#!/usr/bin/env python
# coding: utf-8

# # Carter time machine

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get_ipython().run_line_magic('display', 'latex')


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var('a r')


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var('th', latex_name=r'\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


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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


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graph.save('ker_sign_gpp.pdf')


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


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rp


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rm


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rp(0.9)


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rm(0.9)


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


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s = solve(df==0, r, solution_dict=True)
s


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rmin = s[2][r]
rmin


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df.subs(r=rmin).simplify_full()


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


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fmin = f(rmin.subs(th=pi/2), a, pi/2).simplify_full()
fmin


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