Time does not run slower or faster

[birulami]

In another thread that was locked, Chris Hillmann told me:

Nothing in relativity theory (special or general) says "time runs slower here than there"! That wouldn't even make sense. Rather, when you somehow compare the times kept by observers in different states of motion (you need to say precisely how you do this!), you will generally find discrepancies.

Understanding time still makes me scratch my head

Don't we believe that the twin paradox is true, i.e. if I go on a long journey with high enough speed and come back that I am less old than the twin brother I left behind. This is not just 'times kept by observers'. We are not just talking about mechanical or other clocks. I am biologically provable younger than my brother. If I would have taken along radioactive material, it would have decayed less than the same amount of the same material left behind. Everything that went along aged less than similar things left behind.

May I at least say that SR shows that things age slower on a speedy journey? Well, this is what I see by comparison when I come back from the journey.

But then, how is time different from a measure of age?

(Please no arguments that GR is needed for the twin paradox, see http://www.math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html)


Still puzzled,
Harald.

 

[Sojourner01]

I believe the point Chris was making is that the statement in question implies a 'fact' quite separate from the points of view of the people making the claim. It is very important in relativity to understand that there is no 'true' state of affairs and that you could conceivably observe any possible relationship in time between two events, if you were to travel in a certain direction at a certain speed relative to these events.

As for the twins paradox - it's a paradox of special relativity, because modelling the two twins in special relativity yields a nonsensical and contradictory result - you are supposed to be confused by SR's answer because the idea is that it doesn't work for an accelerating frame. It is a means of proving that special relativity is inadequate in some real scenarios, and shows that general relativity is required.

 

[chroot]

One observer can observe a distant clock running slow. That does not imply that that clock appears to be running slowly from any other perspective. Indeed, for a person traveling with that clock, the clock will appear to run completely normally.

If you measure time only by looking at your own wristwatch -- which moves with you -- then you will never experience time slowing down or speeding up at all. Your wristwatch will look to you as it always does, regardless of how you move.

- Warren

 

[JesseM]

As for the twins paradox - it's a paradox of special relativity, because modelling the two twins in special relativity yields a nonsensical and contradictory result - you are supposed to be confused by SR's answer because the idea is that it doesn't work for an accelerating frame. It is a means of proving that special relativity is inadequate in some real scenarios, and shows that general relativity is required.

I don't think it shows SR is inadequate to the physical problem since you can perfectly well use SR to predict how much the accelerating twin aged, it's just that if you want to use the basic (non-tensor) equations of SR you have to calculate this from the point of view of an inertial frame, you can't use the same equations in a non-inertial coordinate system where the accelerating twin was at rest the whole time.

My guess is that Chris was just making the point that in relativity it only makes sense to compare the rate that two clocks are ticking at any given moment if they are right next to each other, for separated clocks there is no physical meaning to the question of which is ticking slower, since different coordinate systems give different answers. However, all coordinate systems will agree on the accumulated time on each clock between the time they depart from a common point in spacetime and the time they reunite at a different point in spacetime, so you can say that one twin "aged less" over the course of the entire trip, you just can't say that twin was "aging slower" at any particular moment during the trip.

 

[Chris Hillman]

Not even close!

In another thread that was locked

For good reason, BTW, so everyone please be careful to avoid trying to simply start up a locked thread all over again.

I believe the point Chris was making

in post # 14 of https://www.physicsforums.com/showthread.php?t=171946

is that the statement in question implies a 'fact' quite separate from the points of view of the people making the claim. It is very important in relativity to understand that there is no 'true' state of affairs and that you could conceivably observe any possible relationship in time between two events, if you were to travel in a certain direction at a certain speed relative to these events.

That's not even remotely close what I meant, and indeed seems to directly contrary to what I was trying to tell Harald/birulami. This is discouraging, since I thought I expressed myself reasonably clearly! (JesseM seemed to understand what I wrote.)

Sojourner, I was in fact trying to debunk the sloppy notion that "time slows down" is a useful way to think about anything in either str or gtr. Rather, in these theories, one can compute what various ideal observers will measure in specific scenarios, including elapsed times between events on their world lines. The relationship between such measurements turns out to be incompatible with the Galileo's kinematics, but is described in flat spacetime by Einstein's relativistic kinematics. (In curved spacetime things are bit more complex, but on small scales, relativistic kinematics works fine.)

I was also trying to debunk the sloppy notion that relativity theory says "everything is relative" or that "there are no absolute facts". In fact, as I pointed out, one should rather look for statements which do not depend upon the coordinate description. These will have physical meaning and can be interpreted in terms of the results of actual measurements made by actual observers in a specified state of motion (having a specified world line) in the spacetime of interest.

As for the twins paradox - it's a paradox of special relativity, because modelling the two twins in special relativity yields a nonsensical and contradictory result - you are supposed to be confused by SR's answer because the idea is that it doesn't work for an accelerating frame. It is a means of proving that special relativity is inadequate in some real scenarios, and shows that general relativity is required.

That's also wrong, as I have often pointed out in various forums on many past occasions. Rather, we can treat accelerated observers in special relativity, but we need to use a tool appropriate to treat observers who are not inertial observers. The appropriate tool is the kinematic decomposition of a vector field, which works the same way in any Lorentzian spacetime, but this concerns the mathematics of congruences, not physics at all--- in particular, this is completely independent of any theory of gravitation.

Don't we believe that the twin paradox is true, i.e. if I go on a long journey with high enough speed and come back that I am less old than the twin brother I left behind. This is not just 'times kept by observers'. We are not just talking about mechanical or other clocks. I am biologically provable younger than my brother. If I would have taken along radioactive material, it would have decayed less than the same amount of the same material left behind. Everything that went along aged less than similar things left behind.

That is correct. I don't understand why you think this is incompatible with anything I said. Have you read Geroch, General Relativity from A to B? This is also a nice introduction to str which should clarify your confusion.

May I at least say that SR shows that things age slower on a speedy journey?

That's why you are confused, Harald--- "things age slower"? Slower with respect to what?! A clock? If not a clock, then what? "A speedy journey"? Speedy in what sense? (Recall that "distance in the large" is tricky. Recall also that in the conventional twin paradox, one twin suffers an impulsive blow and has nonconstant velocity.)

As you just said yourself, ideal clocks are assumed not to be affected by accelerations. Thus, ideal clocks all run at the same rate everywhere and everywhen, by definition. The elapsed time between events A, B as measured by clock C is the distance measured along the world line of C from A to B. Note that the events A, B are on the world line of C, by definition, as JesseM pointed out--- any other scenario involving timing of events distant from C requires carefully defining a notion of "distance in the large"; there is no unique choice, so whatever conclusion you draw in such a scenario may only be valid for a specific notion of "distance in the large".

As is so often the case, the basic confusion here is between local and global structure. The twin paradox involves a global comparison. I think JesseM understands this--- at a guess, because he has a stronger background in manifold theory than Sojourner or Harald. None of us can do anything about the fact that understanding relativistic physics requires grappling with a bunch of subtle points from manifold theory. The best thing I can do, I think, is to try to recommend some good books which address some of these issues, such as Geroch, The Geometry of Physics.

 

[pervect]

I:

Don't we believe that the twin paradox is true, i.e. if I go on a long journey with high enough speed and come back that I am less old than the twin brother I left behind. This is not just 'times kept by observers'.

Consider the statement: distances (specifically E-W distances) on the Earth are shorter at higher lattitudes (closer to the North pole) than at lower lattiutdes (closer to the equator).

Would you agree with this statement, or would you say "distances are the same, no matter where you are on Earth"?

 

[birulami]

I said

May I at least say that SR shows that things age slower on a speedy journey?

and Chris Hillman answered

That's why you are confused, Harald--- "things age slower"? Slower with respect to what?! A clock? If not a clock, then what? "A speedy journey"? Speedy in what sense?

Ok, that makes sense. Without reference, the comparative "slower" has no meaning. In the twin paradox setup, the acceleration singles out the reference frame of the observer who stays at home for comparison.

Aside from all physical theory I find it disturbing to imagine the process that causes me and the stuff I take along to age less over the journey as compared to home. If the lag is a few years it would be truly shocking to get back. Something happens during the journey, something works different on the journey than at home. And I wouldn't mind getting a bit more insight in how the mental picture of the process could be. There is more here to understand than how to apply the Lorentz transformation. But sometimes I have the impression that there is a ban on thinking in that direction. Maybe it is to philosophical and beyond physics?

Thanks for your input,
Harald.

 

[birulami]

Have you read Geroch, General Relativity from A to B? This is also a nice introduction to str which should clarify your confusion.

Looks like I should have taken the book more seriously. It is dusting on the bookshelf. I'll try it again then.

Harald.

 

[Chris Hillman]

Ok, that makes sense. Without reference, the comparative "slower" has no meaning. In the twin paradox setup, the acceleration singles out the reference frame of the observer who stays at home for comparison.

Right.

Aside from all physical theory I find it disturbing to imagine the process that causes me and the stuff I take along to age less over the journey as compared to home. If the lag is a few years it would be truly shocking to get back. Something happens during the journey,

Yes. Do you see what it was? (Answer below).

something works different on the journey than at home.

