CoCalc Shared FilesBHLectures / sage / Kerr_sign_gpp.ipynbOpen in CoCalc with one click!
Author: Eric Gourgoulhon
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Carter time machine

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%display latex
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var('a r')
(a,r)\left(a, r\right)
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var('th', latex_name=r'\theta')
θ{\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
(r,a,θ)  2a2rsin(θ)2+(a2cos(θ)2+r2)(a2+r2)\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)}
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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
a  a2+1+1a \ {\mapsto}\ \sqrt{-a^{2} + 1} + 1
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rm
a  a2+1+1a \ {\mapsto}\ -\sqrt{-a^{2} + 1} + 1
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rp(0.9)
1.435889894354071.43588989435407
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rm(0.9)
0.5641101056459330.564110105645933
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df = diff(f(r,a,th), r).simplify_full() df
4a2r+4r32(a2ra2)sin(θ)24 \, a^{2} r + 4 \, r^{3} - 2 \, {\left(a^{2} r - a^{2}\right)} \sin\left({\theta}\right)^{2}
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s = solve(df==0, r, solution_dict=True) s
[{r:(sin(θ)22)a2(i3+1)12(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)1312(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13(i3+1)},{r:(sin(θ)22)a2(i3+1)12(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)1312(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13(i3+1)},{r:(sin(θ)22)a26(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13+(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13}]\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]
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rmin = s[2][r] rmin
(sin(θ)22)a26(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13+(14a2sin(θ)2+11223a2sin(θ)6+(4a2+9)sin(θ)48a2sin(θ)2+163a2a2)13\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()
00
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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
a8+4(11223a2+9a214a2)23a61823a2+9a6+54a672(11223a2+9a214a2)43a4+864(11223a2+9a214a2)53a2+432(11223a2+9a214a2)83432(11223a2+9a214a2)43\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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