Twin Paradox: Special Relativity Theory Explained

[keith_21]

Hi all, I am taking a grade 12 physics course and we just covered special relativity theory however one thing troubles me; the twin paradox. The thought experiment proposes that a one twin travels to a distant star and back at a speed approaching that of light while the other twin remains on Earth. The twin on Earth should see his twin in the spaceship age slower, but wouldn't the twin on the ship think the same thing seeing Earth recede at high speed and then return. According to my textbook the answer is NO because the special theory of relativity applies only to inertial frames (in this case the Earth). The situation is not symmetrical since the spaceships velocity must change at the turn around point meaning it is a non-inertial reference frame.

Here is my question; could the twin in the spaceship not interpret the event as Earth moving away and then returning, from the frame of reference of the ship does Earth not appear to change its velocity at a turn around point making it non-inertial? Why is this thought experiment not symmetrical?

 

[robphy]

According to my textbook... the special theory of relativity applies only to inertial frames (in this case the Earth).

What textbook said that? It's not accurate.

 

[moe darklight]

I was having the same problem, here's a website that explains it pretty well:

http://www.phys.vt.edu/~jhs/faq/twins.html

and here's the thread I started where some people expanded some more on what the website explains:

https://www.physicsforums.com/showthread.php?t=150894

 

[jtbell]

The twin paradox has been discussed here many times. A forum search on "twin" should turn up plenty of reading material. You might be interested in two detailed descriptions of the same scenario which both show that both twins must agree on what is happening, if they do it correctly:

Using the relativistic Doppler effect to analyze what each twin sees if he watches the other twin through a telescope:

https://www.physicsforums.com/showpost.php?p=510214&postcount=3

Using the Lorentz transformation equations:

https://www.physicsforums.com/showpost.php?p=1178108&postcount=3

 

[JesseM]

What textbook said that? It's not accurate.

It's accurate in the sense that the ordinary algebraic equations of SR like $$\tau = t \sqrt{1 - v^2/c^2}$$ can only be used in inertial frames, although as I understand it you can put SR into tensor form so it'll apply in any frame, or you can figure out a different set of algebraic equations for an accelerating frame.

 

[robphy]

It's accurate in the sense that the ordinary algebraic equations of SR like $$\tau = t \sqrt{1 - v^2/c^2}$$ can only be used in inertial frames, although as I understand it you can put SR into tensor form so it'll apply in any frame, or you can figure out a different set of algebraic equations for an accelerating frame.

As you've demonstrated, the original statement is inaccurate because you had to elaborate on the restrictions. More specifically, while equations-often-seen-in-SR [merely a subset of SR's equations] apply only to inertial frames, SR, itself, can apply to any frame (inertial or noninertial [accelerating]). Implicitly, I'm using the modern interpretation of SR as "relativity on R⁴ with a flat Minkowskian metric".

IMHO, that "SR applies only to inertial frames" is akin to the inaccurate thinking in kinematics that "velocity is defined as distance over time"... in the sense that a special case or application of a concept is being inappropriately generalized.

 

[quantum123]

The twin paradox melts away in GR.
The unaccelerated twin has a world-line between two events with the shape of a geodesic which maximizes the proper time - time measured by the twin's clock. Hence the unaccelerated(earth) twin is a very special observer in spacetime. The Earth twin 's clock will run faster than the space twin's clock. There is no symmetry in this case.

 

[JesseM]

As you've demonstrated, the original statement is inaccurate because you had to elaborate on the restrictions. More specifically, while equations-often-seen-in-SR [merely a subset of SR's equations] apply only to inertial frames, SR, itself, can apply to any frame (inertial or noninertial [accelerating]). Implicitly, I'm using the modern interpretation of SR as "relativity on R⁴ with a flat Minkowskian metric".

Yeah, but I don't think a high school textbook really needs to elaborate on this mathematically more sophisticated definition of relativity in terms of a metric (which often would not even be presented to college undergraduates--I wasn't taught it anyway); the statement can basically be taken to mean "the form of relativity we've presented in this textbook can only be used in inertial frames". Anyway, saying "special relativity can be applied to accelerated frames" would at least be equally inaccurate/unclear without detailed elaboration.

