Orodruin
Staff Emeritus
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If you put an accelerometer on the accelerating object, it will show a non-zero value.

hi,

Einstein did not even consider the twin paradox as problematic at all, he argued that it is a simple consequence of his special relativity?

obviously he never gave a explanation of why the two twins don't age the same he instead left it to others to do so.

was Einstein just having a hunch witch turned out to be just a lucky guess?
Despite the presentation in textbooks of a "twin paradox" in special relativity (as a student exercise), there never was anything really paradoxical about the "twin paradox" in that context. SR only accepts inertial coordinate systems as reference for the Lorentz transformations as Einstein already illustrated in 1905, with his clock prediction. Langevin presented in 1911 the example with a space traveller -from both perspectives- to illustrate how a change of velocity is "absolute" in SR. You can read that example starting from p.50 here:
http://en.wikisource.org/wiki/Translation:The_Evolution_of_Space_and_Time

The twin paradox only appeared with Einstein's development of general relativity. According to original, 1916 GR, acceleration is relative in the sense that 'coordinate systems in arbitrary states of motion are qualified' so that the traveller can rightly claim to be "in rest" all the time. You can read Einstein's 1918 answer to this problem as advanced by critics here:

If you put an accelerometer on the accelerating object, it will show a non-zero value.
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

I When he reunites with the earth twin, they find that the earth twin has aged 25 years while the traveling twin has aged 16 years. The dots on this diagram mark off one-year increments of time for both twins:

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

I
For the next two IRF's and their diagrams, I'm going to refer to the earth twin as the inertial blue twin and the traveling twin as the non-inertial red twin.

First we're going to transform to the IRF in which the non-inertial red twin is at rest during the first part of the scenario. The diagram looks like this:

Note how the inertial blue twin is moving at -0.6c during the entire scenario and so his clock is Time Dilated by 1.25 the entire time. During the first part of the scenario, the non-inertial red twin's clock is not Time Dilated because he is not moving. But at his time of 8 years, he starts moving at -0.882c where his Time Dilation is now 2.125 and after 8 more years he catches up to the inertial blue twin who has aged 25 years by the time they reunite.

Finally we're going to transform to the IRF in which the non-inertial red twin is at rest during the last part of the scenario:

This is similar to the previous IRF so we can use the same numbers but in different orders but the net result is that the twins age by the same amounts.

I hope this is clear and removes all your confusion. If not, ask.

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')?

tom.stoer
adoion, the only thing which is paradoxical in the twin paradox are the explanations. The proper time each twin measures on his own wrist watch is related to the "path length" of his/her trip through spacetime. for different paths we expect different "length". Consider two different paths with different lengths from New York to Boston. Why would you talk about a paradox at all?

As stevendaryl mentioned if you have two point particles and nothing else then they will travel inertially only..

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.

• 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
While I don't entirely follow your arguments, an accelerometer is indeed insufficient. Note that identically the same problem occurs in classical mechanics. See my clarifications in posts #17 and #25 here [edit: replaced by direct links]:

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• The twin paradox only appeared with Einstein's development of general relativity

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

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 ). [..]
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.

stevendaryl
Staff Emeritus
ok so what ingredient needs to be added??

A particle can't accelerate without a force. So to have two particles, one of which accelerates and the other doesn't, you have to have a force that applies to one and not the other.

Most thought experiments involving Special Relativity just assume contact forces: A rocket is accelerated by throwing matter behind it.

stevendaryl
Staff Emeritus
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.

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 travelling twin] as the main reason...". 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.

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
Staff Emeritus
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
Staff Emeritus
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.

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?

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
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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.

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
Staff Emeritus
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.

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stevendaryl
Staff Emeritus
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
Gold Member
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 realise that circular motion, around the equator in this case, is non-inertial motion.)

stevendaryl
Staff Emeritus
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
Gold Member
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
Homework Helper
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2020 Award
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.

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.