No! We keep trying to tell you that physics is based on the premise that physical law works the same everywhere and everywhen.

And I wouldn't mind getting a bit more insight in how the mental picture of the process could be. There is more here to understand than how to apply the Lorentz transformation. But sometimes I have the impression that there is a ban on thinking in that direction. Maybe it is to philosophical and beyond physics?

Instead of muttering darkly about conspiracy theories, go back to the first bit I quoted above and see if you can take it further.

What changed? Model the scenario using two world lines which are intially and finally coincident. One represents the world line of an inertial observer, so it is a straight line. The other bends away then bends back. Note that there are at least three places where acceleration is needed since path curvature nonzero: first, the traveling twin needs to start moving away from his brother, second he has to slow down and then start approaching his brother, third he has to slow down again so that his world line is coincident with his brother's world line. Now it should be easy to see that the Lorentzian distance measured along the traveler's world line is smaller (don't forget the sign in the line element!).

HTH

 

[JesseM]

Aside from all physical theory I find it disturbing to imagine the process that causes me and the stuff I take along to age less over the journey as compared to home. If the lag is a few years it would be truly shocking to get back. Something happens during the journey, something works different on the journey than at home. And I wouldn't mind getting a bit more insight in how the mental picture of the process could be. There is more here to understand than how to apply the Lorentz transformation. But sometimes I have the impression that there is a ban on thinking in that direction. Maybe it is to philosophical and beyond physics?

Conceptually, I think the best approach is to ditch the idea that time "flow" in any real sense, and accept the spacetime viewpoint where the universe is a 4D spacetime manifold with various worldlines embedded in it like pieces of string frozen in a block of ice. Then the fact that one twin ages less can be understood in a "geometric" sense, in terms of the lengths of different paths through spacetime which are determined by the spacetime geometry. Just as the distance between two points on a 2D piece of paper (the length of a straight line connecting them) is given by the 2D pythagorean theorme \sqrt{dx^2 + dy^2}, and the distance between two points in 3D space is given by the 3D pythagorean theorem \sqrt{dx^2 + dy^2 + dz^2}, so the "distance" in spacetime between points in 4D spacetime (the 'proper time' along a straight worldline connecting them, where 'proper time' is the time interval measured by a clock which has that worldline) is given by \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}. And just as you can find the length of a non-straight path in space by approximating the path as a bunch of line segments and using the pythagorean theorem to find the length of each (and taking the limit as the number of segments approaches infinity in the case of a smooth curve), so you can find the proper time along a non-straight worldline by approximating it as a bunch of straight (inertial) segments and using the above formula to find the proper time along each. Just as it is always true that a straight path between points in 2D or 3D space is always shorter than a non-straight path between the same two points, so it is always true that a straight worldline between two events in the flat spacetime of SR will always have a greater amount of proper time than a non-inertial worldline which goes between the same two events; this is why the twin that accelerates will always have aged less than the inertial twin.

I took the analogy between paths on paper and worldlines in spacetime a bit further in post #9 on this thread in an effort to explain why there needn't be an objective truth about which of the two twins is aging slower at a given moment even if there is an objective truth about which of the twins aged less in total when they reunite; you may or may not find it helpful too:

Here's an analogy--on a 2D sheet of paper, draw two points, a "starting point" A and a "finishing point" B, and then draw two paths between them, one a straight line and the other a bent line. Now draw x and y coordinate axes, with the y-axis parallel to the the straight line. To get some specific numbers, let's say the starting point A is at x=0, y=0 and the finishing point B is at x=0, y=8, and the bent path consists of two straight line segments at different angles, the first of which of which goes from A to a point C at x=3, y=4, while the second line segment goes from C to B. Note the y-coordinates of the two points A and B, in this case y=0 and y=8, and then for any y-coordinate in between these two values, like y=4, there will be a unique point on each path with this y-coordinate. So you can ask about the distance along each path that you'd need to travel to get to the point on the path that has that y-coordinate; let's invent a term for that distance, like "partial path length". For example, at coordinate y=4, the "partial path length" along the straight path would have to be 4, while the "partial path length" on the bent path would larger, in this case 5 (the distance from point A to point C). If you look at the y-coordinate of the finishing point B, y=8, then the "partial path length" at y=8 would just be equal to the total length of the path from the starting point to the finishing point. In this case the "partial path length" for the straight path at y=8 would be 8, while the "partial path length" for the bent path would be 10.

Now, keep the same two paths between the same two points, but redraw your x and y axes so the y-axis is no longer parallel to the straight path--for example, we might draw the y-axis so it's parallel to the line segment joining A and C. Now the coordinates of the starting point A and the finishing point B for each path won't be the same--if we place the origin so that A still has coordinatex x=0, y=0, then the finishing point B will now have coordinates x=0, y=5.12. It's still true that "partial path length" for each path at the y-coordinate of the finishing point, y=5.12, must just be the total length of each path, which won't have changed just because we picked a different coordinate system, so it'll still be 8 for the straight path and 10 for the bent path. But at some earlier y-coordinate, since the lines of constant y are now at different angles, they'll intersect the two paths at different points so the "partial path length" at this y-coordinate will be different--for example, at y-coordinate y=2.56 in this coordinate system, the "partial path length" on the straight path would be 4 (just like the partial path length at y=4 in the previous coordinate system), while the "partial path length" on the bent path would be 2.56. Notice that while in the previous coordinate system the "partial path length" of the straight path was always smaller than the bent path at a given y-coordinate, in this coordinate system the "partial path length" of the straight path can actually be larger for certain values of y, although both coordinate systems agree that the total path length between A and B is shorter for the straight path.

All of this is pretty closely analogous to the situation in relativity, with different coordinate systems on the paper being analogous to different inertial reference frames in relativity, the y-coordinate being analogous to the coordinate time t in a given frame, and the "partial path length" at a given y-coordinate being analogous to the proper time T accumulated by a particular clock at a given coordinate time t. Just as both coordinate systems agreed on the value of the "partial path length" at the y-coordinate of point B where the two paths reunite, so different frames in relativity will always agree on the value of the proper time read by each twin's clock at the t-coordinate where they reunite at a single point in space. But hopefully you would agree that there is no single true answer to the question of which path is accumulating "partial path length" faster before they reach point B--this is entirely coordinate-dependent, you can get different answers depending on how you orient your y-axis and none is more "objectively true" than any other. In the same way, I'd say there's no single true answer to the question of which twin is accumulating proper time faster (or 'aging faster') before they reunite at a single point in space.

 

[daniel_i_l]

You can also look at it like this: When the traveling twin A looks at the twin B and sees that B looks older, that means that in A's frame the event that B turns 50 (for example) is simultaneous to the event that A turned 10. It's not that time "moved slower" for A, but rather when A suddenly changed frame when he turned around to come home his "line of simultaneity" flipped around (making B look older) even though A sees B;s clock running slower than his. So in his frame, the two events mentioned above happen at the same time. And in A's frame B is also younger due to time dilation (which can be explained in a similar way) . That's how they can both agree on the age without "time slowing down".

 

[birulami]

Thanks for the extensive answers. Both, Martin Hillman and JesseM, suggest that looking at worldlines helps best to understand the issue. It comes down to the simple rule to remember: "When two worldlines connecting two events are compared, the longer worldline will show the slower clock".

While this "the longer the slower" rule is easy to remember, easy to apply and is a perfect device to predict outcomes of experiments, it explains nothing. In a bit harsh of a comparison it is like describing a TV by saying: "When you push the ON button, you will see moving pictures on the screen". This provides for nice predictions. But it does not help to understand how a TV works. In the same sense the worldlines describe what happens, but not how.

Isn't it a fair question too ask what is going differently in two clocks on differently long worldlines such that at the end they show different times?

One answer to my original question was that the physical laws are the same on both worldlines. Of course they are, but nevertheless the outcome is different, so something is different. Ok, the difference is the length of the worldlines. But hey, how can I phrase it properly to say that this is not an explanation. Maybe the different length of the worldlines is the cause of the different clock readings. Still this leaves the question how the cause causes the effect.

Harald.

 

[robphy]

Thanks for the extensive answers. Both, Martin Hillman and JesseM, suggest that looking at worldlines helps best to understand the issue. It comes down to the simple rule to remember: "When two worldlines connecting two events are compared, the longer worldline will show the slower clock".

While this "the longer the slower" rule is easy to remember, easy to apply and is a perfect device to predict outcomes of experiments, it explains nothing. In a bit harsh of a comparison it is like describing a TV by saying: "When you push the ON button, you will see moving pictures on the screen". This provides for nice predictions. But it does not help to understand how a TV works. In the same sense the worldlines describe what happens, but not how.

Isn't it a fair question too ask what is going differently in two clocks on differently long worldlines such that at the end they show different times?

One answer to my original question was that the physical laws are the same on both worldlines. Of course they are, but nevertheless the outcome is different, so something is different. Ok, the difference is the length of the worldlines. But hey, how can I phrase it properly to say that this is not an explanation. Maybe the different length of the worldlines is the cause of the different clock readings. Still this leaves the question how the cause causes the effect.

Harald.

Give my page http://www.phy.syr.edu/courses/modules/LIGHTCONE/LightClock/ a try. It's purpose is to give a physical explanation [a "physics first!" explanation] (consistent with more mathematical approaches) of why the lightclocks tick the way they do. In other words, WHERE are the ticks marked off on each worldline, and WHY are they where we claim they are? (For more technical details, consult the papers and the posters near the bottom of the page.)

 

[Chris Hillman]

This is growing tiresome

Thanks for the extensive answers. Both, Martin Hillman and JesseM, suggest that looking at worldlines helps best to understand the issue.

I think you mean me, Chris Hillman (I would have thought that my transparent choice of PF username would have prevented any possibility of confusing me with anyone else, but apparently not...)

It comes down to the simple rule to remember: "When two worldlines connecting two events are compared, the longer worldline will show the slower clock".

Say rather:

"When two clocks whose worldlines are mutually tangent at two distinct common events A,B, but follow different paths in between those two events, the clock with the longer worldline (meaning, arc length integrated as per the metric structure, which is part of the definition of a Lorentzian manifold) will show the greater elapsed time."

Note two critical additions: first, I specifically said the two world lines must share the same unit tangent vector at two points where they coincide, and second, I specifically said we are comparing the "elapsed proper time" recorded by the two clocks. I avoided saying either clock "runs slower"; they are both ideal clocks, so by definition they run at the same rate under all circumstances.

While this "the longer the slower" rule is easy to remember, easy to apply and is a perfect device to predict outcomes of experiments, it explains nothing. In a bit harsh of a comparison it is like describing a TV by saying: "When you push the ON button, you will see moving pictures on the screen". This provides for nice predictions. But it does not help to understand how a TV works. In the same sense the worldlines describe what happens, but not how.

Take it up with Issac Newton.

(Newton's greatest contribution to math/sci may have been his insight that rather than trying to answer such imponderables as "what causes gravity?", one should focus on simply trying to mathematically describe gravitational interactions. IOW, one should focus on constructing mathematical models within some mathematically formulated physical theory. If you want to argue that physics should not conform to this Newtonian vision, you should probably move discussion to another forum at PF.)

Isn't it a fair question too ask what is going differently in two clocks on differently long worldlines such that at the end they show different times?

Nothing is "going differently". That's the whole point. That's why several posters have made specific corrections to what you wrote, emphasizing that ideal clocks have identical properties and function the same way under all circumstances.

One answer to my original question was that the physical laws are the same on both worldlines. Of course they are, but nevertheless the outcome is different, so something is different.

But Harald, you know what is different! The length of the two world lines between A,B!

Ok, the difference is the length of the worldlines.

See, you do know that. Although the restrictions I mentioned are required to make your "principle" valid.

But hey, how can I phrase it properly to say that this is not an explanation. Maybe the different length of the worldlines is the cause of the different clock readings. Still this leaves the question how the cause causes the effect.

Imagine two people who start walking at point A and take different paths to point B, and compare their odometer readings and find that one odometer shows a greater change. Your complaint is exactly analogous to insisting that the fact that one path is longer fails to explain the discrepancy in odometer readings. This is why the notion of Lorentzian manifolds is so valuable, conceptually speaking--- it offers a vivid geometric picture which makes it thousands of times easier to think about relativistic kinematics, as (after some initial resistance) Einstein himself recognized.

Maybe your real question is: why should we "believe" that Lorentzian manifolds offer the best way to think about relativistic kinematics. (Not the same thing at all as "offer the best way to think about time in all circumstances and all scales".) If so, your question should definitely move elsewhere.

I can't help noticing that even casual inspection suggests considerable similarity between your views and those of one "Harold Ellis Ensle" who was known to me years ago as a relativity denier over at the unmoderated UseNet newsgroup sci.physics.relativity, a place my own world line has avoided for many years by my clock.

 

[birulami]

I think you mean me, Chris Hillman

Sorry for that. And thanks for your continued patience with my stubborn questions.

"When two clocks whose worldlines are mutually tangent at two distinct common events A,B, but follow different paths in between those two events, the clock with the longer worldline (meaning, arc length integrated as per the metric structure, which is part of the definition of a Lorentzian manifold) will show the greater elapsed time."

I see where I went wrong. The usual graphical depiction of worldlines makes worldlines with the smaller elapsed time look longer, because their graphical length is euclidean. But the length of the worldline is in fact defined to be the elapsed time on the clock (times c) if you want.

If you want to argue that physics should not conform to this Newtonian vision, you should probably move discussion to another forum at PF.)

Maybe I should. Not that I want to prove old Isaac wrong, but my question may indeed be more philosophical than physical.

Imagine two people who start walking at point A and take different paths to point B, and compare their odometer readings and find that one odometer shows a greater change. Your complaint is exactly analogous to insisting that the fact that one path is longer fails to explain the discrepancy in odometer readings.

Except that the odometer measures euclidian and is therefore much more intuitive, whereas length of a worldline, defined by means of ds=sqrt(c2dt^2-dx^2-dy^2-dz^2), has these hard to imagine opposite signs in it. Of course I know that "easier to imagine" is not the best criterium for scientific results, but without a mental picture, progress in understanding hard. The strange thing is, that after rearranging into ds^2+dx^2+dy^2+dz^2=c^2dt^2, imagination snaps back and I see a simple tradeoff between "moving through space" (dxyz) and "experiencing time" (ds), where ds is the time I feel and measure with my clock while I move along space dxyz.

I can't help noticing that even casual inspection suggests considerable similarity between your views and those of one "Harold Ellis Ensle" who was known to me years ago as a relativity denier ...

I am not this person, honestly. But writing down ds^2+dx^2+dy^2+dz^2=c^2dt^2 certainly makes me look like a heretic, because from there it seems to be easy to arrive at a view of relativity where relativity is an emergent property of an absolute background on which the magnitude of v=(ds,dx,dy,dz)/dt is constant, i.e. |v(t)|=c for all t (and yes, t is the flywheel that keeps this this absolute theater running).

I am not trying to tell you that this is the right way to look at it. Rather I am trying to understand why it is wrong. One argument I heard elsewhere was that Occams Razor favors getting rid of the absolute background, because it is not necessary for any reasoning. In a way this was an implicit concession that the math does not give any different results. But I think there should be a stronger argument than Occams Razor, given the fact that the euclidian view is so much easier to imagine.

Feel free to ignore my musings. You gave already some great advice and I keep reading Geroch.

Thanks,
Harald.

 

[Chris Hillman]

Required intuition for str: visualizing Minkowski geometry

Sorry for that. And thanks for your continued patience with my stubborn questions.

Well, it's been wearing thin.

I am not [Harald Ellis Ensle], honestly.

However, I do appreciate this information.

I see where I went wrong. The usual graphical depiction of worldlines makes worldlines with the smaller elapsed time look longer, because their graphical length is euclidean. But the length of the worldline is in fact defined to be the elapsed time on the clock (times c) if you want.

Right, and books like Taylor & Wheeler should help you fix your intuition so that when you look at curves, you can visualize either Euclid geometry (shouldn't be a problem from the sound of it) or Minkowski geometry as needed.

Except that the odometer measures euclidian and is therefore much more intuitive, whereas length of a worldline, defined by means of ds=sqrt(c2dt^2-dx^2-dy^2-dz^2), has these hard to imagine opposite signs in it.

Thousands if not tens of thousands of students master this every year, so it is by no means impossible to obtain the necessary intuition. When I was learning str (from Taylor & Wheeler), I found it very helpful to make two column tables comparing in detail E^2 and E^(1,1) trigonometry, path curvature, and so on.

Of course I know that "easier to imagine" is not the best criterium for scientific results, but without a mental picture, progress in understanding hard.

OK, but that's your problem, since tens of thousands of physicists around the world use the correct intuition on a regular basis. Minkowski geometry is a tried and true way to think about relativistic physics. Indeed, a hundred years of experience (well, ninety nine) shows that it is the best way. In any case, it is the universal standard, so if you want to discuss relativistic physics you must master this geometric intuition.

But writing down ds^2+dx^2+dy^2+dz^2=c^2dt^2 certainly makes me look like a heretic, because from there it seems to be easy to arrive at a view of relativity where relativity is an emergent property of an absolute background on which the magnitude of v=(ds,dx,dy,dz)/dt is constant, i.e. |v(t)|=c for all t

Well, that's completely incorrect as I think you realize, but in any case, this is not the appropriate place to propose new ideas which you yourself suspect may be "heretical". Please take to the "Original Research" subforum.

I am not trying to tell you that this is the right way to look at it. Rather I am trying to understand why it is wrong.

OK, good, you know it's wrong. All you need to do is become comfortable with Minkowski geometry. Then you will at least be able to understand what str says, which will be good progress towards resolving whatever philosophical issues might be troubling you.

 

[rqr]

Imagine two people who start walking at point A and take different paths to point B, and compare their odometer readings and find that one odometer shows a greater change. Your complaint is exactly analogous to insisting that the fact that one path is longer fails to explain the discrepancy in odometer readings.

Imagine two people (or two clocks) traveling the same path
between points A and B, but moving at different speeds.
The paradox: Even though they take equal paths, they end
up having different ages (or recording different times).

The odometer analogy no longer applies.

So what is it about being in different frames that causes
people to age differently (and clocks to record different
amounts of time over the same path)?

