Danny Ross Lunsford
Sep16-04, 07:09 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nThis occured to me at lunch, and is sort of fun...\n\nWe know that the Lorentz transformation can be derived from a\ngroup-theoretic analysis of space and time, based on simple\nassumptions of isotropy, linearity etc. There exist both synthetic and\nanalytic versions of this derivation that have been posted here. The\nend result is, there is a parameter with the dimensions of a velocity\nthat is either finite, or infinite, that characterizes the geometry.\nOf course, it is C.\n\nOne might ask, is Euclidean geometry so characterized? The answer is\nyes!\n\nMetric geometry sits inside projective geometry by positing a\nfundamental quadric. In relativity it is of course the light cone - in\n1+1 dimensions\n\nx^2 - (ct)^2 = 0\n\nwhich can be factored\n\n(x - ct)(x + ct) = 0\n\nso x/t = +-c - this shows how the fundamental quadric is related to\nthe characteristic parameter.\n\nWhat about Euclidean plane geometry? The fundamental quadric is\n\nx^2 + y^2 = 0\n\nwhich seems like an empty statement, but makes sense in the context of\nprojective geometry as the "circular points at infinity". We can now\nfactor this as\n\n(x - iy)(x + iy) = 0\n\nThe characteristic parameter of Euclidean geometry is the imaginary\nunit! So "i" plays the role of the "speed of imaginary light" in\nEuclidean geometry :)\n\nThis has a beautiful interpretation. Intuitively, one knows that, on\nthe Euclidean plane, one can imagine a thing called "infinity" which\ncan never be got closer to, from which all regular points are\n"infinitely" distant. No matter how far you go, you\'re always\n"infinitely" far away from "infinity". This is the *exactly analogous*\nresult to the impossibility of attaining the speed C in relativity.\n\nThis may be the most basic way complex numbers enter into physics.\n\n-drl\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>This occured to me at lunch, and is sort of fun...
We know that the Lorentz transformation can be derived from a
group-theoretic analysis of space and time, based on simple
assumptions of isotropy, linearity etc. There exist both synthetic and
analytic versions of this derivation that have been posted here. The
end result is, there is a parameter with the dimensions of a velocity
that is either finite, or infinite, that characterizes the geometry.
Of course, it is C.
One might ask, is Euclidean geometry so characterized? The answer is
yes!
Metric geometry sits inside projective geometry by positing a
fundamental quadric. In relativity it is of course the light cone - in
1+1 dimensions
x^2 - (ct)^2 =
which can be factored
(x - ct)(x + ct) =
so x/t = +-c - this shows how the fundamental quadric is related to
the characteristic parameter.
What about Euclidean plane geometry? The fundamental quadric is
x^2 + y^2 =
which seems like an empty statement, but makes sense in the context of
projective geometry as the "circular points at infinity". We can now
factor this as
(x - iy)(x + iy) =
The characteristic parameter of Euclidean geometry is the imaginary
unit! So "i" plays the role of the "speed of imaginary light" in
Euclidean geometry :)
This has a beautiful interpretation. Intuitively, one knows that, on
the Euclidean plane, one can imagine a thing called "infinity" which
can never be got closer to, from which all regular points are
"infinitely" distant. No matter how far you go, you're always
"infinitely" far away from "infinity". This is the *exactly analogous*
result to the impossibility of attaining the speed C in relativity.
This may be the most basic way complex numbers enter into physics.
-drl
We know that the Lorentz transformation can be derived from a
group-theoretic analysis of space and time, based on simple
assumptions of isotropy, linearity etc. There exist both synthetic and
analytic versions of this derivation that have been posted here. The
end result is, there is a parameter with the dimensions of a velocity
that is either finite, or infinite, that characterizes the geometry.
Of course, it is C.
One might ask, is Euclidean geometry so characterized? The answer is
yes!
Metric geometry sits inside projective geometry by positing a
fundamental quadric. In relativity it is of course the light cone - in
1+1 dimensions
x^2 - (ct)^2 =
which can be factored
(x - ct)(x + ct) =
so x/t = +-c - this shows how the fundamental quadric is related to
the characteristic parameter.
What about Euclidean plane geometry? The fundamental quadric is
x^2 + y^2 =
which seems like an empty statement, but makes sense in the context of
projective geometry as the "circular points at infinity". We can now
factor this as
(x - iy)(x + iy) =
The characteristic parameter of Euclidean geometry is the imaginary
unit! So "i" plays the role of the "speed of imaginary light" in
Euclidean geometry :)
This has a beautiful interpretation. Intuitively, one knows that, on
the Euclidean plane, one can imagine a thing called "infinity" which
can never be got closer to, from which all regular points are
"infinitely" distant. No matter how far you go, you're always
"infinitely" far away from "infinity". This is the *exactly analogous*
result to the impossibility of attaining the speed C in relativity.
This may be the most basic way complex numbers enter into physics.
-drl