Was Einstein lucky when not considering twin paradox as paradox?

[stevendaryl]

Inertial motion is what Special Relativity is based on, and Einstein obtained the time dilation conclusion on this basis only (without considering any symmetry-breaking accelerations etc.) . That was the OP's point.

SR was developed by considering inertial FRAMES, not inertial MOTIONS. Motions are described relative to a frame, but the motions themselves are not confined to be inertial in SR.

The assumptions that led to SR were that:

1. The laws of physics in their simplest form look the same when described from the point of view of any inertial frame.
2. The speed of light has the same speed in any inertial frame.
3. Empty space is the same in all directions and at all locations and at all times.

These (possibly together with the assumption that Newtonian physics works in the limit of small velocity) allow you to derive the laws of SR, and those rules (possibly together with assumptions about the nature of idealized clocks) allow you to predict what happens when a clock undergoes noninertial motion.

SR is not in any way restricted to inertial motion--it (or more precisely, the usual mathematical formulation of it) is restricted to using inertial frames to describe motion, but the motion itself is not required to be inertial.

The situation is no different from in Newtonian physics. The whole point of Newton's laws (and SR are intended to be a replacement of those laws) is to describe how objects move when acted upon by forces.

 

[Fantasist]

Sorry if my clarification was not clear enough. As I as well as others here remarked, there was no such paradox known in the context of SR alone, and it was still not paradoxical in that context in 1911 - the so-called "twin paradox" of textbooks is just an SR student exercise. Note also that Einstein started developing GR from about 1907.

It may be merely a student exercise today, but exactly what we are discussing here was a serious issue for Einstein already before he published his GR, which was discussed by leading scientists at the time. I quote from the Wikipedia article

Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the traveling twin] as the main reason...".[1] Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se.[2]

 

[Fantasist]

SR is not in any way restricted to inertial motion--it (or more precisely, the usual mathematical formulation of it) is restricted to using inertial frames to describe motion, but the motion itself is not required to be inertial.s.

Nobody said that SR is necessarily restricted to inertial motion, but its conclusions do not in any way depend on non-inertial motion. Einstein derived his results (including time dilation) using inertial motion only.

 

[stevendaryl]

That's clearly incorrect. The 'twin paradox' problem implied by the inertial frame scenario in Einstein's theory was already known since about 1911 (still several years after Einstein's 1905 paper appeared; see http://en.wikipedia.org/wiki/Twin_paradox ). It seems more like Einstein developed GR in order to be able to include non-inertial scenarios and thus 'get out of jail' with the twin paradox issue here.
I found a further interesting article in this respect here http://www.iisc.ernet.in/~currsci/dec252005/2009.pdf

It is completely false to say that the Twin Paradox required General Relativity for its resolution. It's also false that General Relativity is needed to be able to describe noninertial coordinate systems (such as the coordinate system of the traveling twin). Additional mathematics is required, but no additional physics is required. Mathematically, if you have a description of the laws of physics in an inertial coordinate system, then calculus alone will allow you to get a description in a noninertial coordinate system. That's true in exactly the same way that Newtonian physics, described in rectangular coordinates, is sufficient to figure out what physics looks like in spherical coordinates. There are no additional physical principles involved, just calculus.

So the "resolution" to the twin paradox described in the paper isn't, from the point of view of modern understanding, a "General Relativity" solution. It's a Special Relativity solution using generalized (non-inertial) coordinates. Einstein falsely believed that "general covariance"--the principle that the laws of physics have the same form in any coordinate system, whatsoever--would uniquely imply what that laws must be. That isn't true. You can take any laws (Newtonian physics, for example) and rewrite them in a generally covariant form.

But what you find when you rewrite the laws of physics in terms of general coordinates is that there are additional terms in the equations that were not present in inertial coordinates. These are terms that are sometimes called "inertial forces" and they look like position-dependent forces that affect the motion of all objects (regardless of their physical composition) in the same way. These "inertial forces" look like gravitational fields. Einstein's insight was to suppose that real gravitational fields are similarly inertial forces due to using noninertial coordinates. Working out how this could be the case leads to General Relativity.