As an analogy, would you object to a high school textbook on classical mechanics which said "an inertial frame is one where Newton's laws of motion hold", when technically Newton's laws can also be stated in tensor form so that they work in non-inertial frames?

 

[robphy]

Anyway, saying "special relativity can be applied to accelerated frames" would at least be equally inaccurate/unclear without detailed elaboration.

That's the point of my remark... trying to dispel the
often heard misconception that "special relativity can't handle accelerated frames" (which it can!)
which is implied by

"the special theory of relativity applies only to inertial frames"

Yeah, but I don't think a high school textbook really needs to elaborate on this mathematically more sophisticated definition of relativity in terms of a metric (which often would not even be presented to college undergraduates--I wasn't taught it anyway); the statement can basically be taken to mean "the form of relativity we've presented in this textbook can only be used in inertial frames". Anyway, saying "special relativity can be applied to accelerated frames" would at least be equally inaccurate/unclear without detailed elaboration.

As an analogy, would you object to a high school textbook on classical mechanics which said "an inertial frame is one where Newton's laws of motion hold", when technically Newton's laws can also be stated in tensor form so that they work in non-inertial frames?

Contrary to your implication, I'm not advocating including all of the technical details (e.g. a metric, etc...) in a statement.

I am advocating more correct statements.

Ideally, a statement (a "blurb" or "slogan", if you will) should stand alone.

IMHO, it is better to make an incomplete-but-correct statement... rather than one that is incorrect-without-additional-remarks. (An example of a statement that is incorrect-without-additional-remarks is saying "velocity=distance/time" without specifying the restrictive condition when that is true.)

In the incomplete-but-correct statement, you have a correct statement without all of the details (which will enlighten you later).

In the incorrect-without-additional-remarks statement, you have to have to unlearn an incorrect statement and any other misconceptions derived from it (which will possibly annoy you later).

 

[JesseM]

Anyway, saying "special relativity can be applied to accelerated frames" would at least be equally inaccurate/unclear without detailed elaboration.

That's the point of my remark... trying to dispel the
often heard misconception that "special relativity can't handle accelerated frames" (which it can!)
which is implied by

How is what I said the point of your remark? What I said was that it would be equally inaccurate to say "special relativity can be applied to accelerated frames"...i.e. if you're going to say "special relativity can't be applied to accelerated frames" is wrong, then you should also say "special relativity can be applied to accelerated frames" is wrong too, because in neither case have you specified clearly what you mean by "applying special relativity" (obviously you can't apply the algebraic equations of SR like the time dilation equation to an accelerated frame).

Contrary to your implication, I'm not advocating including all of the technical details (e.g. a metric, etc...) in a statement.

Well, what you seem to be arguing is that if a certain statement is true but only with certain unstated assumptions--namely, that what the textbook means by "special relativity can't be applied" is just that you can't use the equations of SR presented in the textbook itself, not that you can't use some more mathematically sophisticated equations which professional physicists would use as a way of stating the theory of special relativity--then the statement is incorrect. I would say it is perhaps incomplete, but not incorrect, and in practice students will understand from this that they can't use the equations they've been given in non-inertial frames, which is correct.

Again, what would you say about the statement, often seen in textbooks, that an inertial frame in classical mechanics can be defined as one where Newton's laws hold? The laws of Newtonian mechanics can be stated in tensor form just like SR, and in this form they hold in accelerated frames too, no?

In the incomplete-but-correct statement, you have a correct statement without all of the details (which will enlighten you later).

In the incorrect-without-additional-remarks statement, you have to have to unlearn an incorrect statement and any other misconceptions derived from it (which will possibly annoy you later).

I don't see a clear distinction can be made between "incomplete" and "incorrect without additional remarks" in this case. Would you disagree that the statement "SR can be applied to accelerated frames" could be seen as "incorrect without additional remarks", since plenty of specific equations in SR cannot be used in accelerated frames?

 

[MeJennifer]

Well, running the risk of getting stuck in the middle of this discussion, strictly speaking acceleration is not handled by SR for the simple reason that acceleration is mitigated by curved space-time.

 

[JesseM]

Well, running the risk of getting stuck in the middle of this discussion, strictly speaking acceleration is not handled by SR for the simple reason that acceleration is mitigated by curved space-time.

