# Carter time machine

In [2]:
%display latex

In [3]:
var('a r')

$\left(a, r\right)$
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var('th', latex_name=r'\theta')

${\theta}$
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f(r,a,th) = (r^2+a^2)*(r^2+a^2*cos(th)^2) + 2*a^2*r*sin(th)^2
f

$\left( r, a, {\theta} \right) \ {\mapsto} \ 2 \, a^{2} r \sin\left({\theta}\right)^{2} + {\left(a^{2} \cos\left({\theta}\right)^{2} + r^{2}\right)} {\left(a^{2} + r^{2}\right)}$
In [69]:
g0 = plot(f(r,0.9,0), (r,-1.5,1.5), legend_label=r'$\theta=0$',
thickness=2, linestyle=':', color='red')
g1 = plot(f(r,0.9,pi/4), (r,-1.5,1.5), legend_label=r'$\theta=\pi/4$',
thickness=2, linestyle='-.', color='grey')
g2 = plot(f(r,0.9,pi/3), (r,-1.5,1.5), legend_label=r'$\theta=\pi/3$',
thickness=2, linestyle='--', color='blue')
g3 = plot(f(r,0.9,pi/2), (r,-1.5,1.5), legend_label=r'$\theta=\pi/2$',
thickness=2, color='violet')
graph = g0+g1+g2+g3
graph.axes_labels([r'$r/m$', r'$\rho^2 (r^2+a^2) + 2 a^2 m r \, \sin^2\theta$'])
graph.set_legend_options(loc='upper right')
graph

In [70]:
graph.save('ker_sign_gpp.pdf')

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

In [52]:
rp

$a \ {\mapsto}\ \sqrt{-a^{2} + 1} + 1$
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rm

$a \ {\mapsto}\ -\sqrt{-a^{2} + 1} + 1$
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rp(0.9)

$1.43588989435407$
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rm(0.9)

$0.564110105645933$
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df = diff(f(r,a,th), r).simplify_full()
df

$4 \, a^{2} r + 4 \, r^{3} - 2 \, {\left(a^{2} r - a^{2}\right)} \sin\left({\theta}\right)^{2}$
In [84]:
s = solve(df==0, r, solution_dict=True)
s

$\left[\left\{r : -\frac{{\left(\sin\left({\theta}\right)^{2} - 2\right)} a^{2} {\left(-i \, \sqrt{3} + 1\right)}}{12 \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}} - \frac{1}{2} \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}} {\left(i \, \sqrt{3} + 1\right)}\right\}, \left\{r : -\frac{{\left(\sin\left({\theta}\right)^{2} - 2\right)} a^{2} {\left(i \, \sqrt{3} + 1\right)}}{12 \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}} - \frac{1}{2} \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}} {\left(-i \, \sqrt{3} + 1\right)}\right\}, \left\{r : \frac{{\left(\sin\left({\theta}\right)^{2} - 2\right)} a^{2}}{6 \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}\right\}\right]$
In [87]:
rmin = s[2][r]
rmin

$\frac{{\left(\sin\left({\theta}\right)^{2} - 2\right)} a^{2}}{6 \, {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}} + {\left(-\frac{1}{4} \, a^{2} \sin\left({\theta}\right)^{2} + \frac{1}{12} \, \sqrt{-\frac{2}{3} \, a^{2} \sin\left({\theta}\right)^{6} + {\left(4 \, a^{2} + 9\right)} \sin\left({\theta}\right)^{4} - 8 \, a^{2} \sin\left({\theta}\right)^{2} + \frac{16}{3} \, a^{2}} a^{2}\right)}^{\frac{1}{3}}$
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df.subs(r=rmin).simplify_full()

$0$
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plot(rmin.subs(th=pi/2), (a,0.01, 0.9))

In [107]:
fmin = f(rmin.subs(th=pi/2), a, pi/2).simplify_full()
fmin

$\frac{a^{8} + 4 \, {\left(\frac{1}{12} \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{2} - \frac{1}{4} \, a^{2}\right)}^{\frac{2}{3}} a^{6} - 18 \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{6} + 54 \, a^{6} - 72 \, {\left(\frac{1}{12} \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{2} - \frac{1}{4} \, a^{2}\right)}^{\frac{4}{3}} a^{4} + 864 \, {\left(\frac{1}{12} \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{2} - \frac{1}{4} \, a^{2}\right)}^{\frac{5}{3}} a^{2} + 432 \, {\left(\frac{1}{12} \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{2} - \frac{1}{4} \, a^{2}\right)}^{\frac{8}{3}}}{432 \, {\left(\frac{1}{12} \, \sqrt{\frac{2}{3} \, a^{2} + 9} a^{2} - \frac{1}{4} \, a^{2}\right)}^{\frac{4}{3}}}$
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