Alex Green
Jul23-04, 06:35 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>\nUncle Al <UncleAl0@hate.spam.net> wrote in message news:<40FFDFA8.80B797A6@hate.spam.net>...\n> David Park wrote:\n> >\n> > "Alex Green" <dralexgreen@yahoo.co.uk> wrote in message\n> > news:42c8441.0407220559.6ad09eb2@posting.google.com...\n> > >\n> > > Can anyone tell me about the true physical significance of the\n> > > negative coefficient in the metric tensor of \'flat\' space-time?\n> > >\n> >\n> > There is a very good recent pedagogical paper on this topic, "Spacetime and\n> > Euclidean Geometry" by Dieter Brill and Ted Jacobson, arXiv:gr-qc/0407044.\n> > Using the invariance of the speed of light and the equivalence of inertial\n> > frames, they derive the analog of the Pythagorean theorem in Minkowskian\n> > space, T^2 = t^2 - x^2. This is an elegant and geometrical presentation.\n>\n> http://arXiv.org/abs/gr-qc/0407022\n>\n\nThanks for this link, I liked their derivation of the square on the\nhypoteneuse where the hypoteneuse is a space-time interval.\n\nThe problem with deriving the \'flat metric\' from Einstein\'s original\nassumptions is that the assumption that the speed of light is constant\ndoes not seem to be as fundamental as assuming that the universe is a\n(3+1)D or 4D manifold.\n\nIf the universe is assumed to be four dimensional then the invariance\nof the space-time interval is just a restatement of Pythagoras\'\ntheorem and the constancy of the speed of light is a simple result of\nthe existence of the zero space-time interval for finite values of\ndistance and time (*). In other words the constancy of the speed of\nlight is a consequence of a four dimensional manifold. The constancy\nof the speed of light is evidence for the manifold rather than an\nassumption in it\'s own right (ie: velocity is a combination of\ndistance and time).\n\nHowever, the \'Pythagoras Theorem\' used in SR is not quite the same as\nthe original. It includes what Weyl called a \'negative dimension\'\nexpressed by -(ct)^2. It is interesting to examine the use of:\n\ns^2 = x^2 + y^2 + z^2 - (ct)^2\n\nas a form of Pythagoras\' Theorem (where x etc are intervals). The\nmodern derivation of this equation uses the metric tensor \'g\' so that\ns^2 = g[x][x]. I have no problem where the coefficients in the tensor\ncater for departures from Euclidean geometry and the expansion of\ng[x][x] is x*x + y*y + z*z + (sqrt-1)ct*(sqrt-1)ct in flat spacetime.\nBut I do have a problem with a metric tensor that gives rise to x*x +\ny*y + z*z + (-1)ct*(+1)ct because the physical significance of -ct and\n+ct seems to be dynamical, ie: a motion back and forth in time. It\nmixes space as a phenomenon with time as a process.\n\nWhen a thing moves from the origin to a position \'x\' there does not\nseem to be a physical process that moves back from x to the origin\nback through time. If the metric tensor (-1,1,1,1) is a statement that\nsuch an event actually takes place then this should be clearly stated\nas an added assumption. However, processes cannot really be\nassumptions, all processes must contain causes and effects, so what is\nthe cause of -ct and +ct as the foundation of the metric?\n\nBest Wishes\n\nAlex Green\n\n(*) Prediction of constancy of speed of light from (3+1)D or 4D\nmanifold:\n\nIf x and t are intervals, c is a constant that converts seconds to\nmetres and suppressing y and z then the spacetime interval is:\ns^2 = x^2 - (ct)^2\nand, given that the manifold has only 4 dimensions, this is invariant\nfor all observers.\nif s=0\n0 = x^2 - (ct)^2 which is also invariant for all observers.\nif x is traversed at velocity \'v\' then vt = ct and this ONLY occurs\nwhen v=c. So all observers will observe the same velocity for a thing\nmoving at \'c\' m/sec and \'c\' is a universal constant.\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>Uncle Al <UncleAl0@hate.spam.net> wrote in message news:<40FFDFA8.80B797A6@hate.spam.net>...
> David Park wrote:
> >
> > "Alex Green" <dralexgreen@yahoo.co.uk> wrote in message
> > news:42c8441.0407220559.6ad09eb2@posting.google.com...
> > >
> > > Can anyone tell me about the true physical significance of the
> > > negative coefficient in the metric tensor of 'flat' space-time?
> > >
> >
> > There is a very good recent pedagogical paper on this topic, "Spacetime and
> > Euclidean Geometry" by Dieter Brill and Ted Jacobson, arXiv:http://www.arxiv.org/abs/gr-qc/0407044.
> > Using the invariance of the speed of light and the equivalence of inertial
> > frames, they derive the analog of the Pythagorean theorem in Minkowskian
> > space, T^2 = t^2 - x^2. This is an elegant and geometrical presentation.
>
> http://arXiv.org/abs/http://www.arxiv.org/abs/gr-qc/0407022
>
Thanks for this link, I liked their derivation of the square on the
hypoteneuse where the hypoteneuse is a space-time interval.
The problem with deriving the 'flat metric' from Einstein's original
assumptions is that the assumption that the speed of light is constant
does not seem to be as fundamental as assuming that the universe is a
(3+1)D or 4D manifold.
If the universe is assumed to be four dimensional then the invariance
of the space-time interval is just a restatement of Pythagoras'
theorem and the constancy of the speed of light is a simple result of
the existence of the zero space-time interval for finite values of
distance and time (*). In other words the constancy of the speed of
light is a consequence of a four dimensional manifold. The constancy
of the speed of light is evidence for the manifold rather than an
assumption in it's own right (ie: velocity is a combination of
distance and time).
