Understanding the Twin Paradox: Exploring Contradictions in Special Relativity

[Fredrik]

As far as I am aware acceleration has no direct effect on clock rates, we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration.

What do you think of Mach's principle that were it not for the mass in the rest of the universe, and an experiment like this were performed in isolation, the ship's twin would feel no acceleration, and inertia would not even exist?

I like talking about these things too, but I would like to point out that ideas like "Mach's principle" or "ideal clocks" have no place in a discussion about the twin paradox. The twin paradox is the (false) claim that special relativity predicts two contradictory things about the twins' ages when they meet again. Special relativity is just Minkowski space, and the twins are just three straight lines. If you find a way to eliminate the contradiction that involves the properties of clocks (ideal or not), or some "principle" that isn't a part of SR, then you haven't solved the problem. You still wouldn't have any idea if there really is a contradiction in SR, or what SR really says! If you want to really solve the problem, you have to do it using the properties of Minkowski space, and nothing else.

 

[DrGreg]

If we had real acceleration instead of instantaneous, it would be obvious that, from the ship's twin's view, the Earth and space station do not stay at rest with each other. So as the ship starts slowing down, the Earth and space station are getting farther apart. If the ship decelerates at 1 G proper acceleration, is it important that his coordinate velocity and acceleration relative to Earth will be different than relative to the space station?

I haven't seen this addressed in textbooks or resolutions on the net, since they all either use instantaneous acceleration, or just split the ship frames to keep the math simple. Is there a resolution on the net that shows the math for realistic acceleration?

Despite what others have said, it is possible to set up a coordinate system for an accelerating observer. But that coordinate system is not an inertial frame so it doesn't behave like inertial frames do.

Warning: the following requires a knowledge of hyperbolic functions and calculus.

If $(t, x)$ are the inertial coordinates of an inertial observer I (ignore y and z as being constant), consider a new coordinate system $(T, X)$ defined by

$$x + \frac{c^2}{g} - x_0 = \left( X + \frac{c^2}{g} \right) \cosh \frac {g(T-T_0)}{c}$$
$$t - t_0 = \left( X + \frac{c^2}{g} \right) \sinh \frac {g(T-T_0)}{c}$$​

These are Rindler coordinates. T is the proper time of an accelerating observer A located at X = 0, and with a constant proper acceleration of g. Two events with the same value of T are simultaneous in A's co-moving inertial frame and X measures distance from A in the co-moving inertial frame. The velocity of A relative to I is

$$\frac{dx}{dt} = c \tanh \frac {g(T-T_0)}{c}$$​

with Lorentz factor

$$\gamma = \cosh \frac {g(T-T_0)}{c}$$​

The coordinate acceleration of A relative to I is

$$\frac{d^2x}{dt^2} = g \, sech^3 \frac {g(T-T_0)}{c} = \frac {g}{\gamma^3}$$​

Choose values of T₀, t₀ and x₀ to synchronise clocks and distances between the A and I frames in the way you want. When $T = T_0$, A is stationary relative to I at $t = t_0$, $x = x_0$. (So I is the co-moving inertial frame at that moment.)

References

Rindler, W. (2006 2nd ed), Relativity: Special, General and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5, Section 3.8, pp.71-73 and Section 12.4, pp.267-272.

Gibbs, P. and Koks, D. (2006), http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html , Usenet Physics FAQ, accessed 19 June 2008.

Anonymous, (undated), "Born rigidity, Acceleration, and Inertia", MathPages, accessed 19 June 2008.

 

[matheinste]

Hello Fredrik.

Quote:-

---we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration.---

I only included this comment because some people might think that a clock in the mechanical sense might be affected by acceleration via its mechanism. To make a point an ordinary clockwork clock may well be affected either slowing down or speeding up due to acceleration. Also people may not know whether or not atomic clocks are directly affected by acceleration by way of the physics involved. The use of an ideal clock is just to remove any possible objections of this sort from a situation which for some reason many find confusing.

Of course Mach's views on inertia are irelevant to a universe peopled by earth, space stations, twins and the like, i.e. matter.

Matheinste.

 

[Fredrik]

Despite what others have said, it is possible to set up a coordinate system for an accelerating observer.

I didn't mean that it's impossible to define a coordinate system that takes the accelerated observer's world line to be its time axis. (I said something to that effect in another thread, and you were right to correct me then). What I meant is that it doesn't make much sense to think such coordinates as representing the accelerating observer's point of view. I'm sure there are lots of ways to slice up space-time into a one-parameter family of space-like hypersurfaces that we can (if we want to) think of as representing space at different times. Why should the choice defined by Rindler coordinates be the "correct" choice?

