[Commits] [svn:einsteintoolkit] IDAnalyticBH/trunk/ (Rev. 175)

[knarf at cct.lsu.edu]

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) */