CoCalc Shared FilesBHLectures / sage / Kerr_primitives_BL_Kerr.ipynbOpen in CoCalc with one click!
Author: Eric Gourgoulhon
Views : 11

From Boyer-Lindquist to Kerr coordinates

This worksheet provides the primitives of the functions r(r2+a2)/Δr\mapsto (r^2+a^2)/\Delta and r1/Δr\mapsto 1/\Delta, which appear in the relations between Boyer-Lindquist coordinates and Kerr ones.

We set m=1m=1.

In [1]:
%display latex
In [2]:
var('r a', domain='real')
(r,a)\left(r, a\right)
In [3]:
assume(a<1)
In [4]:
assume(a>=0)
In [5]:
f = (r^2 + a^2)/(r^2 - 2*r + a^2) f
a2+r2a2+r22r\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}
In [6]:
s = integrate(f, r) s
r+log(ra2+11r+a2+11)a2+1+log(a2+r22r)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()
a2+r2a2+r22r\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}
In [8]:
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+(a2+11)log(12r+12a2+112)+(a2+1+1)log(12r12a2+112)a2+1r + \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
a2+r2a2+r22r\frac{a^{2} + r^{2}}{a^{2} + r^{2} - 2 \, r}
In [11]:
bool(dFdr == f)
True\mathrm{True}
In [12]:
g = 1/(r^2 - 2*r + a^2) g
1a2+r22r\frac{1}{a^{2} + r^{2} - 2 \, r}
In [13]:
integrate(g,r)
log(ra2+11r+a2+11)2a2+1\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
log(ra2+11r+a2+11)2a2+1\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
1a2+r22r\frac{1}{a^{2} + r^{2} - 2 \, r}
In [16]:
bool(dGdr == g)
True\mathrm{True}
In [ ]: