<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>\n\nThe \'flat metric\' (*) of modern Special Relativity is conventionally\ndescribed by a metric tensor that has a principle diagonal given by\n(-1,1,1,1) or equivalently (1,-1,-1,-1).\n\nI have always been troubled by the change of sign in the principle\ndiagonal of the metric tensor, not because it is mathematically\nsuspect but because it seems to have a physical significance that is\ndynamical rather than geometrical. The negative coefficient in the\nmetric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2\n+ dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than\nsquaring (sqrt -1)*dt. This use of -dt*+dt seems to imply a movement\nthrough time and back again rather than a simple geometric phenomenon\nsuch as occurs in Pythagoras\' Theorem (which expresses the spherical\nsymmetry of space).\n\nCan anyone tell me about the true physical significance of the\nnegative coefficient in the metric tensor of \'flat\' space-time?\n\n\n* See for instance "An Introduction to General Relativity" by Hughston\n& Todd.p7.\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>The 'flat metric' (*) of modern Special Relativity is conventionally
described by a metric tensor that has a principle diagonal given by
(-1,1,1,1) or equivalently (1,-1,-1,-1).
I have always been troubled by the change of sign in the principle
diagonal of the metric tensor, not because it is mathematically
suspect but because it seems to have a physical significance that is
dynamical rather than geometrical. The negative coefficient in the
metric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2+ dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than
squaring (\sqrt -1)*dt. This use of -dt*+dt seems to imply a movement
through time and back again rather than a simple geometric phenomenon
such as occurs in Pythagoras' Theorem (which expresses the spherical
symmetry of space).
Can anyone tell me about the true physical significance of the
negative coefficient in the metric tensor of 'flat' space-time?
* See for instance "An Introduction to General Relativity" by Hughston
& Todd.p7.
David Park
Jul22-04, 11:23 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>\n"Alex Green" <dralexgreen@yahoo.co.uk> wrote in message\nnews:42c8441.0407220559.6ad09eb2@posting.google.com...\n>\n>\n> The \'flat metric\' (*) of modern Special Relativity is conventionally\n> described by a metric tensor that has a principle diagonal given by\n> (-1,1,1,1) or equivalently (1,-1,-1,-1).\n>\n> I have always been troubled by the change of sign in the principle\n> diagonal of the metric tensor, not because it is mathematically\n> suspect but because it seems to have a physical significance that is\n> dynamical rather than geometrical. The negative coefficient in the\n> metric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2\n> + dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than\n> squaring (sqrt -1)*dt. This use of -dt*+dt seems to imply a movement\n> through time and back again rather than a simple geometric phenomenon\n> such as occurs in Pythagoras\' Theorem (which expresses the spherical\n> symmetry of space).\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> * See for instance "An Introduction to General Relativity" by Hughston\n> & Todd.p7.\n\nThere is a very good recent pedagogical paper on this topic, "Spacetime and\nEuclidean Geometry" by Dieter Brill and Ted Jacobson, arXiv:gr-qc/0407044.\nUsing the invariance of the speed of light and the equivalence of inertial\nframes, they derive the analog of the Pythagorean theorem in Minkowskian\nspace, T^2 = t^2 - x^2. This is an elegant and geometrical presentation.\n\nDavid Park\ndjmp@earthlink.net\nhttp://home.earthlink.net/~djmp/\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>"Alex Green" <dralexgreen@yahoo.co.uk> wrote in message
news:42c8441.0407220559.6ad09eb2@posting.google.com...
>
>
> The 'flat metric' (*) of modern Special Relativity is conventionally
> described by a metric tensor that has a principle diagonal given by
> (-1,1,1,1) or equivalently (1,-1,-1,-1).
>
> I have always been troubled by the change of sign in the principle
> diagonal of the metric tensor, not because it is mathematically
> suspect but because it seems to have a physical significance that is
> dynamical rather than geometrical. The negative coefficient in the
> metric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2
> + dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than
> squaring (\sqrt -1)*dt. This use of -dt*+dt seems to imply a movement
> through time and back again rather than a simple geometric phenomenon
> such as occurs in Pythagoras' Theorem (which expresses the spherical
> symmetry of space).
>
> Can anyone tell me about the true physical significance of the
> negative coefficient in the metric tensor of 'flat' space-time?
>
>
> * See for instance "An Introduction to General Relativity" by Hughston
> & Todd.p7.
