David Norton
Nov25-04, 03:37 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>I\'ve got a little problem that I\'d appreciate some help on. I\'m trying\nto demonstrate that the equations of motion produced by the ADM\nHamiltonian are the same as those produced by the Lagrangian for GR.\nSpecifically, I take the Lagrangian for GR to be\n\n\\begin{equation}\nL = \\int d^3x N\\sqrt{g}( R + (trK)^2 - K_{ij}K^{ij} )\n\\end{equation}\n\nwhere \\$R\\$ is the scalar curvature of a three manifold, \\$K_{ij}\\$ is the\nextrinsic curvature of the three manifold embedded in spacetime, and\n\\$trK=g^{ij}K_{ij}\\$ is the trace of the extrinsic curvature. Producing\nan expression for \\$\\dot{g}_{ij}\\$ is no problem, but I\'m stuck when\ntrying to get \\$\\dot{\\pi}^{ij}\\$. What I want to do is to show that the\nequation for \\$\\dot{\\pi}^{ij}\\$ that I get from varying the Lagrangian\nwith respect to \\$g_{ij}\\$ is the same as Eq. (21.115) in Misner, Thorne,\nand Wheeler, thereby proving that the Lagrangian and Hamiltonian\npictures predict the same dynamics. For the benefit of those who don\'t\nhave a copy of MTW to hand, I\'ll write out the equation here (apologies\nfor the \\LaTeX):\n\n\\begin{align}\n\\dot{\\pi}^{ij}\n&= -N\\sqrt{g}(R^{ij} - \\frac{1}{2}g^{ij}R) +\n\\frac{N}{2\\sqrt{g}}g^{ij}(\\pi^{mn}\\pi_{mn} - \\frac{1}{2}(tr\\pi)^2)\n\\nonumber \\\\\n& - \\frac{2N}{\\sqrt{g}}(\\pi^{im}\\pi_m^j - \\frac{1}{2}\\pi^{ij}tr\\pi)\n+ \\sqrt{g}(N^{|ij} - g^{ij}N^{|m}_{|m}) \\nonumber \\\\\n& +(\\pi^{ij}N^m)_{|m} - N^i_{|m}\\pi^{mj} - N^j_{|m}\\pi^{mi}\n\\end{align}\n\nNow, I can produce everything here apart from the last three terms. I\nsuspect that these terms arise through variation of \\$K_{ij}\\$, but can\'t\nimmediately see how to show this. Can anyone help point me in the right\ndirection?\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've got a little problem that I'd appreciate some help on. I'm trying
to demonstrate that the equations of motion produced by the ADM
Hamiltonian are the same as those produced by the Lagrangian for GR.
Specifically, I take the Lagrangian for GR to be
\begin{equation}L = \int d^{3x} N\sqrt{g}( R + (trK)^2 - K_{ij}K^{ij} )\end{equation}
where $R$ is the scalar curvature of a three manifold, $K_{ij}$ is the
extrinsic curvature of the three manifold embedded in spacetime, and
$trK=g^{ij}K_{ij}$ is the trace of the extrinsic curvature. Producing
an expression for $\dot{g}_{ij}$ is no problem, but I'm stuck when
trying to get $\dot{\pi}^{ij}$. What I want to do is to show that the
equation for $\dot{\pi}^{ij}$ that I get from varying the Lagrangian
with respect to $g_{ij}$ is the same as Eq. (21.115) in Misner, Thorne,
and Wheeler, thereby proving that the Lagrangian and Hamiltonian
pictures predict the same dynamics. For the benefit of those who don't
have a copy of MTW to hand, I'll write out the equation here (apologies
for the \LaTeX):\begin{align}\dot{\pi}^{ij}&= -N\sqrt{g}(R^{ij} - \frac{1}{2}g^{ij}R) +\frac{N}{2\sqrt{g}}g^{ij}(\pi^{mn}\pi_{mn} - \frac{1}{2}(tr\pi)^2)\nonumber \\& - \frac{2N}{\sqrt{g}}(\pi^{im}\pi_m^j - \frac{1}{2}\pi^{ij}tr\pi)+ \sqrt{g}(N^{|ij} - g^{ij}N^{|m}_{|m}) \nonumber \\& +(\pi^{ij}N^m)_{|m} - N^{i_}{|m}\pi^{mj} - N^{j_}{|m}\pi^{mi}\end{align}
Now, I can produce everything here apart from the last three terms. I
suspect that these terms arise through variation of $K_{ij}$, but can't
immediately see how to show this. Can anyone help point me in the right
direction?
Thanks in advance.
to demonstrate that the equations of motion produced by the ADM
Hamiltonian are the same as those produced by the Lagrangian for GR.
Specifically, I take the Lagrangian for GR to be
\begin{equation}L = \int d^{3x} N\sqrt{g}( R + (trK)^2 - K_{ij}K^{ij} )\end{equation}
where $R$ is the scalar curvature of a three manifold, $K_{ij}$ is the
extrinsic curvature of the three manifold embedded in spacetime, and
$trK=g^{ij}K_{ij}$ is the trace of the extrinsic curvature. Producing
an expression for $\dot{g}_{ij}$ is no problem, but I'm stuck when
trying to get $\dot{\pi}^{ij}$. What I want to do is to show that the
equation for $\dot{\pi}^{ij}$ that I get from varying the Lagrangian
with respect to $g_{ij}$ is the same as Eq. (21.115) in Misner, Thorne,
and Wheeler, thereby proving that the Lagrangian and Hamiltonian
pictures predict the same dynamics. For the benefit of those who don't
have a copy of MTW to hand, I'll write out the equation here (apologies
for the \LaTeX):\begin{align}\dot{\pi}^{ij}&= -N\sqrt{g}(R^{ij} - \frac{1}{2}g^{ij}R) +\frac{N}{2\sqrt{g}}g^{ij}(\pi^{mn}\pi_{mn} - \frac{1}{2}(tr\pi)^2)\nonumber \\& - \frac{2N}{\sqrt{g}}(\pi^{im}\pi_m^j - \frac{1}{2}\pi^{ij}tr\pi)+ \sqrt{g}(N^{|ij} - g^{ij}N^{|m}_{|m}) \nonumber \\& +(\pi^{ij}N^m)_{|m} - N^{i_}{|m}\pi^{mj} - N^{j_}{|m}\pi^{mi}\end{align}
Now, I can produce everything here apart from the last three terms. I
suspect that these terms arise through variation of $K_{ij}$, but can't
immediately see how to show this. Can anyone help point me in the right
direction?
Thanks in advance.