Second variations of the Hilbert action? [Archive]

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David Norton

Sep9-04, 04:18 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>\nI\'m aware that Einstein\'s equations can be derived through requiring\nthat the first-order variation of the Hilbert action vanish. However,\nbeing interested mainly in mathematics, I\'d like to know what would\nhappen if we considered the second-order variation of the Hilbert\naction and required it to be strictly positive, i.e.,\n\nd^2 I_Hilbert >= 0\n\nSurely this gives one a condition for the action to be a strict\nminimum, but I\'m unable to work out the details. Can someone tell me\nwhether (a) the second variation is important, and (b) are the details\nworked out anywhere? I know that physicists are usually only interested\nin obtaining stationary points of an action, but I\'d like to see what\nhappens when we go minima-hunting.\n\nThanks in advance.\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'm aware that Einstein's equations can be derived through requiring
that the first-order variation of the Hilbert action vanish. However,
being interested mainly in mathematics, I'd like to know what would
happen if we considered the second-order variation of the Hilbert
action and required it to be strictly positive, i.e.,

d^2 I_{Hilbert} >=

Surely this gives one a condition for the action to be a strict
minimum, but I'm unable to work out the details. Can someone tell me
whether (a) the second variation is important, and (b) are the details
worked out anywhere? I know that physicists are usually only interested
in obtaining stationary points of an action, but I'd like to see what
happens when we go minima-hunting.

Thanks in advance.

Robert Low

Sep12-04, 02:23 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>"David Norton" <david_norton80@yahoo.co.uk> wrote in message news:<cho42o\\$kmu@odak26.prod.google.com>...\n> I\'m aware that Einstein\'s equations can be derived through requiring\n> that the first-order variation of the Hilbert action vanish. However,\n> being interested mainly in mathematics, I\'d like to know what would\n> happen if we considered the second-order variation of the Hilbert\n> action and required it to be strictly positive, i.e.,\n>\n> d^2 I_Hilbert >= 0\n>\n> Surely this gives one a condition for the action to be a strict\n> minimum, but I\'m unable to work out the details.\n\nI think that Nigel Bishop proved that Einstein vacua are\nalways saddle points. The paper I\'m thinking of was published\nin "General Relativity and Gravitation" some time ago---late\nseventies or early eighties: sorry, I don\'t have any\nmore details than this, but it might give you a starting\npoint.\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>"David Norton" <david_norton80@yahoo.co.uk> wrote in message news:<cho42o$kmu@odak26.prod.google.com>...
> I'm aware that Einstein's equations can be derived through requiring
> that the first-order variation of the Hilbert action vanish. However,
> being interested mainly in mathematics, I'd like to know what would
> happen if we considered the second-order variation of the Hilbert
> action and required it to be strictly positive, i.e.,
>
> d^2 I_{Hilbert} >=
>
> Surely this gives one a condition for the action to be a strict
> minimum, but I'm unable to work out the details.

I think that Nigel Bishop proved that Einstein vacua are
always saddle points. The paper I'm thinking of was published
in "General Relativity and Gravitation" some time ago---late
seventies or early eighties: sorry, I don't have any
more details than this, but it might give you a starting
point.

carlip-nospam@physics.ucdavis.edu

Sep14-04, 12:17 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>\nRobert Low <mtx014@coventry.ac.uk> wrote:\n\n> I think that Nigel Bishop proved that Einstein vacua are\n> always saddle points. The paper I\'m thinking of was published\n> in "General Relativity and Gravitation" some time ago---late\n> seventies or early eighties: sorry, I don\'t have any\n> more details than this, but it might give you a starting\n> point.\n\nNT Bishop NT, The variational principle of general relativity,\nGRG 14 (1982) 31. The abstract is:\n\nIt is shown that the field equations of general relativity never\nafford a minimum or maximum-not even locally-to the action integral\nI. Solutions of the field equations always represent a stationary\nvalue of I.\n\nSteve Carlip\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>Robert Low <mtx014@coventry.ac.uk> wrote:

> I think that Nigel Bishop proved that Einstein vacua are
> always saddle points. The paper I'm thinking of was published
> in "General Relativity and Gravitation" some time ago---late
> seventies or early eighties: sorry, I don't have any
> more details than this, but it might give you a starting
> point.

NT Bishop NT, The variational principle of general relativity,
GRG 14 (1982) 31. The abstract is:

It is shown that the field equations of general relativity never
afford a minimum or maximum-not even locally-to the action integral
I. Solutions of the field equations always represent a stationary
value of I.

Steve Carlip