You can certainly have acceleration in flat spacetime. Also, flat spacetime is something that will be agreed upon by all coordinate systems--if I'm an inertial observer in flat spacetime and I see an accelerating observer, then even though that observer can have his own coordinate system where the G-forces he experiences are due to a uniform gravitational field rather than acceleration, he'll agree that the curvature of spacetime is flat.

 

[MeJennifer]

You can certainly have acceleration in flat spacetime.

Remember "spacetime tells mass how to move, while mass tells spacetime how to curve"?
Are you saying that things simply accelerate by themselves without a need for space-time to curve?
That seems to me a clear violation of the equivalence principle.

 

[pervect]

Maybe the textbook could just say "accelerated frames are outside the scope of this textbook"? I think that might make everyone happy.

It seems to me that the definition of "special relativity" is what's basically being argued about. From a purist POV, whatever one can deduce without using the equivalence principle or the Einstein field equations would be considered to be "special relativity". From a pedagogical POV, one wants to separate material that requires advanced mathematics to handle from material that does not require advanced mathematics. Hence, one classifies material that requires tensors or in this case differential geometry to handle as "General Relativity", even though the difference is only the mathematical treatment and not the basic physical assumptions.

 

[nakurusil]

Well, running the risk of getting stuck in the middle of this discussion, strictly speaking $$acceleration is not handled by SR$$ for the simple reason that acceleration is mitigated by curved space-time.

This is definitely incorrect, quite afew universities teach accelerated motion in SR. Look up "hyperbolic motion".

 

[pervect]

I think the main issue consists of the defintion of 'frame'. You don't need to consider the notion of the "frame" of an acclerated observer to calculate hyperbolic motion, so that is not especially problematical.

Some of the trickier technical issues involving frames are really only fully resolved with differential geometry.

Unfortunately, this does tend to leave beginning students with strange ideas. The frame-field of an accelreating obserer is really not that much different from the frame of a non-accelerating observer as long as one is sufficiently close to the accelrating observer. Differences only start to creep in as a second order effect of magnitude approximately (1+gL)/c^2.

 

[robphy]

Would you disagree that the statement "SR can be applied to accelerated frames" could be seen as "incorrect without additional remarks", since plenty of specific equations in SR cannot be used in accelerated frames?

"SR can be applied to accelerated frames" is a correct statement.
No additional remarks are needed. (Caveat: This does not mean that you can use every equation in SR in accelerated frames. Indeed, not every equation in SR applies in all cases treated by SR. ...Just like: not every equation in Galilean kinematics applies in all cases treated by Galilean kinematics. [e.g. Velocity is not always distance/time.] None of these statements is in conflict with the truth of the statement above.)

"SR can't be applied to accelerated frames" is an incorrect statement. You may add remarks to restrict the condition when that statement would be true... for example, "when using equations derived for inertial frames".

 

[robphy]

Well, running the risk of getting stuck in the middle of this discussion, strictly speaking acceleration is not handled by SR for the simple reason that acceleration is mitigated by curved space-time.

[4-]acceleration (which causes a worldline not to be a geodesic) is associated with a [nongravitational] 4-force. Spacetime curvature tells us which worldlines are geodesics.

Remember "spacetime tells mass how to move, while mass tells spacetime how to curve"?
Are you saying that things simply accelerate by themselves without a need for space-time to curve?

Although "spacetime [curvature] tells mass how to move", this does not mean that all motion [or all acceleration] is due to curvature. Elaborating on this quote, and using what I said above, it is more correct to say "spacetime [curvature] tells mass how to move [inertially]"... 4-forces can further contribute to [influence] the motion.

 

[MeJennifer]

Although "spacetime [curvature] tells mass how to move", this does not mean that all motion is due to curvature.

That is right and that is not what I was talking about, I am talking about acceleration.

So are you saying that if something accelerates there is no space-time curvature involved? So it just accelerates by itself?

Sorry but that does not make any sense.

You realize that you need energy to accelerate something right and that energy causes curvature right?

 

[robphy]

So are you saying that if something accelerates there is no space-time curvature involved? So it just accelerates by itself?

Sorry but that does not make any sense.

