# Carter-Penrose diagram of Schwarzschild spacetime¶

This worksheet demonstrates a few capabilities of SageMath in computations regarding the Carter-Penrose diagram of Schwarzschild spacetime. It is used to illustrate the lectures Geometry and physics of black holes. The corresponding tools have been developed within the SageManifolds project (version 1.2, as included in SageMath 8.2).

Click here to download the worksheet file (ipynb format). To run it, you must start SageMath with the Jupyter notebook, via the command sage -n jupyter

NB: a version of SageMath at least equal to 7.5 is required to run this worksheet:

In [1]:
version()

Out[1]:
'SageMath version 8.2, Release Date: 2018-05-05'

First we set up the notebook to display mathematical objects using LaTeX formatting:

In [2]:
%display latex


## Spacetime¶

We declare the spacetime manifold $M$:

In [3]:
M = Manifold(4, 'M')
print(M)

4-dimensional differentiable manifold M


## The Schwarzschild-Droste domain¶

The domain of Schwarzschild-Droste coordinates is $M_{\rm SD} = M_{\rm I} \cup M_{\rm II}$:

In [4]:
M_SD = M.open_subset('M_SD', latex_name=r'M_{\rm SD}')
M_I = M_SD.open_subset('M_I', latex_name=r'M_{\rm I}')
M_II = M_SD.open_subset('M_II', latex_name=r'M_{\rm II}')
M_SD.declare_union(M_I, M_II)


The Schwarzschild-Droste coordinates $(t,r,\theta,\phi)$:

In [5]:
X_SD.<t,r,th,ph> = M_SD.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
m = var('m', domain='real') ; assume(m>=0)
X_SD

Out[5]:
In [6]:
X_SD_I = X_SD.restrict(M_I, r>2*m) ; X_SD_I

Out[6]:
In [7]:
X_SD_II = X_SD.restrict(M_II, r<2*m) ; X_SD_II

Out[7]:
In [8]:
M.default_chart()

Out[8]:
In [9]:
M.atlas()

Out[9]:

## Kruskal-Szekeres coordinates¶

In [10]:
X_KS.<T,X,th,ph> = M.chart(r'T X th:(0,pi):\theta ph:(0,2*pi):\phi')
X_KS

Out[10]:
In [11]:
X_KS_I = X_KS.restrict(M_I, [X>0, T<X, T>-X]) ; X_KS_I

Out[11]:
In [12]:
X_KS_II = X_KS.restrict(M_II, [T>0, T>abs(X)]) ; X_KS_II

Out[12]:
In [13]:
SD_I_to_KS = X_SD_I.transition_map(X_KS_I, [sqrt(r/(2*m)-1)*exp(r/(4*m))*sinh(t/(4*m)),
sqrt(r/(2*m)-1)*exp(r/(4*m))*cosh(t/(4*m)),
th, ph])
SD_I_to_KS.display()

Out[13]:
In [14]:
SD_II_to_KS = X_SD_II.transition_map(X_KS_II, [sqrt(1-r/(2*m))*exp(r/(4*m))*cosh(t/(4*m)),
sqrt(1-r/(2*m))*exp(r/(4*m))*sinh(t/(4*m)),
th, ph])
SD_II_to_KS.display()

Out[14]:

### Plot of Schwarzschild-Droste grid on $M_{\rm I}$ in terms of KS coordinates¶

In [15]:
graph = X_SD_I.plot(X_KS, ambient_coords=(X,T), fixed_coords={th:pi/2,ph:pi},
ranges={t:(-10,10), r:(2.001,5)}, steps={t:1, r:0.5},
style={t:'--', r:'-'}, color='blue', parameters={m:1})


Adding the Schwarzschild horizon to the plot:

In [16]:
hor = line([(0,0), (4,4)], color='black', thickness=2) \
+ text(r'$\mathscr{H}$', (3, 2.7), fontsize=20, color='black')

In [17]:
hor2 = line([(0,0), (4,4)], color='black', thickness=2) \
+ text(r'$\mathscr{H}$', (2.95, 3.2), fontsize=20, color='black')
region_labels = text(r'$\mathscr{M}_{\rm I}$', (2.4, 0.4), fontsize=20, color='blue')
graph2 = graph + hor2 + region_labels
show(graph2, xmin=-3, xmax=3, ymin=-3, ymax=3)


Adding the curvature singularity $r=0$ to the plot:

In [18]:
sing = X_SD_II.plot(X_KS, fixed_coords={r:0, th:pi/2, ph:pi}, ambient_coords=(X,T),
color='brown', thickness=4, style='--', parameters={m:1}) \
+ text(r'$r=0$', (2.5, 3), rotation=45, fontsize=16, color='brown')

In [19]:
graph += X_SD_II.plot(X_KS, ambient_coords=(X,T), fixed_coords={th:pi/2,ph:pi},
ranges={t:(-10,10), r:(0.001,1.999)}, steps={t:1, r:0.5},
style={t:'--', r:'-'}, color='steelblue', parameters={m:1})
region_labels = text(r'$\mathscr{M}_{\rm I}$', (2.4, 0.4), fontsize=20, color='blue') + \
text(r'$\mathscr{M}_{\rm II}$', (0, 0.5), fontsize=20, color='steelblue')
graph += hor + sing + region_labels
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)


## Extension to $M_{\rm III}$ and $M_{\rm IV}$¶

In [20]:
M_III = M.open_subset('M_III', latex_name=r'M_{\rm III}', coord_def={X_KS: [X<0, X<T, T<-X]})
X_KS_III = X_KS.restrict(M_III) ; X_KS_III

Out[20]:
In [21]:
M_IV = M.open_subset('M_IV', latex_name=r'M_{\rm IV}', coord_def={X_KS: [T<0, T<-abs(X)]})
X_KS_IV = X_KS.restrict(M_IV) ; X_KS_IV

Out[21]:

Schwarzschild-Droste coordinates in $M_{\rm III}$ and $M_{\rm IV}$:

In [22]:
X_SD_III.<t,r,th,ph> = M_III.chart(r't r:(2*m,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
X_SD_III

Out[22]:
In [23]:
SD_III_to_KS = X_SD_III.transition_map(X_KS_III, [-sqrt(r/(2*m)-1)*exp(r/(4*m))*sinh(t/(4*m)),
- sqrt(r/(2*m)-1)*exp(r/(4*m))*cosh(t/(4*m)),
th, ph])
SD_III_to_KS.display()

Out[23]:
In [24]:
X_SD_IV.<t,r,th,ph> = M_IV.chart(r't r:(0,2*m) th:(0,pi):\theta ph:(0,2*pi):\phi')
X_SD_IV

Out[24]:
In [25]:
SD_IV_to_KS = X_SD_IV.transition_map(X_KS_IV, [-sqrt(1-r/(2*m))*exp(r/(4*m))*cosh(t/(4*m)),
-sqrt(1-r/(2*m))*exp(r/(4*m))*sinh(t/(4*m)),
th, ph])
SD_IV_to_KS.display()

Out[25]:

## Standard compactified coordinates¶

The coordinates $(\hat T, \hat X, \theta, \varphi)$ associated with the conformal compactification of the Schwarzschild spacetime are

In [26]:
X_C.<T1,X1,th,ph> = M.chart(r'T1:(-pi/2,pi/2):\hat{T} X1:(-pi,pi):\hat{X} th:(0,pi):\theta ph:(0,2*pi):\varphi')
X_C

Out[26]:

The chart of compactified coordinates plotted in terms of itself:

In [27]:
X_C.plot(X_C, ambient_coords=(X1,T1), number_values=100)

Out[27]:

The transition map from Kruskal-Szekeres coordinates to the compactified ones:

In [28]:
KS_to_C = X_KS.transition_map(X_C, [atan(T+X)+atan(T-X),
atan(T+X)-atan(T-X),
th, ph])
print(KS_to_C)
KS_to_C.display()

Change of coordinates from Chart (M, (T, X, th, ph)) to Chart (M, (T1, X1, th, ph))

Out[28]:

### Transition map between the Schwarzschild-Droste chart and the chart of compactified coordinates¶

The transition map is obtained by composition of previously defined ones:

In [29]:
SD_I_to_C = KS_to_C.restrict(M_I) * SD_I_to_KS
print(SD_I_to_C)
SD_I_to_C.display()

Change of coordinates from Chart (M_I, (t, r, th, ph)) to Chart (M_I, (T1, X1, th, ph))

Out[29]:
In [30]:
SD_II_to_C = KS_to_C.restrict(M_II) * SD_II_to_KS
print(SD_II_to_C)
SD_II_to_C.display()

Change of coordinates from Chart (M_II, (t, r, th, ph)) to Chart (M_II, (T1, X1, th, ph))

Out[30]:
In [31]:
SD_III_to_C = KS_to_C.restrict(M_III) * SD_III_to_KS
print(SD_III_to_C)
SD_III_to_C.display()

Change of coordinates from Chart (M_III, (t, r, th, ph)) to Chart (M_III, (T1, X1, th, ph))

Out[31]:
In [32]:
SD_IV_to_C = KS_to_C.restrict(M_IV) * SD_IV_to_KS
print(SD_IV_to_C)
SD_IV_to_C.display()

Change of coordinates from Chart (M_IV, (t, r, th, ph)) to Chart (M_IV, (T1, X1, th, ph))

Out[32]:

## Carter-Penrose diagram¶

The diagram is obtained by plotting the curves of constant Schwarzschild-Droste coordinates with respect to the compactified chart.

