Doug Sweetser
Apr8-04, 02:27 PM
<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>Hello:\n\nI was thinking about this sort of thing recently:\n\n> the geodesic is timelike <=> mass^2 > 0 (tardyons)\n> the geodesic is lightlike <=> mass^2 = 0 (luxons)\n> the geodesic is spacelike <=> mass^2 < 0 (tachyons)\n\nSee, there are perfectly fine paths in spacetime that are spacelike\nseparated from an observer:\n\n\\t| / x\n\\|/ |\nR--------|\n/|\\ x\n/ | \\\n\nThe arbitrary choice of the origin makes all the events on that\nworldline spacelike separated from the origin. The relativistic\nvelocity of the x--x worldline is zero, and could be created by a real\nparticle.\n\nWhat happens if this spacetime graph is transformed to the classical\nrealm? The 45 degree lines end up going flat. In the limit of this\nprocess, the nice defined slope of the x--x worldline becomes\nundefined. Uncool.\n\nI had an alternate idea, and want to see if someone else has thought of\nthis before. The Minkowski metric is an indefinite metric. It is that\ndarn negative distance squared that doesn\'t make sense, particularly\nfor a pure mathematician. So let\'s try and aid the mathematicians in\nthe audience. We apply a simple rule: if |t| > |R|, the point gets\nplotted in spacetime as always. This should fill up the past and\nfuture timelike light cones. If |t| < |R|, then we plot the points in\nthe complex-valued tangent space:\n\nit| / x\n\\|/ |\niR--------|\n/|\\ x\n/ | \\\n\nNow the metric will be a positive definite number because\n\n(it)^2 - (iR)^2 = -|t|^2 + |R|^2 > 0\n\nNote, the observer cannot travel a distance iR to get to these points.\nYet gamma and beta are well defined real numbers because they are\nratios of two imaginary numbers.\n\nThe Minkowski metric is a metric, not a pseudo metric, so long as this\nrule of accounting in enforced for timelike events graphed in\nspacetime, and spacelike events graphed in the complex-valued tangent\nspace.\n\n\ndoug\nquaternions.com\n\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>Hello:
I was thinking about this sort of thing recently:
> the geodesic is timelike <=> mass^2 > (tardyons)
> the geodesic is lightlike <=> mass^2 = (luxons)
> the geodesic is spacelike <=> mass^2 < (tachyons)
See, there are perfectly fine paths in spacetime that are spacelike
separated from an observer:
\t| / x[/itex]
\|/ |
R--------|
/|\ x
[itex]/ | \
The arbitrary choice of the origin makes all the events on that
worldline spacelike separated from the origin. The relativistic
velocity of the x--x worldline is zero, and could be created by a real
particle.
What happens if this spacetime graph is transformed to the classical
realm? The 45 degree lines end up going flat. In the limit of this
process, the nice defined slope of the x--x worldline becomes
undefined. Uncool.
I had an alternate idea, and want to see if someone else has thought of
this before. The Minkowski metric is an indefinite metric. It is that
darn negative distance squared that doesn't make sense, particularly
for a pure mathematician. So let's try and aid the mathematicians in
the audience. We apply a simple rule: if |t| > |R|, the point gets
plotted in spacetime as always. This should fill up the past and
future timelike light cones. If |t| < |R|, then we plot the points in
the complex-valued tangent space:
it| / x
\|/ |
iR--------|
/|\ x
/ | \
Now the metric will be a positive definite number because
(it)^2 - (iR)^2 = -|t|^2 + |R|^2 >
Note, the observer cannot travel a distance iR to get to these points.
Yet \gamma and \beta are well defined real numbers because they are
ratios of two imaginary numbers.
The Minkowski metric is a metric, not a pseudo metric, so long as this
rule of accounting in enforced for timelike events graphed in
spacetime, and spacelike events graphed in the complex-valued tangent
space.
doug
quaternions.com
I was thinking about this sort of thing recently:
> the geodesic is timelike <=> mass^2 > (tardyons)
> the geodesic is lightlike <=> mass^2 = (luxons)
> the geodesic is spacelike <=> mass^2 < (tachyons)
See, there are perfectly fine paths in spacetime that are spacelike
separated from an observer:
\t| / x[/itex]
\|/ |
R--------|
/|\ x
[itex]/ | \
The arbitrary choice of the origin makes all the events on that
worldline spacelike separated from the origin. The relativistic
velocity of the x--x worldline is zero, and could be created by a real
particle.
What happens if this spacetime graph is transformed to the classical
realm? The 45 degree lines end up going flat. In the limit of this
process, the nice defined slope of the x--x worldline becomes
undefined. Uncool.
I had an alternate idea, and want to see if someone else has thought of
this before. The Minkowski metric is an indefinite metric. It is that
darn negative distance squared that doesn't make sense, particularly
for a pure mathematician. So let's try and aid the mathematicians in
the audience. We apply a simple rule: if |t| > |R|, the point gets
plotted in spacetime as always. This should fill up the past and
future timelike light cones. If |t| < |R|, then we plot the points in
the complex-valued tangent space:
it| / x
\|/ |
iR--------|
/|\ x
/ | \
Now the metric will be a positive definite number because
(it)^2 - (iR)^2 = -|t|^2 + |R|^2 >
Note, the observer cannot travel a distance iR to get to these points.
Yet \gamma and \beta are well defined real numbers because they are
ratios of two imaginary numbers.
The Minkowski metric is a metric, not a pseudo metric, so long as this
rule of accounting in enforced for timelike events graphed in
spacetime, and spacelike events graphed in the complex-valued tangent
space.
doug
quaternions.com