Conformal completion of Minkowski spacetime

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

It makes use of SageMath differential geometry tools developed through the SageManifolds project.

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

In [1]:
version()
Out[1]:
'SageMath version 9.3.beta5, Release Date: 2020-12-27'

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

In [2]:
%display latex

Spherical coordinates on Minkowski spacetime

We declare the spacetime manifold $M$:

In [3]:
M = Manifold(4, 'M')
print(M)
4-dimensional differentiable manifold M

and the spherical coordinates $(t,r,\theta,\phi)$ as a chart on $M$:

In [4]:
XS.<t,r,th,ph> = M.chart(r't r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):\phi')
XS
Out[4]:
In [5]:
XS.coord_range()
Out[5]:

In term of these coordinates, the Minkowski metric is

In [6]:
g = M.lorentzian_metric('g')
g[0,0] = -1
g[1,1] = 1
g[2,2] = r^2
g[3,3] = r^2*sin(th)^2
g.display()
Out[6]:

Null coordinates

Let us introduce the null coordinates $u=t-r$ (retarded time) and $v=t+r$ (advanced time):

In [7]:
XN.<u,v,th,ph> = M.chart(r'u v th:(0,pi):\theta ph:(0,2*pi):\phi')
XN.add_restrictions(v-u>0)
XN
Out[7]:
In [8]:
XN.coord_range()
Out[8]:
In [9]:
XS_to_XN = XS.transition_map(XN, [t-r, t+r, th, ph])
XS_to_XN.display()
Out[9]:
In [10]:
XS_to_XN.inverse().display()
Out[10]:

In terms of the null coordinates $(u,v,\theta,\phi)$, the Minkowski metric writes

In [11]:
g.display(XN.frame(), XN)
Out[11]:

Let us plot the coordinate grid $(u,v)$ in terms of the coordinates $(t,r)$:

In [12]:
graph = XN.plot(XS, ambient_coords=(r,t), fixed_coords={th: pi/2, ph: pi}, 
                number_values=17, plot_points=200, color='green', 
                style={u: '-', v: ':'}, thickness={u: 1, v: 2})
show(graph)
In [13]:
show(graph, xmin=0, xmax=4, ymin=0, ymax=4, aspect_ratio=1)
In [14]:
graph.save("glo_null_coord.pdf", xmin=0, xmax=4, ymin=0, ymax=4, 
           aspect_ratio=1)

Compactified null coordinates

Instead of $(u,v)$, which span $\mathbb{R}$, let consider the coordinates $U = \mathrm{atan}\, u$ and $V = \mathrm{atan}\, v$, which span $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$:

In [15]:
graph = plot(atan(u), (u,-6, 6), thickness=2, axes_labels=[r'$u$', r'$U$']) \
        + line([(-6,-pi/2), (6,-pi/2)], linestyle='--') \
        + line([(-6,pi/2), (6,pi/2)], linestyle='--')
show(graph, aspect_ratio=1)
In [16]:
graph.save('glo_atan.pdf', aspect_ratio=1)
In [17]:
XNC.<U,V,th,ph> = M.chart(r'U:(-pi/2,pi/2) V:(-pi/2,pi/2) th:(0,pi):\theta ph:(0,2*pi):\phi')
XNC.add_restrictions(V-U>0)
XNC
Out[17]:
In [18]:
XNC.coord_range()
Out[18]:
In [19]:
XN_to_XNC = XN.transition_map(XNC, [atan(u), atan(v), th, ph])
XN_to_XNC.display()
Out[19]:
In [20]:
XN_to_XNC.inverse().display()
Out[20]:

Expressed in terms of the coordinates $(U,V,\theta,\phi)$, the metric tensor is

In [21]:
g.display(XNC.frame(), XNC)
Out[21]:

Let us call $\Omega^{-2}$ the common factor:

In [22]:
Omega = M.scalar_field({XNC: 2*cos(U)*cos(V)}, name='Omega', latex_name=r'\Omega')
Omega.display()
Out[22]:
In [23]:
Omega.display(XS)
Out[23]:

Conformal metric

We introduce the metric $\tilde g = \Omega^2 g$:

In [24]:
gt = M.lorentzian_metric('gt', latex_name=r'\tilde{g}')
gt.set(Omega^2*g)
gt.display(XNC.frame(), XNC)
Out[24]:

