Minkowski Space Time Briefly Revisited

I introduced rigid rods as a term of art. In my conception, the attempt to "scoop up" a collector in a butterfly net is modeled as two interpenetrating rigid rods in 1 spatial dimension such that the latter totally contains the former. Thus all that is required is a certain mathematical relationship of the endpoints without worrying about the likely physiological results of being smacked in the face with silk netting at a relative velocity of \(\frac{12}{13} c\).

Working in only one spatial dimension and ignoring the details of how dueling butterfly collectors could mutually ensnare each other even if there nets were arbitrarily large is an abstraction, because I have no interest in the trivia of how one sets up the details when the core concept is how is the physics described in a frame where a particular man is at rest versus how is the physics described in a frame where the corresponding capturing netting is at rest.

Is your analogy with the title characters of the folk tales in that you think physicists ignore or give no voice to relativity in current practice so that relativity is waiting for a prince/rescuer? Or is your analogy with the successful Disney movies based on those stories which did well at the box office and are cherished bits of Western culture? Or is your analogy with Disney's cynical practice of keeping them off the market for years at a time so that they can have re-releases of their aging intellectual property postured as special opportunities for consumers rather than as a long-term strategy?

Parochial Englishman -- the Australians have not changed -- they just adopted a coordinate system where you are not upright.

Post #95 is the last in a series where I unify the (L,v) frame-dependent description of moving rigid rods with the relativistic Doppler shift for wavelength and equivalently pulses of light measured by length. The same transform
\( \begin{pmatrix} L' \\ v' \end{pmatrix} = \frac{c^2}{c^2 - u \, v} \begin{pmatrix} \sqrt{1 - \frac{u^2}{c^2}} L \\ v - u \end{pmatrix}\)
is valid for \(-c \leq v \leq c\) and \( -c < u < c \) and applies to both rigid ponderable rods with an associated rest frame and finite-duration flashes of light which have no associated rest frame. Having unified the phenomena of ponderable bodies with a natural rest frame and flashes of light with none, there is no philosophical barrier to freeing oneself from always talking about ponderable objects as measured in their natural rest frame, which is great, because in the topic you introduced in post #51 you specifically ask about comparing the measurements of two objects (barn and pole OR butterfly net and man) which do not share a state of motion.

Such a comparison is permissible only if one never gets distracted from the fact that we are talking about frame dependent quantities and thus L and v transform together. Likewise in the gory collision prysk envisions, it is important to note that the energy and momentum of net and man are frame dependent quantities and thus E and p transform together. The man doesn't stop worrying about the net because in its own rest frame it has p=0.

That is a parochial viewpoint and at odds with general relativity which admits many descriptions of the physics as equivalent.

 

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Well, to be clear, you champion Farsight-Relativity, an idea that you freely admit cannot be used to solve physics problems or do any applications, unlike the relativity theory developed by and after Einstein.

No measurement of any rod changes and no rod changes just because some other thing changes its velocity. In a given system of coordinates, the length of the rod is always the same if that rod does not change its state of motion. The length of the rod simply is different in different systems of coordinates, all the physics that we do in that system of coordinates must respect that difference, and it is not the measurement that changes it is whether or not we care to do measurements according to a specific system of coordinates or not.

 

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I think you are mixing notation, I have restored the notation of post #95.

u = 0, L = 1 meter, v = c ⇒ (L', v', n') = (1 meter, c, n)

-c < u < -0.9 c , L = 1 meter, v = c ⇒ (L', v', n') = (\(\sqrt{\frac{c+u}{c-u}}\) meter < 1 meter, c, n)

0.9 c < u < c , L = 1 meter, v = c ⇒ (L', v', n') = (\(\sqrt{\frac{c+u}{c-u}}\) meter > 1 meter, c, n)

No. This is very much contrary to the way things work for material objects at rest. But the case (L,v) = (1 meter, 0) is different from (L,v) = (1 meter, 0.99 c)

But for material objects with are introduced as measured at a high relative velocity:

u = 0.9 c, L = 1 meter, v = 0.99 c ⇒ \(L' = \frac{\sqrt{1 - \frac{u^2}{c^2}}}{1 - \frac{u \, v}{c^2}} L = \frac{100 \sqrt{19}}{109} L \approx 3.998990 \, \textrm{meters}\), \(v' = \frac{90}{109} c \approx 0.825688 c\)

u = 0.9 c, L = 1 meter, v = c ⇒ \(L' = \frac{\sqrt{1 - \frac{u^2}{c^2}}}{1 - \frac{u \, v}{c^2}} L = \sqrt{19} L \approx 4.358899 \, \textrm{meters}\), \(v' = c\)

Not so different, eh?

