Is special relativity sufficient to treat the twin paradox?

[Loren Booda]

I cannot conceive of a complete path which the "away" twin can follow, relative to the "home" twin, that does not involve significant acceleration. A straight line from home to star then back home in reverse, a centrifugal circle from home (0 degrees) to star (180 degrees) to home (360 degrees), or perhaps any geodesic intersecting home and star introduces acceleration, and apparently requires general relativity.

 

[bcrowell]

The idea of restricting special relativity (SR) to inertial frames is basically an anachronism that dates back to the period from 1905 to 1915 when SR was available but general relativity (GR) wasn't. During that period, nobody really had a clear picture of what the boundaries of applicability of SR were, so the default was to assume that SR was inapplicable to any situation involving accelerating objects. The modern attitude is that SR can handle accelerating objects just fine, and only requires a restriction to flat spacetime (i.e., no gravitational fields). There's a nice description of this in Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975).

There are a bunch of anachronisms like this that have taken an amazingly long time to die out in textbook treatments of SR:

- restriction to non-accelerating bodies
- relativistic variation of mass
- depiction of the "c" in relativity as being fundamentally related to light

 

[Loren Booda]

bcrowell,

Thanks for relating the history, insight and correction regarding the applicability of special relativity in flat spacetime.

Doesn't Einstein's principle of equivalence correlate "accelerating objects" with "gravitational fields"? Thus inertial acceleration of an object is undifferentiated from its gravitational acceleration.

 

[Fredrik]

The only thing you need to know to see that SR predicts that the astronaut twin is younger is that the statement "A clock measures the proper time of the curve in Minkowski spacetime that represents its motion" is an axiom of the theory.

The equivalence principle is the idea that experiments performed in a small enough region of spacetime can't distinguish between acceleration in flat spacetime and an external gravitational field. For example, a clock attached to your ceiling will run faster than a clock attached to your floor, and if you can't look outside your room, you won't be able to determine if the reason is gravity or acceleration.

The importance of this principle is often overstated, just like Einstein's postulates. (I think they all belong on that list that bcrowell started). They were useful as guidelines that helped Einstein guess the structure of two new theories, but they're not axioms of the theories he did find. (The theories do however make predictions that agree with these principles and postulates...because he was specifically looking for theories that would make such predictions).

Note also that there's no such thing as a "gravitational field" in GR. There's just the metric tensor.

Instead of talking about the "equivalence principle", I prefer to talk about how we can't define acceleration until we have specified what's not accelerating. In both SR, and GR, we define the timelike and null geodesics of the metric tensor to represent inertial motion (of massive and massless particles respectively), and then we define acceleration as a kind of deviation from inertial motion. The experiments you perform in a closed room will tell you how much you deviate from inertial (i.e. geodesic) motion, but they can't tell you what the geodesics look like. That's the statement you should try to understand.

 

[lalbatros]

Special Relativity is a generalisation of Newtonian mechanics.
Therefore it is obvious that SR can describe accelerated objects.
Therefore also for their proper time evolution.

In addtion, proper time is a purely kinematic concept (see Einstein1905).
This is why it is possible to "clean" the twin story so as to avoid the additional effect of finite accelerations.
Simply suppress the twin from the story and replace it by two rockets.
One rocket for the forward travel and one for the backward travel.
A proper synchronization protocol can then illustrate the twin paradox without any additional confusion.

I believe the misconception came with the widespread teaching of GR and the unavoidable misunderstandings.
The web propagates such misconceptions, just like spelling errors that are never corrected.

 

[bcrowell]

Re the points raised by Fredrik about the role and importance of the equivalence principle, one very important thing it does is organize our thoughts about possible theories of gravity and how to test them. Consider the following four theories of gravity:

1. GR
2. Brans-Dicke gravity
3. GR with gravitational torsion
4. Østvang's quasi-metric relativity

1 and 2 satisfy the e.p., while 3 and 4 do not. That means that if you want to test these theories, experiments that test the e.p. can distinguish the two groups. That's why the UW Eot-Wash group uses e.p.-testing experiments to test 3, and one of the central points in the analysis of the Pioneer anomaly is whether it can be explained by a gravitational effect that obeys the e.p.

Personally, I also find that the e.p. is a very good mental organizing tool for understanding the physical interpretation of the mathematical depiction of GR. It's also nice to be able to demonstrate certain things didactically (e.g., gravitational redshifts) based solely on the e.p., without having to teach people the whole machinery of tensors and differential geometry (which isn't possible except for the tiny fraction of people who become grad students in physics).

 

[Loren Booda]

Thanks all as I will pore over your contributions.

 

[Ich]

The idea of restricting special relativity (SR) to inertial frames is basically an anachronism that dates back to the period from 1905 to 1915 when SR was available but general relativity (GR) wasn't.

