User: knarf Date: 2010/10/07 11:42 AM

Added: /trunk/doc/ Kerr.txt

Modified: /trunk/doc/ documentation.tex /trunk/src/ Kerr.c

Log: correct computation of shift vector in Kerr metric, along with the documentation and a proof that this is right

File Changes:

Directory: /trunk/doc/ ======================

File [added]: Kerr.txt Delta lines: +48 -0 =================================================================== --- trunk/doc/Kerr.txt (rev 0) +++ trunk/doc/Kerr.txt 2010-10-07 16:42:08 UTC (rev 175) @@ -0,0 +1,48 @@ +This document shows that the denominator of the definition of shift_phi +in the code (p2*Sigma) is equal to the denominator of equation (11) in +Phys. Rev., D54, 1403­1416 + +Lines starting with '>' are commands for sagemath, line starting with +>> are sagemath output, everything else is comments. + + +Variable definition: +> Sigma,a,r,m,R,Delta,st2,ct2=var('Sigma,a,r,m,R,Delta,st2,ct2') + +This is p2*Sigma from the code, st2 being sin^2(theta) = rho^2/R^2: + +> code_denom=(a^2+r^2)*Sigma+2*m*a^2*r*st2 + +Now substitute Sigma, with ct2 being cos^2(theta) = z^2/R^2: + +> code_denom=code_denom.substitute(Sigma=r^2+a^2*ct2) + +and use that st2+ct2=1: + +> code_denom=code_denom.substitute(ct2=1-st2) + +Now look at the denominator in the paper: + +> paper_denom=(r^2+a^2)^2-a^2*st2*Delta + +Substitute Delta: + +> paper_denom=paper_denom.substitute(Delta=r^2-2*m*r+a^2) + +And look at the difference between code_denom and paper_denom: + +> (code_denom-paper_denom).expand() +>> 0 + +qed + +And here is the notebook as textblock: + +sage: Sigma,a,r,m,R,Delta,st2,ct2=var('Sigma,a,r,m,R,Delta,st2,ct2') +sage: code_denom=(a^2+r^2)*Sigma+2*m*a^2*r*st2 +sage: code_denom=code_denom.substitute(Sigma=r^2+a^2*ct2) +sage: code_denom=code_denom.substitute(ct2=1-st2) +sage: paper_denom=(r^2+a^2)^2-a^2*st2*Delta +sage: paper_denom=paper_denom.substitute(Delta=r^2-2*m*r+a^2) +sage: (code_denom-paper_denom).expand() +0

File [modified]: documentation.tex Delta lines: +1 -1 =================================================================== --- trunk/doc/documentation.tex 2010-07-12 14:50:35 UTC (rev 174) +++ trunk/doc/documentation.tex 2010-10-07 16:42:08 UTC (rev 175) @@ -194,7 +194,7 @@ in which case the formulas \begin{eqnarray} \alpha & = &\sqrt{\frac{\Delta}{p^2}}, \ - \beta^\phi & = & -2 m r_k a / p^2, + \beta^\phi & = & -2 m r_k a / (p^2 \Sigma), \end{eqnarray} where \begin{equation}

Directory: /trunk/src/ ======================

File [modified]: Kerr.c Delta lines: +1 -1 =================================================================== --- trunk/src/Kerr.c 2010-07-12 14:50:35 UTC (rev 174) +++ trunk/src/Kerr.c 2010-10-07 16:42:08 UTC (rev 175) @@ -132,7 +132,7 @@ p2=a_2+r_2-a*beta_phi; /* drdR=sqrt_Delta/R; */ lapse=sqrt_Delta/sqrt(p2); - shift_phi=-2*m*rK*a/p2; + shift_phi=-2*m*rK*a/(p2*Sigma);

/* Kerr metric in quasi-isotropic coordinates */ /* ds^2=Phi4*(dR^2+R^2*dth^2+R^2*Chi^2*sin(th)^2*dphi^2) */