This notebook details some computations in the extremal Reissner-Nordström spacetime.

Path: extremal_Reissner_Nordstrom.ipynb

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Kernel: SageMath 9.8

Extremal Reissner-Nordström spacetime

This SageMath notebook accompanies the article Peeling at extreme black hole horizons by Jack Borthwick, Eric Gourgoulhon and Jean-Philippe Nicolas, arXIv:2303.14574. It involves differential geometry tools implemented in SageMath through the SageManifolds project.

In [1]:

version()

'SageMath version 9.8, Release Date: 2023-02-11'

In [2]:

%display latex

Spacetime manifold

We declare the spacetime manifold $\mathscr{M}$ as a 4-dimensional Lorentzian manifold, with the keyword signature='negative'to indicate that the metric signature is chosen to be $(+,-,-,-)$ :

In [3]:

M = Manifold(4, 'M', latex_name=r'\mathscr{M}', structure='Lorentzian', 
             signature='negative')
print(M)
M

4-dimensional Lorentzian manifold M

$\displaystyle \mathscr{M}$

We consider $\mathscr{M}$ to be a part of the maximally extended extreme Reissner-Nordström spacetime, namely $\mathscr{M} = E_+ \cup E_-,$ where $E_+$ and $E_-$ are the open subsets covered by the outgoing and ingoing Eddington-Finkelstein coordinates, respectively:

In [4]:

Ep = M.open_subset('Ep', latex_name='E_+')
Em = M.open_subset('Em', latex_name='E_-')
M.declare_union(Ep, Em)

In [5]:

list(M.open_covers())

$\displaystyle \left[\{\mathscr{M}\}, \{E_-, E_+\}\right]$

The Schwarzschild-Droste coordinates $(t,r,\theta,\varphi)$ on $E_+$ :

In [6]:

XSp.<t,r,th,ph> = Ep.chart(r"t r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
XSp

$\displaystyle \left(E_+,(t, r, {\theta}, {\varphi})\right)$

In [7]:

XSp.coord_range()

$\displaystyle t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( 0 , +\infty \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\varphi} :\ \left[ 0 , 2 \, \pi \right] \mbox{(periodic)}$

Notation: in this notebook, all coordinate charts have a Python name starting with X, like XSp above (S standing for Schwarzschild and p for $+$ ).

The Schwarzschild-Droste coordinates $(t,r,\theta,\varphi)$ on $E_-$ :

In [8]:

XSm.<t,r,th,ph> = Em.chart(r"t r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
XSm

$\displaystyle \left(E_-,(t, r, {\theta}, {\varphi})\right)$

In [9]:

XSm.coord_range()

$\displaystyle t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( 0 , +\infty \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\varphi} :\ \left[ 0 , 2 \, \pi \right] \mbox{(periodic)}$

The black hole exterior is $\mathrm{Ext} = E_+ \cap E_-$ We therefore declare it as follows:

In [10]:

Ext = Ep.intersection(Em, name='Ext', latex_name=r'\mathrm{Ext}')
Ext

$\displaystyle \mathrm{Ext}$

The mass parameter of the extreme Reissner-Nordström spacetime:

In [11]:

m = var('m', domain='real')
assume(m>0)

The Schwarzschild-Droste coordinates $(t,r,\theta,\varphi)$ on $\mathrm{Ext}$ are defined from the restriction of the chart XSp to $r>m$ :

In [12]:

XSE = XSp.restrict(Ext, r>m)
XSE

$\displaystyle \left(\mathrm{Ext},(t, r, {\theta}, {\varphi})\right)$

In [13]:

XSE.coord_range()

$\displaystyle t :\ \left( -\infty, +\infty \right) ;\quad r :\ \left( m , +\infty \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\varphi} :\ \left[ 0 , 2 \, \pi \right] \mbox{(periodic)}$

The chart XSE coincides with the restriction of the chart XSm to $r>m$ :

In [14]:

XSE is XSm.restrict(Ext, r>m)

$\displaystyle \mathrm{True}$

In [15]:

list(M.subsets())

