Kernel: SageMath 7.2.rc1

Kruskal-Szekeres coordinates and Carter-Penrose diagram of Schwarzschild spacetime

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

These computations are based on SageManifolds (v0.9)

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

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

In [1]:

%display latex

Spacetime

We declare the spacetime manifold $M$ :

In [2]:

M = Manifold(4, 'M')
print(M)

4-dimensional differentiable manifold M

The ingoing Eddington-Finkelstein domain

The domain of ingoing Eddington-Finkelstein coordinates $(\tilde t, r, \theta, \phi)$ :

In [3]:

M_EF = M.open_subset('M_EF', latex_name=r'M_{\rm EF}')

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_EF.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') ; assume(m>=0)
X_SD.add_restrictions(r!=2*m)
X_SD

In [6]:

X_SD_I = X_SD.restrict(M_I, r>2*m) ; X_SD_I

In [7]:

X_SD_II = X_SD.restrict(M_II, r<2*m) ; X_SD_II

In [8]:

M.default_chart()

Eddington-Finkelstein coordinates

The ingoing Eddington-Finkelstein chart:

In [9]:

X_EF.<te,r,th,ph> = M_EF.chart(r'te:\tilde{t} r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi') 
X_EF

In [10]:

SD_to_EF = X_SD.transition_map(X_EF, [t+2*m*ln(abs(r/(2*m)-1)), r, th, ph])
SD_to_EF.display()

In [11]:

SD_to_EF.inverse().display()

In [12]:

X_EF_I = X_EF.restrict(M_I, r>2*m) ; X_EF_I

In [13]:

X_EF_II = X_EF.restrict(M_II, r<2*m) ; X_EF_II

In [14]:

M.atlas()

In [ ]:

Kruskal-Szekeres coordinates

In [15]:

X_KS.<T,X,th,ph> = M.chart(r'T X th:(0,pi):\theta ph:(0,2*pi):\phi')
X_KS.add_restrictions(T^2 < 1 + X^2)
X_KS

In [16]:

X_KS_I = X_KS.restrict(M_I, [X>0, T<X, T>-X]) ; X_KS_I

In [17]:

X_KS_II = X_KS.restrict(M_II, [T>0, T>abs(X)]) ; X_KS_II

In [18]:

X_KS_EF = X_KS.restrict(M_EF, X+T>0) ; X_KS_EF

In [19]:

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()

In [20]:

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()

In [21]:

EF_to_KS = X_EF.transition_map(X_KS_EF, [exp(r/(4*m))*(cosh(te/(4*m))-r/(4*m)*exp(-te/(4*m))), 
                                         exp(r/(4*m))*(sinh(te/(4*m))+r/(4*m)*exp(-te/(4*m))), 
                                         th, ph])
EF_to_KS.display()

Plot of the IEF grid in terms of KS coordinates:

In [22]:

graph = X_EF.plot(X_KS, ambient_coords=(X,T), fixed_coords={th:pi/2,ph:pi}, 
                  ranges={te:(-16,6), r:(1e-6,5)}, steps={te:1, r:0.5}, 
                  style={te:'--', r:'-'}, parameters={m:1})

In [23]:

graph += text(r'$\tilde{t}=0$', (2.65, 0.25), fontsize=16, color='red', rotation=-24)
graph += text(r'$\tilde{t}=2m$', (2.8, 1.6), fontsize=16, color='red')
graph += text(r'$\tilde{t}=-2m$', (2.68, -0.9), fontsize=16, color='red', rotation=-35)
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)

Adding the Schwarzschild horizon to the plot:

In [24]:

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

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

In [25]:

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')
graph += sing
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)

In [26]:

graph.save("sch_IEF_KS.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

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

In [27]:

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})

In [28]:

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)

In [29]:

graph2.save("sch_SD_I_KS.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

In [30]:

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='blue', 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='blue') 
graph += hor + sing + region_labels
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)

In [31]:

graph.save("sch_SD_KS.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

Radial null geodesics

The outgoing family:

In [32]:

var('u')
outgeod = M.curve({X_EF: [r + 4*m*ln(abs(r/(2*m)-1)) + u, r, pi/2, pi]}, (r, 0, +oo))
outgeod.display()

The ingoing family:

In [33]:

var('v')
ingeod = M.curve({X_EF: [-r + v, r, pi/2, pi]}, (r, 0, +oo))
ingeod.display()

In [34]:

graph = Graphics()
for u0 in range(-10, 10, 2):
    graph += outgeod.plot(chart=X_KS, ambient_coords=(X,T), prange=(0.01, 1.99), 
                          parameters={m: 1, u: u0}, color='green', style='-')
    graph += outgeod.plot(chart=X_KS, ambient_coords=(X,T), prange=(2.01, 5), 
                          parameters={m: 1, u: u0}, color='green', style='-')
    graph += ingeod.plot(chart=X_KS, ambient_coords=(X,T), prange=(0.01, 5), 
                         parameters={m: 1, v: u0}, color='green', style='--')
graph += hor
graph += sing
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)

In [35]:

graph.save("sch_rad_null_geod_KS.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

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

In [36]:

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

In [37]:

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

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

In [38]:

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

In [39]:

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()

In [40]:

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

In [41]:

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()

Plot of the maximal extension

In [42]:

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})
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})
graph += X_SD_III.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='chocolate', parameters={m:1})
graph += X_SD_IV.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='gold', parameters={m:1})

In [43]:

bifhor = line([(-4,-4), (4,4)], color='black', thickness=3) + \
         line([(-4,4), (4,-4)], color='black', thickness=3) + \
         text(r'$\mathscr{H}$', (3, 2.7), fontsize=20, color='black')
sing2 = X_SD_IV.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')
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') + \
                text(r'$\mathscr{M}_{\rm III}$', (-2.4, 0.4), fontsize=20, color='chocolate') + \
                text(r'$\mathscr{M}_{\rm IV}$', (0, -0.5), fontsize=20, color='gold')
graph += bifhor + sing + sing2 + region_labels
show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)

In [44]:

graph.save("sch_kruskal_diag.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

In [45]:

xi = M.vector_field(name='xi', latex_name=r'\xi')
xi[X_KS.frame(), 0, X_KS] = X/(4*m)
xi[X_KS.frame(), 1, X_KS] = T/(4*m)
xi.display(X_KS.frame(), X_KS)

In [46]:

xi.display(X_KS_I.frame(), X_SD_I)

In [47]:

xi.display(X_SD_I.frame())

In [48]:

#graph_xi = xi.plot(X_KS, chart_domain=X_KS, ambient_coords=(X,T), fixed_coords={th:pi/2,ph:pi},
#        max_range=4, nb_values=19, color='red', parameters={m:1})
#graph = graph_xi + bifhor + sing + sing2 
#show(graph, xmin=-3, xmax=3, ymin=-3, ymax=3)
# graph.save("sch_xi_extend.pdf", xmin=-3, xmax=3, ymin=-3, ymax=3)

Carter-Penrose diagram

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

In [49]:

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

The chart of compactified coordinates plotted in terms of itself:

In [50]:

X_C.plot(X_C, ambient_coords=(X1,T1), nb_values=25)

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

In [51]:

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))

The Kruskal-Szekeres chart plotted in terms of the compactified coordinates:

In [52]:

graph = X_KS.plot(X_C, ambient_coords=(X1,T1), fixed_coords={th:pi/2,ph:0}, 
                    ranges={T:(-15,15), X:(-8,8)}, nb_values=33, plot_points=150, 
                    style={T:'--', X:'-'})
show(graph)

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 [53]:

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))

In [54]:

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))

In [55]:

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))

In [56]:

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))

Carter-Penrose diagram

The diagram is obtained by plotting the Schwarzschild-Droste charts with respect to the compactified chart:

In [63]:

graph = X_SD_I.plot(X_C, ranges={t:(-16,16), r:(2.001,12)}, fixed_coords={th:pi/2,ph:0}, 
                    ambient_coords=(X1,T1), nb_values=25, style={t:'--', r:'-'},
                    color='blue', parameters={m:1})
graph += X_SD_II.plot(X_C, ranges={t:(-16,16), r:(0.001,1.999)}, fixed_coords={th:pi/2,ph:0},
                      ambient_coords=(X1,T1), nb_values=25, style={t:'--', r:'-'}, 
                      color='steelblue', parameters={m:1})
graph += X_SD_III.plot(X_C, ranges={t:(-16,16), r:(2.001,12)}, fixed_coords={th:pi/2,ph:0},
                       ambient_coords=(X1,T1), nb_values=25, style={t:'--', r:'-'}, 
                       color='chocolate', parameters={m:1})
graph += X_SD_IV.plot(X_C, ranges={t:(-16,16), r:(0.001,1.999)}, fixed_coords={th:pi/2,ph:0},
                      ambient_coords=(X1,T1), nb_values=25, style={t:'--', r:'-'},
                      color='gold', parameters={m:1})

In [64]:

sing = X_SD_II.plot(X_C, fixed_coords={r:0, th:pi/2, ph:pi}, ambient_coords=(X1,T1),
                    max_range=20, color='brown', thickness=4, 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=20, color='brown', thickness=4, style='--',
                    parameters={m:1}) + text(r"$r'=0$", (0.4, -1.7), fontsize=16, color='brown')
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')
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')
graph += sing+sing2+bifhor+region_labels
show(graph)

In [65]:

graph.save('sch_carter-penrose.pdf')

In [ ]:

Kruskal-Szekeres coordinates and Carter-Penrose diagram of Schwarzschild spacetime

Spacetime

The ingoing Eddington-Finkelstein domain

The Schwarzschild-Droste domain

Eddington-Finkelstein coordinates

Kruskal-Szekeres coordinates

Plot of the IEF grid in terms of KS coordinates:

Plot of Schwarzschild-Droste grid on MIM_{\rm I}MI​ in terms of KS coordinates

Radial null geodesics

Extension to MIIIM_{\rm III}MIII​ and MIVM_{\rm IV}MIV​

Plot of the maximal extension

Carter-Penrose diagram

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

Carter-Penrose diagram

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

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