In retrospect, General Relativity was not needed to describe things from the point of view of an accelerated coordinate system. That description is derivable from SR alone. And that description has terms that are "gravity-like", but all within SR. GR is only needed if you want to describe real gravity, due to the presence of massive objects.

 

[stevendaryl]

Nobody said that SR is necessarily restricted to inertial motion, but its conclusions do not in any way depend on non-inertial motion. Einstein derived his results (including time dilation) using inertial motion only.

I just explained why your phrasing is not the best way to say it. Einstein derived his results using inertial FRAMES only. The results themselves describe both inertial and noninertial motion. So it is a fact that a clock which (from the point of view of any inertial frame) accelerates away and then accelerates back to its original location will show less elapsed time than a clock that remains stationary in that frame. That is a fact that depends on noninertial motion (since it's a fact ABOUT noninertial motion), and it follows from Einstein's SR. It depends on noninertial motion, but it doesn't depend on a noninertial FRAME.

 

[Fantasist]

I just explained why your phrasing is not the best way to say it. Einstein derived his results using inertial FRAMES only. The results themselves describe both inertial and noninertial motion. So it is a fact that a clock which (from the point of view of any inertial frame) accelerates away and then accelerates back to its original location will show less elapsed time than a clock that remains stationary in that frame. That is a fact that depends on noninertial motion (since it's a fact ABOUT noninertial motion), and it follows from Einstein's SR. It depends on noninertial motion, but it doesn't depend on a noninertial FRAME.

Where in Einstein's 1905 paper do you read that time dilation results from non-inertial motion?
I read the following there

From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by 1/2*t*v^2/c^2.

where v (according to the earlier definitions) is constant (i,e, the motion is inertial).

 

[Dale]

ok so what ingredient needs to be added??

An external field which would accelerate one particle and not the other or a third particle which interacts with one and not the other.

 

[adoion]

An external field which would accelerate one particle and not the other or a third particle which interacts with one and not the other.

I already added this in my last example as you can see, but this doesn't help.

 

[stevendaryl]

I found a further interesting article in this respect here http://www.iisc.ernet.in/~currsci/dec252005/2009.pdf

I consider that paper deeply misleading. It's possible that the confusion in that paper is an accurate reflection of the confusion of physicists (including Einstein himself) in the early days of relativity. But just because people were confused about it in the past doesn't mean that we need to confuse ourselves in the same way.

The paper has the following line:

Einstein needed the general relativistic physics to resolve the twin paradox in special relativity, and admitted so.

Einstein may have believed that he needed general relativity to describe things from the point of view of the traveling twin, but if so, he was mistaken. The mistake was probably caused by the fact that the relationship between general covariance (which is pure mathematics) and general relativity (which is a theory of physics) was not clearly understood.

The so-called "general relativistic" solution to the twin paradox proceeds as follows:

1. Describe the situation from the point of view of the accelerating twin.
2. From the point of view of this twin, there are inertial forces involved when the twin turns around.
3. Invoking the equivalence principle, these inertial forces are equivalent to a gravitational field.
4. According to General Relativity, clocks within a gravitational field experience gravitational time-dilation.
5. Using gravitational time dilation, you can work out the differential elapsed times on the clocks of the two twins.

What's convoluted and downright circular about this argument is that time dilation due to inertial forces is derivable from pure Special Relativity. As a matter of fact, gravitational time dilation was discovered by Einstein several years before he even completed GR. Einstein, using his "Elevator" thought-experiment, deduced that there had to be gravitational time dilation and gravitational bending of light from SR and the equivalence principle. The logical order was this: In the noninertial frame of an elevator accelerating in empty space, there is apparent position-dependent time dilation and bending of light. If we assume that a gravitational field on the surface of a planet is equivalent to the apparent gravitational field inside an accelerating elevator, then there must be position-dependent time dilation and bending of light due to a gravitational field, as well.

So Einstein derived gravitational time dilation from considering noninertial frames, not the other way around. So it's completely circular to invoke a theory of gravity to explain effects aboard an accelerating rocket. It's not wrong, but it's ridiculously convoluted.

1. You derive gravitational time dilation for a rocket at rest on a planet by invoking the equivalence principle and transforming to the case of a rocket accelerating in empty space.
2. Then you derive time dilation on board an accelerating rocket by transforming it to the case of a rocket at rest on a planet and using gravitational time dilation.