IMHO it's not an anachronism, it is how special relativity is defined: incorporating the principle of special covariance, namely concerning inertial frames. At least that's what I've read in Wald IIRC (sorry, can't remember the page).
Modern usage of the word therefore seems to be a deviation from "correct" terminology.

Edit:
But most importantly, one inertial frame is enough tho handle the twins. You can easily integrate proper time along a curved path without using GR. Einstein showed that explicitlly in the very first paper on SR.

 

[GRDixon]

I cannot conceive of a complete path which the "away" twin can follow, relative to the "home" twin, that does not involve significant acceleration. A straight line from home to star then back home in reverse, a centrifugal circle from home (0 degrees) to star (180 degrees) to home (360 degrees), or perhaps any geodesic intersecting home and star introduces acceleration, and apparently requires general relativity.

The away twin must accelerate in order to return, but the difference in aging is primarily (if not wholly) a matter of his motion relative to the "home" twin. Consider the case where the away twin steps from IRF K (which he initially shared with the home twin) into IRF K'. Using the measuring rods and clocks of K', he measures the home twin to be aging more slowly than he is. But when he steps from K' back into K (at the furthest distance from the home twin), he reconsiders. Back in K it's clear that it was HE who was aging more slowly. He now steps from K into K'' and returns to the home twin. Again, now using the measuring rods and clocks of K'', he measures the home twin to be aging more slowly than he is. However, upon returning home and stepping from K'' back into K, he again concludes that it was actually he who was aging more slowly.

The question now becomes: do the accelerations experienced in "stepping" from one IRF to another have any affect on the away twin's rate of aging? Experiments with the half lives of particles in synchrotrons indicate that the time dilation of the away particles is strictly speed-dependent. In brief, the radial accelerations have no measurable effect on the rate of away particle aging. (The linear accelerations in synchrotrons is practically zero, since the particles are injected into the "accelerator" already traveling at near-light speeds.) I don't know whether linear accelerations have been investigated in a linear accelerator such as the one at Stanford. To my hazy knowledge, particles are inserted into such linear "accelerators" also traveling at near-light speed.

I am aware that the rate of aging is dependent on gravitational-field-strength/linear-acceleration-magnitude in GRT. But I'm not qualified to discuss the difference in aging (if any) that would occur if the turn-around acceleration was infinite, but lasted for only an infinitesimal time.

 

[bcrowell]

Note also that there's no such thing as a "gravitational field" in GR. There's just the metric tensor.

The Christoffel symbol plays the role of the gravitational field in GR, and I think the early relativists explicitly thought of it this way. See, e.g., the bottom of page 1 of Scwarzschild's 1916 paper: http://arxiv.org/abs/physics/9905030 , but note that he puts it in quotes. It's just that the Christoffel symbol isn't frame-independent and locally observable the way the acceleration of gravity is in Newtonian physics. It plays the role of a gauge field in GR.

 

[Al68]

I cannot conceive of a complete path which the "away" twin can follow, relative to the "home" twin, that does not involve significant acceleration. A straight line from home to star then back home in reverse, a centrifugal circle from home (0 degrees) to star (180 degrees) to home (360 degrees), or perhaps any geodesic intersecting home and star introduces acceleration, and apparently requires general relativity.

The only thing we can't do in SR is use the lorentz transformations in a non-inertial reference frame, such as one defined with the ship at rest during its acceleration. If this shortcoming is important to you, you can use GR.

Better yet, analyze the scenario in SR from the accelerating ship's perspective using a series of co-moving inertial frames, and calculate Earth's clock reading for each ship clock reading simultaneous with the ship being momentarily at rest in each co-moving frame. Then calculate the difference in Earth's clock's reading relative to the difference in the ship's clock reading between any two of those successive co-moving frames as the number of such frames approaches infinity and the interval between them approaches zero.

After you're done, you will have derived the (uniform-linear) gravitational time dilation formula. This derivation (for the rate of a non-local clock relative to an accelerated reference frame) combined with the equivalence principle provided the basis for Einstein's prediction that a clock at a higher gravitational potential will run slow relative to a clock at a lower potential in a gravitational field.

 

[GRDixon]

Better yet, analyze the scenario in SR from the accelerating ship's perspective using a series of co-moving inertial frames, and calculate Earth's clock reading for each ship clock reading simultaneous with the ship being momentarily at rest in each co-moving frame. Then calculate the difference in Earth's clock's reading relative to the difference in the ship's clock reading between any two of those successive co-moving frames as the number of such frames approaches infinity and the interval between them approaches zero.

After you're done, you will have derived the (uniform-linear) gravitational time dilation formula. This derivation (for the rate of a non-local clock relative to an accelerated reference frame) combined with the equivalence principle provided the basis for Einstein's prediction that a clock at a higher gravitational potential will run slow relative to a clock at a lower potential in a gravitational field.

Fascinating. You've inspired me to attempt following Einstein's footsteps.