# The Twin Paradox

1. Jan 14, 2007

### 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 space ship 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?

2. 3. Jan 14, 2007

### robphy What textbook said that? It's not accurate.

4. Jan 14, 2007

### moe darklight 5. Jan 15, 2007

### jtbell ### Staff: Mentor

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

6. Jan 15, 2007

### JesseM 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.

7. Jan 15, 2007

### robphy 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.

Last edited: Jan 15, 2007
8. Jan 15, 2007

### 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.

9. Jan 15, 2007

### JesseM 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?

10. Jan 15, 2007

### robphy 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

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).

Last edited: Jan 15, 2007
11. Jan 15, 2007

### JesseM 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).
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?
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?

12. Jan 15, 2007

### 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.

13. Jan 15, 2007

### JesseM 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.

14. Jan 15, 2007

### MeJennifer 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.

15. Jan 15, 2007

### pervect Staff Emeritus
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.

16. Jan 15, 2007

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

17. Jan 15, 2007

### pervect Staff Emeritus
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.

18. Jan 15, 2007

### robphy "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".

19. Jan 15, 2007

### robphy [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.

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.

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20. Jan 15, 2007

### MeJennifer 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?

21. Jan 15, 2007

### robphy 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).