Clearly the metric components ${\tilde g}_{\theta\theta}$ and ${\tilde g}_{\phi\phi}$ can be simplified further. Let us do it by hand, by extracting the symbolic expression via expr():

In [25]:
g22 = gt[XNC.frame(), 2, 2, XNC].expr()
g22
Out[25]:
In [26]:
g22.factor().reduce_trig()
Out[26]:
In [27]:
g33st = gt[XNC.frame(), 3, 3, XNC].expr() / sin(th)^2
g33st
Out[27]:
In [28]:
g33st.factor().reduce_trig()
Out[28]:
In [29]:
gt.add_comp(XNC.frame())[2,2, XNC] = g22.factor().reduce_trig()
gt.add_comp(XNC.frame())[3,3, XNC] = g33st.factor().reduce_trig() * sin(th)^2

Hence the final form of the conformal metric in terms of the compactified null coordinates:

In [30]:
gt.display(XNC.frame(), XNC)
Out[30]:

In terms of the non-compactified null coordinates $(u,v,\theta,\phi)$:

In [31]:
gt.display(XN.frame(), XN)
Out[31]:

and in terms of the default coordinates $(t,r,\theta,\phi)$:

In [32]:
gt.display()
Out[32]:

Einstein cylinder coordinates

Let us introduce some coordinates $(\tau,\chi)$ such that the null coordinates $(U,V)$ are respectively half the retarded time $\tau -\chi$ and half the advanced time $\tau+\chi$:

In [33]:
XC.<tau,ch,th,ph> = M.chart(r'tau:(-pi,pi):\tau ch:(0,pi):\chi th:(0,pi):\theta ph:(0,2*pi):\phi')
XC.add_restrictions([tau<pi-ch, tau>ch-pi])
XC
Out[33]:
In [34]:
XC.coord_range()
Out[34]:
In [35]:
XC_to_XNC = XC.transition_map(XNC, [(tau-ch)/2, (tau+ch)/2, th, ph])
XC_to_XNC.display()
Out[35]:
In [36]:
XC_to_XNC.inverse().display()
Out[36]:

The conformal metric takes then the form of the standard metric on the Einstein cylinder $\mathbb{R}\times\mathbb{S}^3$:

In [37]:
gt.display(XC.frame(), XC)
Out[37]:

The square of the conformal factor expressed in all the coordinates introduced so far:

In [38]:
(Omega^2).display()
Out[38]:
In [39]:
XS_to_XC = M.coord_change(XNC,XC) * M.coord_change(XN, XNC) * M.coord_change(XS, XN)
XS_to_XC.display()
Out[39]:
In [40]:
XC_to_XS = M.coord_change(XN, XS) * M.coord_change(XNC, XN) * M.coord_change(XC,XNC)
XC_to_XS.display()
Out[40]:

The expressions for $t$ and $r$ can be simplified:

In [41]:
tc = XC_to_XS(tau,ch,th,ph)[0]
tc
Out[41]:
In [42]:
tc.reduce_trig()
Out[42]:
In [43]:
rc = XC_to_XS(tau,ch,th,ph)[1]
rc
Out[43]:
In [44]:
rc.reduce_trig()
Out[44]:
In [45]:
XS_to_XC.set_inverse(tc.reduce_trig(), rc.reduce_trig(), th, ph)
XC_to_XS = XS_to_XC.inverse()
Check of the inverse coordinate transformation:
  t == t  *passed*
  r == r  *passed*
  th == th  *passed*
  ph == ph  *passed*
  tau == arctan((sin(ch) + sin(tau))/(cos(ch) + cos(tau))) + arctan(-(sin(ch) - sin(tau))/(cos(ch) + cos(tau)))  **failed**
  ch == arctan((sin(ch) + sin(tau))/(cos(ch) + cos(tau))) - arctan(-(sin(ch) - sin(tau))/(cos(ch) + cos(tau)))  **failed**
  th == th  *passed*
  ph == ph  *passed*
NB: a failed report can reflect a mere lack of simplification.
In [46]:
XC_to_XS.display()
Out[46]:

Conformal Penrose diagram

Let us draw the coordinate grid $(t,r)$ in terms of the coordinates $(\tau,\chi)$:

