View Full Version : algebraic reformulation of GR
Thomas Larsson
Mar9-05, 02:02 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>mihai wrote:\n\n>Has any reseach been done on whether, by replacing associativity with\n>another algebraic property, the corresponding equation for the\n>generalized Maurer-Cartan form is GR?\n\nMaybe you should tell which other algebraic property you have in mind.\nAssociativity can be relaxed in many ways, e.g. sha (strong homotopy\nassociativy), 2-associativity, octonions, Malcev algebras, ... AFAIK,\nthe only non-associative algebraic structures that ever played a role in\nphysics are those closely relatated to associativity, e.g. Lie algebras,\nJordan algebras, octonions. It seems like the reason why such structures\nare interesting is exactly their close relation to associativity.\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>mihai wrote:
>Has any reseach been done on whether, by replacing associativity with
>another algebraic property, the corresponding equation for the
>generalized Maurer-Cartan form is GR?
Maybe you should tell which other algebraic property you have in mind.
Associativity can be relaxed in many ways, e.g. sha (strong homotopy
associativy), 2-associativity, octonions, Malcev algebras, ... AFAIK,
the only non-associative algebraic structures that ever played a role in
physics are those closely relatated to associativity, e.g. Lie algebras,
Jordan algebras, octonions. It seems like the reason why such structures
are interesting is exactly their close relation to associativity.
repstsb@yahoo.ca
Mar16-05, 02:06 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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0503080428.1a1276eb@posting.google. com>...\n\n> Maybe you should tell which other algebraic property you have in mind.\n\nIf S is a smooth algebraic structure with identity and inverses, and s\nan element of S, then left multiplication by s^-1 induces a map T(s)\n-> T(e). If we equip T(e) with a metric, then the metric pulls back to\na metric on T(s). For example, this would give, for the multiplication\nof quaternions, the metric g_ii = 1 / (x1^2 + x2^2 + x3^2 + x4^2).\nThis metric is not a solution of GR.\n\nI was wondering wheter for some algebraic properties, like ((ab)b)a =\n((ba)a)b, the metrics constructed by the above procedure are the\nsolutions of GR.\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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0503080428.1a1276eb@posting.google.com>...
> Maybe you should tell which other algebraic property you have in mind.
If S is a smooth algebraic structure with identity and inverses, and s
an element of S, then left multiplication by s^-1 induces a map T(s)
-> T(e). If we equip T(e) with a metric, then the metric pulls back to
a metric on T(s). For example, this would give, for the multiplication
of quaternions, the metric g_{ii} = 1 / (x1^2 + x2^2 + x3^2 + x4^2).
This metric is not a solution of GR.
I was wondering wheter for some algebraic properties, like ((ab)b)a =
((ba)a)b, the metrics constructed by the above procedure are the
solutions of GR.
repstsb@yahoo.ca
Mar16-05, 10:18 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>\nthomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0503080428.1a1276eb@posting.google. com>...\n\n> Maybe you should tell which other algebraic property you have in mind.\n\nIf S is a smooth algebraic structure with identity and inverses, and s\nan element of S, then left multiplication by s^-1 induces a map T(s)\n-> T(e). If we equip T(e) with a metric, then the metric pulls back to\na metric on T(s). For example, this would give, for the multiplication\nof quaternions, the metric g_ii = 1 / (x1^2 + x2^2 + x3^2 + x4^2).\nThis metric is not a solution of GR.\n\nI was wondering wheter for some algebraic properties, like ((ab)b)a =\n((ba)a)b, the metrics constructed by the above procedure are the\nsolutions of GR.\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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0503080428.1a1276eb@posting.google.com>...
> Maybe you should tell which other algebraic property you have in mind.
If S is a smooth algebraic structure with identity and inverses, and s
an element of S, then left multiplication by s^-1 induces a map T(s)
-> T(e). If we equip T(e) with a metric, then the metric pulls back to
a metric on T(s). For example, this would give, for the multiplication
of quaternions, the metric g_{ii} = 1 / (x1^2 + x2^2 + x3^2 + x4^2).
This metric is not a solution of GR.
I was wondering wheter for some algebraic properties, like ((ab)b)a =
((ba)a)b, the metrics constructed by the above procedure are the
solutions of GR.
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