From 8b1c7440296e61926ce950995b6d7c2b8e9e49fb Mon Sep 17 00:00:00 2001
From: knarf <knarf@6a3ddf76-46e1-4315-99d9-bc56cac1ef84>
Date: Thu, 7 Oct 2010 16:42:08 +0000
Subject: correct computation of shift vector in Kerr metric, along with the
 documentation and a proof that this is right

git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/IDAnalyticBH/trunk@175 6a3ddf76-46e1-4315-99d9-bc56cac1ef84
---
 doc/Kerr.txt          | 48 ++++++++++++++++++++++++++++++++++++++++++++++++
 doc/documentation.tex |  2 +-
 src/Kerr.c            |  2 +-
 3 files changed, 50 insertions(+), 2 deletions(-)
 create mode 100644 doc/Kerr.txt

diff --git a/doc/Kerr.txt b/doc/Kerr.txt
new file mode 100644
index 0000000..43433d6
--- /dev/null
+++ b/doc/Kerr.txt
@@ -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
diff --git a/doc/documentation.tex b/doc/documentation.tex
index 0de53c7..45727f0 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -194,7 +194,7 @@ The gauge can be set to the Kerr lapse and shift with the parameters
 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}
diff --git a/src/Kerr.c b/src/Kerr.c
index ae39fd6..d7966ca 100644
--- a/src/Kerr.c
+++ b/src/Kerr.c
@@ -132,7 +132,7 @@ void KerrID(CCTK_ARGUMENTS)
     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) */
-- 
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