Aren't all inertial frames supposed to be equivlent?

rqr

 

[JesseM]

Imagine two people (or two clocks) traveling the same path
between points A and B, but moving at different speeds.
The paradox: Even though they take equal paths, they end
up having different ages (or recording different times).

The odometer analogy no longer applies.

You're misunderstanding the analogy I think, I believe Chris was comparing distance along paths through space with proper time along paths through spacetime (worldlines). Even if two people travel the same spatial path between spatial positions A and B as defined by a particular coordinate system (other coordinate systems will not agree that they both stop at the same position in space B), their paths through spacetime, as seen on a spacetime diagram, would be different, and the paths would not even end at the same point in spacetime. Just as the geometry of space is such that a straight path will always be the shortest distance between two points, so the geometry of flat spacetime is such that if you have multiple worldlines between two events in spacetime, a straight worldline (corresponding to inertial motion) will always have the maximum proper time.

 

[rqr]

(to JeeseM)

Thanks for the response, JesseM. (But where's Chris? ;-))
Re your "straight worldline" comment, didn't both of my people have such lines?
If there is no acceleration, then why would people in different (inertial) frame age
differently?

(Also, I could have said that I am not using worldlines, but am simply using equal (meaning fully equivalent - same direction, no acceleration)
distances through space. This may sound like I am using absolutes, but consider
this: If two people travel between the same points A and B of a given inertial
frame's x axis, then they must have traveled equal distances through space.
And since our people and clocks do not actually travel along worldlines, but
float in space, I have indeed shown that the odometer example is invalid.
And think about this: Clocks and people in space do not have roads to
"rub against" as do a car's tires in the odometer case.)

In SR, it is given that all inertial frames are equivalent. There is no preferred
frame. There are no preferred frames. However, experiment tell us that people
in different inertial frames age differently. (All acceleration can be eliminated by
simply adding a third person to the Twin Paradox case.) Forget about worldlines,
odometers, and Minkowskian geometry; all I want to know is

Why do people in different inertial frames age differently?

Here I am floating around in inertial Frame A, and there you are floating
around in (a supposedly equivalent) inertial Frame B, but we are aging
differently! What's up wid dat?

rqr

 

[JesseM]

Thanks for the response, JesseM. (But where's Chris? ;-))
Re your "straight worldline" comment, didn't both of my people have such lines?
If there is no acceleration, then why would people in different (inertial) frame age
differently?

Yes, but your two people don't meet at a single point in spacetime, you just had them end up at the same point in space (in one coordinate system). So you're not comparing the proper time on two worldlines which go between the same two points in spacetime, so the analogy between two paths between points in Euclidean space doesn't work. This would be more analogous to two straight paths in space which start from the same point but go in different directions, and both end at positions that have the same x-coordinate as viewed in some coordinate system. One of these paths can be longer than the other, just as one wordline had a greater proper time in your example.

(Also, I could have said that I am not using worldlines, but am simply using equal
distances through space.

But then you're missing the point of the analogy, which was to relate the notion of lengths of two paths between a pair of points in Euclidean space to the notion of proper times of two worldlines between a pair of events in spacetime.

This may sound like I am using absolutes, but consider
this: If two people travel between the same points A and B of a given inertial
frame's x axis, then they must have traveled equal distances through space.

Only in that frame's coordinate system. In other frames they traveled different distances through space.

And since our people and clocks do not actually travel along worldlines, but
float in space

This is meaningless, a worldline is just a set of points in spacetime, and all it means to "travel along a worldline" is to occupy each of those points.

I have indeed shown that the odometer example is invalid.

The odometer was an analogy, and it's a perfectly valid one. Just as the length of a path in space is independent of your spatial coordinate system, and just as the length of a straight-line path between two points in space is always shorter than the length of a non-straight paths between those points, so the proper time of a worldline is independent of your inertial coordinate system in relativity, and the proper time of a straight worldline between two events in spacetime is always greater than the proper time of a non-straight wordline between those same two events.

And think about this: Clocks and people in space do not have roads to
"rub against" as do a car's tires in the odometer case.)

No, but they have clocks. Again, it's an analogy where features of one case are mapped to features of the second, with the length of paths in space (as measured by an odometer) mapped to the proper time of worldlines in spacetime (as measured by a clock).

In SR, it is given that all inertial frames are equivalent. There is no preferred
frame. There are no preferred frames. However, experiment tell us that people
in different inertial frames age differently. (All acceleration can be eliminated by
simply adding a third person to the Twin Paradox case.)

If you use only inertial clocks, there is no objective frame-independent truth about which of any pair of clocks was ticking faster--disagreements over simultaneity mean different frames disagree about what time is showing one a given clock "at the same time" that a clock at a different location is showing a certain reading.

Here I am floating around in inertial Frame A, and there you are floating
around in (a supposedly equivalent) inertial Frame B, but we are aging
differently! What's up wid dat?

Again, not aging differently in any objective sense. In my frame you're aging slower, in your frame I'm aging slower, and in a frame where our speeds are equal we're both aging at the same rate.

 

[ZapperZ]

This question on the Twin "Paradox" (there really isn't a paradox anymore) seems to pop up rather frequently. May I suggest that one refers to http://www.oberlin.edu/physics/dstyer/Einstein/SRBook.pdf on this? It addresses practically all the FAQs on Special Relativity that we often get on here.

Zz.

 

[atulrajendran66]

hi! i hav just joined ths fabulous place let's see if i could do any help

In another thread that was locked, Chris Hillmann told me:


Understanding time still makes me scratch my head

Don't we believe that the twin paradox is true, i.e. if I go on a long journey with high enough speed and come back that I am less old than the twin brother I left behind. This is not just 'times kept by observers'. We are not just talking about mechanical or other clocks. I am biologically provable younger than my brother. If I would have taken along radioactive material, it would have decayed less than the same amount of the same material left behind. Everything that went along aged less than similar things left behind.

May I at least say that SR shows that things age slower on a speedy journey? Well, this is what I see by comparison when I come back from the journey.

But then, how is time different from a measure of age?

(Please no arguments that GR is needed for the twin paradox, see http://www.math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html)


Still puzzled,
Harald.

well! you seem to disagree that how biologically the traveling twin seem to be younger than the other. well imagine a clock, you are at a normal speed of 20 mph the one tick of the seconds hand will not seem different to you that you see them normally. while now if u r moving at a speed comparable to that of the speed of light then no for a person whose stationary the ticking of a clock would be the same but for u traveling with such a great speed one tick of the clock kept stationary on Earth may be a year or many years for you. hope it helped you clear your doubt.
please have a look at my question related to wormholes or if you ask your friends to have a look at it bcoz i have a presentation on this topic

 

[JesseM]

well! you seem to disagree that how biologically the traveling twin seem to be younger than the other. well imagine a clock, you are at a normal speed of 20 mph the one tick of the seconds hand will not seem different to you that you see them normally. while now if u r moving at a speed comparable to that of the speed of light then no for a person whose stationary the ticking of a clock would be the same

The word "stationary" has no absolute meaning in relativity, all speeds can only be defined relative to something else. If your speed relative to the Earth is close to the speed of light, then in the Earth's frame you are moving fast and your clock is ticking slow, while in your frame the Earth is moving fast and the Earth's clock is ticking slow.

 

[robphy]

This question on the Twin "Paradox" (there really isn't a paradox anymore) seems to pop up rather frequently. May I suggest that one refers to http://www.oberlin.edu/physics/dstyer/Einstein/SRBook.pdf on this? It addresses practically all the FAQs on Special Relativity that we often get on here.

Zz.

It might be a little more "illustrative" if it used spacetime diagrams.

 

[vivesdn]

From the initial question I think that the issue was how can it be that time passes at different paces here and there (moving fast or slow). As my level of physics and mathematics is not high, I would answer this in terms of a postulate: it is assumed that the speed of light must be the same observed by anyone, anywhere, anywhen, ..
Imagine you're traveling at 1km/h less than speed of light. You're looking the wings of your spacecraft , where there are position lights. You should be able to see the light slowly leaving it's source as it is only 1km/h faster than the spececraft. But if the postulate is true, then you will measure its speed as being 'c', so you have to be living in a timeframe slower than if you were at rest. And this is relative as you will never know if something is really at rest.
The most interesting: there are a number of observations that support this very first postulate (atomic clocks traveling in opposite orbits or high speed particles formed in the upper atmosphere that should not reach lower layers due to its short life at rest). So, the answer to the thread is: Yes, the time runs slower or faster depending on your frame as 'c' must not depend on it.

 

[Thrice]

Model the scenario using two world lines which are intially and finally coincident. One represents the world line of an inertial observer, so it is a straight line. The other bends away then bends back. Note that there are at least three places where acceleration is needed since path curvature nonzero: first, the traveling twin needs to start moving away from his brother, second he has to slow down and then start approaching his brother, third he has to slow down again so that his world line is coincident with his brother's world line. Now it should be easy to see that the Lorentzian distance measured along the traveler's world line is smaller (don't forget the sign in the line element!).

HTH

I sometimes wonder if I'm the only one who finds mathematical arguments easier or somehow more convincing. It's a very short step from the geodesic equations to maximizing proper time. I know some good articles & videos if the OP is interested in math.

