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# Scary relativity Topic Page: 1 2 3 4 5 6

Scary because there is some linear algebra here!

A square matrix as many of you know, is an arrangement of numbers that embodies in some contexts a transformation of a certain sort - for example when I rotate the coordinates in the plane,

x' = x cos a - y sin a
y' = x sin a + y cos a

This is more easily expressed (read brackets as a column vector, and parentheses as a row vector, and double brackets as a matrix with entries in "reading" order)

[x' y'] = M [x y]

where M is the square matrix [[ cos a, -sin a, sin a, cos a ]]

The determinant is the product of the diagonal entries minus that of the off-diagonal entries, here cos^2 a + sin ^2 a = 1.

OK let's introduce four basic matrices (with complex numbers as entries)

Sx = [[ 0, 1, 1, 0]]
Sy = [[ 0, -i, i 0 ]]
Sz = [[ 1, 0, 0, -1 ]]
S0 = [[ 1, 0, 0, 1 ]]

and let's make the matrix object

R = x Sx + y Sy + z Sz + t S0 = [[ z + t, x - iy, x + iy, -z + t ]]

The determinant is

|R| = (z+t)(-z+t) - (x-iy)(x+iy) = -x^2 - y^2 - z^2 + t^2.

You'll see where this is going eventually. Ask questions now.

-drl

For me, Danny, this math is over my head. I would need pictures to explain each step and a glossary (with pictures) to explain the words vector, row vector, column vector, and matrix.

BUT DO NOT DO THIS, at least at this time. Rick and Brentwood also asked you to explain the special relativity thing. It is quite likely they do understand simple math like this and I do not want to hold the three of you up. Later, perhaps y'all can draw pictures for me and stuff like that BUT NOT NOW, or perhaps, even at all.

Thank you for taking this on.

Otto

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Sounds like a job for numpy.

"Scientists aren't perfect, just peer reviewed."
"Eye of Sauron Observatory", featuring "Sauron's Other Eye", 16" dob, conical Royce mirror.

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Rick and Brentwood also asked you to explain the special relativity thing. It is quite likely they do understand simple math like this and I do not want to hold the three of you up.

It's even more likely that at least one of them has no idea what any of it means. I feel like Homer Simpson when someone talks to him, and all he's hearing is "blah, blah, blah". It may as well be runes, or the Voynich manuscript.

Is this degree of immersion in the math really necessary to create an understandable analogy for the car-headlight thing? If so, well, thanks anyway, but there's nothing here for me.

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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The Lorentz transformations would be much easier to follow.

WO Zenithstar 71ED Customized Barska 80ED Triplet Celestron C102gt Skywatcher 100ED AP130GT C8 C11
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Post deleted by The Mighty Mo

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Nothing here for me either. I can understand some of Otto's posts more!
If the answer to the question, "What does an outside observer see when a vehicle travelling at just below c puts on its headlights" can only be answered with an equation, then I would say that there is no answer, it is still an unknown.

I don't believe in astrology. I am a Gemini and we're very skeptical.

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C'mon be patient! You have to see things before you can understand them! It's not a linear process. Ask questions!

-drl

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why do you introduce imaginary numbers?

"We finally figured how to spell the word 'biocentric', now we just have to find out what it means". 95.3 MF 'The Range'
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You'll see! It's nothing unreal - it's very real. You HAVE to be patient. It will all come down to a flag...

-drl

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Will the simplified version come down to solving a right triangle with sides
ct, vt, and (hypotenuse) ct' and eventually solving for t'? I've seen that approach and it is pretty understandable, but requires pics.

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Danny,a&#1082;&#1086; &#1072;&#1079; &#1074;&#1080; &#1077; &#1087;&#1086;&#1080;&#1089;&#1082;&#1072;&#1083; &#1074;&#1098;&#1087;&#1088;&#1086;&#1089; &#1074; &#1041;&#1098;&#1083;&#1075;&#1072;&#1088;&#1080;&#1103;, &#1097;&#1077; &#1084;&#1086;&#1078;&#1077;&#1090;&#1077; &#1076;&#1072; &#1075;&#1086; &#1088;&#1072;&#1079;&#1073;&#1080;&#1088;&#1072;&#1084;&#1077; &#1073;&#1077;&#1079; &#1076;&#1072; &#1080;&#1079;&#1087;&#1086;&#1083;&#1079;&#1074;&#1072;&#1090;&#1077; Google? &#1053;&#1077; &#1084;&#1086;&#1075;&#1072; &#1076;&#1072; &#1087;&#1086;&#1089;&#1090;&#1072;&#1074;&#1103; &#1089;&#1080; &#1087;&#1086;&#1089;&#1090; &#1074; Google &#1080; &#1076;&#1072; &#1089;&#1077; &#1087;&#1088;&#1077;&#1074;&#1077;&#1078;&#1076;&#1072;&#1090;.?

I don't believe in astrology. I am a Gemini and we're very skeptical.

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Quote:

Will the simplified version come down to solving a right triangle with sides
ct, vt, and (hypotenuse) ct' and eventually solving for t'? I've seen that approach and it is pretty understandable, but requires pics.

Nope, 100% rigorous.

-drl

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I haven't the math background or understanding of relativity to be able to see how this turns out. Can light be slowed down or appear slowed down from an observer's perspective at the side of the road? I have a hard time with the idea of a light beam traveling at a foot a minute or looking like it is proceeding this slowly. I thought that the light sent from the headlights would just travel at the "speed of light" as soon as turned on from the car. That it would appear instanteous from the observer in the car and from the observer at the side of the road it would also, regardless of how fast the car moved. Is there enough time and distance between the car and when the light is switched on to make it look like the light comes on before the car appears? Sorry if this sounds really stupid. I'm totally confused by it, but am hoping there will be a way to describe what actually would appear to happen for the observer at the side of the road that I can understand.

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Heck! Was hoping I could follow.

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Tough crowd.

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You'll see where this is going eventually. Ask questions now.

-drl

SL(2,C) to SO+(1,3) to PSL(2,C) to M ?

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Maya, it doesn't sound stupid to me; or else, I'm equally stupid. I can't help feeling that there must be a simple analogy that would give people like us at least a rough picture of the situation.

Danny, I appreciate your attempting to explain the whole thing, and I will follow it and try to understand. But I really hope this ends up with something on the order of "think of it like this..."
But I have no questions now, beyond "what does it mean?"

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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Danny,a&#1082;&#1086; &#1072;&#1079; &#1074;&#1080; &#1077; &#1087;&#1086;&#1080;&#1089;&#1082;&#1072;&#1083; &#1074;&#1098;&#1087;&#1088;&#1086;&#1089; &#1074; &#1041;&#1098;&#1083;&#1075;&#1072;&#1088;&#1080;&#1103;, &#1097;&#1077; &#1084;&#1086;&#1078;&#1077;&#1090;&#1077; &#1076;&#1072; &#1075;&#1086; &#1088;&#1072;&#1079;&#1073;&#1080;&#1088;&#1072;&#1084;&#1077; &#1073;&#1077;&#1079; &#1076;&#1072; &#1080;&#1079;&#1087;&#1086;&#1083;&#1079;&#1074;&#1072;&#1090;&#1077; Google? &#1053;&#1077; &#1084;&#1086;&#1075;&#1072; &#1076;&#1072; &#1087;&#1086;&#1089;&#1090;&#1072;&#1074;&#1103; &#1089;&#1080; &#1087;&#1086;&#1089;&#1090; &#1074; Google &#1080; &#1076;&#1072; &#1089;&#1077; &#1087;&#1088;&#1077;&#1074;&#1077;&#1078;&#1076;&#1072;&#1090;.?

Brentwood, is the above what you intended to post? Looks like a lot of ASCII to me.

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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ask questions: OK; now mind you, I'm not being sassy or trying to be stupid; I just have no more familiarity with this math than I do with a language foreign to me.

What is a "square matrix"?

What do
x' = x cos a - y sin a
y' = x sin a + y cos a
mean?

And what is the meaning of these things in relation to the paradox of 1+1=1 at near light speeds?

I mean, I can do the simple equations of t=t(0) X the-square-root of 1 - v^2/c^2; the same for Length and the same for mass when mass(0) is divided by the-square-root of 1 - v^2/c^2. I can do it. I don't understand why these do what they do. I don't understand why, from the perspective of an outside observer, the combined velocity of headlight beam and the gal-in-the-spaceship-already-traveling-at-near-c is still just less than c? I would like to understand why this is the case. And as one of our colleagues here said, it would be nice if there was a real world example that explained this phenomenon; not just gave an analogical similarity like that silly higgs-boson-like-a-snow-field thing.

But, as I am writing, I am afraid I am getting to a really clear question.

Danny, even though this is special relativity and not quantum mechanics, is this an area of relativity which, like quantum mechanics, it cannot be explained with analogies from the ordinary world of fried eggs, baseball, and why the landlord can't get the furnace working?

Otto

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I can't help feeling that there must be a simple analogy that would give people like us at least a rough picture of the situation.

I sure hope so, Rick.

A square matrix? Otto, I picture a square of sedimentary rock embedded with fossilized sea shells... It's bad when you haven't the math training....

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...

"We finally figured how to spell the word 'biocentric', now we just have to find out what it means". 95.3 MF 'The Range'
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The page of gibberish I posted above was a google translation of a couple of questions in Bulgarian. It looked all right when I posted it, but it was in Cyrillic script so maybe that's why it came through as just numbers.
The point that I was trying to make is the same as Otto's, why post in a format that is unintelligible to most on here.
That maths makes no sense to me, I can't even get enough of it to ask any questions!

I don't believe in astrology. I am a Gemini and we're very skeptical.

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...

Problem is, it's pretty sad too!

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I'm no expert, but the idea is that -- by postulate --ALL observers will always measure the speed of light as the same, regardless of their state of relative motion. SO, since that number c is FORCED to remain a constant, and since
velocity=distance/time, TIME is forced to change. Two postulates (constancy of light speed, and same rules of physics in all systems moving at constant velocity relative to one another) must be accepted, and the rest follows. If you accept the postulates, you must swallow that time is dilated.
My brief summary based on what I'm sure is an incomplete understanding.

It's all quite counterintuitive and hard (impossible?) to swallow.

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Wow, that isn't intuitive. Has there been any physical experimental evidence or must this remain just a result through math?

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If we are correct that the-gal-traveling-in-a-spaceship-at-near-c-velocity (whom hereafter for the sake of not using these silly hyphens, we will call SpaceGalNear-c), SpaceGalNear-c turns on the headlights. It is a dark stormy misty night in the universe. SpaceGalNear-c sees the light from her headlights go out ahead of her into the misty night at the speed of light; i.e. "c".

We, who are observing her, by the side of the road, see SpaceGalNear-c traveling at near-c, and we see the headlights of her spaceship turn on. But, instead of seeing the light of her headlights zooming away in front of her at the speed of light ("c") what we see is the light from the headlights almost statically holding place just in front of her spaceship; maybe just inching ahead a small bit every second.

Am I seeing this correctly? From the outside of observer the light from the headlight would be almost statically standing still in front of the headlight lamp of SpaceGalNear-c?

Otto

This can work, I mean whereas the light beam seems near static to the observer but normal to SpaceGalNear-c; is if SpaceGalNear-c's time has really slowed down as perceived by the outside observer; as in almost down to t= nearly 0; then it would appear to be screeching away at light speed to SpaceGal, but to us it would appear near static.

But why does the time slow down?

Otto

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ALL observers will always measure the speed of light as the same, regardless of their state of relative motion

That in itself is counterintuitive, so why would we start with a counterintuitive postulate?

"We finally figured how to spell the word 'biocentric', now we just have to find out what it means". 95.3 MF 'The Range'
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so a square matrix is a container with 1s and 0s, whose place in the 9 positions within this container is determined by them following that formula related to i's and j's; "delta_ij = {1 if i = j, 0 if i <> j"

What does the "delta_ij" mean?

