From Boyer-Lindquist to Kerr coordinates

This worksheet provides the primitives of the functions $r\mapsto (r^2+a^2)/\Delta$ and $r\mapsto 1/\Delta$ , which appear in the relations between Boyer-Lindquist coordinates and Kerr ones.

We set $m=1$ .

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%display latex

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var('r a', domain='real')

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assume(a<1)

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assume(a>=0)

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f = (r^2 + a^2)/(r^2 - 2*r + a^2)
f

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s = integrate(f, r)
s

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diff(s, r).simplify_full()

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rp = 1 + sqrt(1-a^2)
rm = 1 - sqrt(1-a^2)

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F = r + 1/sqrt(1-a^2)*(rp*ln(abs((r-rp)/2)) - rm*ln(abs((r-rm)/2)))
F

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dFdr = diff(F,r).simplify_full()
dFdr

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bool(dFdr == f)

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g = 1/(r^2 - 2*r + a^2)
g

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integrate(g,r)

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G = 1/(2*sqrt(1-a^2))*ln(abs((r-rp)/(r-rm)))
G

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dGdr = diff(G,r).simplify_full()
dGdr

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bool(dGdr == g)

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