background, boosts are no longer isometries of the metric.
Wasn’t this done in the early ’60s with atomic clocks and airliners? Nobody freaked out, perhaps because it was a matter of nanoseconds. Now if brother 2 had his pay held and invested in compound interest accounts, he would own the world, and hire his aged brother to clean the pool.
On a donut, there is a special reference frame, the reference frame in which the length of the loop is maximum. If you are not in this reference frame (ie, if you are the astronaut twin), something very odd happens. If you follow your plane of simultaneity around the loop, you will find that you are simultaneous with your future and past self.
From the astronaut twin’s perspective, his sibling will appear to be going away in one direction. His sibling will also appear to be coming towards him in the opposite direction. But in that opposite direction, the astronaut is simultaneous with his future self. Therefore, the astronaut’s twin will have had extra time to age, and will be older when they meet.
Stuck on a plane grr
Peter I don’t understand why TVs signal that wraps around the torus is faster on your example….isn’t time dilation independent of whether the twin is coming or going?
On a space-time topology T^1xR, with a locally Lorentzian metric, I believe that the argument of the ArXiv/PhysRevA paper you cite does not go through.
Consider twins, both traveling inertially. When they pass each other, they both start sending a signal once each second (they have clocks, but they don’t have enough measuring rods to measure topologically significant distances). So far, nothing is different from an RxR space-time: both observe an incoming, slower than once per second sequence of signals. At a certain point, however, which will appear to be at the same elapsed proper time for both, they will both start to receive a second signal, from the opposite direction. Assuming they both decide that this is from their twin, the incoming signal runs faster than once per second, so that both will receive and send the same number of slow/fast signals by the time they meet; assuming their biological clocks run at the same rate as their transmission clocks, they will have aged by the same amount when they meet, both still traveling inertially.
Suppose that Twin 1 transmits 2000 signals in the course of the whole trip; Twin 2 will receive, say, 800 signals while transmitting 1000 signals, then will receive the balance of 1200 signals, from the opposite direction, while transmitting 1000 more signals and arriving back at Twin 1.
Meanwhile, if Twin 2 listens only to the incoming signals from Twin 1 from a single direction, s/he will receive only 1600 signals, and be confused when s/he notices Twin 1 pass by, aged exactly the same as them. The Twins have to pay attention to the second signal when it starts to arrive to make sense of the signals they receive.
An argument that invokes a global topology to identify a preferred frame of reference in a space-time that has a locally Lorentzian metric seems instrumentally problematic. Two local observers cannot know whether they are in a globally preferred frame of reference, all they can observe are the signals that the other observer sends. Eventually, they can start taking into account multiple signals from the other observer and indeed from themselves to deduce some of the global topological structure, but the signals that have so far been sent in the twins scenario I describe above do not yet give either twin enough information to deduce much about the global topology.
I find the use of coordinatization in the paper you cite very heavy-handed.
The propagation of light in a T^3xR topology may make the twins receive more than two signals in the course of a single loop trip, potentially many more, from many directions, allowing them to deduce more about the global structure and confusing the analysis, but if the twins carry out the shortest inertial loop trip, they will age the same.
I’ve put a space-time diagram in my Yale webspace temporarily, at http://pantheon.yale.edu/~pwm22/T1xRPic.jpg. The left and right hand sides of the diagram are identified; one of the twins is “stationary”, the other “travels”, but the diagram could be transformed to a coordinate system in which the twins are reversed.
In this diagram, I think the twins both send out 18 signals and receive 18 signals from the other twin. I haven’t put in the signals that “wrap around” that the “traveling” twin will also receive.
My apologies, Barrow and Levin are correct.
In coordinate-free terms, there is a class of privileged inertial observers, which can be defined using only simultaneity and signaling as having the property that one signal sent each way simultaneously will arrive back at the same time. For other inertial observers, two such messages will return at different times.
If one twin is a privileged observer, then everything is as Barrow and Janner describe, the moving twin ages less. If the twins set off at the same speed relative to a privileged observer, but in opposite directions, they will age identically.
From local observations, the twins can only measure their relative speed, not their speed relative to the privileged frames. Only when they start receiving signals that allow them to evaluate global information will they be able to compute their speed relative to the privileged frames.
Yep, I thought about that long ago but in “hyperspherical space” not as a donut. In such a space, there must be a preferred frame. There are several ways to think about it. One demonstration is to send a box around the universe with little test bodies in it. They will follow their own geodesics, and so will oscillate back and forth inside the box if “the box really moves.” (Consider the way that great circles can’t be parallel all the way around, they have to overlap.) Maybe in our universe it would correspond to the CMB.
Ironically, this is even more important than just about physics since it implies that geometry itself is not invariant under time. Even with no “features” or even variations in curvature we can show “actual motion” on a hypersphere by just demanding simple consistency like motion along geodesics – nothing else, no “physics”! In the purely geometric case we don’t even consider Lorentz transformations, gravitation and inertia, or other physical issues. It’s just a case of moving the box with its test particles around the sphere and showing something different from the system sitting at rest. It reminds me of another point I made, that circular motion can’t really be relative: the relative velocities of parts of a rotating circle depend on the rate of rotation. If you believe that at least relative velocities are real, that becomes a problem. Note this point doesn’t depend at all on inertia properties and acceleration taken directly as an indicator.
tyrannogenius
Interesting post! I am part of a group that has wanted to see special relativity “in action.” We have adopted the ever popular arcade game Asteroids to accommodate the theory of special relativity. There is an adaptation that is the “classic” game of asteroids, a sandbox version where you can alter the speed of light and change reference frames, and a new version we call “Nine ‘Roid” that is Asteroids meets nine-ball, meets relativity!
Take a look: referencegames.com
I don’t get it. Are they born at different velocities? Doesn’t one of them have to accelerate to get into outer space or are they both lying on the beach side by side passing each other cold beers while one of them traverses the universe?
If neither of them accelerates and they are both in inertial frames, why aren’t both of them traversing the universe? It would give a whole new meaning to the surfing term tubular.
I guess this is why I’m not a physicist. I can’t imagine how a side by side experience could turn into a gaining distance experience without acceleration.