It works, but you could get the same result without ever mentioning the planet at all. You introduce it only to transform it away.

 

[stevendaryl]

Where in Einstein's 1905 paper do you read that time dilation results from non-inertial motion?

I'm saying that the case of a noninertial clock is a deduction from Einstein's paper. His paper doesn't explicitly derive that case, but that's the whole point of having a "theory". A theory can be used to derive an infinite number of special cases.

The result of the twin paradox, that the traveling twin will be younger than the twin who travels inertially when they reunite, is a special case derivable from the theory introduced in Einstein's 1905 paper.

 

[DrGreg]

Where in Einstein's 1905 paper do you read that time dilation results from non-inertial motion?

In the English translation On the Electrodynamics of Moving Bodies at the top of page 11, the final paragraph of §4.

"Thence we conclude that a balance-clock at the equator must go more slowly, by a very
small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions."​

(I assume you realize that circular motion, around the equator in this case, is non-inertial motion.)

 

[stevendaryl]

if the external force is uniform like for example far away from an electric charge where the lines of force are almost parallel and same in magnitude.

in this case the accelerometer and the object would be accelerated the same and you would conclude that the other object is accelerating.

also how would you be sure that the accelerometer is not acted by a force instead of the objet who's force it is supposed to measure.
also the accelerometer introduces a third object or observer in the system if you have only the accelerometer and one object then once again you wouldn't know witch one is accelerating

If there is a force that affects all objects in exactly the same way, independent of what they are made out of, then you are exactly right--such a force would be unobservable using an accelerometer. It would only be observable by looking at larger-scale phenomena--tidal forces: how that force changes from place to place and from moment to moment. That's what gravity is. I believe that you could lump any such "universal" force in with gravity.

Such universal forces require a treatment that goes beyond Special Relativity. So the development of SR does not take into account such forces. It's not a complete theory, in that sense.

 

[ghwellsjr]

This is not consistent with your diagram, which shows only 20 years on the blue scale, not 25.

Yes, you are correct, I put down the wrong number in post #11, twice, in fact. I guess I was looking at the coordinate time of the reunions for the second and third diagrams. Anyway, thanks for catching this.

You evaluated further above the age difference from the viewpoint of the 'inertial blue twin'. How can you say that the subsequent consideration evaluates the age difference from the viewpoint of the 'non-inertial red twin', when the latter in fact never occupies the reference frames for which you claim time dilation here (you evaluate the time dilation in a third reference frame which moves opposite to the 'non-inertial red twin')?

I never used the term "viewpoint". I used the term "defining IRF" to specify the scenario and then I talked about transforming the coordinates of all the significant events to two other IRF's moving at different speeds with respect to the defining IRF.

Both twins "occupy" all three IRF's. In the defining IRF, the blue twin, who remains inertial is not moving but the red twin is moving at a constant speed, although he changes direction half-way through, making him non-inertial. The second IRF was chosen so that the red twin would not be moving during the first part of the scenario but he starts moving at his time of 8 years making him non-inertial while the blue twin is always moving inertially. The third IRF was chosen so that the red twin would not be moving during the last part of the scenario but he started out moving until his time of 8 years making him non-inertial while the blue twin is always moving inertially. I thought I made all these points clear in post #11.

I don't consider any of these diagrams to be showing the "viewpoint" of either twin. I would have had to draw in light signals going between the twins to show their viewpoints and they would be exactly the same in all three IRF's.

 

[PeroK]

It seems to me that the OP read and understood Einstein's paper very well. What he is asking is, whether there is any information why Einstein did not check the consistency of his time dilation calculation by changing the rest frame to the other observer/clock. Was it deliberate or an oversight?

In my opinion:

a) No one who understands SR would ask whether Einstein was "lucky", with the implication he missed all the potential paradoxes and was fortunate that others resolved these paradoxes and, luckily, left his theory intact.

b) No one who understands SR very well would obsess over the twin paradox and fail to grasp the lack of symmetry vis-a-vis the role played by an accelerating reference frame.

 

[adoion]

To sum up a little bit,

A reference frame is inertial if there are no fiction forces present like coriolis and centrifugal forces, witch are associated with rotating reference frames. Then we also have fiction forces due to linearly accelerating reference frames.

but how does one make sure that a force is or is not fictional? one has to find the source of the force or one has to find other reference frames in witch those forces disappear and one is left with the simples form of laws, especially Newton's second law, the les forces there are to consider the simpler it is.

the point is that we would always have terms in the equation of Newton's second law $F=ma$ witch would be always present, like the coriolis term for example, and one would be forced to state Newton's second law in a more complicated form including those additional terms instead of adding these terms every time a new calculation needs to be made, this is valid for a rotating system.
obviously Newton's first law would have to be restated in a rotating reference frame as "all bodies tend to rotate around at a fixed radius or with uniformly changing radius unless acted upon by an force".
all of this is more complicated.

if the reference frame is linearly accelerating then one wouldn't need to do anything with Newton's laws and they would take their simplest form anyway. objects that move under the influence of the same force as the Reference frame would appear to stand still or move uniformly and objects that would appear to accelerate would be the once that move with different accelerations than the reference frame.
anyways, we would have just a shift in perception of what's accelerating and what's not and not a change in laws.
the law of gravity for example would have the same form, just that the masses in the universe would appear to accelerate a little bit faster or slower in a particular direction, than in another reference frame.

in the twin paradox, motion is uniform (constant velocity and direction) until the turn around where obviously an acceleration happens.
whatever the source of the acceleration might be is nowhere mentioned in the statement of the paradox so it can be anything.

both of the twins must make their measurements from their point of view and since in both cases the laws of physics take their simplest form, both of them are correct in assuming that their reference frame is inertial and that the other one is accelerating.

 

[Fantasist]

It's possible that the confusion in that paper is an accurate reflection of the confusion of physicists (including Einstein himself) in the early days of relativity. But just because people were confused about it in the past doesn't mean that we need to confuse ourselves in the same way.

The OP was asking about the perception of the twin paradox in the early days (starting with Einstein's paper). If discussing this issue still causes confusion today, it shows that probably not everything is as clear-cut here as it is sometimes portrayed. Only a continued discussion of confusing issues can lastly lead to full clarification, not their suppression.

 

[stevendaryl]

The OP was asking about the perception of the twin paradox in the early days (starting with Einstein's paper). If discussing this issue still causes confusion today, it shows that probably not everything is as clear-cut here as it is sometimes portrayed. Only a continued discussion of confusing issues can lastly lead to full clarification, not their suppression.

I'm not talking about suppression. I'm talking about intentionally introducing misconceptions, and then trying to clear them up. I don't see that that's helpful.

 

[Fantasist]

In the English translation On the Electrodynamics of Moving Bodies at the top of page 11, the final paragraph of §4.

"Thence we conclude that a balance-clock at the equator must go more slowly, by a very
small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions."​

(I assume you realize that circular motion, around the equator in this case, is non-inertial motion.)

You possibly misunderstood this paragraph. Einstein's argument is as follows here:

1) a circle can be approximated by a polygon (by making the sides infinitesimally small)
2) a polygon is a piece-wise linear curve, so we have a piece-wise constant velocity vector that only changes in direction (and thus leaves the time dilation factor constant)
3) so for a circle of circumference C, the time dilation is the same as for a straight line of length C (assuming the speed v is the same).

So rather on the contrary, for a circular orbit there is a time dilation as it can be approximated by a (piece-wise) inertial motion (for which Einstein knows how to derive the time dilation).

 

[stevendaryl]

if the reference frame is linearly accelerating then one wouldn't need to do anything with Newton's laws and they would take their simplest form anyway. objects that move under the influence of the same force as the Reference frame would appear to stand still or move uniformly and objects that would appear to accelerate would be the once that move with different accelerations than the reference frame. anyways, we would have just a shift in perception of what's accelerating and what's not and not a change in laws.

I don't think that's correct. If you do Newtonian physics using an accelerated coordinate system, then Newton's laws don't hold.

