ges_null_critical_geod.ipynb

Kernel: SageMath 9.0

Null geodesic in Schwarzschild spacetime with a critical impact parameter

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)

The case $r>3m$ with $\varphi_\infty = 0$

phi_inf = 0

$r(\varphi)$ decaying, $\varphi_0=-\varphi_*$

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")

$r(\varphi)$ increasing, $\varphi_0=\varphi_*$

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")

The case $r<3m$

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")