However, the 'Pythagoras Theorem' used in SR is not quite the same as
the original. It includes what Weyl called a 'negative dimension'
expressed by -(ct)^2. It is interesting to examine the use of:
s^2 = x^2 + y^2 + z^2 - (ct)^2
as a form of Pythagoras' Theorem (where x etc are intervals). The
modern derivation of this equation uses the metric tensor 'g' so that
s^2 = g[x][x]. I have no problem where the coefficients in the tensor
cater for departures from Euclidean geometry and the expansion of
g[x][x] is x*x + y*y + z*z + (\sqrt-1)ct*(\sqrt-1)ct in flat spacetime.
But I do have a problem with a metric tensor that gives rise to x*x +y*y + z*z + (-1)ct*(+1)ct because the physical significance of -ct and
+ct seems to be dynamical, ie: a motion back and forth in time. It
mixes space as a phenomenon with time as a process.
When a thing moves from the origin to a position 'x' there does not
seem to be a physical process that moves back from x to the origin
back through time. If the metric tensor (-1,1,1,1) is a statement that
such an event actually takes place then this should be clearly stated
as an added assumption. However, processes cannot really be
assumptions, all processes must contain causes and effects, so what is
the cause of -ct and +ct as the foundation of the metric?
Best Wishes
Alex Green
(*) Prediction of constancy of speed of light from (3+1)D or 4D
manifold:
If x and t are intervals, c is a constant that converts seconds to
metres and suppressing y and z then the spacetime interval is:
s^2 = x^2 - (ct)^2
and, given that the manifold has only 4 dimensions, this is invariant
for all observers.
if s=0
= x^2 - (ct)^2 which is also invariant for all observers.
if x is traversed at velocity 'v' then vt = ct and this ONLY occurs
when v=c. So all observers will observe the same velocity for a thing
moving at 'c' m/sec and 'c' is a universal constant.
> David Park wrote:
> >
> > "Alex Green" <dralexgreen@yahoo.co.uk> wrote in message
> > news:42c8441.0407220559.6ad09eb2@posting.google.com...
> > >
> > > Can anyone tell me about the true physical significance of the
> > > negative coefficient in the metric tensor of 'flat' space-time?
> > >
> >
> > There is a very good recent pedagogical paper on this topic, "Spacetime and
> > Euclidean Geometry" by Dieter Brill and Ted Jacobson, arXiv:http://www.arxiv.org/abs/gr-qc/0407044.
> > Using the invariance of the speed of light and the equivalence of inertial
> > frames, they derive the analog of the Pythagorean theorem in Minkowskian
> > space, T^2 = t^2 - x^2. This is an elegant and geometrical presentation.
>
> http://arXiv.org/abs/http://www.arxiv.org/abs/gr-qc/0407022
>
Thanks for this link, I liked their derivation of the square on the
hypoteneuse where the hypoteneuse is a space-time interval.
The problem with deriving the 'flat metric' from Einstein's original
assumptions is that the assumption that the speed of light is constant
does not seem to be as fundamental as assuming that the universe is a
(3+1)D or 4D manifold.
If the universe is assumed to be four dimensional then the invariance
of the space-time interval is just a restatement of Pythagoras'
theorem and the constancy of the speed of light is a simple result of
the existence of the zero space-time interval for finite values of
distance and time (*). In other words the constancy of the speed of
light is a consequence of a four dimensional manifold. The constancy
of the speed of light is evidence for the manifold rather than an
assumption in it's own right (ie: velocity is a combination of
distance and time).
However, the 'Pythagoras Theorem' used in SR is not quite the same as
the original. It includes what Weyl called a 'negative dimension'
expressed by -(ct)^2. It is interesting to examine the use of:
s^2 = x^2 + y^2 + z^2 - (ct)^2
as a form of Pythagoras' Theorem (where x etc are intervals). The
modern derivation of this equation uses the metric tensor 'g' so that
s^2 = g[x][x]. I have no problem where the coefficients in the tensor
cater for departures from Euclidean geometry and the expansion of
g[x][x] is x*x + y*y + z*z + (\sqrt-1)ct*(\sqrt-1)ct in flat spacetime.
But I do have a problem with a metric tensor that gives rise to x*x +y*y + z*z + (-1)ct*(+1)ct because the physical significance of -ct and
+ct seems to be dynamical, ie: a motion back and forth in time. It
mixes space as a phenomenon with time as a process.
When a thing moves from the origin to a position 'x' there does not
seem to be a physical process that moves back from x to the origin
back through time. If the metric tensor (-1,1,1,1) is a statement that
such an event actually takes place then this should be clearly stated
as an added assumption. However, processes cannot really be
assumptions, all processes must contain causes and effects, so what is
the cause of -ct and +ct as the foundation of the metric?
Best Wishes
Alex Green
(*) Prediction of constancy of speed of light from (3+1)D or 4D
manifold:
If x and t are intervals, c is a constant that converts seconds to
metres and suppressing y and z then the spacetime interval is:
s^2 = x^2 - (ct)^2
and, given that the manifold has only 4 dimensions, this is invariant
for all observers.
if s=0
= x^2 - (ct)^2 which is also invariant for all observers.
if x is traversed at velocity 'v' then vt = ct and this ONLY occurs
when v=c. So all observers will observe the same velocity for a thing
moving at 'c' m/sec and 'c' is a universal constant.