You realize that you need energy to accelerate something right and that energy causes curvature right?

Regarding particles as test point particles (that have no backreaction on the spacetime), if something accelerates (with a nonzero 4-acceleration... so the worldline is not geodesic) it is due to a 4-force applied to the particle by a nongravitational agent [e.g. another object, like the surface of the earth].

As I said before, the role played by spacetime curvature is to determine which curves are geodesic (i.e. what the inertial motions are) and which are not (i.e. what the noninertial [a.k.a. accelerated] motions are).

 

[JesseM]

Remember "spacetime tells mass how to move, while mass tells spacetime how to curve"?
Are you saying that things simply accelerate by themselves without a need for space-time to curve?
That seems to me a clear violation of the equivalence principle.

Your argument would be right if we were talking about a universe with no laws of physics besides GR, but other forces such as electromagnetism can cause objects to deviate from geodesics. Then again, it's possible the other forces could ultimately have some sort of "curved spacetime" explanation--I think the Kaluza-Klein theory tried to do this for electromagnetism, and I think some kind of quantum version of it was incorporated into string theory. But in terms of the best current theories that can actually be used to make predictions, you can have additional force fields on a curved spacetime which make things move on non-geodesic paths.

 

[MeJennifer]

Regarding particles as test point particles (that have no backreaction on the spacetime), if something accelerates (with a nonzero 4-acceleration... so the worldline is not geodesic) it is due to a 4-force applied to the particle by a nongravitational agent [e.g. another object, like the surface of the earth].

And that other force is a form of energy or not?
If it is then please explain how that force would not curve space-time.

Remember it is not just mass that curves space-time, energy also curves space-time.

 

[JesseM]

"SR can be applied to accelerated frames" is a correct statement.
No additional remarks are needed. (Caveat: This does not mean that you can use every equation in SR in accelerated frames. Indeed, not every equation in SR applies in all cases treated by SR. ...Just like: not every equation in Galilean kinematics applies in all cases treated by Galilean kinematics. [e.g. Velocity is not always distance/time.] None of these statements is in conflict with the truth of the statement above.)

"SR can't be applied to accelerated frames" is an incorrect statement. You may add remarks to restrict the condition when that statement would be true... for example, "when using equations derived for inertial frames".

OK, I guess that makes sense. But it seems to me it's more a matter of being correct about standard definitions than about any physical issues...it's a matter of convention, not physics, that the words "special relativity" are used for both the set of algebraic equation and the set of tensor equations. Of course the convention is a reasonable one, since it would be problematic to treat two different mathematical procedures which make identical physical predictions as "different theories", but the point is that the student won't be led to any incorrect understanding of physics by the statement that SR doesn't apply to non-inertial frames, since they will understand "SR" to mean the formulation they've been presented in the textbook, with no knowledge of the tensor form. And it would be difficult to include your more "correct" version in textbooks without at least a brief mention of the tensor form, since just saying "SR can be applied to accelerated frames" without elaboration would definitely tend to lead to incorrect physical ideas about applying the equations they've been learning to an accelerated frame. Maybe the best balance would be to say something like "The equations of SR presented in this textbook cannot be applied to accelerated frames, although there is a more sophisticated way of describing the theory using tensor mathematics, the details of which are beyond the scope of this book, which will work in accelerated frames as well as inertial ones."

I'm still curious about your answer to my question about classical mechanics--do you think it's incorrect for textbooks to say an inertial frame is one where Newton's laws hold? And even if you think it's technically incorrect, do you think it would be better pedagogically to include the same sort of caveat about formulating Newton's laws in tensor form, or do you think that'd be unecessary information for a high school student?

 

[JesseM]

And that other force is a form of energy or not?
If it is then please explain how that force would not curve space-time.

Remember it is not just mass that curves space-time, energy also curves space-time.

A force field would curve spacetime, but at normal energies it would be by a negligible amount--the deviation from a straight-line path seen when one charged object passes near another one is not primarily due to spacetime curvature, and for the amounts of charge in our ordinary experience I think the changes to object's path due to electromagnetic fields curving spacetime would be far too small to measure (at least not without using some very sophisticated equipment).

 

[MeJennifer]

A force field would curve spacetime, but at normal energies it would be by a negligible amount...