In an inertial coordinate system, you have:

$m \frac{d^2 x^j}{dt^2} = F^j$ (F = ma)

Now, switch to a new coordinate system $x'^j = x^j + A^j t^2$

In this new coordinate system, you have:

$m \frac{d^2 x'^j}{dt^2} - m A^j = F^j$

This does not have the same form. You could try to restore it to the same form by moving the constant acceleration $m A^j$ to the other side:

$m \frac{d^2 x'^j}{dt^2} = F'^j = m A^j + F^j$

so you have a new "ficititious force" $m A^j$. But this new force DOESN'T obey Newton's laws. In particular, it doesn't obey the third law, "For every action, there is an equal and opposite reaction". If that's interpreted to mean that whenever there is a force on one object, that object exerts an equal and opposite force, then that's false for fictitious forces. The mass $m$ has a force $mA^j$ exerted on it, but it doesn't exert an equal and opposite force on anything. Momentum is not conserved in this new coordinate system.

 

[stevendaryl]

You possibly misunderstood this paragraph. Einstein's argument is as follows here:

1) a circle can be approximated by a polygon (by making the sides infinitesimally small)
2) a polygon is a piece-wise linear curve, so we have a piece-wise constant velocity vector that only changes in direction (and thus leaves the time dilation factor constant)
3) so for a circle of circumference C, the time dilation is the same as for a straight line of length C.

So rather on the contrary, for a circular orbit there is a time dilation as it can be approximated by a (piece-wise) inertial motion (for which Einstein knows how to derive the time dilation).

What you're saying is always true. You can always approximate the time dilation for any noninertial motion by breaking it up into small segments, and approximate those segments by constant-velocity segments.

That prescription gives rise to the following formula for computing proper time for noninertial motion:

The proper time for taking a path $x(t)$ from time $t=t_1$ to $t=t_2$ is

$\int_{t_1}^{t_2} \sqrt{1-\frac{(\frac{dx}{dt})^2}{c^2}} dt$

 

[Dale]

Inertial motion is what Special Relativity is based on

This is not quite correct. Special relativity is based on inertial frames, not inertial motion. There is an important difference between the two. Even in the first paper, Einstein's "on the electrodynamics of moving bodies", it was clear how to correctly treat non inertial motion (see section 4). To the OP's question it wasn't luck and it was not unaddressed by Einstein.

EDIT: I see stevendaryl made the same point first!

 

[Dale]

I already added this in my last example as you can see, but this doesn't help.

I didn't see the "last example" you are referring to, but it does completely resolve the issue. The accelerating twin is the one which interacts with the external field or the other particle.

 

[Fantasist]

What you're saying is always true. You can always approximate the time dilation for any noninertial motion by breaking it up into small segments, and approximate those segments by constant-velocity segments.

Still, it is only a geometrical argument. Each of the twins could describe the motion of the other this way. So how do we single out one of the twins on this basis?

 

[adoion]

I don't think that's correct. If you do Newtonian physics using an accelerated coordinate system, then Newton's laws don't hold.

In an inertial coordinate system, you have:

$m \frac{d^2 x^j}{dt^2} = F^j$ (F = ma)

Now, switch to a new coordinate system $x'^j = x^j + A^j t^2$

In this new coordinate system, you have:

$m \frac{d^2 x'^j}{dt^2} - m A^j = F^j$

This does not have the same form. You could try to restore it to the same form by moving the constant acceleration $m A^j$ to the other side:

$m \frac{d^2 x'^j}{dt^2} = F'^j = m A^j + F^j$

so you have a new "ficititious force" $m A^j$. But this new force DOESN'T obey Newton's laws. In particular, it doesn't obey the third law, "For every action, there is an equal and opposite reaction". If that's interpreted to mean that whenever there is a force on one object, that object exerts an equal and opposite force, then that's false for fictitious forces. The mass $m$ has a force $mA^j$ exerted on it, but it doesn't exert an equal and opposite force on anything. Momentum is not conserved in this new coordinate system.

as I mentioned the new coordinate system would be just shifted compared to the old one, your fictious force $mA^j$ would only be present from the old coordinate systems point of view.
but if you fix your new reference frame is accelerating compared to the old one then from the new RF point of view (witch you suppose to be stationary) the old RF is accelerating and you would have fictious forces if you look at the old one from the new RF.
But the laws you can state the same as in the old one.

while when the RF is rotating you would clearly have laws involving rotational forces right away, and if you step out of it you would have Newton's ordinary laws. you would not be able to compensate and have only linear acceleration in the rotating RF-laws.

so clearly there is a sense of linear symmetry in our universe but not symmetry between rotational and linear reference frames, this is a consequence of the way the universe is setup and how things are moving.