In [47]:
graphXS = XS.plot(XC, ambient_coords=(ch, tau), fixed_coords={th: pi/2, ph: pi}, 
                  max_range=30, number_values=51, plot_points=250, 
                  color={t: 'red', r: 'grey'})
graph_i0 = circle((pi,0), 0.05, fill=True, color='grey') + \
           text(r"$i^0$", (3.3, 0.2), fontsize=18, color='grey') 
graph_ip = circle((0,pi), 0.05, fill=True, color='red') + \
           text(r"$i^+$", (0.25, 3.3), fontsize=18, color='red')
graph_im = circle((0,-pi), 0.05, fill=True, color='red') + \
           text(r"$i^-$", (0.25, -3.3), fontsize=18, color='red')
graph_Ip = line([(0,pi), (pi,0)], color='green', thickness=2) + \
           text(r"$\mathscr{I}^+$", (1.8, 1.8), fontsize=18, color='green')
graph_Im = line([(0,-pi), (pi,0)], color='green', thickness=2) + \
           text(r"$\mathscr{I}^-$", (1.8, -1.8), fontsize=18, color='green')
graph = graphXS + graph_i0 + graph_ip + graph_im + graph_Ip + graph_Im
show(graph)
In [48]:
graph.save('glo_conf_diag_Mink.pdf')

Some blow-up near $i^0$:

In [49]:
graph = XS.plot(XC, ambient_coords=(ch, tau), fixed_coords={th: pi/2, ph: pi}, 
                max_range=100, number_values=41, plot_points=200, 
                color={t: 'red', r: 'grey'})
graph += circle((pi,0), 0.005, fill=True, color='grey') + \
         text(r"$i^0$", (pi, 0.02), fontsize=18, color='grey') 
show(graph, xmin=3., xmax=3.2, ymin=-0.2, ymax=0.2, aspect_ratio=1)

To produce a more satisfactory figure, let us use some logarithmic radial coordinate:

In [50]:
XL.<t, rh, th, ph> = M.chart(r't rh:\rho th:(0,pi):\theta ph:(0,2*pi):\phi')
XL
Out[50]:
In [51]:
XS_to_XL = XS.transition_map(XL, [t, ln(r), th, ph])
XS_to_XL.display()
Out[51]:
In [52]:
XS_to_XL.inverse().display()
Out[52]:
In [53]:
XL_to_XC = M.coord_change(XS, XC) * M.coord_change(XL, XS)
XC_to_XL = M.coord_change(XS, XL) * M.coord_change(XC, XS)
In [54]:
graph = XL.plot(XC, ambient_coords=(ch, tau), fixed_coords={th: pi/2, ph: pi}, 
                ranges={t: (-20, 20), rh: (-2, 10)}, number_values=19, 
                color={t: 'red', rh: 'grey'})
graph += circle((pi,0), 0.005, fill=True, color='grey') + \
         text(r"$i^0$", (pi, 0.02), fontsize=18, color='grey') 
show(graph, xmin=3., xmax=3.2, ymin=-0.2, ymax=0.2, aspect_ratio=1)

Null radial geodesics in the conformal diagram

To get a view of the null radial geodesics in the conformal diagram, it suffices to plot the chart $(u,v,\theta,\phi)$ in terms of the chart $(\tau,\chi,\theta,\phi)$. The following plot shows

  • the null geodesics defined by $(u,\theta,\phi) = (u_0, \pi/2,\pi)$ for 17 values of $u_0$ evenly spaced in $[-8,8]$ (dashed lines)
  • the null geodesics defined by $(v,\theta,\phi) = (v_0, \pi/2,\pi)$ for 17 values of $v_0$ evenly spaced in $[-8,8]$ (solid lines)
In [55]:
graphXN = XN.plot(XC, ambient_coords=(ch, tau), fixed_coords={th: pi/2, ph: pi}, 
                  number_values=17, plot_points=150, color='green', 
                  style={u: '-', v: ':'}, thickness={u: 1, v: 2})
graph = graphXN + graph_i0 + graph_ip + graph_im + graph_Ip + graph_Im
show(graph)
In [56]:
graph.save('glo_conf_Mink_null.pdf')

Conformal factor

The conformal factor expressed in various coordinate systems:

In [57]:
Omega.display()
Out[57]:

