CoCalc Shared FilesBHLectures / sage / Schwarz_Kruskal_Szekeres.ipynbOpen in CoCalc with one click!
Author: Eric Gourgoulhon
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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 MM:

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 (t~,r,θ,ϕ)(\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 MSD=MIMIIM_{\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,θ,ϕ)(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
(MSD,(t,r,θ,ϕ))\left(M_{\rm SD},(t, r, {\theta}, {\phi})\right)
In [6]:
X_SD_I = X_SD.restrict(M_I, r>2*m) ; X_SD_I
(MI,(t,r,θ,ϕ))\left(M_{\rm I},(t, r, {\theta}, {\phi})\right)
In [7]:
X_SD_II = X_SD.restrict(M_II, r<2*m) ; X_SD_II
(MII,(t,r,θ,ϕ))\left(M_{\rm II},(t, r, {\theta}, {\phi})\right)
In [8]:
M.default_chart()
(MSD,(t,r,θ,ϕ))\left(M_{\rm SD},(t, r, {\theta}, {\phi})\right)

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
(MEF,(t~,r,θ,ϕ))\left(M_{\rm EF},({\tilde{t}}, r, {\theta}, {\phi})\right)
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()
{t~=2mlog(r2m1)+tr=rθ=θϕ=ϕ\left\{\begin{array}{lcl} {\tilde{t}} & = & 2 \, m \log\left({\left| \frac{r}{2 \, m} - 1 \right|}\right) + t \\ r & = & r \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.
In [11]:
SD_to_EF.inverse().display()
{t=2mlog(2)+2mlog(m)2mlog(2m+r)+t~r=rθ=θϕ=ϕ\left\{\begin{array}{lcl} t & = & 2 \, m \log\left(2\right) + 2 \, m \log\left(m\right) - 2 \, m \log\left({\left| -2 \, m + r \right|}\right) + {\tilde{t}} \\ r & = & r \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.
In [12]:
X_EF_I = X_EF.restrict(M_I, r>2*m) ; X_EF_I
(MI,(t~,r,θ,ϕ))\left(M_{\rm I},({\tilde{t}}, r, {\theta}, {\phi})\right)
In [13]:
X_EF_II = X_EF.restrict(M_II, r<2*m) ; X_EF_II
(MII,(t~,r,θ,ϕ))\left(M_{\rm II},({\tilde{t}}, r, {\theta}, {\phi})\right)
In [14]:
M.atlas()
[(MSD,(t,r,θ,ϕ)),(MI,(t,r,θ,ϕ)),(MII,(t,r,θ,ϕ)),(MEF,(t~,r,θ,ϕ)),(MSD,(t~,r,θ,ϕ)),(MI,(t~,r,θ,ϕ)),(MII,(t~,r,θ,ϕ))]\left[\left(M_{\rm SD},(t, r, {\theta}, {\phi})\right), \left(M_{\rm I},(t, r, {\theta}, {\phi})\right), \left(M_{\rm II},(t, r, {\theta}, {\phi})\right), \left(M_{\rm EF},({\tilde{t}}, r, {\theta}, {\phi})\right), \left(M_{\rm SD},({\tilde{t}}, r, {\theta}, {\phi})\right), \left(M_{\rm I},({\tilde{t}}, r, {\theta}, {\phi})\right), \left(M_{\rm II},({\tilde{t}}, r, {\theta}, {\phi})\right)\right]
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
(M,(T,X,θ,ϕ))\left(M,(T, X, {\theta}, {\phi})\right)
In [16]:
X_KS_I = X_KS.restrict(M_I, [X>0, T<X, T>-X]) ; X_KS_I
(MI,(T,X,θ,ϕ))\left(M_{\rm I},(T, X, {\theta}, {\phi})\right)
In [17]:
X_KS_II = X_KS.restrict(M_II, [T>0, T>abs(X)]) ; X_KS_II
(MII,(T,X,θ,ϕ))\left(M_{\rm II},(T, X, {\theta}, {\phi})\right)
In [18]:
X_KS_EF = X_KS.restrict(M_EF, X+T>0) ; X_KS_EF
(MEF,(T,X,θ,ϕ))\left(M_{\rm EF},(T, X, {\theta}, {\phi})\right)
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()
{T=r2m1e(r4m)sinh(t4m)X=r2m1cosh(t4m)e(r4m)θ=θϕ=ϕ\left\{\begin{array}{lcl} T & = & \sqrt{\frac{r}{2 \, m} - 1} e^{\left(\frac{r}{4 \, m}\right)} \sinh\left(\frac{t}{4 \, m}\right) \\ X & = & \sqrt{\frac{r}{2 \, m} - 1} \cosh\left(\frac{t}{4 \, m}\right) e^{\left(\frac{r}{4 \, m}\right)} \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.
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()
{T=r2m+1cosh(t4m)e(r4m)X=r2m+1e(r4m)sinh(t4m)θ=θϕ=ϕ\left\{\begin{array}{lcl} T & = & \sqrt{-\frac{r}{2 \, m} + 1} \cosh\left(\frac{t}{4 \, m}\right) e^{\left(\frac{r}{4 \, m}\right)} \\ X & = & \sqrt{-\frac{r}{2 \, m} + 1} e^{\left(\frac{r}{4 \, m}\right)} \sinh\left(\frac{t}{4 \, m}\right) \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.
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()
{T=14(re(t~4m)m4cosh(t~4m))e(r4m)X=14(re(t~4m)m+4sinh(t~4m))e(r4m)θ=θϕ=ϕ\left\{\begin{array}{lcl} T & = & -\frac{1}{4} \, {\left(\frac{r e^{\left(-\frac{{\tilde{t}}}{4 \, m}\right)}}{m} - 4 \, \cosh\left(\frac{{\tilde{t}}}{4 \, m}\right)\right)} e^{\left(\frac{r}{4 \, m}\right)} \\ X & = & \frac{1}{4} \, {\left(\frac{r e^{\left(-\frac{{\tilde{t}}}{4 \, m}\right)}}{m} + 4 \, \sinh\left(\frac{{\tilde{t}}}{4 \, m}\right)\right)} e^{\left(\frac{r}{4 \, m}\right)} \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.

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=0r=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 MIM_{\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()
(0,+)Mr(t~,r,θ,ϕ)=(4mlog(r2m1)+r+u,r,12π,π)r(T,X,θ,ϕ)=((22m+rcosh(4mlog(2)+4mlog(m)4mlog(2m+r)ru4m)e(r4m+u4m)r)e(u4m)22m+r,(22m+re(r4m+u4m)sinh(4mlog(2)+4mlog(m)4mlog(2m+r)ru4m)+r)e(u4m)22m+r,12π,π)\begin{array}{llcl} & \left( 0 , +\infty \right) & \longrightarrow & M \\ & r & \longmapsto & \left({\tilde{t}}, r, {\theta}, {\phi}\right) = \left(4 \, m \log\left({\left| \frac{r}{2 \, m} - 1 \right|}\right) + r + u, r, \frac{1}{2} \, \pi, \pi\right) \\ & r & \longmapsto & \left(T, X, {\theta}, {\phi}\right) = \left(\frac{{\left(2 \, {\left| -2 \, m + r \right|} \cosh\left(-\frac{4 \, m \log\left(2\right) + 4 \, m \log\left(m\right) - 4 \, m \log\left({\left| -2 \, m + r \right|}\right) - r - u}{4 \, m}\right) e^{\left(\frac{r}{4 \, m} + \frac{u}{4 \, m}\right)} - r\right)} e^{\left(-\frac{u}{4 \, m}\right)}}{2 \, {\left| -2 \, m + r \right|}}, \frac{{\left(2 \, {\left| -2 \, m + r \right|} e^{\left(\frac{r}{4 \, m} + \frac{u}{4 \, m}\right)} \sinh\left(-\frac{4 \, m \log\left(2\right) + 4 \, m \log\left(m\right) - 4 \, m \log\left({\left| -2 \, m + r \right|}\right) - r - u}{4 \, m}\right) + r\right)} e^{\left(-\frac{u}{4 \, m}\right)}}{2 \, {\left| -2 \, m + r \right|}}, \frac{1}{2} \, \pi, \pi\right) \end{array}

The ingoing family:

In [33]:
var('v') ingeod = M.curve({X_EF: [-r + v, r, pi/2, pi]}, (r, 0, +oo)) ingeod.display()
(0,+)Mr(t~,r,θ,ϕ)=(r+v,r,12π,π)r(T,X,θ,ϕ)=((4mcosh(rv4m)e(r4m+v4m)re(r2m))e(v4m)4m,(4me(r4m+v4m)sinh(rv4m)+re(r2m))e(v4m)4m,12π,π)\begin{array}{llcl} & \left( 0 , +\infty \right) & \longrightarrow & M \\ & r & \longmapsto & \left({\tilde{t}}, r, {\theta}, {\phi}\right) = \left(-r + v, r, \frac{1}{2} \, \pi, \pi\right) \\ & r & \longmapsto & \left(T, X, {\theta}, {\phi}\right) = \left(\frac{{\left(4 \, m \cosh\left(-\frac{r - v}{4 \, m}\right) e^{\left(\frac{r}{4 \, m} + \frac{v}{4 \, m}\right)} - r e^{\left(\frac{r}{2 \, m}\right)}\right)} e^{\left(-\frac{v}{4 \, m}\right)}}{4 \, m}, \frac{{\left(4 \, m e^{\left(\frac{r}{4 \, m} + \frac{v}{4 \, m}\right)} \sinh\left(-\frac{r - v}{4 \, m}\right) + r e^{\left(\frac{r}{2 \, m}\right)}\right)} e^{\left(-\frac{v}{4 \, m}\right)}}{4 \, m}, \frac{1}{2} \, \pi, \pi\right) \end{array}
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 MIIIM_{\rm III} and MIVM_{\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
(MIII,(T,X,θ,ϕ))\left(M_{\rm III},(T, X, {\theta}, {\phi})\right)
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
(MIV,(T,X,θ,ϕ))\left(M_{\rm IV},(T, X, {\theta}, {\phi})\right)

Schwarzschild-Droste coordinates in MIIIM_{\rm III} and MIVM_{\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
(MIII,(t,r,θ,ϕ))\left(M_{\rm III},(t, r, {\theta}, {\phi})\right)
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()
{T=r2m1e(r4m)sinh(t4m)X=r2m1cosh(t4m)e(r4m)θ=θϕ=ϕ\left\{\begin{array}{lcl} T & = & -\sqrt{\frac{r}{2 \, m} - 1} e^{\left(\frac{r}{4 \, m}\right)} \sinh\left(\frac{t}{4 \, m}\right) \\ X & = & -\sqrt{\frac{r}{2 \, m} - 1} \cosh\left(\frac{t}{4 \, m}\right) e^{\left(\frac{r}{4 \, m}\right)} \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.
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
(MIV,(t,r,θ,ϕ))\left(M_{\rm IV},(t, r, {\theta}, {\phi})\right)
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()
{T=r2m+1cosh(t4m)e(r4m)X=r2m+1e(r4m)sinh(t4m)θ=θϕ=ϕ\left\{\begin{array}{lcl} T & = & -\sqrt{-\frac{r}{2 \, m} + 1} \cosh\left(\frac{t}{4 \, m}\right) e^{\left(\frac{r}{4 \, m}\right)} \\ X & = & -\sqrt{-\frac{r}{2 \, m} + 1} e^{\left(\frac{r}{4 \, m}\right)} \sinh\left(\frac{t}{4 \, m}\right) \\ {\theta} & = & {\theta} \\ {\phi} & = & {\phi} \end{array}\right.

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)