 

[Dale]

I'm saying that the case of a noninertial clock is a deduction from Einstein's paper.

Furthermore, it is a deduction which he explicitly makes.

 

[Dale]

but how does one make sure that a force is or is not fictional?

One uses an accelerometer.

if the reference frame is linearly accelerating then one wouldn't need to do anything with Newton's laws and they would take their simplest form

This is not correct. In a linearly accelerating reference frame there is a fictitious force. Accelerometers at rest in linearly accelerating reference frames register the acceleration.

both of the twins must make their measurements from their point of view and since in both cases the laws of physics take their simplest form, both of them are correct in assuming that their reference frame is inertial and that the other one is accelerating.

This is simply false. An accelerometer aboard the traveling twin's ship registers a sharp spike halfway through the journey. He knows that his rest frame is not inertial and there are no "laws of physics take their simplest form" which would explain that accelerometer reading.

 

[Nugatory]

But how does one make sure that a force is or is not fictional? one has to find the source of the force or one has to find other reference frames in which those forces disappear

Yes, that is pretty much it. Of course finding such a reference frame is a purely mathematical exercise - I just need to find a coordinate transformation that makes the coordinate accelerations go away.

If we observe a force that accelerates all objects equally regardless of their mass (as does centrifugal and coriolis force) that's a fairly solid hint that we're dealing with a fictional force and that such a coordinate transformation exists. Indeed, the only non-fictional force with that property is Newtonian gravity - and GR eliminates that special case by providing a mathematical framework in which it is also a fictional force.

 

[stevendaryl]

as I mentioned the new coordinate system would be just shifted compared to the old one, your fictious force $mA^j$ would only be present from the old coordinate systems point of view.

I know that's what you said, but it's not true. That term is in the equations, either as a "force" term (on the F side of F=ma) or as an acceleration term (on the ma side of F=ma). Neither choice leaves Newton's laws unchanged. Either you have to modify the notion of "acceleration" to include terms due to noninertial, non-Cartesian coordinates, or you modify the third law, and allow for forces without a "reaction" counterpart.

but if you fix your new reference frame is accelerating compared to the old one then from the new RF point of view (witch you suppose to be stationary) the old RF is accelerating and you would have fictious forces if you look at the old one from the new RF.

I'm sorry, but that's just not true. In terms of $x^j$, you have:

$m \frac{d^2 x^j}{dt^2} = F^j$ (no fictitious forces)

In terms of $x'^j$, you have:
$m \frac{d^2 x^j}{dt^2} = F^j + m A^j$ (fictitious forces are present)

Some frames have fictitious forces, and some frames do not. The ones that do not are the inertial frames.

 

[Dale]

as I mentioned the new coordinate system would be just shifted compared to the old one, your fictious force $mA^j$ would only be present from the old coordinate systems point of view.

No, this is not true. The fictitious force is intrinsic to the coordinate system itself and not required to be in reference to any other coordinate system's point of view. That fictitious force must be included to match the observations of motion within that frame alone.

but if you fix your new reference frame is accelerating compared to the old one then from the new RF point of view (witch you suppose to be stationary) the old RF is accelerating and you would have fictious forces if you look at the old one from the new RF.

This is also not true. The lack of fictitious force in the inertial reference frame is inherent to the frame itself and does not require any reference to any other frame. The fictitious force must be absent to match the observations of motion within that frame alone. If you include it you would get the wrong motions.

EDIT: or in other words: "What stevendaryl said".

 

[stevendaryl]

Still, it is only a geometrical argument. Each of the twins could describe the motion of the other this way. So how do we single out one of the twins on this basis?

You don't. You approximate each twin's path by a bunch of little constant-velocity paths, and you compute the proper time for each path using the formula: $\delta \tau = \sqrt{\delta t^2 - \frac{1}{c^2} \delta x^2}$ You add up $\delta \tau$ for each segment, and that gives you how much each twin ages along his path. There is no singling out of one twin over the other.

There is a choice that must be made, which is to pick an inertial coordinate system for measuring $\delta x$ and $\delta t$. But every inertial coordinate system will give the same value for $\delta \tau$. It's exactly like computing the length of a line segment in Euclidean geometry. The length is given by: $\delta L= \sqrt{\delta x^2 + \delta y^2}$. You can choose any Cartesian coordinate system you like to measure $\delta x$ and $\delta y$, and you will get the same answer for $\delta L$.

 

[PeterDonis]

if you have only 2 point particles an nothing else, how do you determent witch one is accelerating?

Easy: attach an accelerometer to each particle. The one whose accelerometer reads nonzero for some portion of the trip is the one who accelerated.

DaleSpam and stevendaryl have correctly pointed out that, strictly speaking, for one of the particles to accelerate in the above sense (i.e., for its accelerometer to read nonzero at some point), there must be other "stuff" present in the scenario. The "point particle" whose accelerometer reads nonzero has to exchange momentum with something (for example, a rocket exhaust). But you don't have to know any of the details of how that happens to know which particle accelerated from the accelerometer readings.

 

[harrylin]

It may be merely a student exercise today, but exactly what we are discussing here was a serious issue for Einstein already before he published his GR, which was discussed by leading scientists at the time. I quote from the Wikipedia article
Starting with Paul Langevin in 1911, there have been various explanations of this paradox. [..]

I gave you the link to his paper; you can check for yourself that, contrary to Wikipedia's suggestion*, it was not considered to be an existing paradox. Instead it was an original, non-paradoxical example of predictions based on what Einstein later named "special relativity".

*About Wikipedia, I'm not sure that the person who wrote that intended to make your claim, it may be just poor phrasing. And you can use the back-in-time feature of Wikipedia to find different opinions. ;)

 

[harrylin]

[..] In retrospect, General Relativity was not needed to describe things from the point of view of an accelerated coordinate system. That description is derivable from SR alone. And that description has terms that are "gravity-like", but all within SR. GR is only needed if you want to describe real gravity, due to the presence of massive objects.

"In retrospect"? Why do you think that this was not understood at that time? It had been straightforward in classical mechanics to describe things from the point of view of an accelerated coordinate system, so it seems unlikely that this wasn't understood from the start in SR. (sorry for going slightly off topic, but it fits rather well in the discussion here).

 

[stevendaryl]

"In retrospect"? Why do you think that this was not understood at that time? It had been straightforward in classical mechanics to describe things from the point of view of an accelerated coordinate system, so it seems unlikely that this wasn't understood from the start in SR. (sorry for going slightly off topic, but it fits rather well in the discussion here).

You're certainly right, that noninertial frames came up in Newtonian mechanics. So where did the idea that GR was necessary to handle an accelerated reference frame come from?

I think that part of it is the insistence on relativity. Although Newtonian mechanics also satisfied a principle of (Galilean) relativity, I don't think that it played that much role in the teaching and application of the subject. Nobody bothered (as far as I know) to try to write Newtonian mechanics in a way that treated all coordinate systems equally. I don't think that the latter was developed until after GR (the Newton-Cartan formulation of Newtonian physics).

 

[adoion]

Easy: attach an accelerometer to each particle. The one whose accelerometer reads nonzero for some portion of the trip is the one who accelerated.

DaleSpam and stevendaryl have correctly pointed out that, strictly speaking, for one of the particles to accelerate in the above sense (i.e., for its accelerometer to read nonzero at some point), there must be other "stuff" present in the scenario. The "point particle" whose accelerometer reads nonzero has to exchange momentum with something (for example, a rocket exhaust). But you don't have to know any of the details of how that happens to know which particle accelerated from the accelerometer readings.

you would have to calibrate the accelerometers differently in order for both of them to be zero if they accelerate differently, because if you assume that your accelerating RF is actually stationary then you calibrate your accelerometer to read zero in your reference frame, don't you?

and there is another thing, if you change acceleration in a linear fashion then your laws are not the simplest anymore and maybe its the way to think about the twin paradox.