In [33]:
r_tab = [2.01*m, 2.1*m, 2.5*m, 4*m, 8*m, 12*m, 20*m, 100*m]
curves_t = dict()
for r0 in r_tab:
curves_t[r0] = M.curve({X_SD_I: [t, r0, pi/2, pi]}, (t,-oo,+oo))
curves_t[r0].coord_expr(X_C.restrict(M_I))

In [34]:
graph_t = Graphics()
for r0 in r_tab:
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-150, -10),
parameters={m:1}, plot_points=100, color='blue', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-10, 10),
parameters={m:1}, plot_points=100, color='blue', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(10, 150),
parameters={m:1}, plot_points=100, color='blue', style='--')

In [35]:
t_tab = [-50*m, -20*m, -10*m, -5*m, -2*m, 0, 2*m, 5*m, 10*m, 20*m, 50*m]
curves_r = dict()
for t0 in t_tab:
curves_r[t0] = M.curve({X_SD_I: [t0, r, pi/2, pi]}, (r, 2*m, +oo))
curves_r[t0].coord_expr(X_C.restrict(M_I))

In [36]:
graph_r = Graphics()
for t0 in t_tab:
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(2.0001, 4),
parameters={m:1}, plot_points=100, color='blue')
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(4, 1000),
parameters={m:1}, plot_points=100, color='blue')

In [37]:
bifhor = line([(-pi/2,-pi/2), (pi/2,pi/2)], color='black', thickness=3) + \
line([(-pi/2,pi/2), (pi/2,-pi/2)], color='black', thickness=3) + \
text(r'$\mathscr{H}$', (1, 1.2), fontsize=20, color='black')

In [38]:
sing1 = X_SD_II.plot(X_C, fixed_coords={r:0, th:pi/2, ph:pi}, ambient_coords=(X1,T1),
max_range=200, number_values=30, color='brown', thickness=3,
style='--', parameters={m:1}) + \
text(r'$r=0$', (0.4, 1.7), fontsize=16, color='brown')
sing2 = X_SD_IV.plot(X_C, fixed_coords={r:0, th:pi/2, ph:pi}, ambient_coords=(X1,T1),
max_range=200, number_values=30, color='brown', thickness=3,
style='--', parameters={m:1}) + \
text(r"$r'=0$", (0.4, -1.7), fontsize=16, color='brown')
sing = sing1 + sing2

In [39]:
scri = line([(pi,0), (pi/2,pi/2)], color='green', thickness=3) + \
text(r"$\mathscr{I}^+$", (2.6, 0.9), fontsize=20, color='green') + \
line([(pi/2, -pi/2), (pi,0)], color='green', thickness=3) + \
text(r"$\mathscr{I}^-$", (2.55, -0.9), fontsize=20, color='green') + \
line([(-pi,0), (-pi/2,pi/2)], color='green', thickness=3) + \
text(r"${\mathscr{I}'}^+$", (-2.55, 0.9), fontsize=20, color='green') + \
line([(-pi/2, -pi/2), (-pi,0)], color='green', thickness=3) + \
text(r"${\mathscr{I}'}^-$", (-2.6, -0.9), fontsize=20, color='green')

In [40]:
region_labels = text(r'$\mathscr{M}_{\rm I}$', (2, 0.4), fontsize=20, color='blue',
background_color='white') + \
text(r'$\mathscr{M}_{\rm II}$', (0.4, 1), fontsize=20, color='steelblue',
background_color='white') + \
text(r'$\mathscr{M}_{\rm III}$', (-2, 0.4), fontsize=20, color='chocolate',
background_color='white') + \
text(r'$\mathscr{M}_{\rm IV}$', (0.4, -1), fontsize=20, color='gold',
background_color='white')

In [41]:
graph = graph_t + graph_r
show(graph + bifhor + sing + scri, aspect_ratio=1)

