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deSitter
Still in Old School

Reged: 12/09/04

Scary relativity
#6214651 - 11/25/13 09:14 AM

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

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Otto Piechowski
Pooh-Bah

Reged: 09/20/05

Loc: Lexington, KY
Re: Scary relativity [Re: deSitter]
#6214787 - 11/25/13 10:24 AM

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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llanitedave
Humble Megalomaniac

Reged: 09/26/05

Loc: Amargosa Valley, NV, USA
Re: Scary relativity [Re: Otto Piechowski]
#6214942 - 11/25/13 11:49 AM

Sounds like a job for numpy.

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Rick Woods
Postmaster

Reged: 01/27/05

Loc: Inner Solar System
Re: Scary relativity [Re: Otto Piechowski]
#6214978 - 11/25/13 12:08 PM

Quote:

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.

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PeterR280
Pooh-Bah

Reged: 05/27/13

Re: Scary relativity [Re: Rick Woods]
#6214985 - 11/25/13 12:10 PM

The Lorentz transformations would be much easier to follow.

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The Mighty Mo
professor emeritus

Reged: 10/12/13

Loc: South of North, North of South...
Re: Scary relativity *DELETED* [Re: Rick Woods]
#6215014 - 11/25/13 12:23 PM

Post deleted by The Mighty Mo

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brentwood
Carpal Tunnel

Reged: 11/04/05

Re: Scary relativity [Re: The Mighty Mo]
#6215039 - 11/25/13 12:35 PM

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.

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deSitter
Still in Old School

Reged: 12/09/04

Re: Scary relativity [Re: brentwood]
#6215107 - 11/25/13 01:06 PM

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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dickbill
scholastic sledgehammer

Reged: 09/30/08

Re: Scary relativity [Re: deSitter]
#6215126 - 11/25/13 01:17 PM

why do you introduce imaginary numbers?

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deSitter
Still in Old School

Reged: 12/09/04

Re: Scary relativity [Re: dickbill]
#6215138 - 11/25/13 01:26 PM

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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choran
Carpal Tunnel

Reged: 12/28/12

Re: Scary relativity [Re: deSitter]
#6215200 - 11/25/13 01:56 PM

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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brentwood
Carpal Tunnel

Reged: 11/04/05

Re: Scary relativity [Re: choran]
#6215224 - 11/25/13 02:08 PM

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;.?

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choran
Carpal Tunnel

Reged: 12/28/12

Re: Scary relativity [Re: brentwood]
#6215269 - 11/25/13 02:27 PM

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deSitter
Still in Old School

Reged: 12/09/04

Re: Scary relativity [Re: choran]
#6215320 - 11/25/13 02:43 PM

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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Crow Haven
Pooh-Bah

Reged: 01/09/09

Loc: Oregon USA
Re: Scary relativity [Re: deSitter]
#6215370 - 11/25/13 03:04 PM

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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choran
Carpal Tunnel

Reged: 12/28/12

Re: Scary relativity [Re: deSitter]
#6215379 - 11/25/13 03:07 PM

Heck! Was hoping I could follow.

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Neutrino?
sage

Reged: 12/14/09

Loc: Wasatch Front
Re: Scary relativity [Re: deSitter]
#6215383 - 11/25/13 03:09 PM

Tough crowd.

Quote:

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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Rick Woods
Postmaster

Reged: 01/27/05

Loc: Inner Solar System
Re: Scary relativity [Re: Crow Haven]
#6215398 - 11/25/13 03:14 PM

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?"

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Rick Woods
Postmaster

Reged: 01/27/05

Loc: Inner Solar System
Re: Scary relativity [Re: brentwood]
#6215404 - 11/25/13 03:16 PM

Quote:

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.

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Otto Piechowski
Pooh-Bah

Reged: 09/20/05

Loc: Lexington, KY
Re: Scary relativity [Re: Rick Woods]
#6215414 - 11/25/13 03:20 PM

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

Edited by Otto Piechowski (11/25/13 03:30 PM)

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