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.
David Park
djmp@earthlink.net
http://home.earthlink.net/~djmp/
Franz Heymann
Jul25-04, 09:16 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>\n\n\n\n"Alex Green" <dralexgreen@yahoo.co.uk> wrote in message\nnews:42c8441.0407220559.6ad09eb2@posting.google.com...\n>\n>\n> The \'flat metric\' (*) of modern Special Relativity is conventionally\n> described by a metric tensor that has a principle diagonal given by\n> (-1,1,1,1) or equivalently (1,-1,-1,-1).\n>\n> I have always been troubled by the change of sign in the principle\n> diagonal of the metric tensor, not because it is mathematically\n> suspect but because it seems to have a physical significance that is\n> dynamical rather than geometrical.\n\nTime really *is* different from space. That difference is formalised\nin the negative sign associated with the time component of the metric.\nIf it had not been there, the situation would have been so symmetrical\nthat it would have been impossible to say which of the four components\nreferred to time.\n\n[snip]\n\nFranz\n\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>"Alex Green" <dralexgreen@yahoo.co.uk> wrote in message
news:42c8441.0407220559.6ad09eb2@posting.google.com...
>
>
> The 'flat metric' (*) of modern Special Relativity is conventionally
> described by a metric tensor that has a principle diagonal given by
> (-1,1,1,1) or equivalently (1,-1,-1,-1).
>
> I have always been troubled by the change of sign in the principle
> diagonal of the metric tensor, not because it is mathematically
> suspect but because it seems to have a physical significance that is
> dynamical rather than geometrical.
Time really *is* different from space. That difference is formalised
in the negative sign associated with the time component of the metric.
If it had not been there, the situation would have been so symmetrical
that it would have been impossible to say which of the four components
referred to time.
[snip]
Franz
Peter Battaglino
Aug12-04, 09:31 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>\n\n\n\nMake that 0407022 :)\n\n>It\'s actually arXiv:gr-qc/0407044, I believe.\n>\n>Peter\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>> David Park\n>> djmp@earthlink.net\n>> http://home.earthlink.net/~djmp/\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>Make that 0407022 :)
>It's actually arXiv:http://www.arxiv.org/abs/gr-qc/0407044, I believe.
>
>Peter
>
>> 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.
>>
>> David Park
>> djmp@earthlink.net
>> http://home.earthlink.net/~djmp/
David Park
Aug13-04, 06:41 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>\n"Peter Battaglino" <peterbat@gmail.com> wrote in message\nnews:7a94953a04073110411874db97@mail.gmail.com...\n>\n>\n>\n>\n> Make that 0407022 :)\n>\n> >It\'s actually arXiv:gr-qc/0407044, I believe.\n> >\n> >Peter\n> >\n> >> There is a very good recent pedagogical paper on this topic, "Spacetime\nand\n> >> Euclidean Geometry" by Dieter Brill and Ted Jacobson,\narXiv:gr-qc/0407044.\n> >> Using the invariance of the speed of light and the equivalence of\ninertial\n> >> frames, they derive the analog of the Pythagorean theorem in\nMinkowskian\n> >> space, T^2 = t^2 - x^2. This is an elegant and geometrical\npresentation.\n> >>\n> >> David Park\n> >> djmp@earthlink.net\n> >> http://home.earthlink.net/~djmp/\n\nSorry that I incorrectly gave the reference. Peter is correct.\n\nFor those who have Mathematica, I have written a Mathematica notebook,\nSpacetimeGeometry, which is available on the Mathematica page of my web site\nbelow. It is based heavily on the Brill-Jacobson paper and also has\nmaterial from the Ludvigsen General Relativity: A Geometric Approach book.\nThe notebook contains 23 different animations that illustrate the various\nconcepts. Events, world lines, proper time, Lorentz transformation as light\ncone preserving and area preserving transformations of spacetime,\nmeasurement of distance, relativity of simultaneity, Bondi k factor, Doppler\nshift, Minkowski squares and causal domains, theorem form Brill-Jacobson,\nrelation of Bondi k factor to Lorentz transformation, animated proof of\nspacetime interval, velocity from Bondi k factor, composition of velocities\nusing Bondi k factor, time dilation and locus of points with fixed\nintervals.\n\nDavid Park.\ndjmp@earthlink.net\nhttp://home.earthlink.net/~djmp/\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>"Peter Battaglino" <peterbat@gmail.com> wrote in message
news:7a94953a04073110411874db97@mail.gmail.com...