No. They can be unified. These equations hold for both (freely propagating) ponderable matter (m > 0 ) and energy ( m = 0).
\(E^2 = (mc^2)^2 + (pc)^2 \\ E v = p c^2\)

Assuming m > 0, we can write: \(E = \sqrt{ (mc^2)^2 + (pc)^2 }\) and so \(v = \frac{p c}{\sqrt{ (mc)^2 + p^2 }} =\).
Since we have \(v' = \frac{v - u}{1 - \frac{v \, u}{c^2}}\) we require
\(\frac{\frac{p c}{\sqrt{ (mc)^2 + p^2 }} - u}{1 - \frac{ u \frac{p c}{\sqrt{ (mc)^2 + p^2 }} }{c^2}} = \frac{p' c}{\sqrt{ (mc)^2 + p'^2 }}\)

or \(p' = \frac{p - \frac{u}{c} \sqrt{p^2 + m^2 c^2}}{\sqrt{1 - \frac{u^2}{c^2}}}\)
It follows that \(E' = \sqrt{(mc^2)^2 + (p'c)^2} = \sqrt{ c^4 m^2 + \frac{(p c^2 - u E)^2}{c^2 - u^2}}\).

--- Examples ---

If \(E = mc^2\) and \(p = 0\) this says \(p' = \frac{- m u}{\sqrt{1 - \frac{u^2}{c^2}}}\) and \(E' = \sqrt{ m^2 c^4 + \frac{u^2 m^2 c^4}{c^2 - u^2}} = \frac{m c^2}{\sqrt{1 - \frac{u^2}{c^2}}}\) in agreement with experiment.

If \(E = p c > 0\) and \(m = 0\) this says \(p' = \frac{p ( 1 - \frac{u}{c} ) }{\sqrt{1 - \frac{u^2}{c^2}}} = \sqrt{\frac{ c - u }{c + u}} p\) and \(E' = \sqrt{\frac{(c - u )^2}{c^2 - u^2}} E = \sqrt{\frac{c - u}{c + u}} E\) which is again our relativistic Doppler effect for INVERSE wavelength or frequency.

 

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No problem with that.

It isn't trivia, this cuts right to the heart of length contraction, and you're ducking the issue.

Nearly. I think physicists get relativity wrong and it needs a prince/rescuer to cut through a hundred-year thicket of misunderstanding.

It's nothing to do with Disney. The Sleeping Beauty and Cinderella in the guise of Cendrillon and Ashenputtel predate Disney.

It's a rubbish analogy that totally ducks the issue you're avoiding.

You've gone all round the houses, and you've ended up saying nothing that tackles the issue.*

No, it's crucial to understanding general relativity. Your measurement of photon frequency changes when you descend to a lower altitude. The photon frequency doesn't change, nor does its E=hf energy. You must surely know this. When you send a 511keV photon into a black hole, the black hole mass increases by 511keV/c². Conservation of energy applies.

* You ducked the issue in your previous post too.

 

And do you think that you, who can't do a physics problem, are the right person for this job? How can you tell if a physicist has made a mistake when you can't understnad how they got their result?

 

That was a stupid question. As per the first question... You know he does! And to the second he has read Einstien!

 

Ah, I remember the good old days when Farsight admitted to reading only the first three pages of Einstein's relativity textbook.

 

Sadly, I won't have time to read or respond in detail for hours or perhaps days. I did have one immediate question.

My use of "trivia" called out "how one sets up the details" of "dueling butterfly collectors." If you mean the same thing, then please explain how Special Relativity is all about "dueling butterfly collectors" or int the alternative then please explain first what you are calling "not trivia."

 

I did.

Then same answer for the rods as I originally gave for the men:

There's no such thing as a perfectly rigid rod in the real world. No such thing is even hypothetically possible in relativistic physics, and the resolution to your thought experiment is just going to be crushed rods and/or torn nets. It's really that simple and boring.

 

Farsight, rpenner wins the special relativity comprehension question, and has also completed a spot lacking in my freshman relativity education.

Case 3:

(measuring pulse length while traveling in the same direction at 0.9c)

0.9 c < u < c , L = 1 meter, v = c ⇒ (L', v', n') = (Doppler expression) * meter > 1 meter, c, n)

Yes, and this makes perfect sense to resolve the twin paradox as well. Many thanks for clearing that up.

