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author | knarf <knarf@6a3ddf76-46e1-4315-99d9-bc56cac1ef84> | 2010-10-07 16:42:08 +0000 |
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committer | knarf <knarf@6a3ddf76-46e1-4315-99d9-bc56cac1ef84> | 2010-10-07 16:42:08 +0000 |
commit | 8b1c7440296e61926ce950995b6d7c2b8e9e49fb (patch) | |
tree | 49db6fee5c071c2f32c51980f59d6a625c4ac2c5 | |
parent | 67d93eee204f939ad97713873efcdac5fc9eb8db (diff) |
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
-rw-r--r-- | doc/Kerr.txt | 48 | ||||
-rw-r--r-- | doc/documentation.tex | 2 | ||||
-rw-r--r-- | src/Kerr.c | 2 |
3 files changed, 50 insertions, 2 deletions
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} @@ -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) */ |