geo_timelike_h.ipynb

Kernel: SageMath 9.1.beta8

Timelike and spacelike geodesics

This Jupyter/SageMath notebook is relative to the lectures Geometry and physics of black holes.

%display latex

M = Manifold(4, 'M')
X.<t,x,y,z> = M.chart()
X

h = var('h')
C = M.curve([t, h*t*(1-t), 0, 0], (t,0,1))
C

C.coord_expr()

p = M((0,0,0,0), name='p')
q = M((1,0,0,0), name='q')

graph = Graphics()
for hval in [-0.75, -0.5, -0.25, 0, 0.25, 0.5, 0.75]:
    graph += C.plot(ambient_coords=(x,t), parameters={h: hval},
                    thickness=2)
graph += p.plot(ambient_coords=(x,t), size=60, label_offset=0.04, fontsize=24)
graph += q.plot(ambient_coords=(x,t), size=60, label_offset=0.04, fontsize=24)
show(graph, aspect_ratio=1, xmin=-1/2, xmax=1/2, 
     axes_labels=[r'$x/T$', r'$t/T$'], fontsize=14)

graph.save('geo_timelike_h.pdf', aspect_ratio=1, 
           xmin=-1/2, xmax=1/2, 
           axes_labels=[r'$x/T$', r'$t/T$'], fontsize=14)

C = M.curve([h*x*(1-x), x, 0, 0], (x,0,1))
C.coord_expr()

q = M((0,1,0,0), name='q')
graph = Graphics()
for hval in [-0.75, -0.5, -0.25, 0, 0.25, 0.5, 0.75]:
    graph += C.plot(ambient_coords=(x,t), parameters={h: hval},
                    thickness=2)
graph += p.plot(ambient_coords=(x,t), size=60, label_offset=0.04, fontsize=24)
graph += q.plot(ambient_coords=(x,t), size=60, label_offset=0.03, fontsize=24)
show(graph, aspect_ratio=1, 
     axes_labels=[r'$x/L$', r'$t/L$'], fontsize=14)

graph.save('geo_spacelike_h.pdf', aspect_ratio=1, 
           axes_labels=[r'$x/L$', r'$t/L$'], fontsize=14)