 

[Idjot]

Time dilation is supposed to be merely a theoretical phenomenon rather than an actual difference in physical aging, right? The twins should both appear older than the other to the other, right? That's the paradox, right? Take two rocks broken from the same boulder. If one rock stays in "Earth time" on Earth and the other travels at near c for one million years in "Earth time" and returns, they'll still have the same carbon dating result in "Earth time", right? -Original age before experiment + 1 million years, right?

It's general relativity that in essence says "What's the difference? Who's to say which rock is really moving?" , right?

Both rocks are moving relative to the other, right?

 

[JesseM]

Time dilation is supposed to be merely a theoretical phenomenon rather than an actual difference in physical aging, right?

No, if the twins separate and then reunite later, one will have actually physically aged less.

The twins should both appear older than the other to the other, right? That's the paradox, right?

The paradox is that you might naively think each twin would view himself at rest and the other as moving away and then returning, so that each should predict the other will have aged less, but this fails to take into account that the time dilation equation only works in inertial (non-accelerating reference frames), and one twin has to accelerate to turn around after they've been moving apart so they can reunite. If the other twin moved inertially between the departure and the reuniting, the twin who accelerated will always be the younger one when they reunite.

Take two rocks broken from the same boulder. If one rock stays in "Earth time" on Earth and the other travels at near c for one million years in "Earth time" and returns, they'll still have the same carbon dating result in "Earth time", right?

No, the rock that stayed on Earth will be dated to be older (and just as an irrelevant side-note, you'd actually have to use some other form of radiometric dating besides carbon dating, which according to this page has a maximum age range of 50,000-100,000 years).

It's general relativity that in essence says "What's the difference? Who's to say which rock is really moving?" , right?

General relativity does allow you to use any coordinate system you like, even a non-inertial one, but it has a more complicated way of calculating the elapsed time on a clock which depends on the metric of the spacetime you're using--if you're using a non-inertial coordinate system in flat spacetime, and you express the metric correctly in this coordinate system, then using this metric you'll still find that the accelerating twin aged less, by the same amount as predicted using the ordinary time dilation equation of SR in an inertial coordinate system.

 

[rqr]

(to JesseM again)

JesseM, experiment disagrees with your claim that there's no
objective differential aging.

For example, take a look at the following "pro-SR" web site:
http://mentock.home.mindspring.com/twins.htm
(This site is based on experimental results of course.)

At the very bottom of the page is listed the "secret ingredient"
which cleverly eliminates all accelerations of the Twin Paradox.
As I noted, this ingredient is a third person added to the mix.

Here is what happens during the 3-person experiment:
1. Ann "stays at home" while Bob "travels out into space."
(both are the same age at the start of course)
2. "Messenger" Carl passes Bob when they are the same age.
3. When Carl catches up with Ann, their ages are certainly
objectively different.

Since Wayne Throop (the website's creator) was able to
avoid accelerations, all he had left to "explain" this
objective age differential was (and I quote)

---------"Bob must change his inertial frame"----------

Throop even hammers home his point in his final sentence:

--"... there are three distinct inertial frames involved."--

BUT...

My question remains:
What is there about being in different inertial frames that
causes people to (objectively or really) age differently?

Ann is now a very old lady, whereas Carl is still young!
Gray hair and wrinkles versus teen angst and pimples!
You can't get more objective than that!
And it's certainly not reciprocal! (As in "my clock is
slow to you, yet your clock is slow to me")

Perhaps Ann's frame was painted red, but Carl's was
painted green?

rqr

 

[ZapperZ]

My question remains:
What is there about being in different inertial frames that
causes people to (objectively or really) age differently?

You might as well ask "What is the reason that light has the same velocity in all inertial reference frame". This is because starting from that postulate is how we arrive at all the other consequences of SR, including "time dilation" effects.

The problem here is that we did not know any better. We are so familiar with our world, that we "forgot" that many of the things we take for granted are implicitly built upon our perception that light is instantaneous. This includes our concept of space and time. It is when there is a considerable difference in speed between two different systems does this perception run into trouble, and we begin to realize all the stuff that we have assumed to be right are no longer so. How we define space and time now need to be redefined because we now realize that what we are familiar with is simply a "special case" for v<<c.

So really, what you actually should be asking, in fact, is why would two frame NOT have different time period, rather than why should be have different time periods, because the latter is simply a special case of a more general situation. In our universe, having two frame with almost the same velocity is the exception, rather than the rule.

Zz.

 

[JesseM]

JesseM, experiment disagrees with your claim that there's no
objective differential aging.

I said there's no objective differential aging as long as everyone moves inertially--if one twin moves away and then turns around to return, when they reunite he'll have objectively aged less.

For example, take a look at the following "pro-SR" web site:
http://mentock.home.mindspring.com/twins.htm
(This site is based on experimental results of course.)

At the very bottom of the page is listed the "secret ingredient"
which cleverly eliminates all accelerations of the Twin Paradox.
As I noted, this ingredient is a third person added to the mix.

Here is what happens during the 3-person experiment:
1. Ann "stays at home" while Bob "travels out into space."
(both are the same age at the start of course)
2. "Messenger" Carl passes Bob when they are the same age.
3. When Carl catches up with Ann, their ages are certainly
objectively different.

But how do you think this show anyone objectively aged less? The page you linked to certainly doesn't make that claim. In Bob's frame, Ann ages slower than he does--the reason Carl was younger than Ann when they met was that in Bob's frame, Carl is aging even more slowly than Ann because his speed relative to Bob is higher than Ann's. In Ann's frame, both Bob and Carl aged less than she did, since they both had equal speeds. In Carl's frame Ann was aging more slowly than him, but because of his definition of simultaneity Ann was already much older than he or Bob at the moment he and Bob passed each other, so because of this "head start" she was still older when he and Ann met despite aging more slowly. All three will correctly predict that Carl's age is less than Ann's when they meet, but for different reasons.

My question remains:
What is there about being in different inertial frames that
causes people to (objectively or really) age differently?

Nothing, you're just wrong. In each observer's own inertial frame, everyone else is aging slower than they are, and no matter which frame you use you'll still get the same answer to physical questions like the reading on Carl and Ann's clock when they meet.

Ann is now a very old lady, whereas Carl is still young!

Yes, and in Bob's frame that's because Carl is aging slower than Ann due to his higher speed. Also, according to Bob's definition of simultaneity, he is even older than Ann "at the same moment" that Carl and Ann meet.

On the other hand, in Carl's frame Ann aged more slowly than he did...but he has a different definition of simultaneity which says that at the "same moment" that he was passing Bob and they both momentarily had the same age, Ann was already much older than either of them, so even though she aged less between that moment and the moment Carl reached her, she was still older than him when they met.

I can come up with a numerical example of how this works if you're confused.

 

[Idjot]

No Joke

Before I throw in the kicker here... Is there a unanimously acknowledged difference in this forum between acceleration generated g's and mass generated g's? I hope not. Because there is no difference between them. If there is anyone here that recognizes that a g is a g I'd like to share an insight with you: You've been accelerating your whole life - from the moment of conception. Your rate of acceleration is and HAS been 9.8 meters per second per second since you "came" into existence. Ask yourself how fast you're going right now and then ask yourself how old you would be in "Earth time" if you had been born and raised on the moon in a space station. Here's the kicker: Your velocity would be well over the speed of light by now either way. Can the formula you presently use to find Time Dilation accommodate such speeds? Of course not. It's time to reevaluate time dilation.

 

[JesseM]

Before I throw in the kicker here... Is there a unanimously acknowledged difference in this forum between acceleration generated g's and mass generated g's? I hope not. Because there is no difference between them. If there is anyone here that recognizes that a g is a g I'd like to share an insight with you: You've been accelerating your whole life - from the moment of conception. Your rate of acceleration is and HAS been 9.8 meters per second per second since you "came" into existence. Ask yourself how fast you're going right now and then ask yourself how old you would be in "Earth time" if you had been born and raised on the moon in a space station. Here's the kicker: Your velocity would be well over the speed of light by now either way. Can the formula you presently use to find Time Dilation accommodate such speeds? Of course not. It's time to reevaluate time dilation.

General relativity already deals with the equivalence between gravitationally generated G-forces in curved spacetime and G-forces generated by acceleration in flat spacetime--this is known as the equivalence principle. The equivalence is a "local" one, meaning it's only exact in the limit of a very small region of space for a very short region of time. One can't say that sitting in a gravitational field (curved spacetime) for a long period of time is equivalent to constantly accelerating in flat spacetime and thus continually increasing one's velocity relative to an observer far from the source of gravity; this would imply that the distant observer would measure your clock to tick more and more slowly as time passed, when in fact general relativity predicts that the gravitational time dilation factor will be constant in this situation (and for an observer sitting on Earth, the gravitational time dilation factor is very small, so you can reasonably approximate the Earth-bound twin as an inertial observer). To understand how it can be true that the equivalence principle is always respected locally yet the gravitational time dilation factor is constant, you'd probably need to study the theory of general relativity in a lot more depth.

 

[rqr]

(to JesseM again)

JesseM wrote:
> ... because of his definition of simultaneity Ann was
>already much older than he or Bob ...

I wasn't aware that clock synchronization could affect aging;
it would be a strange universe if one's physical aging process
were somehow affected by the way some dude on a faraway
planet had set his clocks. (And one would have to age in
many different ways if each outside observer set his clocks
differently!)

Perhaps it will help if I rephrase the question as follows:

(Note: My rephrasing is just another view of the cited site.)

Two people of the same age pass in space while moving inertially.
We can quantify by saying that both are 5 years old when they
meet in passing.

Physically speaking (and aging is a physical process), we have
only the following (2) choices after they separate:

(1) They continue to age alike
or
(2) They don't

To keep it simple, let's chose (1).

Let’s say that Ann went to the left, and Bob went to the right.

When Bob turns 10, he is meet by Copy-Bob (or Messenger Carl),
who is also 10.

Since Ann and Bob are aging alike all through the experiment
(because of (1) and the fact that they never change frames),
we know that Ann is "now" also 10. (If you insist otherwise,
then you are saying that Ann somehow actually (really) aged
differently from Bob, and we have the same problem, but sooner!)

When Copy-Bob goes on to catch up with Ann, he's still
only sweet 16, whereas she is a married-with-children 28!

This proves that these two people actually aged differently.
They were both 10 years old at the same time, and yet later
they had actually different ages (16 versus 28).

All we have essentially are direct physical comparisons of
people as they meet in space while moving inertially.

There was no acceleration involved.(No one changed frames.)
There is no need to bring up clock synchronization.
(It cannot affect physical aging anyway!)
There is no need to talk about the optical Doppler effect.
There is no need to mention odometers or worldlines.
There is no need to bring up other frames' points-of-view.

Ann aged 18 years while Copy-Bob aged only 6 years. Copy-Bob
aged slower than Ann. Ann aged faster than Copy-Bob.

Contrast the just-given facts with your following claim:

[quote:]
>I said there's no objective differential aging as long as
>everyone moves inertially ...

Ann and Copy-Bob aged differently even though no one
accelerated during the experiment.

[quote:]
... if one twin moves away and then turns around to return,
when they reunite he'll have objectively aged less.

The only difference between this and the cited site's case
is that the latter eliminated accelerations.

But we can go your way if you wish because (as not many know)
acceleration does not affect a clock's physical rate.

In other words, the objective age difference that you agreed
to (by having a turnaround twin) _cannot_ be explained by
acceleration, so how would you explain it?

rqr

 

[paw]

JesseM, experiment disagrees with your claim that there's no
objective differential aging.

For example, take a look at the following "pro-SR" web site:
http://mentock.home.mindspring.com/twins.htm
(This site is based on experimental results of course.)

At the very bottom of the page is listed the "secret ingredient"
which cleverly eliminates all accelerations of the Twin Paradox.
As I noted, this ingredient is a third person added to the mix.

On the contrary, all experimental evidence to date supports JesseM's analysis completely.

That website conveniently glosses over the fact that there is still an acceleration involved. When "Bob zooms off after Ann at 15/17 light speed..." Bob is of course accelerating. Adding messanger Carl seems to eliminate this acceleration but in fact it doesn't. Bob cannot pass any objective information about Ann's speed or age to Carl. Carl is in a different frame of reference and will not agree with Bob about Anns age. In fact Carl will see the system exactly the same way as Bob would if he accelerated into Carls frame. In other words there is an implicit acceleration here.

 

[JesseM]

JesseM wrote:
> ... because of his definition of simultaneity Ann was
>already much older than he or Bob ...

I wasn't aware that clock synchronization could affect aging;
it would be a strange universe if one's physical aging process
were somehow affected by the way some dude on a faraway
planet had set his clocks. (And one would have to age in
many different ways if each outside observer set his clocks
differently!)

Clock synchronization cannot affect the answer to any genuinely physical question about aging, like how much a person ages between two events on their own worldline (like the event of leaving Earth and the event of returning). But the whole point is that the question of which of two inertial observers is "aging faster" is not a physical one, it's a coordinate-dependent one, and as such it depends on your chosen coordinate system's definition of simultaneity. Since Carl and Ann were not at the same location to compare ages objectively until the moment they first meant, without picking a definition of simultaneity you have no basis for thinking the fact that she was older when she met implies she was aging more quickly.

Perhaps it will help if I rephrase the question as follows:

(Note: My rephrasing is just another view of the cited site.)

Two people of the same age pass in space while moving inertially.
We can quantify by saying that both are 5 years old when they
meet in passing.

Physically speaking (and aging is a physical process), we have
only the following (2) choices after they separate:

(1) They continue to age alike
or
(2) They don't

But which choice we make depends on our coordinate system, there is no real physical answer to the question of whether they age at the same rate or not (in a frame where both were moving at the same speed in opposite directions, they would age at identical rates, but in a frame where their speeds were different, they would age at different rates, and relativity says they're no basis for preferring one inertial frame over another).

To keep it simple, let's chose (1).

OK, then we're choosing a frame where Bob and Ann both have the same speed.

Let’s say that Ann went to the left, and Bob went to the right.

When Bob turns 10, he is meet by Copy-Bob (or Messenger Carl),
who is also 10.

Since Ann and Bob are aging alike all through the experiment
(because of (1) and the fact that they never change frames),
we know that Ann is "now" also 10.

Only if we use this frame's definition of simultaneity.

(If you insist otherwise,
then you are saying that Ann somehow actually (really) aged
differently from Bob, and we have the same problem, but sooner!)

No, I am saying that according to relativity there is no "actual" or "real" answer to whether Ann ages at the same rate as Bob or a different rate, it's totally coordinate-dependent, just like the question of whether Ann's speed is equal to Bob's or different.

When Copy-Bob goes on to catch up with Ann, he's still
only sweet 16, whereas she is a married-with-children 28!

Yes, and in the frame where Ann and Bob age at the same rate, it would be impossibe for copy-Bob to catch up with Ann unless his speed is greater than either of theirs (if his speed was the same as Ann's and he was traveling to the right as well, then the distance between them would remain constant), thus he will be aging slower than them in this frame.

This proves that these two people actually aged differently.

Not if "actually" implies a coordinate-independent statement which doesn't depend on your choice of simultaneity. It is true that in the frame where Ann and Bob age at the same rate, copy-Bob ages slower than them, but there is nothing special about this frame.

They were both 10 years old at the same time

Only using this frame's definition of simultaneity, not in other frames.

and yet later
they had actually different ages (16 versus 28).

Yes, they had different ages when they met, and all frames will agree on this, but they will disagree that the event of copy-Bob turning 10 was simultaneous with Ann turning 10.

All we have essentially are direct physical comparisons of
people as they meet in space while moving inertially.

But to support your notion that copy-Bob aged slower, you must use a definition of simultaneity which tells you that copy-Bob was aged 10 at the same moment Ann was aged 10, and this is not based on any "direct physical comparison of people as they meet in space", you can easily pick a frame which agrees about all local readings when people meet but disagrees that these two widely-separated events happened simultaneously.

There is no need to bring up clock synchronization.

Of course there is, do you think the notion that Ann and copy-Bob both turned 10 simultaneously does not depend on your clock synchronization?

But we can go your way if you wish because (as not many know)
acceleration does not affect a clock's physical rate.

In other words, the objective age difference that you agreed
to (by having a turnaround twin) _cannot_ be explained by
acceleration, so how would you explain it?

rqr

Acceleration does not affect the clock's instantaneous rate of ticking in any given frame, that depends only on its speed in that frame, but since acceleration means a change in velocity (which means the speed must be changing in at least some frames), it does affect the integral of \sqrt{1 - v(t)^2/c^2} which each inertial frame uses to calculate the elapsed proper time on a non-inertial worldline. And if each frame does this integral using the speed as a function of time v(t) in their own coordinates, they will each get the same answer for the proper time along a worldline between two events, and mathematically you can show that this proper time is always smaller than the proper time on an inertial worldline which goes between the same two events.

It might also help if you'd read over my geometric analogy in post #10--your statement that rate of clock ticking depends only on speed rather than on acceleration is analogous to the statement that for the paths on paper in my example, the rate that their "partial path length" is increasing in a particular coordinate system is dependent only on the slope of that line in that coordinate system (analogous to velocity), not on the rate that the slope changes (analogous to acceleration), yet it's still obvious that a straight-line path with constant slope between two points will always have a shorter length than a non-straight path between those points whose slope changes.

 

[RandallB]

Bob is of course accelerating. Adding messanger Carl seems to eliminate this acceleration but in fact it doesn't. Bob cannot pass any objective information about Ann's speed or age to Carl. Carl is in a different frame of reference and will not agree with Bob about Anns age. In fact Carl will see the system exactly the same way as Bob would if he accelerated into Carls frame. In other words there is an implicit acceleration here.

Nonsense, replacing BOB’s turn around with an information transfer to CARL at the “turn around point” rather than having Bob actually turn around does demonstrate that GR accelerations do not have anything to do with the SR twins paradox.
CARL only needs the information from BOB that he has when they are both at the “turn around point”. That does not include any “current” info about ANN; only how old BOB is currently and what BOB remembers about how old both he and ANN where when he was near her at the start of the experiment. Accelerations implied or real & GR have nothing to do with working out the details of the Twins problem, just the three defined SR inertial reference frames.

Being able to establish an “objective” decision as to which one Ann, Bob, or Carl is “really” aging the fastest is a matter establishing a “preferred" reference frame. SR does not necessarily require that such a frame be defined.

 

[JesseM]

Nonsense, replacing BOB’s turn around with an information transfer to CARL at the “turn around point” rather than having Bob actually turn around does demonstrate that GR accelerations do not have anything to do with the SR twins paradox.

paw didn't say anything about "GR" accelerations, you can calculate the proper time on an accelerated worldline in flat spacetime just fine using SR alone. It's certainly true that if you have two worldlines between the same pair of events, and one is inertial while the other involves acceleration, then the one involving acceleration will always have less proper time, so in that sense acceleration is important in understanding the twin paradox. The situation with three twins doesn't actually involve two different worldlines that go between the same two events, although if you have a twin who travels away from Earth inertially and then instantaneously accelerates and returns inertially, then during the outbound leg he'll age the same amount as Bob traveling alongside him, and during the inbound leg he'll age the same amount as Carl traveling alongside him, so this shows you can compute the total proper time on his non-inertial worldline by adding segments of the worldlines of two inertial observers.

 

[pervect]

paw didn't say anything about "GR" accelerations, you can calculate the proper time on an accelerated worldline in flat spacetime just fine using SR alone. It's certainly true that if you have two worldlines between the same pair of events, and one is inertial while the other involves acceleration, then the one involving acceleration will always have less proper time, so in that sense acceleration is important in understanding the twin paradox. The situation with three twins doesn't actually involve two different worldlines that go between the same two events, although if you have a twin who travels away from Earth inertially and then instantaneously accelerates and returns inertially, then during the outbound leg he'll age the same amount as Bob traveling alongside him, and during the inbound leg he'll age the same amount as Carl traveling alongside him, so this shows you can compute the total proper time on his non-inertial worldline by adding segments of the worldlines of two inertial observers.

If there is anyone for some strange reason who still doesn't see that Jesse is right, I would suggest reading the first chapter of "Space-time physics", a textbook on GR, available for download at:

http://www.eftaylor.com/download.html#special_relativity

(This is the first chapter of the first edition that's available for download, later editions can be found at your local library).

The twin paradox is much like the triangle inequality in standard Euclidean geometry. The difference is, that in Euclidean geometry, the shortest distance between two points is a straight line, while in a Lorentz geometry.

In a Lorentz geometry, a curved worldline between two points is shorter than the direct wordline between two events - shorter as measured by the elapsed proper time along the wordline.

 

[RandallB]

paw didn't say anything about "GR" accelerations,

I disagree, when he said "In other words there is an implicit acceleration here." it supported the often seen and incorrect view that the aging differances between twins is based on an "Acceleration" in the turn around not in the SR frames alone as your discriptions show.

 

[JesseM]

I disagree, when he said "In other words there is an implicit acceleration here." it supported the often seen and incorrect view that the aging differances between twins is based on an "Acceleration" in the turn around not in the SR frames alone as your discriptions show.

The fact that the traveling twin's worldine has less proper time is based on the fact that the twin accelerates during the turnaround, in just the same that the fact that a bent path between two points in space has a shorter distance than a straight path is based on the fact that there is a change in its slope. But this doesn't imply you're using GR, you can use an inertial frame in SR to calculate the proper time between two events along an non-inertial worldline by taking the speed as a function of time v(t) in that frame and evaluating the integral \int^{t_1}_{t_0} \sqrt{1 - v(t)^2/c^2} \, dt where t1 and t0 are the times of the two events in this frame. If you have two worldlines between a given pair of events in flat spacetime, and one has a constant v(t) function while the other has a non-constant v(t), that fact alone is enough to absolutely guarantee that the one with the non-constant v(t) (i.e. the one that accelerated) will yield a smaller value when you evaluate this integral.

 

[vivesdn]

The twin that remains on Earth is constantly accelerating, as he is describing a cycloid orbit around the sun and daily following the Earth perimeter.
Honestly, I think that relativistic effects are due to speed. The funny thing, and hence the word relativity, is that it is not possible to know who is really moving and who's at rest, as everybody will measure light speed as being c because his time paces correspondingly. If the word 'rest' means anything at all in this context.

 

[JesseM]

The twin that remains on Earth is constantly accelerating, as he is describing a cycloid orbit around the sun and daily following the Earth perimeter.

In calculations of the twin paradox it's assumed that the so-called "Earth twin" is moving inertially--remember, this is a thought-experiment about special relativity, not an analysis of something that's actually been done in the real world! To adequately deal with the motions of the Earth you'd have to take into account the fact that the Earth is moving in curved spacetime, so you'd need general relativity to deal with this; but as an approximation for this situation, treating the Earth as moving inertially in flat spacetime would probably introduce only a slight error, the velocities and gravity are low enough that the difference in aging between a clock on Earth and a clock arbitrarily far away from it and at rest in the Schwarzschild coordinate system centered on the Sun would be very small. In any case, the point of the thought-experiment was just to show the difference between inertial and non-inertial paths in flat spacetime, you're free to imagine the inertial twin was actually moving inertially in deep space rather than sitting on the Earth.

 

[vivesdn]

JesseM,

my point was that what make time move slower is the velocity, close enough to c if an effect has to be measured, not accelerations.
I've been told that there are particles created by cosmic rays in the upper atmosphere that, given their speed, altitude and life, should not reach the surface. But they actually do, so their lifespan is higher when moving at high speed that when created on an accelerator with lower speeds.
But as I'm not an expert, I may be wrong.

 

[JesseM]

JesseM,

my point was that what make time move slower is the velocity, close enough to c if an effect has to be measured, not accelerations.

It's true that time dilation in any frame depends only on velocity, but velocity is frame-dependent, whereas it's true in any frame that if one clock travels inertially between two points in spacetime while another travels between the same two points on a path that includes some acceleration, the one that accelerated will always have elapsed less time.

I've been told that there are particles created by cosmic rays in the upper atmosphere that, given their speed, altitude and life, should not reach the surface. But they actually do, so their lifespan is higher when moving at high speed that when created on an accelerator with lower speeds.

Yes, this is true of muons, see http://www.glenbrook.k12.il.us/gbssci/phys/Class/relativity/u7l2e2muon.html for example. But the explanation for this depends on your frame--in the Earth's frame, they make it to Earth because time dilation slows their decay rate down, but in the muon's own rest frame, it's the Earth's clocks that are running slow while their own decay rate is unchanged, and the reason they make it through the atmosphere before decaying is that the distance from the upper atmosphere to the surface is shrunk by length contraction. Each frame's point of view is equally valid, there's no reason to prefer one over the other.

 

[pervect]

JesseM,

my point was that what make time move slower is the velocity, close enough to c if an effect has to be measured, not accelerations.
I've been told that there are particles created by cosmic rays in the upper atmosphere that, given their speed, altitude and life, should not reach the surface. But they actually do, so their lifespan is higher when moving at high speed that when created on an accelerator with lower speeds.
But as I'm not an expert, I may be wrong.

If you regard speed as a purely relative quantity, an angle on the space-time graph you are correct. But if you regard speed as an "absolute", you are mistaken.

An analogy can be drawn between the twin paradox and the lengthy of a hypotenuse. The twin paradox is just a triangle in a Minkowski geometry. The point is that what is important in determining the relative lengths of the triangle sides is the angle. In the twin paradox, this angle is just the relative velocity between the twins. This determines the ratio between the length of the hypotenuse (proper time elapsed for the twin that does not accelerate) vs the sum of the lengths of the two sides of a triange (proper time elapsed for a twin that changes direction).

The effect of gravity requires GR to analyze. There is an effect due to height according to GR, but it's very small.

[add]
If you consider a short enough period of time, though, it is still true in GR that the longest elapsed time between two points is a geodesic. You run into trouble if you make the time interval too long (i.e. you have to keep the time less than an orbital period).

If you have a non-orbital geodesic that is at the same height at time t and at time t + delta, what is that geodesic? It's the path of a ball that you throw up into the air at time t, and that lands on the ground at time t+delta. This is the only geodesic with those qualities for short time intervals.

This is the path that maximizes proper time. It will have a longer elapsed time than a clock that's fixed to the surface of the Earth.

 

[paw]

Nonsense, replacing BOB’s turn around with an information transfer to CARL at the “turn around point” rather than having Bob actually turn around does demonstrate that GR accelerations do not have anything to do with the SR twins paradox.
CARL only needs the information from BOB that he has when they are both at the “turn around point”. That does not include any “current” info about ANN; only how old BOB is currently and what BOB remembers about how old both he and ANN where when he was near her at the start of the experiment. Accelerations implied or real & GR have nothing to do with working out the details of the Twins problem, just the three defined SR inertial reference frames.

Being able to establish an “objective” decision as to which one Ann, Bob, or Carl is “really” aging the fastest is a matter establishing a “preferred" reference frame. SR does not necessarily require that such a frame be defined.

Evidently I wasn't clear. I'm not saying you need to include acceleration in the analysis, I'm saying the result of the analysis is the same in either treatment. Inertial Carl is equivalent to accelerated Bob. At some point the twins have to compare clocks to have an objective measure of their respective ages. Whether they physically get back together or a third inertial party carries the information back it still involves a real or implied acceleration.