 

[yuiop]

Hello Fredrik.

Quote:-

---we are of course talking about ideal clocks with no bits that can be affected by the physical forces involved in acceleration.---

Yes.. A grandfather clock with a pendulum would certainly be affected by acceleration, but it also far from an ideal clock.

 

[DrGreg]

I didn't mean that it's impossible to define a coordinate system that takes the accelerated observer's world line to be its time axis. (I said something to that effect in another thread, and you were right to correct me then). What I meant is that it doesn't make much sense to think such coordinates as representing the accelerating observer's point of view. I'm sure there are lots of ways to slice up space-time into a one-parameter family of space-like hypersurfaces that we can (if we want to) think of as representing space at different times. Why should the choice defined by Rindler coordinates be the "correct" choice?

You are right that there are other choices of accelerated coordinate system. And it is debatable as to exactly what the accelerated observer's "point of view" is. Nevertheless it is conventional to consider the co-moving inertial frame to represent the "instantaneous" view, and Rindler coordinates are the only coordinates (I think) that are compatible with this view in the sense that:

- the observer is at fixed spatial coordinates X = Y = Z = 0
- at X = 0 (but not at other positions), T is the proper time of the observer
- every surface of constant T coincides with the plane of simultaneity of the corresponding co-moving inertial frame
- within each such simultaneity plane, the Rindler spatial coordinates X, Y, Z coincide with the co-moving inertial frame's spatial coordinates

That, in my view, makes Rindler coordinates a more "natural" choice than any others. Of course all "points of view" are a mathematical construct, even in inertial frames. They don't reflect what you see with your eyes; the frame point of view is something you have to calculate retrospectively from observations made after the events being measured, and it depends on what conventions you choose to adopt to perform the calculation.

And I think Rindler coordinates would answer the question put in post #65: they give us a way of seamlessly (up to continuous first derivative) interpolating between the two points of view of inertial motion before and after acceleration. The attached left-hand diagram illustrates the accelerated twin's point of view in the Twins Paradox. (The right-hand diagram shows the inertial twin's point of view.)

 

[phyti]

Is this pretty much a consensus view?

What do you think of Mach's principle that were it not for the mass in the rest of the universe, and an experiment like this were performed in isolation, the ship's twin would feel no acceleration, and inertia would not even exist?

Thanks,
Al

Don't know about consensus view, but do know time dilation is an experimentally verified fact, and it explains the differences in observer perceptions.
If a and b are two cities 200 miles apart, and you fly between them at 100 mph, you arrive in 2 hr. IF you fly between them at 200 mph, you arrive in 1 hr. The distance between them did not change, you got there quicker! You can't give an unqualified statement such as 'the space contracted' without explaining how. This is a popular misconception, because SR does not state it. The transformation rules apply to the varied observations/perceptions of different frames so as to preserve the one set of actual physical events. SR is like an accounting method that reconciles the perceptions, but is does not alter the actual events.
There is one event, but many perceptions.
Mach:
If the mass of the universe is on average, uniformly distributed (including the lumps), and considering the vast distances involved, the net gravitational effect is zero. Any inertial effects are the result of local mass, (within the solar system). Two space ships would still resist acceleration because of the ships mass.
Consider, if all matter had local effects, it would be impossible to conduct an isolated experiment, and you would get random variations from distant events.

I would like to clarify... that isolated experiments have a purpose as control elements, but
in hypothetical scenarios, this does not represent a real world situation.

Sorry I didn't respond sooner.

 

[Al68]

I like talking about these things too, but I would like to point out that ideas like "Mach's principle" or "ideal clocks" have no place in a discussion about the twin paradox. The twin paradox is the (false) claim that special relativity predicts two contradictory things about the twins' ages when they meet again.

Well, you're right, most of my questions were about situations very different from the twins paradox. I referenced it just because everyone is familiar with it. Maybe I should have used a different title for the topic.

And yes, Mach's principle is a little off track, but interesting. Einstein was the one who coined the phrase "Mach's principle" while discussing why inertial frames are different from non-inertial frames, and the seemingly circular logic of saying that Newton's laws "hold good" in inertial frames, and we know a frame is inertial (a priori) because Newton's laws "hold good". He considered this a "defect" of SR. I wouldn't call it a defect, just an unanswered question.

DrGreg, I think I should be careful what I ask for. It's been a couple of years since college. I think I'll have to take some time to understand your post.

Thanks,
Al