$\displaystyle \left[E_+, \mathscr{M}, \mathrm{Ext}, E_-\right]$

At this stage, 3 charts have been constructed on $\mathscr{M}$ and one of them is considered as the "default" chart (i.e. the chart that is used if omitted in the arguments of a function):

In [16]:

M.atlas()

$\displaystyle \left[\left(E_+,(t, r, {\theta}, {\varphi})\right), \left(E_-,(t, r, {\theta}, {\varphi})\right), \left(\mathrm{Ext},(t, r, {\theta}, {\varphi})\right)\right]$

In [17]:

M.default_chart()

$\displaystyle \left(E_+,(t, r, {\theta}, {\varphi})\right)$

Metric tensor

We define the metric tensor $g$ from its components in the charts $(E_+,(t,r,\theta,\varphi))$ and $(E_-,(t,r,\theta,\varphi))$ :

In [18]:

g = M.metric()

F = (1 - m/r)^2

gp = g.restrict(Ep)
gp[0,0] = F
gp[1,1] = -1/F
gp[2,2] = -r^2
gp[3,3] = -r^2*sin(th)^2

gm = g.restrict(Em)
gm[0,0] = F
gm[1,1] = -1/F
gm[2,2] = -r^2
gm[3,3] = -r^2*sin(th)^2

In [19]:

g.display()  # display in the default chart, i.e. XSp

$\displaystyle g = {\left(\frac{m}{r} - 1\right)}^{2} \mathrm{d} t\otimes \mathrm{d} t -\frac{1}{{\left(\frac{m}{r} - 1\right)}^{2}} \mathrm{d} r\otimes \mathrm{d} r -r^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -r^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

In [20]:

g.display(XSp) # same as above

In [21]:

g.display(XSm)

Eddington-Finkelstein coordinates

Tortoise coordinate [Eq. (2.4)]

In [22]:

rstar(r) = r - m + 2*m*ln(abs(r - m)/m) - m^2/(r - m)
rstar(r)

$\displaystyle 2 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - m + \frac{m^{2}}{m - r} + r$

Note that the additive constant is chosen so that $r_* = 0$ for $r=2m$ :

In [23]:

rstar(2*m)

$\displaystyle 0$

The chart of the outgoing Eddington-Finkelstein coordinates $(u,r,\theta,\varphi)$ on $E_+$ :

In [24]:

XOEF.<u,r,th,ph> = Ep.chart(r"u r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
XOEF

$\displaystyle \left(E_+,(u, r, {\theta}, {\varphi})\right)$

In [25]:

Sp_to_OEF = XSp.transition_map(XOEF, [t - rstar(r), r, th, ph])  # Eq. (2.5)
Sp_to_OEF.display()

$\displaystyle \left\{\begin{array}{lcl} u & = & -2 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) + m - \frac{m^{2}}{m - r} - r + t \\ r & = & r \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [26]:

Sp_to_OEF.inverse().display()

$\displaystyle \left\{\begin{array}{lcl} t & = & -\frac{2 \, m^{2} \log\left(m\right) - 2 \, {\left(m \log\left(m\right) + m\right)} r + r^{2} - {\left(m - r\right)} u - 2 \, {\left(m^{2} - m r\right)} \log\left({\left| -m + r \right|}\right)}{m - r} \\ r & = & r \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [27]:

gp.display(XOEF)

$\displaystyle g = \left( \frac{m^{2} - 2 \, m r + r^{2}}{r^{2}} \right) \mathrm{d} u\otimes \mathrm{d} u +\mathrm{d} u\otimes \mathrm{d} r +\mathrm{d} r\otimes \mathrm{d} u -r^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -r^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

The transition maps are propagated from $E_+$ to the subset $\mathrm{Ext}$ :

In [28]:

Ep.coord_change(XSp, XOEF).restrict(Ext)
Ep.coord_change(XOEF, XSp).restrict(Ext)

$\displaystyle \left(\mathrm{Ext},(u, r, {\theta}, {\varphi})\right) \rightarrow \left(\mathrm{Ext},(t, r, {\theta}, {\varphi})\right)$

The chart of the ingoing Eddington-Finkelstein coordinates $(v,r,\theta,\varphi)$ on $E_-$ :

In [29]:

XIEF.<v,r,th,ph> = Em.chart(r"v r:(0,+oo) th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
XIEF

$\displaystyle \left(E_-,(v, r, {\theta}, {\varphi})\right)$

In [30]:

Sm_to_IEF = XSm.transition_map(XIEF, [t + rstar(r), r, th, ph])
Sm_to_IEF.display()

$\displaystyle \left\{\begin{array}{lcl} v & = & 2 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - m + \frac{m^{2}}{m - r} + r + t \\ r & = & r \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [31]:

Sm_to_IEF.inverse().display()

$\displaystyle \left\{\begin{array}{lcl} t & = & \frac{2 \, m^{2} \log\left(m\right) - 2 \, {\left(m \log\left(m\right) + m\right)} r + r^{2} + {\left(m - r\right)} v - 2 \, {\left(m^{2} - m r\right)} \log\left({\left| -m + r \right|}\right)}{m - r} \\ r & = & r \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [32]:

gm.display(XIEF)

$\displaystyle g = \left( \frac{m^{2} - 2 \, m r + r^{2}}{r^{2}} \right) \mathrm{d} v\otimes \mathrm{d} v -\mathrm{d} v\otimes \mathrm{d} r -\mathrm{d} r\otimes \mathrm{d} v -r^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -r^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

The transition maps are propagated from $E_-$ to the subset $\mathrm{Ext}$ :

In [33]:

Em.coord_change(XSm, XIEF).restrict(Ext)
Em.coord_change(XIEF, XSm).restrict(Ext)

$\displaystyle \left(\mathrm{Ext},(v, r, {\theta}, {\varphi})\right) \rightarrow \left(\mathrm{Ext},(t, r, {\theta}, {\varphi})\right)$

Principal null directions

The tangent vector $n_+$ to the outgoing principal null geodesics is defined according to Eq. (2.7):

In [34]:

np = Ep.vector_field(1/F, 1, 0, 0, 
                     name='np', latex_name=r'n_+')
np.display()

$\displaystyle n_+ = \frac{1}{{\left(\frac{m}{r} - 1\right)}^{2}} \frac{\partial}{\partial t } +\frac{\partial}{\partial r }$

$n_+$ is a null vector:

In [35]:

g(np, np).expr()

$\displaystyle 0$

and is a geodesic vector, i.e. it obeys $\nabla_{n_+} n_+ = 0$ :

In [36]:

nabla = g.connection()

In [37]:

nabla(np).contract(np).display()

$\displaystyle 0$

Similarly, the tangent vector $n_-$ to the ingoing principal null geodesics is [cf. Eq. (2.7)]:

In [38]:

nm = Em.vector_field(1/F, -1, 0, 0, 
                     name='nm', latex_name=r'n_-')
nm.display()

$\displaystyle n_- = \frac{1}{{\left(\frac{m}{r} - 1\right)}^{2}} \frac{\partial}{\partial t } -\frac{\partial}{\partial r }$

$n_-$ is a null geodesic vector:

In [39]:

g(nm, nm).expr()

$\displaystyle 0$

In [40]:

nabla(nm).contract(nm).display()

$\displaystyle 0$

Compactified pictures of $E_+$ , $E_-$ and $\mathscr{M} = E_+ \cup E_-$

We introduce the Euclidean plane $\mathbb{R}^2$ to draw some pictures:

In [41]:

R2.<T,X> = EuclideanSpace()
X2 = R2.cartesian_coordinates()

and we define a map from $\mathscr{M}$ to a compact region of $\mathbb{R}^2$ by means of the arctangent function:

In [42]:

Psi = M.diff_map(R2, {(XOEF, X2): (atan(u/(2*m)) + atan((u+2*rstar(r))/(2*m)) - pi*unit_step(m - r),
                                   atan((u+2*rstar(r))/(2*m)) - atan(u/(2*m)) - pi*unit_step(m - r)),
                      (XIEF, X2): (atan((v-2*rstar(r))/(2*m)) + atan(v/(2*m)) + pi*unit_step(m - r),
                                   atan(v/(2*m)) - atan((v-2*rstar(r))/(2*m)) - pi*unit_step(m - r))},
                 name='Psi', latex_name=r'\Psi')
Psi.display(XOEF, X2)

$\displaystyle \begin{array}{llcl} \Psi:& \mathscr{M} & \longrightarrow & \mathbb{E}^{2} \\ \mbox{on}\ E_+ : & \left(u, r, {\theta}, {\varphi}\right) & \longmapsto & \left(T, X\right) = \left(-\pi \mathrm{u}\left(m - r\right) + \arctan\left(\frac{4 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - 2 \, m + \frac{2 \, m^{2}}{m - r} + 2 \, r + u}{2 \, m}\right) + \arctan\left(\frac{u}{2 \, m}\right), -\pi \mathrm{u}\left(m - r\right) + \arctan\left(\frac{4 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - 2 \, m + \frac{2 \, m^{2}}{m - r} + 2 \, r + u}{2 \, m}\right) - \arctan\left(\frac{u}{2 \, m}\right)\right) \end{array}$

In [43]:

Psi.display(XIEF, X2)

$\displaystyle \begin{array}{llcl} \Psi:& \mathscr{M} & \longrightarrow & \mathbb{E}^{2} \\ \mbox{on}\ E_- : & \left(v, r, {\theta}, {\varphi}\right) & \longmapsto & \left(T, X\right) = \left(\pi \mathrm{u}\left(m - r\right) + \arctan\left(-\frac{4 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - 2 \, m + \frac{2 \, m^{2}}{m - r} + 2 \, r - v}{2 \, m}\right) + \arctan\left(\frac{v}{2 \, m}\right), -\pi \mathrm{u}\left(m - r\right) - \arctan\left(-\frac{4 \, m \log\left(\frac{{\left| -m + r \right|}}{m}\right) - 2 \, m + \frac{2 \, m^{2}}{m - r} + 2 \, r - v}{2 \, m}\right) + \arctan\left(\frac{v}{2 \, m}\right)\right) \end{array}$

We use this map to get a compactified view of $E_+$ , with the outgoing null geodesics as solid green lines and the curves $r=\mathrm{const}$ depicted in red, via the method plot of the chart XOEF = $(E_+, (u,r,\theta,\varphi))$ :

In [44]:

graphEp = XOEF.plot(X2, mapping=Psi, ambient_coords=(X,T), 
                    fixed_coords={th: pi/2,  ph: pi}, color={u: 'red', r: 'green'}, 
                    ranges={u: (-40, 40), r: (0, 20)}, number_values={u: 17, r: 16},
                    parameters={m: 1})
graphEp

Let us superpose the hypersurfaces

$r=0$ (the curvature singularity): orange dotted line
$r=m$ (the past event horizon $\mathscr{H}^-$ ): solid black line
$r=2m$ (the photon sphere): red dashed line

In [45]:

def plot_const_r_Ep(r0, color='red', linestyle='-', thickness=1, plot_points=300):
    return XOEF.plot(X2, mapping=Psi, ambient_coords=(X,T), 
                     fixed_coords={r: r0, th: pi/2, ph: pi},
                     ranges={u: (-100, 100)}, color=color, style=linestyle,
                     thickness=thickness, plot_points=plot_points, parameters={m: 1})

graphEp += plot_const_r_Ep(0, color='orange', linestyle=':', thickness=5)
graphEp += plot_const_r_Ep(1.001*m, color='black', thickness=3)
graphEp += plot_const_r_Ep(2*m, color='red', linestyle='--', thickness=3)
graphEp += plot_const_r_Ep(0.8*m)
graphEp

The above figure is similar to Fig. 1a of the article tilted by $45^\circ$ .

