Determining metric tensor

Jan25-05, 11:40 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>Anyone know off hand how you experimentally determine\nthe contravariant or covariant metric tensor of GM?\nSeems like a simple question but I don\'t know the\nanswer...\n-- NPC\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>Anyone know off hand how you experimentally determine
the contravariant or covariant metric tensor of GM?
Seems like a simple question but I don't know the
answer...
-- NPC

Jan27-05, 04:01 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>qmagick@yahoo.com wrote:\n\n> Anyone know off hand how you experimentally determine\n> the contravariant or covariant metric tensor of GM?\n> Seems like a simple question but I don\'t know the\n> answer...\n\nThere isn\'t a simple answer to this. One assumes the idea of arc-length\nand the idea of metric is needed for its invariant description. The\nnatural invariant objects to use as reference are then light cones - so\nyou\'d expect something like using the world-lines of massless objects.\nThis turns out to be true - see Shapiro\'s radar delay experiments. But\nit can\'t be all - light cones are conformally invariant structures and\nthe world is (apparently) not conformally invariant. We throw in a\nnormalization of the metric as a tacit assumption. (A well-justified\none it would appear, but an assumption nevertheless.)\n\nThus experimentally one determines the metric divided by a "natural\nvolume element", 9 numbers. The 10th is given by the normalization.\nThis is a kind of reducibility of the metric and was mentioned here by\nA. Jadczyk I believe.\n\n-drl\n\n\n-drl\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>qmagick@yahoo.com wrote:

> Anyone know off hand how you experimentally determine
> the contravariant or covariant metric tensor of GM?
> Seems like a simple question but I don't know the
> answer...

There isn't a simple answer to this. One assumes the idea of arc-length
and the idea of metric is needed for its invariant description. The
natural invariant objects to use as reference are then light cones - so
you'd expect something like using the world-lines of massless objects.
This turns out to be true - see Shapiro's radar delay experiments. But
it can't be all - light cones are conformally invariant structures and
the world is (apparently) not conformally invariant. We throw in a
normalization of the metric as a tacit assumption. (A well-justified
one it would appear, but an assumption nevertheless.)

Thus experimentally one determines the metric divided by a "natural
volume element", 9 numbers. The 10th is given by the normalization.
This is a kind of reducibility of the metric and was mentioned here by
A. Jadczyk I believe.

Jan30-05, 03:42 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>> Thus experimentally one determines the metric divided by a "natural\n> volume element", 9 numbers. The 10th is given by the normalization.\n> This is a kind of reducibility of the metric and was mentioned here\nby\n> A. Jadczyk I believe.\n\nI will have to look into the references you gave but what in the world\nare the 9+1 numbers you are refering to? Are they some kind of\ntriangulation of space-time? or, I don\'t know...\n-- NPC\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>> Thus experimentally one determines the metric divided by a "natural
> volume element", 9 numbers. The 10th is given by the normalization.
> This is a kind of reducibility of the metric and was mentioned here
by
> A. Jadczyk I believe.

I will have to look into the references you gave but what in the world
are the 9+1 numbers you are refering to? Are they some kind of
triangulation of space-time? or, I don't know...
-- NPC

Jan31-05, 12:45 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>qmagick@yahoo.com wrote:\n> > Thus experimentally one determines the metric divided by a "natural\n> > volume element", 9 numbers. The 10th is given by the normalization.\n> > This is a kind of reducibility of the metric and was mentioned here\n> by\n> > A. Jadczyk I believe.\n>\n> I will have to look into the references you gave but what in the world\n> are the 9+1 numbers you are refering to? Are they some kind of\n> triangulation of space-time? or, I don\'t know...\n> -- NPC\n\nYes, you can say it that way. It\'s half of "parallelogramization", that\nis, move a vector that holds itself "fixed" in a well-defined way from\nA to B to C and then from A to B\' to C and compare them, setting up a\nlinear, homogeneous function of B and B\' - a second rank tensor as it\ndevelops. (One definition of a tensor is - a tensor of rank N\nassociates a scalar with any N vectors, in a linear, homogeneous\nmanner.)\n\n-drl\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>qmagick@yahoo.com wrote:
> > Thus experimentally one determines the metric divided by a "natural
> > volume element", 9 numbers. The 10th is given by the normalization.
> > This is a kind of reducibility of the metric and was mentioned here
> by
> > A. Jadczyk I believe.
>
> I will have to look into the references you gave but what in the world
> are the 9+1 numbers you are refering to? Are they some kind of
> triangulation of space-time? or, I don't know...
> -- NPC

Yes, you can say it that way. It's half of "parallelogramization", that
is, move a vector that holds itself "fixed" in a well-defined way from
A to B to C and then from A to B' to C and compare them, setting up a
linear, homogeneous function of B and

second rank tensor as it
develops. (One definition of a tensor is - a tensor of rank N
associates a scalar with any N vectors, in a linear, homogeneous
manner.)

Feb1-05, 02:31 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>On Tue, 25 Jan 2005 qmagick@yahoo.com wrote:\n\n> Anyone know off hand how you experimentally determine the contravariant\n> or covariant metric tensor of GM? Seems like a simple question but I\n> don\'t know the answer...\n\nGM? Do you mean the metric tensor as used in gtr (General Theory of\nRelativity)? If so, try the chapter called "Metric as the Foundation of\nAll" in Misner, Thorne, & Wheeler, Gravitation, Freeman 1973, and then try\n\nauthor = {F. de Felice and C.J.S. Clarke},\ntitle = {Relativity on Curved Manifolds},\npublisher = {Cambridge University Press},\nyear = 1990}\n\nwhich offers a chapter on measurement theory in gtr.\n\n"T. Essel" (spelunking somewhere in cyberspace)\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>On Tue, 25 Jan 2005 qmagick@yahoo.com wrote:

> Anyone know off hand how you experimentally determine the contravariant
> or covariant metric tensor of GM? Seems like a simple question but I
> don't know the answer...

GM? Do you mean the metric tensor as used in gtr (General Theory of
Relativity)? If so, try the chapter called "Metric as the Foundation of
All" in Misner, Thorne, & Wheeler, Gravitation, Freeman 1973, and then try

author

. de Felice and C.J.S. Clarke},
title = {Relativity on Curved Manifolds},
publisher = {Cambridge University Press},
year

which offers a chapter on measurement theory in gtr.

"T. Essel" (spelunking somewhere in cyberspace)

Feb1-05, 02:35 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','to olbar=no,location=no,scrollbars=yes,resizable=yes, status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usene t 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>I was looking for a less mathematical answer and more of a physical\none, like first take a ruler and then... Allthough I guesse I should be\nable to construct what the apparatus for finding the metric tensor is\ngiven the information you have provided. I did know about the metric\ntensor having 10 nonsymmetric components...\n\nThanx for all the answers though.\n-- NPC\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>I was looking for a less mathematical answer and more of a physical
one, like first take a ruler and then... Allthough I guesse I should be
able to construct what the apparatus for finding the metric tensor is
given the information you have provided. I did know about the metric
tensor having 10 nonsymmetric components...

Thanx for all the answers though.
-- NPC