>
>
>
>
> Make that 0407022 :)
>
> >It's actually arXiv:http://www.arxiv.org/abs/gr-qc/0407044, I believe.
> >
> >Peter
> >
> >> 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.
> >>
> >> David Park
> >> djmp@earthlink.net
> >> http://home.earthlink.net/~djmp/
Sorry that I incorrectly gave the reference. Peter is correct.
For those who have Mathematica, I have written a Mathematica notebook,
SpacetimeGeometry, which is available on the Mathematica page of my web site
below. It is based heavily on the Brill-Jacobson paper and also has
material from the Ludvigsen General Relativity: A Geometric Approach book.
The notebook contains 23 different animations that illustrate the various
concepts. Events, world lines, proper time, Lorentz transformation as light
cone preserving and area preserving transformations of spacetime,
measurement of distance, relativity of simultaneity, Bondi k factor, Doppler
shift, Minkowski squares and causal domains, theorem form Brill-Jacobson,
relation of Bondi k factor to Lorentz transformation, animated proof of
spacetime interval, velocity from Bondi k factor, composition of velocities
using Bondi k factor, time dilation and locus of points with fixed
intervals.
David Park.
djmp@earthlink.net
http://home.earthlink.net/~djmp/
Alex Green
Aug14-04, 07:58 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>\n"David Park" <djmp@earthlink.net> wrote in message news:<nxRSc.22724\\$Jp6.13279@newsread3.news.atl.earthlink.net>...\n> "Peter Battaglino" <peterbat@gmail.com> wrote in message\n> news:7a94953a04073110411874db97@mail.gmail.com...\n> >\n> >\n> >\n> >\n> > Make that 0407022 :)\n> >\n> > >It\'s actually arXiv:gr-qc/0407044, I believe.\n> > >\n> > >Peter\n> > >\n> > >> There is a very good recent pedagogical paper on this topic, "Spacetime\n> and\n> > >> Euclidean Geometry" by Dieter Brill and Ted Jacobson,\n> arXiv:gr-qc/0407044.\n> > >> Using the invariance of the speed of light and the equivalence of\n> inertial\n> > >> frames, they derive the analog of the Pythagorean theorem in\n> Minkowskian\n> > >> space, T^2 = t^2 - x^2. This is an elegant and geometrical\n> presentation.\n> > >>\n> > >> David Park\n> > >> djmp@earthlink.net\n> > >> http://home.earthlink.net/~djmp/\n>\n> Sorry that I incorrectly gave the reference. Peter is correct.\n\n(The reference is available as a link at David Park\'s interesting web\npage.)\nIn this reference the authors assume that the constancy of the speed\nof light is fundamental:\n\n"According to relativity, the lightrays through a point p are\ndetermined independent of the motion of any source."\n\nClearly the speed of light is constant but it is a ratio of distance\nand time so it is not fundamental, rather it is a result of some\ngeometrical phenomenon or a process. The metric tensor (-1,1,1,1)\ndescribes the metric in terms of standardised units where the speed of\nlight is unity and it is assumed that \'seconds\' are just an\nalternative way of measuring length (1 second = approx 3*10^8 metres).\nThe fact that the speed of light is a constant for all observers is\nthen a simple consequence of the negative sign in the tensor. The\nconstant speed of light is then a PREDICTION of the theory that\nproposes a (3+1)D geometry of space-time.\n\nThe metric tensor is the theory and the constancy of the speed of\nlight is the prediction of the theory. But what is the physical\nsignificance of the tensor? Is it geometrical or dynamical?\n\nBest Wishes\n\nAlex Green\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>"David Park" <djmp@earthlink.net> wrote in message news:<nxRSc.22724$Jp6.13279@newsread3.news.atl.earthlink.net>...
> "Peter Battaglino" <peterbat@gmail.com> wrote in message
> news:7a94953a04073110411874db97@mail.gmail.com...
> >
> >
> >
> >
> > Make that 0407022 :)
> >
> > >It's actually arXiv:http://www.arxiv.org/abs/gr-qc/0407044, I believe.
> > >
> > >Peter
> > >
> > >> 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.
> > >>
> > >> David Park
> > >> djmp@earthlink.net
> > >> http://home.earthlink.net/~djmp/
>
> Sorry that I incorrectly gave the reference. Peter is correct.
