Jon Absoul
May12-04, 05:05 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 noticed something interesting.\n\nI plotted the standard closed Friedmann Universe, where the universe\'s\n"radius" r(t) is a cycloid.\n(Parametrically, I plotted r(psi),t(psi), where psi=0->2pi.)\n\nI also plotted the spacetime curvature scalar, R(t), (the contraction\nof the Ricci tensor), which turns out to start and end at positive\ninfinity, but which becomes negative near maximum expansions, psi in\n~(2.5,3.8).\n\nI was surprised at first, since I thought I would get a positive R\neverywhere. However, a little reading in various textbooks revealed\nthat it\'s only the *spatial* curvature, R for the 3-d subspace\'s 3x3\nmetric at t=constant, that is always positive.\n\nMY QUESTION.\n\nI have little intuition for the meaning of R (the curvature scalar). I\nchecked three of my favorite textbooks -- MTW, Rindler and\nFoster&Nightingale -- but none of them discuss R beyond the\ndefinition.\n\nDoes anyone have any intuition to lend? Or a reference to a textbook\nthat does discuss R in more depth?\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 noticed something interesting.
I plotted the standard closed Friedmann Universe, where the universe's
"radius" r(t) is a cycloid.
(Parametrically, I plotted r(\psi),t(\psi), where \psi=0->2pi.)
I also plotted the spacetime curvature scalar, R(t), (the contraction
of the Ricci tensor), which turns out to start and end at positive
infinity, but which becomes negative near maximum expansions, \psi in
~(2.5,3.8).
I was surprised at first, since I thought I would get a positive R
everywhere. However, a little reading in various textbooks revealed
that it's only the *spatial* curvature, R for the 3-d subspace's 3x3
metric at t=constant, that is always positive.
MY QUESTION.
I have little intuition for the meaning of R (the curvature scalar). I
checked three of my favorite textbooks -- MTW, Rindler and
Foster&Nightingale -- but none of them discuss R beyond the
definition.
Does anyone have any intuition to lend? Or a reference to a textbook
that does discuss R in more depth?
I plotted the standard closed Friedmann Universe, where the universe's
"radius" r(t) is a cycloid.
(Parametrically, I plotted r(\psi),t(\psi), where \psi=0->2pi.)
I also plotted the spacetime curvature scalar, R(t), (the contraction
of the Ricci tensor), which turns out to start and end at positive
infinity, but which becomes negative near maximum expansions, \psi in
~(2.5,3.8).
I was surprised at first, since I thought I would get a positive R
everywhere. However, a little reading in various textbooks revealed
that it's only the *spatial* curvature, R for the 3-d subspace's 3x3
metric at t=constant, that is always positive.
MY QUESTION.
I have little intuition for the meaning of R (the curvature scalar). I
checked three of my favorite textbooks -- MTW, Rindler and
Foster&Nightingale -- but none of them discuss R beyond the
definition.
Does anyone have any intuition to lend? Or a reference to a textbook
that does discuss R in more depth?