Eliminating the Riemannian metric? [Archive]

Very cryptic

Aug16-04, 12:55 PM

grelbr

Aug17-04, 01:31 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>\n\nvery_cryptic@hotmail.com (Very cryptic) wrote in message news:<3cb3ea56.0408152143.12149cfb@posting.google. com>...\n> A thought just occured it me. Isn\'t it possible to eliminate the\n> metric field g in general relativity in a simple way? Just introduce a\n> GL(4,R) connection with the tangent bundle TM acted upon by GL(4,R)\n> and insist every point has at most an SO(3,1) holonomy. For generic\n> connections, this would determine g up to a global rescaling factor.\n> In fact, since we\'re making restrictions on the holonomy, this\n> formulation would suggest we work directly with Wilson loops and lines\n> instead. Also, generic connections would give rise to torsion.\n\nWell, you probably still have a metric. It\'s hard to see how to have\na gravity theory that recovers special relativity in the appropriate\nlimit, and Newtonian gravity in the appropriate limit, and that does\nnot in some way use a metric. Or, at least, it looks a lot like\ngeneral relativity in the appropriate limit, while the microscopic\nlimit may be something very different.\n\nHowever, you don\'t automatically get rid of a metric by moving to\nGL(4,R). You do get quite a few interesting things happening.\nIIRC (and it\'s been a while, and it was never my speciality) there\nhas been some work done along these lines. If you look up Moffat\nand his theory and collaborators, you should see one approach.\nI\'m pretty sure there are others.\ngrelbr\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>very_cryptic@hotmail.com (Very cryptic) wrote in message news:<3cb3ea56.0408152143.12149cfb@posting.google.com>...
> A thought just occured it me. Isn't it possible to eliminate the
> metric field g in general relativity in a simple way? Just introduce a
> GL(4,R) connection with the tangent bundle TM acted upon by GL(4,R)
> and insist every point has at most an SO(3,1) holonomy. For generic
> connections, this would determine g up to a global rescaling factor.
> In fact, since we're making restrictions on the holonomy, this
> formulation would suggest we work directly with Wilson loops and lines
> instead. Also, generic connections would give rise to torsion.

Well, you probably still have a metric. It's hard to see how to have
a gravity theory that recovers special relativity in the appropriate
limit, and Newtonian gravity in the appropriate limit, and that does
not in some way use a metric. Or, at least, it looks a lot like
general relativity in the appropriate limit, while the microscopic
limit may be something very different.

However, you don't automatically get rid of a metric by moving to
GL(4,R). You do get quite a few interesting things happening.
IIRC (and it's been a while, and it was never my speciality) there
has been some work done along these lines. If you look up Moffat
and his theory and collaborators, you should see one approach.
I'm pretty sure there are others.
grelbr