(The reference is available as a link at David Park's interesting web
page.)
In this reference the authors assume that the constancy of the speed
of light is fundamental:
"According to relativity, the lightrays through a point p are
determined independent of the motion of any source."
Clearly the speed of light is constant but it is a ratio of distance
and time so it is not fundamental, rather it is a result of some
geometrical phenomenon or a process. The metric tensor (-1,1,1,1)
describes the metric in terms of standardised units where the speed of
light is unity and it is assumed that 'seconds' are just an
alternative way of measuring length (1 second = approx 3*10^8 metres).
The fact that the speed of light is a constant for all observers is
then a simple consequence of the negative sign in the tensor. The
constant speed of light is then a PREDICTION of the theory that
proposes a (3+1)D geometry of space-time.
The metric tensor is the theory and the constancy of the speed of
light is the prediction of the theory. But what is the physical
significance of the tensor? Is it geometrical or dynamical?
Best Wishes
Alex Green
Igor
Aug14-04, 07:58 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>\n\ndralexgreen@yahoo.co.uk (Alex Green) wrote in message news:<42c8441.0407220559.6ad09eb2@posting.google.com>...\n> The \'flat metric\' (*) of modern Special Relativity is conventionally\n> described by a metric tensor that has a principle diagonal given by\n> (-1,1,1,1) or equivalently (1,-1,-1,-1).\n>\n> I have always been troubled by the change of sign in the principle\n> diagonal of the metric tensor, not because it is mathematically\n> suspect but because it seems to have a physical significance that is\n> dynamical rather than geometrical. The negative coefficient in the\n> metric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2\n> + dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than\n> squaring (sqrt -1)*dt. This use of -dt*+dt seems to imply a movement\n> through time and back again rather than a simple geometric phenomenon\n> such as occurs in Pythagoras\' Theorem (which expresses the spherical\n> symmetry of space).\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> * See for instance "An Introduction to General Relativity" by Hughston\n> & Todd.p7.\n\n\n\nWe essentially have a choice of how we wish to represent the Lorentz\nmetric. We can use real coordinates (x,y,z,ct) and then the metric\ncan have either the signature (-1,1,1,1) or (1,-1,-1,-1). Or we can\nuse Minkowski coordinates (x,y,x,ict), and then the metric is\nEuclidean. In either case, it\'s pretty obvious that the time\ncoordinate is singled out as being fundamentally different than the\nspatial ones, as indeed it is. The physical significance is in having\nthe two oppositely signed so that propagation of light (the null\nscalar product) is identical in both frames of reference.\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>dralexgreen@yahoo.co.uk (Alex Green) wrote in message news:<42c8441.0407220559.6ad09eb2@posting.google.com>...
> The 'flat metric' (*) of modern Special Relativity is conventionally
> described by a metric tensor that has a principle diagonal given by
> (-1,1,1,1) or equivalently (1,-1,-1,-1).
>
> I have always been troubled by the change of sign in the principle
> diagonal of the metric tensor, not because it is mathematically
> suspect but because it seems to have a physical significance that is
> dynamical rather than geometrical. The negative coefficient in the
> metric tensor implies that -(ct)^2 in the metric ( ds^2 = dx^2 + dy^2
> + dz^2 - (cdt)^2 ) originates in multiplying -dt by +dt rather than
> squaring (\sqrt -1)*dt. This use of -dt*+dt seems to imply a movement
> through time and back again rather than a simple geometric phenomenon
> such as occurs in Pythagoras' Theorem (which expresses the spherical
> symmetry of space).
>
> Can anyone tell me about the true physical significance of the
> negative coefficient in the metric tensor of 'flat' space-time?
>
>
> * See for instance "An Introduction to General Relativity" by Hughston
> & Todd.p7.
We essentially have a choice of how we wish to represent the Lorentz
metric. We can use real coordinates (x,y,z,ct) and then the metric
can have either the signature (-1,1,1,1) or (1,-1,-1,-1). Or we can
use Minkowski coordinates (x,y,x,ict), and then the metric is
Euclidean. In either case, it's pretty obvious that the time
coordinate is singled out as being fundamentally different than the
spatial ones, as indeed it is. The physical significance is in having
the two oppositely signed so that propagation of light (the null
scalar product) is identical in both frames of reference.