Ok, so we go from I am wrong to it's neglible.

--the deviation from a straight-line path seen when one charged object passes near another one is not primarily due to spacetime curvature, and for the amounts of charge in our ordinary experience I think the changes to object's path due to electromagnetic fields curving spacetime would be far too small to measure (at least not without using some very sophisticated equipment).

So let me get this right; are you claiming that the total amount of energy applied to accelerate an object is not equal to the amount of space-time curvature induced?

 

[JesseM]

Ok, so we go from I am wrong to it's neglible.

Your original post was wrong in saying "acceleration is not handled by SR for the simple reason that acceleration is mitigated by curved space-time" because an object's path is not only determined by the curvature of spacetime (and your later posts seemed to argue that it was). Anyway, in practice it's standard to treat a negligibly-curved spacetime as equivalent to a flat one--otherwise the only flat spacetime would be one devoid of all particles (even without force fields, their masses would curve it slightly), and the equivalence principle couldn't be used in any finite-sized region.

So let me get this right; are you claiming that the total amount of energy applied to accelerate an object is not equal to the amount of space-time curvature induced?

I'm not sure about the technical details of how spacetime curvature is quantified or whether it is proportional to "the amount of energy applied to accelerate an object" even in pure GR without other forces. (Since everything moves on geodesics in pure GR, does it even make sense to talk about an object 'accelerating' if its path is always locally inertial?) But put it this way: if an object is accelerated by non-gravitational forces it takes far less energy than it would to alter its path in a similar way using only gravity. You can use the muscles in your legs to jump upwards, but you'd need a vast density of energy over your head to pull you away from the Earth at the same speed based only on the gravitational force. Also, consider the fact that a force fields will curve spacetime the same way for everyone, so why is it that two particles with identical masses and initial positions and velocities but different charges will move in different directions in an electromagnetic field? Why is it that a neutrino can travel straight through the entire planet as if it were empty space, while a proton or electron cannot? If objects' paths were determined only by spacetime curvature, then even if particles still generated fields with the same energy densities as real fields in our universe (but with the fields having no other effects besides curving spacetime), then every particle would be even more "ghostly" than the neutrino, since the neutrino does at least interact with matter via the weak nuclear force (but the electromagnetic and strong forces have no effect on it, apart from how they curve spacetime of course).

 

[MeJennifer]

But put it this way: if an object is accelerated by non-gravitational forces it takes far less energy than it would to alter its path in a similar way using only gravity.

Sorry but that defies all common sense!
Instead it takes exactly the same amount of energy!

 

[JesseM]

Sorry but that defies all common sense!
Instead it takes exactly the same amount of energy!

I guess it depends how you define the "energy used" to move something. I was thinking in terms of the total energy of the system doing the attracting, including the energy due to its mass. If a paperclip on a table would move upward at the same speed in response to the electromagnetic pull of a small magnet held above it as it would to the gravitational pull of a miniature black hole at the same distance, doesn't this mean more energy is being used in the second case? If not, how do you define the "amount of energy it takes"?

In any case, do you agree that the path of objects is not due only to the curvature of spacetime, but also to the effects of non-gravitational forces? If not, could you address the example of two particles with opposite charges but identical masses and identical initial positions and velocities, and why they take different paths in identically-curved spacetimes?

 

[MeJennifer]

IIn any case, do you agree that the path of objects is not due only to the curvature of spacetime, but also to the effects of non-gravitational forces? If not, could you address the example of two particles with opposite charges but identical masses and identical initial positions and velocities, and why they take different paths in identically-curved spacetimes?

In GR, when an object accelerates it means that the space-time curvature is modified in that region. Mass, but also energy, changes the curvature of space-time. Technically you would want to inspect the stress-energy tensor to see how.
For instance, if you launch a rocket you have a lot of EM energy density, and what happens when you have a lot of energy density with flux and momentum, you curve space-time, alot!

In this context it might be interesting to lookup Noether's theorem.

 

[JesseM]

In GR, when an object accelerates it means that the space-time curvature is modified in that region.

Not if you are including non-gravitational forces such as EM.

edit: sorry, depends what you mean. I assumed you meant that the object could only be caused to accelerate by a change in spacetime curvature, but perhaps you meant that the motion of the object itself would change the curvature of spacetime, which is of course true unless we consider a "test particle" with infinitesimal mass (and anyway, for an object on the human scale the changes in curvature caused by the object's motion is, again, negligible).

For instance, if you launch a rocket you have a lot of EM energy density, and what happens when you have a lot of energy density with flux and momentum, you curve space-time, alot!

Are you claiming that a rocket moving away from the Earth is following a geodesic path in curved spacetime caused by the "EM energy density"? If so you are badly mistaken. And you still haven't answered my question about why you think two particles identical in every way except their charge would follow different paths in an electromagnetic field, if their paths were determined only by the curvature of spacetime.

In this context it might be interesting to lookup Noether's theorem.

I'm aware of Noether's theorem, what relevance do you think it has to this debate?

 

[MeJennifer]

Are you claiming that a rocket moving away from the Earth is following a geodesic path in curved spacetime caused by the "EM energy density"?

I am not saying that, and hopefully you are not thinking that a body moved by an EM force is traveling in ambient space.
It is traveling on the manifold, and the EM force provides exactly the correct change in curvature to explain the acceleration.

Now I am not saying that anybody has provided a complete and verifiable theory on how to calculate the induced space-time curvatures for EM, weak and strong forces. But if we can't or if we can demonstrate it is impossible then GR is in serious trouble.
But we simply cannot have some attitute of: "well, EM forces just ignore the background theorized by GR, but that's ok, it will still work, EM forces just fly over the manifolds".

We have attempts made, for instance in the Kaluza-Klein theory and of course Einstein tried for a long time to find it as well and then there is string theory.

 

[JesseM]

I am not saying that, and hopefully you are not thinking that a body moved by an EM force is traveling in ambient space.
It is traveling on the manifold, and the EM force provides exactly the correct change in curvature to explain the acceleration.

I don't understand what you mean by "traveling in ambient space" vs. "traveling on the manifold". When you say "manifold", don't you mean the manifold of curved spacetime?

To rephrase the question: do you believe that the accelerating rocket is following a geodesic in curved spacetime (whose curvature of course includes the slight contribution from the EM field)? Or would you agree that EM fields can cause charged particles to follow non-geodesic paths? (in terms of our current most successful theories, not more speculative ideas like Kaluza-Klein).

 

[JesseM]

Now I am not saying that anybody has provided a complete and verifiable theory on how to calculate the induced space-time curvatures for EM, weak and strong forces. But if we can't or if we can demonstrate it is impossible then GR is in serious trouble.
But we simply cannot have some attidute of: "well, EM forces just ignore the background theorized by GR but that's ok, it will still work, it'll just fly over the manifolds".

What are you talking about? When physicists use EM forces in GR, they don't ignore the background theorized by GR or have the EM fields "fly over" it somehow, they define the EM fields on the curved spacetime. The distance in curved spacetime between any two points would presumably affect the EM force between charges at those points, for example, and EM waves would always locally move at c as measured by nearby freefalling observers, which explains how EM waves can get "trapped" at the event horizon of a black hole, without violating the rule that EM waves can never "stop".

And why do you think that any theory which doesn't explain other forces in terms of curved spacetime would cause GR to be in "serious trouble"? There needn't be any violation of the equivalence principle, as you seemed to suggest earlier--as long as the non-geodesic paths seen by an observer in a room falling through a gravitational field look just like the non-geodesic paths seen by an observer in a room moving inertially in empty space, the equivalence principle is fine.

 

[MeJennifer]

Well it is clear to me that I am certainly not the person to explain this to you.

 

[JesseM]

Well it is clear to me that I am certainly not the person to explain this to you.

The things you're saying seem to be your own idiosyncratic notions, not standard ideas or problems that would be recognized by physicists, so if you're not willing to explain what you mean I don't think anyone else can either. If you think your ideas or arguments would be recognized by physicists, can you cite any sources? I'm still not sure if you disagree that the standard understanding is that EM fields cause charged objects to follow non-geodesic paths, but if you do I looked up some sources on arxiv.org which might help convince you otherwise, like this one which discusses the non-geodesic worldlines in the neighborhood of a collapsing magnetized medium.