Variable speed of light

Originally posted by Ivan Seeking
We still have no preferred observers in the inertial frame.

I'm not sure what point you're trying to make.

If one accelerates, that frame is no longer valid under SR, and we can then make a distinction between the two systems.

However, acceleration is not necessary to distinguish between the two systems. As I said, the only thing that determines whether two twins age asymmetrically is whether the spacetime lengths of their worldlines are different. The symmetry can be broken by acceleration, or by other means.

We can resolve this using pure SR, by the way: my calculation was performed entirely in an SR inertial frame (that of the Earth twin).

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chroot
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Originally posted by Ivan Seeking
We still have no preferred observers in the inertial frame.
I have no idea what this means.
If one accelerates, that frame is no longer valid under SR, and we can then make a distinction between the two systems.
This is also incorrect. Special relativity is all that's necessary to understand the twin paradox; Ambitwistor explained it nicely. The only thing you need general relativity to explain is gravitation.

- Warren

Ivan Seeking
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Originally posted by chroot
I have no idea what this means.

There is no absolute reference frame. There is no absolute state of rest.

This is also incorrect. Special relativity is all that's necessary to understand the twin paradox; Ambitwistor explained it nicely. The only thing you need general relativity to explain is gravitation.

- Warren [/B]

How do we determine which twin is younger? One of them has to accelerate in order to leave earth; unless he was born in a state of relative motion as compared to his twin.

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Originally posted by Ivan Seeking
There is no absolute reference frame. There is no absolute state of rest.

So?

How do we determine which twin is younger?

The younger twin is the one whose worldline is shorter. Which twin that is depends on the physical situation. I gave an example.

Ivan Seeking
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One of them has to accelerate in order to leave earth; unless he was born in a relative state of motion as compared to his twin.

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Well, Ambitwistor's is the best explanation of the twin paradox I have run into. It's the first time I even understood the importance of the "Spacetime Interval". (And, as Chroot said, the graphics were killer.)

This gives me a much better sence of what people are saying is the case. I am fairly certain I don't grasp it yet, but the "spacetime interval" must surely have been the missing link I needed to start putting this together in my mind.

The problem for me has always been that if the two people in relative motion measure each others clocks as slow it strikes me as proof positive both clocks are fine and would agree on the total time elapsed if compared later in the same frame. The difference in length of the spacetime interval finally introduces the asymetry that accounts for the differences in the time elapsed in the two different frames.

Originally posted by Ivan Seeking One of them has to accelerate in order to leave earth; unless he was born in a state of relative motion as compared to his twin.

Yes. So?

chroot
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Originally posted by zoobyshoe
It's the first time I even understood the importance of the "Spacetime Interval".
The interval is a very important quantity in relativistic physics because it is invariant. No matter what coordinate system you use, or which observers you consider to be at rest, the interval they will measure for some path $$\Gamma$$ is always the same. The interval is independent of observers and is a fixed quantity for any particular path through spacetime.

- Warren

Originally posted by zoobyshoe
The problem for me has always been that if the two people in relative motion measure each others clocks as slow it strikes me as proof positive both clocks are fine and would agree on the total time elapsed if compared later in the same frame. The difference in length of the spacetime interval finally introduces the asymetry that accounts for the differences in the time elapsed in the two different frames.

It's good to think in terms of geometry. Special relativity is just Euclidean geometry in disguise (with a slightly modified Pythagorean theorem).

Lorentz transformations are analogous to rotations. (Because of the minus sign in the Pythagorean theorem for distance, Lorentz transformations trace out hyperbolas in spacetime instead of circles in space.) So the statement that "there are no preferred inertial frames in SR" is really a disguised version of the statement, "there are no preferred directions in Euclidean geometry".

You can see that in this version of the twin paradox, the two worldlines make a triangle: the worldline of the Earth twin is one side, and the worldline of the travelling twin is the other two. People say, "why isn't the travelling twin's frame the same as the Earth twin's?" In Euclidean geometry, the analog of switching inertial frames is rotating. But you can see that just by a rotation, you can't turn the two bent line segments of the travelling twin into one line segment like the Earth twin's worldline: they are not geometrically equivalent to each other. Rotations don't affect lengths, so the two sides of the triangle will never be equal in length to the other side of the triangle, no matter what you do to it --- unless you start deforming it, but then that doesn't describe the same situation anymore.

Ivan Seeking
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I have no idea why anyone has objected to anything I have said here. I will read the link and pick this up later.

Originally posted by zoobyshoe
Well, Ambitwistor's is the best explanation of the twin paradox I have run into. It's the first time I even understood the importance of the "Spacetime Interval".

Here's a secret: I almost never perform Lorentz transformations. I just work with 4-vectors, and extract physical information from them by finding their spacetime lengths or projections onto other 4-vectors.

Originally posted by Ivan Seeking
I have no idea why anyone has objected to anything I have said here.

It hasn't been clear to me what your points have been, but it has seemed at times that you have been implying that acceleration is necessary for the twins to experience different elapsed proper times, and/or general relativity is required to resolve the twin paradox in the presence of acceleration, neither of which is true.

Originally posted by chroot If you think about it, it has to be that way... if it weren't, then some cosmic ray particle moving at 0.9c with respect to you somewhere in the depths of space would somehow affect YOUR clock!
This part, I don't get. I thought cosmic rays were photons, and as such, could never be observed going less than C.

