This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.
Click here to download the notebook file (ipynb format). To run it, you must start SageMath with sage -n jupyter
.
version()
'SageMath version 9.0, Release Date: 2020-01-01'
%display latex
phi = var('phi')
phi0 = var('phi_0')
R(phi, phi0) = 2/(tanh((phi - phi0)/2)^2 - 1/3)
R(phi, phi0)
phi_s = ln((sqrt(3) + 1)/(sqrt(3) - 1))
phi_s
n(phi_s)
phi_inf = 0
phi_inf = 0
phi_0 = phi_inf - phi_s
phi_0
graph = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)
graph += polar_plot(R(phi, phi_inf - phi_s), (phi, 0.35, 30), color='green',
axes_labels=[r'$x/m$', r'$y/m$'],
plot_points=400)
graph
graph.save("ges_null_b_crit_from_inf_L_pos.pdf")
graph = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)
graph += polar_plot(R(phi, phi_inf + phi_s), (phi, -30, -0.35), color='green',
axes_labels=[r'$x/m$', r'$y/m$'],
plot_points=400)
graph
graph.save("ges_null_b_crit_from_inf_L_neg.pdf")
R2(phi, phi0) = 2/(coth((phi - phi0)/2)^2 - 1/3)
R2(phi, phi0)
graph = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)
graph += polar_plot(R2(phi, 0), (phi, -30, 0), color='green',
axes_labels=[r'$x/m$', r'$y/m$'],
plot_points=400)
graph
graph.save("ges_null_b_crit_intern_1.pdf")
graph = circle((0, 0), 2, edgecolor='black', fill=True, facecolor='grey', alpha=0.5)
graph += polar_plot(R2(phi, 0), (phi, 0, 30), color='green',
axes_labels=[r'$x/m$', r'$y/m$'],
plot_points=400)
graph
graph.save("ges_null_b_crit_intern_2.pdf")
phim = 8
rmax = 10
graph = plot(R(phi, 0), (phi, -phim, -1.001*phi_s), thickness=1.5,
ymin=0, ymax=rmax,
axes_labels=[r'$\varphi-\varphi_0$', r'$r/m$']) + \
plot(R(phi, 0), (phi, 1.001*phi_s, phim), thickness=1.5, ymin=0, ymax=rmax) + \
plot(R2(phi, 0), (phi, -phim, phim), color='red', thickness=1.5)
graph += line([(-phim, 3), (phim, 3)], linestyle=':', color='black')
graph += line([(phi_s, 0), (phi_s, rmax)], linestyle='dashed', color='black') + \
text(r'$\varphi_{\!*}$', (phi_s, -0.5), fontsize=16, color='black')
graph += line([(-phi_s, 0), (-phi_s, rmax)], linestyle='dashed', color='black') + \
text(r'$-\varphi_{\!*}$', (-0.8*phi_s, -0.5), fontsize=16, color='black')
graph
graph.save("ges_null_r_phi_bcrit.pdf")