Similarly, we get a compactified view of $E_-$ via the method plot of the chart XIEF= $(E_-, (v,r,\theta,\varphi))$ , with the ingoing null geodesics as dashed green lines, the curves $r=\mathrm{const}$ depicted in red and the following hypersurfaces:

$r=0$ (the curvature singularity): orange dotted line
$r=m$ (the future event horizon $\mathscr{H}^+$ ): solid black line
$r=2m$ (the photon sphere): red dashed line

In [46]:

graphEn = XIEF.plot(X2, mapping=Psi, ambient_coords=(X,T), fixed_coords={th: pi/2,  ph: pi}, 
                  color={v: 'red', r: 'green'}, ranges={v: (-40, 40), r: (0, 20)},
                  number_values={v: 17, r: 16}, style={v: 'solid', r: 'dashed'},
                  parameters={m: 1})

def plot_const_r_En(r0, color='red', linestyle='-', thickness=1, plot_points=300):
    return XIEF.plot(X2, mapping=Psi, ambient_coords=(X,T), 
                     fixed_coords={r: r0, th: pi/2, ph: pi},
                     ranges={v: (-100, 100)}, color=color, style=linestyle,
                     thickness=thickness, plot_points=plot_points, parameters={m: 1})

graphEn += plot_const_r_En(0, color='orange', linestyle=':', thickness=5)
graphEn += plot_const_r_En(1.001*m, color='black', thickness=3)
graphEn += plot_const_r_En(2*m, color='red', linestyle='--', thickness=3)
graphEn += plot_const_r_En(0.8*m)
graphEn

The above figure is similar to Fig. 1b of the article tilted by $-45^\circ$ .

A compactified view of $\mathscr{M}$ , with the outgoing (resp. ingoing) null geodesics as solid (resp. dashed) green lines and the curves $r=\mathrm{const}$ depicted in red:

In [47]:

graphM = graphEp + graphEn
graphM

The above figure is similar to Fig. 2 of the article.

At this stage, 7 charts have been constructed on $\mathscr{M}$ :

In [48]:

M.atlas()

$\displaystyle \left[\left(E_+,(t, r, {\theta}, {\varphi})\right), \left(E_-,(t, r, {\theta}, {\varphi})\right), \left(\mathrm{Ext},(t, r, {\theta}, {\varphi})\right), \left(E_+,(u, r, {\theta}, {\varphi})\right), \left(\mathrm{Ext},(u, r, {\theta}, {\varphi})\right), \left(E_-,(v, r, {\theta}, {\varphi})\right), \left(\mathrm{Ext},(v, r, {\theta}, {\varphi})\right)\right]$

The conformally compactified exterior

Let us introduce the coordinate $R = 1/r$ on $\mathrm{Ext}$ via a new chart:

In [49]:

XR.<u, R, th, ph> = Ext.chart(r"u R:(0,1/m) th:(0,pi):\theta ph:(0,2*pi):periodic:\varphi")
XR

$\displaystyle \left(\mathrm{Ext},(u, R, {\theta}, {\varphi})\right)$

In [50]:

XR.coord_range()

$\displaystyle u :\ \left( -\infty, +\infty \right) ;\quad R :\ \left( 0 , \frac{1}{m} \right) ;\quad {\theta} :\ \left( 0 , \pi \right) ;\quad {\varphi} :\ \left[ 0 , 2 \, \pi \right] \mbox{(periodic)}$

In [51]:

XOEF_to_XR = XOEF.restrict(Ext).transition_map(XR, (u, 1/r, th, ph))
XOEF_to_XR.display()

$\displaystyle \left\{\begin{array}{lcl} u & = & u \\ R & = & \frac{1}{r} \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [52]:

XOEF_to_XR.inverse().display()

$\displaystyle \left\{\begin{array}{lcl} u & = & u \\ r & = & \frac{1}{R} \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

We get the change of coordinates $(t,r,\theta,\varphi) \to (u,R,\theta,\varphi)$ by combining previously defined changes of coordinates via the operator *:

In [53]:

XSE_to_XR = XOEF_to_XR * Ext.coord_change(XSE, XOEF.restrict(Ext))
XSE_to_XR.display()

$\displaystyle \left\{\begin{array}{lcl} u & = & \frac{2 \, m^{2} \log\left(m\right) - 2 \, {\left(m \log\left(m\right) + m\right)} r + r^{2} + {\left(m - r\right)} t - 2 \, {\left(m^{2} - m r\right)} \log\left({\left| -m + r \right|}\right)}{m - r} \\ R & = & \frac{1}{r} \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

In [54]:

XSE_to_XR.inverse().display()

$\displaystyle \left\{\begin{array}{lcl} t & = & -\frac{2 \, R^{2} m^{2} \log\left(R\right) - 2 \, {\left(R \log\left(R\right) + R\right)} m - {\left(R^{2} m - R\right)} u + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(m\right) - 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left({\left| R m - 1 \right|}\right) + 1}{R^{2} m - R} \\ r & = & \frac{1}{R} \\ {\theta} & = & {\theta} \\ {\varphi} & = & {\varphi} \end{array}\right.$

From now on, we use $(u,R,\theta,\varphi)$ as the default coordinates on $\mathrm{Ext}$ :

In [55]:

Ext.set_default_chart(XR)
Ext.set_default_frame(XR.frame())

The metric $g$ in terms of the "compactified" coordinates $(u,R,\theta,\varphi)$ :

In [56]:

gE = g.restrict(Ext)
gE.display()

$\displaystyle g = \left( R^{2} m^{2} - 2 \, R m + 1 \right) \mathrm{d} u\otimes \mathrm{d} u -\frac{1}{R^{2}} \mathrm{d} u\otimes \mathrm{d} R -\frac{1}{R^{2}} \mathrm{d} R\otimes \mathrm{d} u -\frac{1}{R^{2}} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\frac{\sin\left({\theta}\right)^{2}}{R^{2}} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

Conformal metric $\hat{g}$

We use $R^2$ as the conformal factor, defining $\hat{g}$ by

In [57]:

gh = Ext.lorentzian_metric('gh', latex_name=r'\hat{g}', signature='negative')
gh.set(R^2 * gE)
gh.display()

$\displaystyle \hat{g} = \left( R^{4} m^{2} - 2 \, R^{3} m + R^{2} \right) \mathrm{d} u\otimes \mathrm{d} u -\mathrm{d} u\otimes \mathrm{d} R -\mathrm{d} R\otimes \mathrm{d} u -\mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

In [58]:

gh.apply_map(factor, keep_other_components=True)
gh.display()

$\displaystyle \hat{g} = {\left(R m - 1\right)}^{2} R^{2} \mathrm{d} u\otimes \mathrm{d} u -\mathrm{d} u\otimes \mathrm{d} R -\mathrm{d} R\otimes \mathrm{d} u -\mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\sin\left({\theta}\right)^{2} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

This expression agrees with Eq. (2.9).

Couch-Torrence inversion

We define the Couch-Torrence inversion $\Phi$ according to Eq. (3.1):

In [59]:

Phi = Ext.diffeomorphism(Ext, {(XSE, XSE): (t, m*r/(r - m), th, ph)}, 
                         name='Phi', latex_name=r'\Phi')
Phi.display(XSE, XSE)

$\displaystyle \begin{array}{llcl} \Phi:& \mathrm{Ext} & \longrightarrow & \mathrm{Ext} \\ & \left(t, r, {\theta}, {\varphi}\right) & \longmapsto & \left(t, -\frac{m r}{m - r}, {\theta}, {\varphi}\right) \end{array}$

The Couch-Torrence inversion takes a simple form in terms of the tortoise coordinate $r_*$ :

In [60]:

Phi_r(r) = Phi.expr(XSE, XSE)[1]
Phi_r

$\displaystyle r \ {\mapsto}\ -\frac{m r}{m - r}$

In [61]:

(rstar(Phi_r(r)) + rstar(r)).simplify_log()

$\displaystyle 0$

The inverse of $\Phi$ is

In [62]:

Phi.inverse().display(XSE, XSE)

$\displaystyle \begin{array}{llcl} \Phi^{-1}:& \mathrm{Ext} & \longrightarrow & \mathrm{Ext} \\ & \left(t, r, {\theta}, {\varphi}\right) & \longmapsto & \left(t, -\frac{m r}{m - r}, {\theta}, {\varphi}\right) \end{array}$

$\Phi$ is an involution:

In [63]:

Phi.inverse() == Phi

$\displaystyle \mathrm{True}$

In [64]:

Phi.display(XR, XR)

$\displaystyle \begin{array}{llcl} \Phi:& \mathrm{Ext} & \longrightarrow & \mathrm{Ext} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \left(-\frac{4 \, R^{2} m^{2} \log\left(R\right) - 4 \, {\left(R \log\left(R\right) + R\right)} m - {\left(R^{2} m - R\right)} u + 4 \, {\left(R^{2} m^{2} - R m\right)} \log\left(m\right) - 4 \, {\left(R^{2} m^{2} - R m\right)} \log\left({\left| R m - 1 \right|}\right) + 2}{R^{2} m - R}, -\frac{R m - 1}{m}, {\theta}, {\varphi}\right) \end{array}$

The Couch-Torrence inversion as a conformal isometry of $g$

The pullback of $g$ by $\Phi$ is

In [65]:

Pg = Phi.pullback(gE)
Pg.display(XSE)

$\displaystyle {\Phi}^*g = \frac{m^{2}}{r^{2}} \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{m^{2} r^{2}}{m^{4} - 4 \, m^{3} r + 6 \, m^{2} r^{2} - 4 \, m r^{3} + r^{4}} \right) \mathrm{d} r\otimes \mathrm{d} r + \left( -\frac{m^{2} r^{2}}{m^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \left( -\frac{m^{2} r^{2} \sin\left({\theta}\right)^{2}}{m^{2} - 2 \, m r + r^{2}} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

In [66]:

Pg.apply_map(factor, frame=XSE.frame(), chart=XSE, keep_other_components=True)
Pg.display(XSE)

$\displaystyle {\Phi}^*g = \frac{m^{2}}{r^{2}} \mathrm{d} t\otimes \mathrm{d} t -\frac{m^{2} r^{2}}{{\left(m - r\right)}^{4}} \mathrm{d} r\otimes \mathrm{d} r -\frac{m^{2} r^{2}}{{\left(m - r\right)}^{2}} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} -\frac{m^{2} r^{2} \sin\left({\theta}\right)^{2}}{{\left(m - r\right)}^{2}} \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

We notice that $\Phi^* g = \frac{m^2}{(r - m)^2} g$ Indeed:

In [67]:

Pg == m^2/(r - m)^2 * gE

$\displaystyle \mathrm{True}$

Hence, $\Phi$ is a conformal isometry of $\left. g \right| _{\rm Ext}$ , with conformal factor $\Omega = \frac{m}{r - m}$ .

The Couch-Torrence inversion as an isometry of $\hat{g}$

Let us check that $\Phi^*\hat{g} = \hat{g}$ [Eq. (3.3)]:

In [68]:

Phi.pullback(gh) == gh

$\displaystyle \mathrm{True}$

Computations for the peeling at infinity

The conformal metric on ${\rm Ext}$ is

In [69]:

gh.display()

Its inverse is [unumbered equation above Eq. (4.1)]

In [70]:

gh.inverse().display()

$\displaystyle \hat{g}^{-1} = -\frac{\partial}{\partial u }\otimes \frac{\partial}{\partial R }-\frac{\partial}{\partial R }\otimes \frac{\partial}{\partial u } + \left( -R^{4} m^{2} + 2 \, R^{3} m - R^{2} \right) \frac{\partial}{\partial R }\otimes \frac{\partial}{\partial R } -\frac{\partial}{\partial {\theta} }\otimes \frac{\partial}{\partial {\theta} } -\frac{1}{\sin\left({\theta}\right)^{2}} \frac{\partial}{\partial {\varphi} }\otimes \frac{\partial}{\partial {\varphi} }$

Its scalar curvature is [Eq. (4.1)]

In [71]:

gh.ricci_scalar().expr()

$\displaystyle 12 \, R^{2} m^{2} - 12 \, R m$

The volume 4-form of $\hat{g}$ is [Eq. (4.2)]

In [72]:

gh.volume_form().display()

$\displaystyle \epsilon_{\hat{g}} = \sin\left({\theta}\right) \mathrm{d} u\wedge \mathrm{d} R\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi}$