In other words, I have been under the impression that even if I am traveling at 0.9c all photons whose speed I measure going in any direction relative to mine will be clocked going at C. Is this not the case?

chroot
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Originally posted by zoobyshoe
This part, I don't get. I thought cosmic rays were photons
No, sorry for the confusion -- some cosmic rays are photons -- but many are massive particles like protons and electrons. The photons, of course, will always be observed travelling at c, while the massive particles will always be < c.

- Warren

Originally posted by chroot some cosmic rays are photons -- but many are massive particles like protons and electrons.
Interesting. How did they all get the same name, being such different things?

"Cosmic ray" is just a catch-all term for "particles that come from outer space".

In Six Easy Pieces he mentions cosmic rays as the highest energy photons we are aware of. Is there a term that can be used to differentiate these from the "riff raff" cosmic rays?

Originally posted by zoobyshoe
In Six Easy Pieces he mentions cosmic rays as the highest energy photons we are aware of. Is there a term that can be used to differentiate these from the "riff raff" cosmic rays?

Ultrasuperduper high energy cosmic rays.

Actually, we've detected cosmic rays with much higher energies than the highest-energy cosmic-ray photons we've detected. The most energetic photons we've seen are from gamma-ray bursts, maybe on the order of 10 TeV (possibly 100's of TeV's; I'm not sure what the state of the art is). But we've seen charged cosmic rays up beyond 100 million TeV.

Ivan Seeking
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Originally posted by Ambitwistor
It hasn't been clear to me what your points have been, but it has seemed at times that you have been implying that acceleration is necessary for the twins to experience different elapsed proper times, and/or general relativity is required to resolve the twin paradox in the presence of acceleration, neither of which is true.

Well, somehow we got off track I think...perhaps you are expecting Ivan the Terrible?

First, I was explaining the mechanics of the paradox; that's all.

Next, I addressed the issue of preferred observers; and I still think correctly so. Perhaps this language is out of favor, but specifically I meant that no absolute state of rest or motion exists. This is a still significant concept of SR; no?

Finally, I keep addressing the issue that if we wish to describe one frame of reference as preferred, this in response to Zooby's question about whose clocks run slowly - meaning to prefer one system over the other - a frame of rest must be defined. I meant this all within the context of SR. In all cases we must still define a frame of rest by which we determine who will age less quickly; true? With our twins, we know who is in motion - the one who leaves earth...and this requires acceleration. The page linked makes this assumption immediately

Originally posted by Ivan Seeking
Next, I addressed the issue of preferred observers; and I still think correctly so. Perhaps this language is out of favor, but specifically I meant that no absolute state of rest or motion exists. This is a still significant concept of SR; no?

Yes, it's why people find the twin paradox puzzling: they think that since there are no preferred frames, each observer should observe the same thing.

(Actually, there are senses in which states of motion are absolute in SR: you can absolutely determine the magnitude of your proper acceleration.)

Finally, I keep addressing the issue that if we wish to describe one frame of reference as preferred, this in response to Zooby's question about whose clocks run slowly - meaning to prefer one system over the other - a frame of rest must be defined.

Well, you don't have to work in the rest frame of some observer --- if you wanted, you could choose some bizarre curvilinear coordinate system --- but it's easiest to do so when such a global inertial frame exists (as in Minkowski spacetime).

In all cases we must still define a frame of rest by which we determine who will age less quickly; true? With our twins, we know who is in motion - the one who leaves earth...and this requires acceleration. The page linked makes this assumption immediately

I wouldn't say that we know that the Earth twin is "at rest", if that's what you're implying by defining a rest frame, but it is true that we know that acceleration breaks the symmetry between the two twins in this variant of the twin paradox. (There are other variants in which this is not the case.)

Originally posted by Ivan Seeking
When the car is in motion, it is shorter in the frame of the garage. Likewise, from the frame of the garage, the car's clocks are running more slowly. When the car stops, that is, when the frame of the car coincides with the frame of the garage, the two lengths L and L0 agree. Likewise, it we compare the ticks of the clocks, we find that again they agree – they occur at the same rate. However, and this was a key test of relativity, we find that while the frames of the car and garage did not coincide, ie, while the car is in motion as viewed from the garage, the clocks in the car really were running more slowly...just as observed and predicted.

This was finally verified I think in the early sixties using two atomic clocks; one on the ground, and one in a jet. After flying one of the clocks around for a while, and after accounting for the effects of gravity, the clock on the plane had indeed lost time as predicted to within the accepted margins of error. This has since be replicated in many other ways. Also, we see the lifespan of subatomic particles increase according to Relativity and their relative speed – since their clocks run more slowly, we see them live longer. This is seen in particle accelerators as well as in nature.

To the extent that isotopes have a 1/2t, does that in relativity,
an accelerating particle runs more slowly than than at rest, factor in when calculating the 1/2t of an isotope?

jcsd
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Yes, a particle will appear to have a longer half-life when it is travelliong at relativistic speeds in some rest frame, indeed this is a famous example of a relativtic effect.

Ivan Seeking
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Originally posted by Ambitwistor
I wouldn't say that we know that the Earth twin is "at rest", if that's what you're implying by defining a rest frame, but it is true that we know that acceleration breaks the symmetry between the two twins in this variant of the twin paradox. (There are other variants in which this is not the case.)

This was a poor choice of words on my part.

There are other variants in which acceleration does not break the symmetry between the two twins? I may not know what you mean. Could you give an example or two?

There are other variants in which acceleration does not break the symmetry between the two twins? I may not know what you mean. Could you give an example or two?

I already gave two examples in my first post to this thread. (Note that, as I clarified later, I am speaking of proper acceleration.)