In [42]:
r_tab = [0.1*m, 0.5*m, m, 1.25*m, 1.5*m, 1.7*m, 1.9*m, 1.98*m]
curves_t = dict()
for r0 in r_tab:
curves_t[r0] = M.curve({X_SD_II: [t, r0, pi/2, pi]}, (t,-oo,+oo))
curves_t[r0].coord_expr(X_C.restrict(M_II))

In [43]:
graph_t = Graphics()
for r0 in r_tab:
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-150, -2),
parameters={m:1}, plot_points=50, color='steelblue', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-2, 2),
parameters={m:1}, plot_points=50, color='steelblue', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(2, 150),
parameters={m:1}, plot_points=50, color='steelblue', style='--')

In [44]:
t_tab = [-20*m, -10*m, -5*m, -2*m, 0, 2*m, 5*m, 10*m, 20*m]
curves_r = dict()
for t0 in t_tab:
curves_r[t0] = M.curve({X_SD_II: [t0, r, pi/2, pi]}, (r, 0, 2*m))
curves_r[t0].coord_expr(X_C.restrict(M_II))

In [45]:
graph_r = Graphics()
for t0 in t_tab:
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(0.001, 1.9999),
parameters={m:1}, plot_points=100, color='steelblue')

In [46]:
graph += graph_t + graph_r
show(graph + bifhor + sing + scri + region_labels, aspect_ratio=1)

In [47]:
r_tab = [2.01*m, 2.1*m, 2.5*m, 4*m, 8*m, 12*m, 20*m, 100*m]
curves_t = dict()
for r0 in r_tab:
curves_t[r0] = M.curve({X_SD_III: [t, r0, pi/2, pi]}, (t,-oo,+oo))
curves_t[r0].coord_expr(X_C.restrict(M_III))

In [48]:
graph_t = Graphics()
for r0 in r_tab:
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-150, -10),
parameters={m:1}, plot_points=100, color='chocolate', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-10, 10),
parameters={m:1}, plot_points=100, color='chocolate', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(10, 150),
parameters={m:1}, plot_points=100, color='chocolate', style='--')

In [49]:
t_tab = [-50*m, -20*m, -10*m, -5*m, -2*m, 0, 2*m, 5*m, 10*m, 20*m, 50*m]
curves_r = dict()
for t0 in t_tab:
curves_r[t0] = M.curve({X_SD_III: [t0, r, pi/2, pi]}, (r, 2*m, +oo))
curves_r[t0].coord_expr(X_C.restrict(M_III))

In [50]:
graph_r = Graphics()
for t0 in t_tab:
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(2.0001, 4),
parameters={m:1}, plot_points=100, color='chocolate')
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(4, 1000),
parameters={m:1}, plot_points=100, color='chocolate')

In [51]:
graph += graph_t + graph_r
show(graph + bifhor + sing + scri + region_labels, aspect_ratio=1)

In [52]:
r_tab = [0.1*m, 0.5*m, m, 1.25*m, 1.5*m, 1.7*m, 1.9*m, 1.98*m]
curves_t = dict()
for r0 in r_tab:
curves_t[r0] = M.curve({X_SD_IV: [t, r0, pi/2, pi]}, (t,-oo,+oo))
curves_t[r0].coord_expr(X_C.restrict(M_IV))

In [53]:
graph_t = Graphics()
for r0 in r_tab:
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-150, -2),
parameters={m:1}, plot_points=50, color='gold', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(-2, 2),
parameters={m:1}, plot_points=50, color='gold', style='--')
graph_t += curves_t[r0].plot(X_C, ambient_coords=(X1,T1), prange=(2, 150),
parameters={m:1}, plot_points=50, color='gold', style='--')

In [54]:
t_tab = [-20*m, -10*m, -5*m, -2*m, 0, 2*m, 5*m, 10*m, 20*m]
curves_r = dict()
for t0 in t_tab:
curves_r[t0] = M.curve({X_SD_IV: [t0, r, pi/2, pi]}, (r, 0, 2*m))
curves_r[t0].coord_expr(X_C.restrict(M_IV))

In [55]:
graph_r = Graphics()
for t0 in t_tab:
graph_r += curves_r[t0].plot(X_C, ambient_coords=(X1,T1), prange=(0.001, 1.9999),
parameters={m:1}, plot_points=100, color='gold')

In [56]:
graph += graph_t + graph_r
graph += bifhor + sing + scri + region_labels
show(graph, aspect_ratio=1)

In [57]:
graph.save('max_carter-penrose-std.pdf', aspect_ratio=1)

In [ ]: