The Twin Paradox

by Luke Muehlhauser on November 17, 2010 in Science

One objection to the Kalam Cosmological Argument is that it requires an A Theory of time, but most physicists reject the A theory because of special relativity. Though the main idea of special relativity – that the laws of physics are the same for all observers (in uniform motion), so a microwave oven works just fine on a planet that is moving very fast relative to Earth – is simple, it has strange consequences for our concept of time. When we understand these consequences, we can see why special relativity undermines the A Theory of time.

Last time, I explained the phenomenon of time dilation, which has been experimentally confirmed many times. This leads us to the well-known twin paradox.

Imagine two twins, born at the same time. In my head they look like this:

I’ll call them “Mary-Kate” and “Ashley.”

Mary-Kate is flighty and independent, so she decides to go on a trip to Sirius, a nearby star. Ashley is down-to-earth and very close to her family, so she stays behind on Earth.

Mary-Kate travels at close to the speed of light for over a decade to reach Sirius, but after a few months she is already bored and decides to return home. Many years later, she arrives at Earth and is excited to see Ashley again.

Strangely, we find that Mary-Kate is now many years younger than Ashley, and all the clocks on Mary-Kate’s starship are the same number of years behind the clocks on Earth. And this isn’t because the clocks are bad or because Mary-Kate took better care of her body than Ashley did. It’s because those clocks and Mary-Kate experienced less time during their travels than Ashley did back on Earth.

This story is fictional, but it could happen if we had the right technology. It’s a simple consequence of the time dilation we explored earlier, and that has been experimentally confirmed many times.

Why isn’t their aging reciprocal?

Once you accept this consequence of special relativity, you might ask: “But wait a minute. Yes, it’s true that Mary-Kate traveled for many years at nearly the speed of light while Ashley moved slowly on Earth in Ashley’s reference frame. But it’s just as true that from Mary-Kate’s reference frame, her spaceship was at rest as the Earth sped away at near the speed of light, and then late came back to her at the same speed. So why was it Mary-Kate who ended up younger? Why isn’t the effect reciprocal?”

This is where the “in uniform motion” part of special relativity comes into play.1 Remember that special relativity says that the laws of physics are the same for all observers in uniform motion. You can play tennis on a cruise ship moving at a steady speed in a straight line on perfectly calm waters, but you’d find it rather difficult if the ship was accelerating and decelerating all the time, or in stormy waters. The trajectory of each hit of the ball wouldn’t be quite what you’re used to.

The asymmetry between Mary-Kate and Ashley comes from the fact that while Ashley was in relatively “uniform” motion on Earth, Mary-Kate was in rather extreme non-uniform motion in accelerating and decelerating and turning around.

Time travel

Time dilation provides for the possibility of time travel. If Mart-Kate had the technology to travel to the Andromeda galaxy and back at very, very close to the speed of light, she could make it so that Earth would have experienced a few million years of time, and she would have experienced only a few decades of time. Thus, Ashley could travel into the distant future.

Anyway, we are now ready to explain why special relativity poses a problem for the Kalam Cosmological Argument.

  1. Things change in general relativity, but that’s not important for our discussion. []

Previous post:

Next post:

{ 43 comments… read them below or add one }

Hermes November 17, 2010 at 6:01 am

Prediction: The examples given here will be casually accepted or just ignored, then a scene similar to this will occur when the follow on post is presented.

  (Quote)

Bebok November 17, 2010 at 6:10 am

It’s because those clocks and Ashley experienced less time during their travels than Ashley did back on Earth.

It should be Mary-Kate in the first part, I suppose?

  (Quote)

Taranu November 17, 2010 at 6:35 am

Luke,
will your post series on Time and the Kalam also tackle the tension between Special Relativity and Quantum Entanglement?

I found an interesting article in the february 2009 issue of Scientific American by David Z Albert and Rivka Galchen entitled: Was Einstein Wrong? A Quantum Threat to Special Relativity. The link: http://courses.washington.edu/philo482/wrong.pdf

This article from ScienceDaily is also interesting: NASA Researchers Put New Spin On Einstein’s Relativity Theory. The link: http://www.sciencedaily.com/releases/2003/04/030410073215.htm

  (Quote)

Methodissed November 17, 2010 at 6:38 am

This post is correct in a general sense, but could use some clarity.

* If Mary-Kate is traveling near the speed of light, she wouldn’t need 10+ years to reach Siruis or to see her sister age, i.e., in relativity, it’s not just time that is relative – space is too. Granted, you didn’t specify how close she was to the speed of light. To help people understand the absurdity of relativity, it’s more helpful to speak in terms of hours rather than decades.

* The difference in aging between Mary-Kate and Ashley is not due to the “extreme non-uniform motion in accelerating and decelerating and turning around.” Rather, it’s due to the difference in speed (one is going really fast, one isn’t).

For a great learning resource on special relativity, I highly recommend an MP3 lecture series from The Teaching Company called Einstein’s Relativity and the Quantum Revolutionhttp://www.teach12.com/tgc/courses/course_detail.aspx?cid=153

  (Quote)

myprecious November 17, 2010 at 7:11 am

Mary-Kate was moving relative to Asley just as fast as Asley was moving relative to Mary-Kate. But what matters here is the movement relative to spacetime (not to earth or any other point of reference).

  (Quote)

Methodissed November 17, 2010 at 7:18 am

Myprecious said: “…what matters here is the movement relative to spacetime (not to earth or any other point of reference). ”

When discussing the difference in aging between Mary-Kate and Ashley, what matters is the difference in their uniform motion, i.e., not a special point of reference.

  (Quote)

Charles November 17, 2010 at 7:20 am

Methodissed: The difference in aging between Mary-Kate and Ashley is not due to the “extreme non-uniform motion in accelerating and decelerating and turning around.”Rather, it’s due to the difference in speed (one is going really fast, one isn’t).

Who is moving “really fast” depends on the reference frame. That’s why it’s a paradox.

  (Quote)

Methodissed November 17, 2010 at 7:41 am

Charles said: “Who is moving “really fast” depends on the reference frame. That’s why it’s a paradox. ”

In this discussion we’re using a frame of reference – the speed of light. Both observers would be aware that Mary-Kate is moving faster than Ashley.

  (Quote)

Bryan Elliott November 17, 2010 at 7:44 am

Specifying a destination keeps the paradox, but changes the time frame.

If Mary Kate is travelling at 0.86 c, to reach Sirius, Mary Kate’s trip to Sirius will take 10 years from Ashley’s frame of reference (8.6 ly / 0.86 c). Conventional physics simply works if you are an observer at rest, and you don’t care about the state of the observed, just its position and velocity.

From Mary Kate’s perspective, however, Lorenz confuses things a bit.
We’ll declare a variable to equal the Lorenz factor for this trip; that is to say, the velocity between the two bodies.
y = 1/sqrt(v^2/c^2)

Since we’re expressing velocity in terms of c, we can even simplify it:
y = 1/sqrt(1 – v^2)
y ~= 1.95

For time dilation, simply multiply this Lorenz factor by the time against acceleration (i.e., Mary-Kate accelerated away from Ashley, and experienced differential intertia; you actually have to get into General relativity to calculate this effect, which is much more complex; so we apply the Lorenz factor to the at-rest time (not really*) to get the accelerated time).

t’ = y t’
t’ = 19.5 years

So Mary Kate experienced more time than her twin. Why is this? The same Lorenz factor applies to distance along the axis of travel; space literally appears to stretch out ahead of MK, making the 8.6 ly appear to be 16.8 for her. From both sisters’ measure, MK is covering distance at 0.86 c – but for MK, this is a much longer distance.

You might well ask, if MK’s going to take 20 years from her perspective anyway, why not just go half as fast and take 20 years from both girls’ perspectives? At 0.43c, the Lorenz factor is 1.107; MK’s time is a mere 22.14 years.

Well, in fact, we can solve for the minimum of sqrt(1 – v^2)/v to learn the optimal cruising speed for a vessel, that balances passenger aging versus travel time. It’s exactly sqrt(2)/2 c (y=sqrt(2)), ~423,970 km/s, or 948,395,132 mph. To travel 1 light year at that rate would take (1.41 years [rest] / 2 years [flight]).

” The difference in aging between Mary-Kate and Ashley is not due to the “extreme non-uniform motion in accelerating and decelerating and turning around.” Rather, it’s due to the difference in speed (one is going really fast, one isn’t).”

Actually, no. Luke is correct. Special relativity doesn’t talk a lot about inertia, because the equations for it are a lot more difficult (that’s general). When you say, “one is going really fast; one isn’t”, what are you specifying speed in relation to?

You can detect acceleration without knowing anything about your surroundings, correct? If you were in a closed box, without the ability to sense anything else, if that box were to accelerate, and you with it, you’d notice. Not from vibration, not from anything but the differential inertia you experience.

That is you actively changing your frame of reference.

  (Quote)

Methodissed November 17, 2010 at 7:59 am

Bryan Elliot said: “Actually, no. Luke is correct. Special relativity doesn’t talk a lot about inertia, because the equations for it are a lot more difficult (that’s general). When you say, “one is going really fast; one isn’t”, what are you specifying speed in relation to?”

It sounds like I may have been mistaken about inertia (I don’t do math). Though it seems that if Mary-Kate is traveling for two decades at near the speed of light, her speed (relative to the speed of light) would have a larger effect than acceleration and deceleration. I’m sure that depends on different variables.

As for their difference in speed, Luke’s article references their speed relative to the speed of light, which is why I said that one is going faster than the other.

  (Quote)

polymeron November 17, 2010 at 8:21 am

One interesting thing here, though…
If MK goes to Sirius with one acceleration and one deceleration each way, some time passes on Earth – equal to the amount of time MK experienced multiplied by some factor.

If she goes to Andromeda and back with the same parameters, the same would be true: Ashley would age by the same factor, only the base time is longer.

But, here’s the question: Why would Ashley still age faster during the time the ship was traveling to Andromeda at constant speed? After all, in both trips there were the same acceleration and deceleration intervals which, supposedly, make all the difference. So why does the time spent in uniform motion still matter?

Expanding on what Bryan said, it would seem that since acceleration does change the frame of reference, one probably *could* say a “universal” point of view exists – not in space, and not even as a vector of motion, but as a fraction of the speed of light. Could it be that objects do after all have some fixed speed as a portion of the speed of light? I don’t know the math but from what I know of special relativity the answer is “no”. In that case, I don’t understand why it would work this way.

  (Quote)

Charles November 17, 2010 at 8:33 am

Methodissed: ”In this discussion we’re using a frame of reference – the speed of light. Both observers would be aware that Mary-Kate is moving faster than Ashley.

The choice of reference frame is completely arbitrary. You can look at it from MK’s point of view or from Ashley’s point of view. From MK’s point of view Ashley is moving away from her at near the speed of light, but from Ashley’s point of view the same is true. Hence, the paradox.

  (Quote)

Methodissed November 17, 2010 at 9:03 am

Thanks Charles. I do understand the paradox, though I’m not sure why the objection given the context of this discussion. In this example, the two would not view each other equally, i.e., they both know that the rocketship accelerated away from earth (earth being an arbitrary reference frame) and traveled at xx% the speed of light. Both observers would say that MK is moving faster than Ashley.

The speed of light, being a constant, seems like an excellent reference frame to use when explaining concepts like time dilation.

  (Quote)

Rob November 17, 2010 at 9:05 am

Methodissed,

You have no idea what you are talking about. The key here is that Mary-Kate experiences accelerations that her twin does not.

  (Quote)

Methodissed November 17, 2010 at 9:29 am

Rob said: “You have no idea what you are talking about. ”

I’m open to that possibility. Though your post doesn’t give me enough information to understand where I’m mistaken.

If you’re talking about my original objection about inertia, then I’ve already conceded that I may be wrong. If you’re talking about something else, please elaborate, i.e., Mary-Kate experiences accelerations that her twin does not, therefore…

  (Quote)

Polymeron November 17, 2010 at 9:44 am

Rob, can you then explain my question from above? Why does the acceleration affect time in which it isn’t happening?

  (Quote)

Patrick November 17, 2010 at 9:51 am

Methodissed: I’m not an expert on this. But I’m pretty sure that what you’re doing wrong is that you’re treating speed as if it exists as an objective standard that can let it function as its own frame of reference. Actually, speed only exists as a relation between two perspectives.

Imagine that the universe consists of nothing except vast, empty space, and you, and me.

Scenario 1: You are holding still and I zip past you at nearly the speed of light.
Scenario 2: I am holding still and you zip by me going the other direction at nearly the speed of light.
Scenario 3: We both zip past each other at nearly half the speed of light.

The key to understanding this is that these are the same scenarios. Each is just measuring the speed of the participant from a different perspective.

  (Quote)

Methodissed November 17, 2010 at 10:27 am

Thank you Patrick. I agree with your post, though I’m struggling with the comparison as it relates to Luke’s original post. As for your example, we’re not talking about two people zipping past each other, but rather one person going away from the other, and then coming back and comparing how much they’re aged.

As I understand it, the closer that Mary-Kate gets to the speed of light, the greater the effect of time dilation. Isn’t the speed of light (a constant) a valid reference to explain time dilation in that scenario?

To everyone, I apologize for misspeaking in my original post. As Brian noted I was speaking from my understanding of special relativity and I didn’t understand the implications of general relativity, i.e., I didn’t know what I didn’t know.

  (Quote)

Chris Hallquist November 17, 2010 at 10:36 am

You are my hero for using the Olsen twins to illustrate the twin paradox.

That is all.

  (Quote)

thecos November 17, 2010 at 10:39 am

I’ll try to explain the way I like to think about the twin paradox, although it might be really hard to articulate via text without being able to point to space time diagrams, so bear with me.

It is true and experimentally verified that for the duration of the traveling twin’s journey out to her destination, earth’s clocks are running slow relative to her ship.

It is also true and experimentally verified that for the duration of the traveling twin’s journey back to earth, earth’s clocks are still running slow relative to her ship.

It is also true and experimentally verified that when the traveling twin returns to earth, less time has passed for her compared to the elapsed time on earth.

It sure seems like there’s a contradiction here, but luckily for the sake of relativity, there is some “funny” stuff happening as the traveling twin is turning around at her destination (when she’s accelerating)

Remember that in relativity, simultanaeity is relative, meaning it depends on your reference frame. As the twin is turning around, she’s moving from a reference frame that is moving away from earth into a reference frame that is moving towards earth. Even though the observed time dilation effects on earth are the same for these two frames (since they both have the same relative speed to earth), these two reference frames have a difference notion of “now”. So, as she’s turning around, her notion of “now” on earth is moving very rapidly into the future, which makes it look like a LOT of time is passing very quickly on earth in a very short time for the direction-changing twin.

So, basically, there’s two factors at work here. There’s the time dilation effect of the twin’s relative speeds that is causing both of them to see the other’s clocks as running slow, AND there’s the effect of shifting the meaning of “now” as the traveling twin changes reference frames, and THIS effect only happens to the traveling twin, since she’s the one who is accelerating.

I think someone also brought up the idea that if the traveler turns around at pluto, or alpha centauri, they’re both doing identical acceleration, so why should the aging effects be different based on how far away the destination is? To figure this out, I think you really need to draw a space-time diagram, which gets a bit hairy to describe in words. But basically, if you were to draw a diagram from earth’s point of view with an x-axis and a t-axis, you could plot an objects path through space time. For earth, everything happening on a horizontal line can be said to be happening at the same time. However, for a ship moving relative to earth, that ship’s “simultaneaity line” would havea non-zero slope. In fact, on the journey away from earth (assuming the ship moves to the right) the simultanaeity line would have a positive slope. On the way back, it would have a negative slope. So, during the turnaround, you can imagine the “now” line rotating about the ship’s current point in space time and sort of sweeping out an arc back at earth. The geometry works out so that the amount of time that passes due to the “now” line sweeping up along the time axis is proportional to how far away the turn-around point is.

Blargh! Oops. That post got really long, and the last part probably didn’t make much sense! I hope it was helpful to someone =/ If not, look up minkowski diagrams and learn to draw them and all the crazy special relativity weirdness just pops right out of the geometry! Its really wonderful!

  (Quote)

thecos November 17, 2010 at 10:45 am

As I understand it, the closer that Mary-Kate gets to the speed of light, the greater the effect of time dilation. Isn’t the speed of light (a constant) a valid reference to explain time dilation in that scenario?

So, the reason this doesn’t maek sense to me is that when we say the speed of light is constant in all reference frames, we really mean that. As in, Mary-kate sees the speed of light as c relative to her, and Ashley sees the speed of light relative to her. Mary-Kate sees Ashley moving with some speed greater than zero and less than c. Ashley sees Mary-Kate moving with some speed greater than zero and less than c. Even worse, imagine some little kid on an meteor who has no idea that Mary-Kate took off from earth. From his point, they’re both just moving at some speed greater than zero and less than c. And of course, in the asteroid’s reference frame, the speed of light is still c. So I just don’t see at all how you’re trying to use the speed of light as a reference for anything…

  (Quote)

Charles November 17, 2010 at 11:35 am

Methodissed: we’re not talking about two people zipping past each other, but rather one person going away from the other, and then coming back and comparing how much they’re aged.

Ah … but just who is moving away from whom. Perhaps it is the Earth that moves away and the ship that’s standing still.

  (Quote)

Tarun November 17, 2010 at 1:29 pm

Hi Luke,

I mentioned this on a comment on a previous post, but your account of why the time dilation isn’t reciprocal isn’t entirely accurate. The relevant difference is not the uniformity vs. non-uniformity of the twins’ motions. It is the difference in the space-time lengths (or proper times) of the paths they both follow. The point has been made by previous commenters. In the special theory of relativity, space and time are not absolute, but space-time is absolute.

To see this, think of a cylindrical space-time, one where space “wraps around” so that if I keep heading in a particular spatial direction I will return to where I started. Such a space-time is within the domain of special relativity (it can be assigned a metric that is everywhere Minkowski). In such a space-time, Mary-Kate could set out from Earth on a spaceship, travel for a while, and return to Earth without ever having to turn around. In other words, her motion will be uniform. But according to the special theory it will still be the case that Mary-Kate ages less than Ashley. Both of their motions are uniform, but Mary-Kate’s is along a longer space-time path. Somewhat counter-intuitively, travelling along a longer path between two points takes a smaller proper time, so less aging.

This may seem kind of nitpicky, but it is an extremely common misconception about the special theory, so it is worth getting right.

  (Quote)

Luke Muehlhauser November 17, 2010 at 3:37 pm

Bebok,

Thanks, fixed.

  (Quote)

Luke Muehlhauser November 17, 2010 at 3:40 pm

Methodissed,

Re: your second point. No, it isn’t the speed. There is no sense in which one is “really” going fast and the other isn’t. That’s the whole point of relativity. In fact, this very point is covered explicitly in the TTC course you recommended.

  (Quote)

thecos November 17, 2010 at 3:42 pm

Tarun,

Do you have any good resources for special relativity in cylindrical space time? Because that seems really screwy to me, even by physics standards.

For one thing, just saying that one has a smaller spacetime interval doesn’t really seem to resolve the paradox to me. In your example, are you saying that Ashley’s clocks are NOT running slowly from Mary-Kate’s POV on the spaceship? If Ashley’s clocks are running slowly from Mary-Kate’s frame and Mary-Kate’s clocks are running slowly from Ashley’s frame, and both of them stay in intertial frames for the entire trip, how is the paradox resolved?

Also, what the heck happens to relativity of simultanaeity in this universe? It seems like a reference frame’s surface of simultaneity (all points in spacetime with zero time-like distance) would be almost like a helix, where somebody’s “present” actually spans into the future as it wraps around the cylinder. It also seems like in a cylindrical universe there would be a “preferred” reference frame, namely the frame whose surface of simultaneity is a closed surface, aka, the only reference frame in this universe in which I can even come close wrapping my head around what simultaneous means.

Jees… what the heck is going on here? Lol… I’m trying to draw this and it seems like as Mary-Kate travels away from earth, she sees “multiple” earths (which are actually the same earth separated by time), behind her and in front of her, but even after accounting for the time it takes for light to reach her from each earth to figure out what time it is “now” on the Earths, the Earth in front of her is in a sense further along in Ashley’s future than the Earth behind her is, even though in a certain sense both Earth’s are happening “now” from Mary-Kate’s POV? So Ashley is both young and old (and really old and dead and not even born) at the same “time”, but Mary-Kate is traveling towards the “older”, “future” Ashley by traveling around the universe? So is the time dilation counter-acted by the idea that Mary-Kate is traveling into Ashley’s future in a way that seems fundamentally different than the way we normally travel into the future?

I guess maybe that would resolve the paradox??? Maybe? Haha… in the course of writing this post I went from thinking you were probably wrong to just having my mind completely boggled. Can you recommend any good books or papers about this? I don’t even know how to talk about this without putting quotes every time I say “time”, “future”, “now”, etc…

Regardless, thank you just for bringing this up and for making my afternoon considerably more interesting =P

  (Quote)

Luke Muehlhauser November 17, 2010 at 3:43 pm

Chris,

The full photo actually shows the two of them with nothing but a little sticker on their nipples. But people got mad the last time I showed photos of women so scantily clad. So I have nicely cropped out their sexy-bits. :)

  (Quote)

Silver Bullet November 17, 2010 at 3:49 pm

Fact: the younger one will be hotter.

  (Quote)

Luke Muehlhauser November 17, 2010 at 4:09 pm

Tarun,

Yes, but the difference in their traveled spacetime lengths is due to Mary-Kate’s acceleration and deceleration and turning around. So my post is still correct, right? I was trying to avoid discussing space-time lengths until later.

  (Quote)

myprecious November 17, 2010 at 4:40 pm

Thank you Tarun:)
All others should, at least, read “The Fabric of the Cosmos” by Greene. He says that incorporating spacetime into special relativity “provided an absolute criterion – one that all observers, regardless of their constant relative velocities, would agree on – for deciding whether or not something is accelerating.”

  (Quote)

Tarun November 17, 2010 at 4:49 pm

Luke,

The whole point of the cylindrical space-time example is that Mary-Kate would not have to either accelerate or decelerate. Instead of imagining Mary-Kate being born on Earth and then taking off, imagine that there is spaceship that just travels in a straight line with constant velocity. Since space-time is cylindrical, the ship will periodically pass by Earth. Let’s say Mary-Kate is born on the spaceship one of the times it is passing by Earth, and Ashley is born on Earth at the exact same space-time point. (Of course it can’t be the exact same space-time instant, since they can’t literally overlap and neither of them is point-like, even though they are pretty skinny, but this is a useful approximation that can be relaxed without distorting the argument). The spaceship continues its flight and passes by the Earth again at some point (no acceleration or deceleration involved). Mary-Kate and Ashley compare ages at this point. It will turn out to be the case that Mary-Kate has aged less.

  (Quote)

Tarun November 17, 2010 at 5:14 pm

thecos,

Your first question is a good one, and it points out what I consider a huge conceptual problem with the way special relativity is usually presented, not just to laypeople but also to physics students. The intuition that is instilled in people is that special relativity tells us that if we have two inertial frames we have to treat them completely symmetrically. Any claim that is true in one must be true in the other if we switch the observers. In particular, if in frame A Ashley sees Mary-Kate as aging more slowly then in frame M Mary-Kate must see Ashley as aging more slowly. But this is not the case in general!

The frames can only be treated completely symmetrically if they are completely symmetrical, but there are different ways in which the symmetry can be broken. One possible way for the symmetry to be broken is that one of the frames is accelerating and the other isn’t. In this case one of the frames is not inertial. But the symmetry can still be broken even if both frames are inertial, if there is some objective difference in the nature of their paths through space-time. A path that winds around a cylinder is objectively different from a path that goes up the cylinder in a straight line. The difference arises from the multiply connected topology of a cylinder. Think about it this way: in a regular (non-cylindrical) space, if we have two paths connecting two points it is always possible to deform one of the paths continuously (without changing its end-points) until it coincides with the other point. On a cylinder, however, there is no way to deform a winding path so it coincides with a non-winding path. This objective asymmetry in the paths means we are no longer licensed to apply the intuition that both frames can be treated symmetrically.

Really, the problem has to do with the term “relativity”. It leads to the fallacious notion that in some sense everything is relative to a frame of reference in special relativity. But this is incorrect. There is an absolute background structure: Minkowski space-time.

  (Quote)

thecos November 17, 2010 at 5:15 pm

Tarun,

Right. Accelerating/Decelerating isn’t the only reason why a their space-time intervals could be different. But in Luke’s example, in a non-cylindrical spacetime, if Mary-Kate flies away from Earth, turns around, and flies back, her trajectory through space time is causing her to traverse a longer interval. So I don’t think I’d say Luke is wrong.

Also, stating that one twin is younger because her space-time interval is longer is a completely true statement, but it doesn’t explicitly address time dilation, and so it doesn’t really address the paradox at all.

I think Luke is trying to explain that the assertion that time is passing slowly for Ashley from Mary-Kate’s reference frame is only true when Ashley is moving at a constant velocity, which is correct. As Mary-Kate is in the process of turning around, very interesting and unusual things are happening from her point of view as a result of her acceleration that causes Ashley’s clocks on earth to jump forward like crazy (from Mary-Kate’s POV).

  (Quote)

Methodissed November 17, 2010 at 5:19 pm

Thanks everyone for the education and for helping to set me straight. I obviously have a lot to learn.

  (Quote)

Tarun November 17, 2010 at 5:25 pm

thecos,

Your claim about the surfaces of simultaneity in cylindrical spacetime is correct. There is in a sense a privileged frame in cylindrical spacetime, the one whose temporal axis does not wind around the cylinder. But this sort of privileging is not incompatible with special relativity. The laws are still the same in all frames, but the topology of the spacetime introduces an asymmetry that privileges one of the frames. The kind of privileging that would be problematic from a relativistic point of view is one where the laws take a particularly simple form in one frame as opposed to others. That is not what is happening here.

  (Quote)

Tarun November 17, 2010 at 5:37 pm

thecos,

Luke’s analysis accurately describes the issue in his particular example. There the non-uniformity of Mary-Kate’s motion is what causes the asymmetry. I was just pointing out that the deeper (and in some sense more accurate) explanation is the different path lengths. One way to see that this explanation is deeper is to see that there are cases where the uniformity/non-uniformity of motion is irrelevant but the twin paradox still occurs. The path length explanation is more general.

You say:
“Also, stating that one twin is younger because her space-time interval is longer is a completely true statement, but it doesn’t explicitly address time dilation, and so it doesn’t really address the paradox at all. ”

This I don’t agree with. The path length argument does address the paradox. It shows that the proper times, the times measured by clocks carried by the two separate observers will be different even though they start and end at the same point. What else is needed to address the paradox? Maybe an additional reminder that time dilation in general is not a fundamental effect but a feature of co-ordinate transformation in a very specific type of case: between two inertial frames in a simply-connected space-time.

  (Quote)

thecos November 17, 2010 at 5:56 pm

Tarun,

Yes, I think that reminder is exactly what is needed to resolve the paradox =P

The paradox is that people believe roughly the following two statements:
1. Time dilation symmetrically occurs for both Mary-Kate and Ashley for the duration of the journey.
2. When Mary-Kate gets back, Ashley is older.

These statements are clearly conflicting with each other, thus an apparent paradox.

If you just add the (absolutely true) statement:
3. Mary-Kate traversed a longer space-time path, has a shorter proper time, so Ashley is older.

You still have a “paradox”, because this conflicts with statement 1. To resolve the two conflicting statements, its not enough just to say which one is right, you need to explain why the conflicting statement 1 (statement 1) is wrong (which of course it is). Your reminder about time dilation not being a general effect does this.

  (Quote)

Ken Jacobs November 17, 2010 at 6:32 pm

Maybe this isn’t right, but I think of this in terms of the actual observed physical effects, and I really don’t see a paradox. If one twin is rocketing away at NLS then she and the rocket are time dilated, but there is an additional effect of the her and the rocket’s light being red-shifted (longer wavelength and thus slowed information transfer). Same goes for the red-shifted apearance of the Earth. However this implies that the real time dilation effects are that time slows for the rocket, added to the red-shift light makes it appear that the time on board is even slower from Earth’s perspective than time slowing alone on the rocket.

From the rocket’s perspective, Earth too is red shifted and appears to slow for that reason alone, but since the rocket’s speed is actually near light speed, both the red shift effect and the time dilation is at least lessened or neutralized, and the time dilation experienced in the rocket at least partly cancels out the red-shift effect (slow on board clock and perception must mean stationary things outside seem to run/age at a faster rate relative to the rocket). When the rocket returns, the opposite effects are observed, regarding the light (each are blue shifted). In this case Earth now is blue shifted and time must appear to accelerate and doubly so because of the relatively slowed time on board.

I can’t see it any other way since an observer twin who continually monitors the Earth and her twin for the whole trip (or the rocket/twin from Earth) must see these accelerated or decelerated time effects until they meet up once again.

The bottom line is that on the Rocket, she must observe her Earth twin aging faster than normal at some stage.

So I think it’s only the light coming from the objects or other observable effects that are relative, and not the absolute observable time frame.

  (Quote)

Zeb November 17, 2010 at 7:22 pm

You can detect acceleration without knowing anything about your surroundings, correct?If you were in a closed box, without the ability to sense anything else, if that box were to accelerate, and you with it, you’d notice.Not from vibration, not from anything but the differential inertia you experience.That is you actively changing your frame of reference.  

How does one detect acceleration without reference to some other object? I only detect acceleration when my car speeds up because the car is accelerating relative to my body. In free fall I am accelerating, but because every atom touching my body and inside my body is accelerating uniformly, I can’t tell I am accelerating. At least that’s what I thought; I though free fall was supposed to feel just like being at rest in space.

  (Quote)

Zeb November 17, 2010 at 7:31 pm

I thought the key to the twin paradox was the doppler effect. That’s how it was explained in the first book on relativity I read, anyway. Here’s how it was presented (using easy but inaccurate numbers): say it takes 10 days at 90% the speed of light to reach a destination. On the way to the destination, Mary Kate sees Ashley slow down incredibly; in the 10 days it takes here to get there, Ashley only ages one hour. On the other hand Ashley sees Mary Kate slow down incredibly; by the time she sees her sister go through all 10 day of the trip, Ashley has aged 10 years. On the way back Mary Kate sees Ashley speed up; over the ten days it takes MK to reach Earth, she sees Ashley age 10 years. Meanwhile Ashley sees Mary Kate speed up; in 1 hour Ashley watches MK go through all 10 days of the trip. So when they reunite, Mary Kate has aged 20 days (what they both observed MK experiencing, in line with her speed/distance to the destination), while Ashley has aged 10 years and 1 hour (which they both observed, but Ashley saw it as 10 years during the trip out and 1 hour during the return, while MK observed the opposite).

  (Quote)

Chip November 19, 2010 at 8:14 pm

Ack! Too much going on in this thread. Must… not… get drawn in…

I would like to offer the following link:
http://www.av8n.com/physics/twins.htm

John Denker is a beacon of clarity on the internet. If you’re interested in the twin “paradox” — if you really want to understand what’s going on — everything you need is right there. Doing it right means learning how the spacetime diagrams work, and slowly and patiently working them out by hand. (I’d expect it to take quite a while, especially if you’re not already familiar with the math!)

  (Quote)

Luke Muehlhauser November 19, 2010 at 9:48 pm

Chip,

Thanks!

  (Quote)

Anon November 21, 2010 at 2:40 am

Ummm…From the frame of reference attached to the traveller, doesn’t the Earth accelerate and decelerate in the same way? Or are we saying acceleration is absolute in some sense?

  (Quote)

Leave a Comment