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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
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