The expression in terms of $(\tau,\chi,\theta,\phi)$ can be simplified:

In [58]:
Omega.expr(XC)
Out[58]:
In [59]:
s = Omega.expr(XC) - cos(tau) - cos(ch)
s.trig_reduce()
Out[59]:

Hence we set

In [60]:
Omega.add_expr(cos(tau) + cos(ch), XC)
Omega.display()
Out[60]:

A plot of $\Omega$ in terms of the coordinates $(\tau,\chi)$:

In [61]:
graph = plot3d(Omega.expr(XC), (tau,-pi,pi), (ch,0,pi)) \
        + plot3d(0, (tau,-pi,pi), (ch,0,pi), color='yellow', opacity=0.7)
show(graph, aspect_ratio=1, axes_labels=['tau', 'chi', 'Omega'])
In [62]:
show(graph, aspect_ratio=1, viewer='tachyon')

Differential of the conformal factor

The 1-form $\mathrm{d}\Omega$ is:

In [63]:
dOmega = Omega.differential()
print(dOmega)
1-form dOmega on the 4-dimensional differentiable manifold M
In [64]:
dOmega.display()
Out[64]:
In [65]:
dOmega.display(XNC.frame(), XNC)
Out[65]:
In [66]:
M.set_default_chart(XNC)
M.set_default_frame(XNC.frame())
In [67]:
dOmega.display()
Out[67]:
In [68]:
dOmega1 = M.one_form()
dOmega1[0] = -2*cos(V)*sin(U)
dOmega1[1] = -2*cos(U)*sin(V)
dOmega1.display()
Out[68]:
In [69]:
dOmega1.display(XC.frame(), XC)
Out[69]:

Einstein static universe

In [70]:
E = Manifold(4, 'E')
print(E)
4-dimensional differentiable manifold E
In [71]:
XE.<tau,ch,th,ph> = E.chart(r'tau:\tau ch:(0,pi):\chi th:(0,pi):\theta ph:(0,2*pi):\phi')
XE
Out[71]:
In [72]:
XE.coord_range()
Out[72]:
In [73]:
XC.coord_range()
Out[73]:

Embedding of $M$ in $E$

In [74]:
Phi = M.diff_map(E, {(XC, XE): [tau, ch, th, ph]},
                 name='Phi', latex_name=r'\Phi')
print(Phi)
Phi.display()
Differentiable map Phi from the 4-dimensional differentiable manifold M to the 4-dimensional differentiable manifold E
Out[74]:
In [75]:
XS.plot(XE, mapping=Phi, ambient_coords=(ch, tau), fixed_coords={th: pi/2, ph: pi}, 
        plot_points=200, color={t: 'red', r: 'grey'})
Out[75]:

Embedding of $E$ in $\mathbb{R}^5$

In [76]:
R5 = Manifold(5, 'R^5', latex_name=r'\mathbb{R}^5')
print(R5)
5-dimensional differentiable manifold R^5
In [77]:
X5.<tau,W,X,Y,Z> = R5.chart(r'tau:\tau W X Y Z')
X5
Out[77]:
In [78]:
Psi = E.diff_map(R5, {(XE, X5): [tau,
                                 cos(ch),
                                 sin(ch)*sin(th)*cos(ph), 
                                 sin(ch)*sin(th)*sin(ph), 
                                 sin(ch)*cos(th)]},
                 name='Psi', latex_name=r'\Psi')
print(Psi)
Psi.display()
Differentiable map Psi from the 4-dimensional differentiable manifold E to the 5-dimensional differentiable manifold R^5
Out[78]:

The Einstein cylinder:

In [79]:
graphE = XE.plot(X5, ambient_coords=(W,X,tau), mapping=Psi, 
                 fixed_coords={th:pi/2, ph:0.001}, max_range=4, 
                 number_values=9, color='silver', thickness=0.5,
                 label_axes=False)  # phi = 0 
graphE += XE.plot(X5, ambient_coords=(W,X,tau), mapping=Psi, 
                  fixed_coords={th:pi/2, ph:pi}, max_range=4, 
                  number_values=9, color='silver', thickness=0.5,
                  label_axes=False)  # phi = pi
show(graphE, aspect_ratio=1, axes_labels=['W', 'X', 'tau'])