Time dilation or malfunctioning clock?

[Upchurch]

Acceleration is not important for the Twin Paradox. The really important point is that the twin at home is always at rest in the same reference system. You need two reference systems to have the travelling twin at rest during the whole trip. Even if he didn't have to accelerate and could magically jump to his cruise speed the effect would be there.

That magical jump would be acceleration. Infinite acceleration, to be precise. The acceleration is the key to the whole phenomenon.*

One of my undergrad research projects in college was determining the effect of acceleration duration on the age difference between the two twins in the Twin Paradox. I'd have to really dig out my old notes if you want me to prove it to you, but the longer the traveling twin spends accelerating the larger the age difference becomes. Instant jumps show less age difference then gradually speeding up and gradually slowing down.


* eta: In fact, it is only the symmetry of situation that lets us make the calculation using Special Relativity. If it got any more complicated, it would become a General Relativity problem.

 

[Ziggurat]

As has been explained by others, special relativity only deals with motion at constant velocity.

No. This is wrong. Special relativity can handle acceleration - the math can get ugly, and it's usually taught without getting into acceleration, but there's absolutely nothing about acceleration which precludes its treatment under special relativity. The idea that it cannot handle acceleration comes from a misunderstanding of the equivalence principle. If special relativity were unable to handle acceleration, it would mean that general relativity would not be derivable from the equivalence principle.

It is a simple model that does not accurately describe the real world. In general relativity there is no twins paradox.

There's no twin paradox in special relativity either, because it's not actually a paradox. There is the traveling twin problem, but it can be solved fully (INCLUDING using finite acceleration if you want to wade through the math) using only special relativity. The only thing special relativity cannot accomodate is gravity.

 

[Cuddles]

Special relativity can handle acceleration

The only thing special relativity cannot accomodate is gravity.

Gravity is an acceleration. There is no way to tell the difference between acceleration due to gravity and any other acceleration. Your post contradicts itself.

 

[Ziggurat]

eta: In fact, it is only the symmetry of situation that lets us make the calculation using Special Relativity. If it got any more complicated, it would become a General Relativity problem.

As I mentioned to Cuddles, this is wrong. You can easily handle finite acceleration, curved worldlines, unsymmetric journeys, etc. with special relativity. Making the journey symmetric and assuming infinite acceleration at the turnaround makes the calculation easier (it can be done with simple algebra as opposed to using calculus to get a path integral of the metric along the worldline), but special relativity can accomodate finite accelerations and comples paths. The ONLY addition to the problem which would require the use of general relativity is gravity.

 

[Thabiguy]

Gravity is an acceleration. There is no way to tell the difference between acceleration due to gravity and any other acceleration. Your post contradicts itself.

Ziggurat is correct. Special relativity can deal with acceleration.

General relativity does not explain gravity as acceleration. Quite on the contrary, it assumes that a free-falling observer is at rest not in an accelerating, but in an inertial frame of reference. The consequence is that, unlike in special relativity, inertial frames of reference do not exist globally. - Another way to understand the difference is to realize that an object with mass that is accelerating experiences force, by definition. General relativity however says that gravity is not due to any force but due to space-time curvature.

When gravity is present, special relativity is no longer applicable because inertial reference frames are no longer global in space where gravitational field (or space-time curvature) varies. This means that the outcome of the "twin paradox" will differ in general relativity from special relativity not depending on the acceleration of the travelling twin, but depending on the variations of the gravitational field that the travelling twin goes through.

 

[Yllanes]

That magical jump would be acceleration. Infinite acceleration, to be precise. The acceleration is the key to the whole phenomenon.*

One of my undergrad research projects in college was determining the effect of acceleration duration on the age difference between the two twins in the Twin Paradox. I'd have to really dig out my old notes if you want me to prove it to you, but the longer the traveling twin spends accelerating the larger the age difference becomes. Instant jumps show less age difference then gradually speeding up and gradually slowing down.

I know that. I was only pointing out that you don't need a finite period of acceleration to have an age difference or to be able to distinguish the two twins. One is all the time in the same reference frame, the other isn't. Of course, changing reference frames is accelerating, so you are right, but the possible side effects of acceleration (ynot's concern) do not enter into the derivation.

Also, Ziggurat is right in saying that you can have accelerations in SR. For example, it is very easy to prove that the relativistic uniformly accelerated motion is

[latex]\footnotesize
\begin{align*}
x &=\frac{c^2}{a}\left(\sqrt{1+\frac{a^2t^2}{c^2}}-1\right), &
v &= \frac{a t}{\sqrt{1+\frac{a^2t^2}{c^2}}}
\end{align*}
[/latex]

(for at << c, this formulas are v ~ at, x ~ at²/2). Uniformly accelerated is here a motion such that the acceleration a in the proper reference frame (at each instant of time) is a constant.

 

[Ziggurat]

Gravity is an acceleration. There is no way to tell the difference between acceleration due to gravity and any other acceleration. Your post contradicts itself.

Wrong. As I stated before, this belief comes from a misunderstanding of the equivalence principle.

If acceleration could not be handled within the context of special relativity, then the statement that gravity is locally equivalent to acceleration would do you absolutely no good, and could not be used as a basis for forming general relativity. It would basically be a statement that two things which existing theory cannot deal with are equivalent - but that wouldn't help you understand either. But that's not the case: the equivalence principle is significant precisely because one side of that equivalence IS understandable within the context of special relativity.

Standing still in a uniform gravitational field is locally equivalent to acceleration, that is true. But the rather relevant point is that gravity is NOT uniform, and that to explain it we need a theory which works on more than just a local level. And the ability to handle acceleration in special relativity cannot be applied to gravity directly without accounting for that non-uniformity and non-locality, and it is accounting for those non-local and non-uniform aspects which lead to general relativity.

Acceleration itself can be handled quite easily in special relativity. Calculating the time taken by the traveling twin with finite acceleration is simply the path integral along his worldline of the Minkowski metric. The math may be ugly, but saying it can't handle such cases is equivalent to saying that Euclidean geometry cannot handle curves.