We need to revamp the old pole and barn paradox to something akin to Star Wars (light saber and barn paradox?) This would nicely introduce the different relativistic scenarios for matter, energy.

 

We have measured length contraction, time dilation, and Doppler shift for an energy pulse traveling at the speed of light (and for measuring instruments at rest, traveling +/- 0.9c) for three cases:

Case 1; At rest relative to frame that produced the pulse, frame v = 0
Pulse: length contraction or expansion = 0, time dilation = 0, Doppler = 0; v = c
Matter (instead of pulse): length contraction or expansion = 0, time dilation = 0, Doppler = 0

Case 2: Moving at relativistic speed, direction opposite pulse, frame v = -0.9c
Pulse: length contraction or expansion = 0, time dilation = 0, Doppler = Blue; v = -c
Matter (instead of pulse): length contraction = YES, time dilation = YES, Doppler N/A

Case 3: Moving at relativistic speed, direction parallel to pulse
Pulse: length EXPANSION, time dilation = 0?, Doppler = RED; v = c
Matter (instead of pulse): length contraction = YES, time dilation = YES, Doppler N/A Assume measured at 0.8c (necessary to for 0.9c to catch up) These are slight effects at 0.1 c relative speed

Did we just notice time EXPANSION of an energy pulse from a moving frame, or am I imagining this?
Time is not an independent variable you can separate from energy as easily as relativity does from moving matter. Do you understand this?

 

I've got the idea to email Queen Elizabeth and ask her to make me a princess.

Hey, I'm also a British citizen even though I've never been there.



Oh and Gravitational positioning satellites make Farsight wrong.

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Energy without time ceases to propagate. If you could increase the velocity of the measuring instrument that is in front of the pulse to c for Case 3, the pulse would never catch up to it, and it would be impossible to observe any energy at all from directly opposite the direction of motion. It would be as though the pulse were red shifted to infinite wavelength, or equivalently, that time had stopped. The intuition would be that other directions would not be affected, but this is not the case because matter is also energy. Within each particle of matter, a Doppler red shift similar to the one applied to the pulse takes place, so that if you can slow time in a single direction, it is the same as slowing down time in all directions. Although bound, the energy content in matter spins wildly, seemingly in all directions at once.

 

You obviously meant GPS (GLOBAL Positioning Satellites), BWS. They employ corrections due to gravitational time dilation, but most of the time dilation corrections for GPS satellites are actually due to velocities relative to fixed points on the surface of the Earth.

 

Yeah, but I was fantasizing about being a princess, so I was a bit distracted.

I do, however, get the feeling people really like correcting me. Also, cause I think I'm close to getting banned, I don't feel much effort is worth it. (I've never boasted having a Ph D in physics.)

 

When did this happen?

Assuming you intended that as some kind of direct measurement. Length contraction is implied by time dilation which has persistent results, meaning clocks retain their time dilated record of time when changing frames of reference. Length contraction has no corresponding persistent character. It is predicted by theorey and calculated mathematically, but measured in and direct way...?

 

Other than the Doppler shifting of wavelength, as measured by spectra obtained via diffraction gratings, I don't think there has been a measurement of length change. I believe posts #81 and #95 make a good argument that Doppler shifting of wavelength and length contraction are so closely related that they can be unified in one formula, but we have very few macroscopic examples of relativistic objects in our laboratories.

 

Huh? The issue here is all to do with the reality of length contraction, and you're evading it. When you move fast you see things as length contracted, but they didn't change one iota. You see the other guy as being 1m long rather than 2m long, so you might think you can scoop him in your net, but he thinks the same about you, and you can't both be right.

He's totally ducked the issue, dan.

So have you.

 

How would you feel as a father, so egoistical, teaching you daughter the wrong things?

Would you punch your daughter in the face because she questioned you?

Report my post, I don't care.

You already have that moderator Billy T preaching the wrong things.

 

You know Farsight, you are really crackers!

You sit in the audience at a magic show and believe the guy really disappeared, because you saw it?

I just questioned the measurement of length contraction. Rpenner responded. So he has addressed the issue not just here but in earlier posts and threads.

I do challenge the measurement, now and again. That is not the same as saying that just because it cannot be directly measured is does not occur. There is a great deal of indirect evidence that supports length contraction.., and you know that. You have even made reference to some of that evidence in past discussions.

Do I know it to be true, no. But I do believe that existing evidence supports the reality of length contraction.