Science > Physics > Independent/Dependent Phases 10.1: Special Relativity 2
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Science > Physics |
| User: |
"OsherD" |
| Date: |
08 Oct 2005 11:18:09 AM |
| Object: |
Independent/Dependent Phases 10.1: Special Relativity 2 |
From Osher Doctorow
We now have:
1) sqrt(1 - v^2/c^2) > 1/2 iff v/c < (sqrt(3))/2 = sin(pi/3)
Thus v < sin(pi/3)c versus v > sin(pi/3)c discriminates sqrt(1 -
v^2/c^2) > 1/2 vs < 1/2.
The simplicity and recognizability and more frequent occurrence of
pi/2, pi/3, pi/4, and pi/6 than other "closed form" proper fractions of
pi in both mathematics and physics suggests that the corresponding
analog of fX(x) > 1/2 in (1) may have a deeper explanation. We can of
course take a sequence of v values that approach c or even a monotone
function g(t) = v^2/c^2 that increases in time up to v = c, so that
g(t) = sin(t) from t = 0 to t = pi/2 and v^2 = sin(t)c^2 there.
If v > c were an additional phase, or v^2/c^2 > 1, then the sine
function would obviously be replaced by some other function on that
interval, and we'd have an interval with two phase changes, but it
wouldn't be either surprising or counter-intuitive except for the
change from real to imaginary axis.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 10.1: Special Relativity 2 |
08 Oct 2005 12:30:17 PM |
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From Osher Doctorow
I meant sin(pi/6) = 1/2, cos(pi/6) = sqrt(3)/2, and of course cos(pi/3)
= 1/2, sin(pi/3) = sqrt(3)/2. Also, as we all know, sin(pi/4) =
cos(pi/4) = sqrt(2)/2 = 1/sqrt(2).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 10.1: Special Relativity 2 |
08 Oct 2005 12:34:43 PM |
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From Osher Doctorow
This is even stranger: sqrt(1 - v^2/c^2) > 1/2 = sin(pi/6) iff v/c <
sqrt(3)/2 = cos(pi/6).
Osher Doctorow
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