JesseM gives a much more lucid explanation but I believe I'm saying the same thing. It's meaningless to say who is aging faster unless and until the twins can objectively compare ages.

 

[rqr]

to JesseM

I repeat my prior question:
In other words, the objective age difference that you agreed
to (by having a turnaround twin) _cannot_ be explained by
acceleration, so how would you explain it?

Your reply relied on coordinate systems, but there are none
in the given experiment. Also, even if there were any, then
their coordinate readings certainly could not physically
affect either one's aging or an atomic clock's rate.

Just as you age in only one way as long as you do not change
your speed, any given ideal clock will tick at only one rate
as long as its speed does not change.

Contrast this with the fact that various coordinate systems
will find _different_ "tick rates" for one and the same clock
that is moving at an unchanging speed.

There is no way that the slope of some coordinate line can
have any physical effect upon the intrinsic atomic rate of
an atomic clock or upon one's aging rate.

Math (geometry) cannot affect one's aging rate.
Math (geometry) cannot affect an atomic clock's atomic rate.

We can also look at it in this way:
You have agreed that the Two-people case contains real or
objective differential aging. Even though the Three-people
case has no accelerations, and the Two-people case does have
accelerations, acceleration per se has no physical effect
upon aging rates (or atomic clock rates). Clearly, as far
as aging is concerned, there are no physical differences
between the two cases. Therefore, if the Two-people case
has objective differential aging, then so does other case.

What is the physical cause of objective differential aging?
(It can occur even without accelerations.)
(It cannot be affected by clock synchronization.)
(It cannot be affected by coordinate values.)
(There are no coordinate systems in the given experiment.)
(There's not even a definition of clock synchronization.)
(There are only people passing/meeting in space.)

[added]
Originally Posted by rqr:
>>There is no need to bring up clock synchronization.

quoting JesseM:
>Of course there is, do you think the notion that Ann
>and copy-Bob both turned 10 simultaneously does not
>depend on your clock synchronization?

It cannot depend upon clock synchronization because there
are no clocks in the given experiment. What is does depend
on is the simple given fact that Ann & Bob aged alike.
(This was chosen as one of only two physically possible
paths, (i) Ann & Bob do not age alike, or (ii) they do.)
This simple given fact tell us that whenever Bob is 10,
so is Ann. Therefore, whenever Copy-Bob & Bob are 10, so
is Ann. However, since Ann & Copy-Bob have different ages
at the end, they must have aged differently.

We now must wonder what can be given as the physical
cause of objective differential aging when there is no
acceleration and no clock synchronization to fall back on.

If you pick choice (ii), then you have Bob and Ann aging
differently (objectively) without acceleration, so you
still have the same problem of finding a physical cause
for objective differential aging for inertially-moving
people.

rqr

 

[JesseM]

I repeat my prior question:
In other words, the objective age difference that you agreed
to (by having a turnaround twin) _cannot_ be explained by
acceleration, so how would you explain it?

Of course it can be explained by acceleration, in exactly the same way that the difference in lengths between two paths between a pair of points on a piece of paper can be explained by the fact that one is straight and the other is non-straight (which is the same as saying one has a constant slope and the other has a changing slope, in terms of any cartesian coordinate system). Did you read my geometric analogy, and if so do you see anything wrong with it?

Your reply relied on coordinate systems, but there are none
in the given experiment.

A coordinate system isn't part of the physical facts of an experiment, it's part of how you describe the experiment and calculate things about it. Again think of the analogy with lines on paper--you can calculate the length of a path using a particular cartesian coordinate system, but the actual length of the path is independent of your choice of coordinate system, it's part of the physical facts about the path itself.

Also, even if there were any, then
their coordinate readings certainly could not physically
affect either one's aging or an atomic clock's rate.

Of course they don't, but they are used to calculate the elapsed age on a worldline in just the same way a cartesian coordinate system is used to calculate the elapsed distance on a path through space.

Just as you age in only one way as long as you do not change
your speed, any given ideal clock will tick at only one rate
as long as its speed does not change.

Why do you believe this? There is no physical reason to believe there is an objective truth about the "rate of ticking" of a particular clock that's independent of your choice of coordinate system, any more than there needs to be an absolute truth about an object's "velocity". Do you believe in absolute velocity? If not, why do you think clocks must have an absolute ticking rate?

There is no way that the slope of some coordinate line can
have any physical effect upon the intrinsic atomic rate of
an atomic clock or upon one's aging rate.

According to relativity there is no such thing as "the intrinsic rate" of a clock's ticking, it's observer-dependent just like velocity. There are no physical problems with this view, since there is no need to postulate an absolute ticking rate in order to make predictions about the outcome of any physical event, like what two clocks will read when they pass each other. If you have a problem with this, it would seem that it is based on your philosophical preconceptions, not on scientific issues.

Math (geometry) cannot affect one's aging rate.
Math (geometry) cannot affect an atomic clock's atomic rate.

Again, the "rate" of a clock's ticking at a particular moment has no objective value in relativity. The total time elapsed between two points on a clock's worldline does have an objective value, and this is a function of the geometry of spacetime in just the same way that the total distance between two points on a path through space is a function of the geometry of space.

You have agreed that the Two-people case contains real or
objective differential aging.

Differential aging between two events which lie on both twins' worldlines, yes. Different objective "rate of aging" at any particular moment in time, no.

Even though the Three-people
case has no accelerations, and the Two-people case does have
accelerations, acceleration per se has no physical effect
upon aging rates (or atomic clock rates).

The three-people case does not involve a comparison of different worldlines which travel through the same pair of points in spacetime. You can only talk about an objective amount of aging if you pick two particular points on a given worldline and ask how much a clock which has that worldline will have ticked between those two points. Again, talking about a "rate of aging" at a single moment is meaningless in relativity.

Therefore, if the Two-people case
has objective differential aging, then so does other case.

Objective differential aging between what pair of events in the three-person case? In this case, none of the worldlines intersect any other worldline more than once, so you don't have any pair of events where two different worldlines go between the pair and you can compare how much time elapsed on each worldline between those same two events.

What is the physical cause of objective differential aging?
(It can occur even without accelerations.)

Not if you restrict yourself to talking about the amount a clock/observer ages between two events on its own worldline, and avoid the idea that there is an objective truth about the "rate a clock is ticking relative to absolute time" at any given instant. This is analogous to the fact that there is an objective truth about the length of a path on a piece of paper between two points along that path, but no coordinate-independent truth about the "rate of partial path length increase relative to increasing y-coordinate" (again, please read over my 2D geometric analogy if you haven't already) at a single point on the path.

 

[JesseM]

(continued)

[added]
Originally Posted by rqr:
>>There is no need to bring up clock synchronization.

quoting JesseM:
>Of course there is, do you think the notion that Ann
>and copy-Bob both turned 10 simultaneously does not
>depend on your clock synchronization?

It cannot depend upon clock synchronization because there
are no clocks in the given experiment.

Fine, replace "clock synchronization" with "simultaneity". Your claims about who aged at what rate do depend on an assumption about simultaneity, namely that the event of Ann turning 10 and copy-Bob turning 10 both happened simultaneously. Just as a person might believe there is an objective truth about the "length" of an object even if there are no physical rulers in an experiment, for your claims to make sense you must believe there is an objective truth about simultaneity even if there are no synchronized clocks in the experiment.

What is does depend
on is the simple given fact that Ann & Bob aged alike.
(This was chosen as one of only two physically possible
paths, (i) Ann & Bob do not age alike, or (ii) they do.)

And what made you choose (ii)? An arbitrary whim? Why is it that you think Bob and copy-Bob, who passed each other at constant velocities just like Ann and Bob, do not age alike? And most importantly, why do you not consider the possibility that there is no objective truth about whether two separated observers are aging at the same rate or different rate? Would you say there is an objective truth about whether two observers moving apart move apart at "the same speed" or "different speeds", and if not, why should "rate of aging" be any more of an objective notion than speed?

This simple given fact tell us that whenever Bob is 10,
so is Ann.

And here you are making a claim about simultaneity.

We now must wonder what can be given as the physical
cause of objective differential aging when there is no
acceleration and no clock synchronization to fall back on.

There is no "objective differential aging" here because any claims about aging depend on your choice of simultaneity, which is totally arbitrary. I could just as easily pick a different definition of simultaneity which would tell me that copy-Bob aged more than Ann between the moment copy-Bob passed Bob and the moment he passed Ann, and you'd have no physical reason to say your definition of simultaneity is "more correct" than mine (your arbitrary whim is not a reason)

If you pick choice (ii)

A choice made by an arbitrary whim, not based on any of the observable physical facts of the experiment. Say we decide whether to go with your choice (i) or (ii) by flipping a coin--would you say that the "real physical truth" about who aged more depends on the outcome of this coinflip?

then you have Bob and Ann aging
differently (objectively) without acceleration

No, you have provided no argument whatsoever as to why we should think that there is any objective truth about who ages more, you have only shown that if we pick a totally arbitrary definition of simultaneity then there will be a truth about who ages more according to that definition.