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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 |