## Main Question or Discussion Point

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?

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robphy
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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.

jtbell
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

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
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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.

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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
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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).

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JesseM
JesseM said:
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).
robphy said:
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?
robphy said:
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?

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.

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
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.

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
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.

robphy
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Gold Member
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
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Gold Member
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.

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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
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Gold Member
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.

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).

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? 