CoCalc Shared FilesBHLectures / sage / Kerr_primitives_BL_Kerr.ipynb
Author: Eric Gourgoulhon
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# 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 $m=1$.

In [1]:
%display latex

In [2]:
var('r a', domain='real')

$\left(r, a\right)$
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assume(a<1)

In [4]:
assume(a>=0)

In [5]:
f = (r^2 + a^2)/(r^2 - 2*r + a^2)
f

$\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}$
In [6]:
s = integrate(f, r)
s

$r + \frac{\log\left(\frac{r - \sqrt{-a^{2} + 1} - 1}{r + \sqrt{-a^{2} + 1} - 1}\right)}{\sqrt{-a^{2} + 1}} + \log\left(a^{2} + r^{2} - 2 \, r\right)$
In [7]:
diff(s, r).simplify_full()

$\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}$
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rp = 1 + sqrt(1-a^2)
rm = 1 - sqrt(1-a^2)

In [9]:
F = r + 1/sqrt(1-a^2)*(rp*ln(abs((r-rp)/2)) - rm*ln(abs((r-rm)/2)))
F

$r + \frac{{\left(\sqrt{-a^{2} + 1} - 1\right)} \log\left({\left| \frac{1}{2} \, r + \frac{1}{2} \, \sqrt{-a^{2} + 1} - \frac{1}{2} \right|}\right) + {\left(\sqrt{-a^{2} + 1} + 1\right)} \log\left({\left| \frac{1}{2} \, r - \frac{1}{2} \, \sqrt{-a^{2} + 1} - \frac{1}{2} \right|}\right)}{\sqrt{-a^{2} + 1}}$
In [10]:
dFdr = diff(F,r).simplify_full()
dFdr

$\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}$
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bool(dFdr == f)

$\mathrm{True}$
In [12]:
g = 1/(r^2 - 2*r + a^2)
g

$\frac{1}{a^{2} + r^{2} - 2 \, r}$
In [13]:
integrate(g,r)

$\frac{\log\left(\frac{r - \sqrt{-a^{2} + 1} - 1}{r + \sqrt{-a^{2} + 1} - 1}\right)}{2 \, \sqrt{-a^{2} + 1}}$
In [14]:
G = 1/(2*sqrt(1-a^2))*ln(abs((r-rp)/(r-rm)))
G

$\frac{\log\left({\left| \frac{r - \sqrt{-a^{2} + 1} - 1}{r + \sqrt{-a^{2} + 1} - 1} \right|}\right)}{2 \, \sqrt{-a^{2} + 1}}$
In [15]:
dGdr = diff(G,r).simplify_full()
dGdr

$\frac{1}{a^{2} + r^{2} - 2 \, r}$
In [16]:
bool(dGdr == g)

$\mathrm{True}$
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