Stupider-er Twins Question

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  • #76
DrGreg
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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.)
 

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  • #77
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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.
 
  • #78
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 :biggrin: of years since college. I think I'll have to take some time to understand your post.

Thanks,
Al
 

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