What does the "and i,j = 1 to 3" mean? I mean, what is the relevance to this statement (and i,j = 1 to 3) to the container with 1s and 0s?

Otto

"If you are familiar with any programming languages,
a square matrix is a two-dimensional array with the
same numbers of rows & columns.

For example in VB,

Dim array(10, 10) as Integer

Clear as mud? "

Unfortunately, I had one class of fortran thirty some years ago. That's it. I really appreciate you trying to help me. But its over my head. Might as well be that Rosetta Stone and I have a baby's understanding of the one language.

I fear it would be a waste of your time to try to get me up to speed.

Otto

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One can argue endlessly about the results and what they mean, but yes, there are experiments that verify some of SR's predictions. Some that seem to contradict the predictions in Einstein's first paper on SR (Sagnac experiment, Dayton Miller experiments) are explained away due to problems/misunderstanding of the nature of the reference frame under consideration, or experimental error. It will drive you crazy, trust me, to try to understand it all. Most physicists accept SR as true. Argue against it at your own risk.

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From the outside of observer the light from the headlight would be almost statically standing still in front of the headlight lamp of SpaceGalNear-c?

That's what comes to mind, Otto, when I try to visualize it. However, if this can happen, what is going on with light we observe from a star moving through space if we could observe it from "the side of the road" for example? Would that light also look like it was slowed down and not reaching out to infinity? Or what if we used a laser beam?

Sorry I don't know what to do with the square matrix, other than maybe using it for a good beading pattern....

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OK, EJN I got that. I understand.

Danny, EJN, now that I understand (for the moment) what a square matrix is, what does the sin and cosine stuff mean, and why does it relate to the c+c=c headlight paradox?

Otto

I did use sines and cosines in trig once upon a time; but I taught myself trig; its been forty years; I know they have something to do with angles in a triangle; something about representing the angles numerically, maybe?

Otto

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OK, a star moving at near light speed, spits out light.

I'm thinking.

Let's imagine a quasar traveling, because of universe expansion, relative to us at say (being silly), near-c. If its light has reached us, it has also had the same amount of time to go-out sideways from us; so, though it took forever from our perspective to get to us, it has also had forever to go sidewise to our point of view.

Maybe that explains what the quasar's perpendicular-to-us light is so far away from the quasar, but the gal's headlight light isn't; it's a matter of the total time of the scenario.

?

I'm guessing.

Otto

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Thanks for that info on the experiments. Yes, I don't think I'm going to be able to wrap my head around this very well, but it is interesting to try to get a bit of understanding of it.

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The "C+C paradox" that you refer to is simply a consequence of the first postulate: "The speed of light will always be measured at C, regardless of the state of motion of the light source, or the state of motion of the observer."
You have to accept the postulates and work from there to the consequences.
The matrices are tools used in linear algebra, (they are n x m arrays of numbers which can be manipulated and to which operations can be done). Linear algebra is a course offered a year or so after calculus, or at least that was the case when I was in school.

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Post deleted by The Mighty Mo

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OK, a star moving at near light speed, spits out light.

I'm thinking.

Let's imagine a quasar traveling, because of universe expansion, relative to us at say (being silly), near-c. If its light has reached us, it has also had the same amount of time to go-out sideways from us; so, though it took forever from our perspective to get to us, it has also had forever to go sidewise to our point of view.

Maybe that explains what the quasar's perpendicular-to-us light is so far away from the quasar, but the gal's headlight light isn't; it's a matter of the total time of the scenario.

?

I'm guessing.

Otto

"It's a matter of the total time of the scenario." That sounds reasonable...but it's hard to ignore a picture of a light beam just hanging there in slo-motion, it would be fun to see it.

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If its light has reached us, it has also had the same amount of time to go-out sideways from us; so, though it took forever from our perspective to get to us, it has also had forever to go sidewise to our point of view.

I'm having a lot of trouble following much of this discussion, mostly because of a sort of omniscient observer viewpoint that is apparent here. How can I see light that is going out sideways from a distant object? Well, I can't. The light has to reach my eyeball before I can see it.

Greg - Celestron SkyScout 90mm refractor & planetarium

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