The d'Alembertian w.r.t. $\hat{g}$ of a generic function $f$ is

In [73]:

fs = Ext.scalar_field(function('f')(u,R,th,ph))
Df = fs.dalembertian(gh).coord_function()
Df

$\displaystyle -\frac{2 \, {\left(2 \, R^{3} m^{2} - 3 \, R^{2} m + R\right)} \sin\left({\theta}\right)^{2} \frac{\partial\,f}{\partial R} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \frac{\partial^2\,f}{\partial R ^ 2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial^2\,f}{\partial u\partial R} + \cos\left({\theta}\right) \sin\left({\theta}\right) \frac{\partial\,f}{\partial {\theta}} + \sin\left({\theta}\right)^{2} \frac{\partial^2\,f}{\partial {\theta} ^ 2} + \frac{\partial^2\,f}{\partial {\varphi} ^ 2}}{\sin\left({\theta}\right)^{2}}$

In [74]:

Df.expand()

$\displaystyle -R^{4} m^{2} \frac{\partial^2\,f}{\partial R ^ 2} - 4 \, R^{3} m^{2} \frac{\partial\,f}{\partial R} + 2 \, R^{3} m \frac{\partial^2\,f}{\partial R ^ 2} + 6 \, R^{2} m \frac{\partial\,f}{\partial R} - R^{2} \frac{\partial^2\,f}{\partial R ^ 2} - 2 \, R \frac{\partial\,f}{\partial R} - \frac{\cos\left({\theta}\right) \frac{\partial\,f}{\partial {\theta}}}{\sin\left({\theta}\right)} - \frac{\frac{\partial^2\,f}{\partial {\varphi} ^ 2}}{\sin\left({\theta}\right)^{2}} - 2 \, \frac{\partial^2\,f}{\partial u\partial R} - \frac{\partial^2\,f}{\partial {\theta} ^ 2}$

Check of Eq. (4.5):

In [75]:

Df + 2*diff(f(u,R,th,ph), u, R) + diff(R^2*(1 - m*R)^2*diff(f(u,R,th,ph), R), R)

$\displaystyle -\frac{\cos\left({\theta}\right) \sin\left({\theta}\right) \frac{\partial\,f}{\partial {\theta}} + \sin\left({\theta}\right)^{2} \frac{\partial^2\,f}{\partial {\theta} ^ 2} + \frac{\partial^2\,f}{\partial {\varphi} ^ 2}}{\sin\left({\theta}\right)^{2}}$

The Levi-Civita connection of $\hat{g}$ :

In [76]:

nablah = gh.connection()

Morawetz vector field

The Morawetz vector field is defined according to Eq. (4.6):

In [77]:

K = Ext.vector_field(u^2, -2*(1 + u*R), 0, 0, name='K')
K.display()

$\displaystyle K = u^{2} \frac{\partial}{\partial u } + \left( -2 \, R u - 2 \right) \frac{\partial}{\partial R }$

Its Killing form is [Eq. (4.7)]:

In [78]:

KilK = nablah(K.down(gh)).symmetrize()
KilK.display()

$\displaystyle \left( -4 \, R^{3} m^{2} + 6 \, R^{2} m - 2 \, {\left(R^{4} m^{2} - R^{3} m\right)} u \right) \mathrm{d} u\otimes \mathrm{d} u$

In [79]:

KilK.apply_map(factor)
KilK.display()

$\displaystyle -2 \, {\left(R^{2} m u + 2 \, R m - R u - 3\right)} R^{2} m \mathrm{d} u\otimes \mathrm{d} u$

Energy-momentum tensor of a generic scalar field $\phi$

In [80]:

phis = Ext.scalar_field(function('phi')(u,R,th,ph),
                        name='phi', latex_name=r'\phi')
phis.display(XR)

$\displaystyle \begin{array}{llcl} \phi:& \mathrm{Ext} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & \phi\left(u, R, {\theta}, {\varphi}\right) \end{array}$

The covariant derivative of $\hat{\nabla}\phi$ :

In [81]:

nab_phi = nablah(phis)

Since $\phi$ is a scalar field, we have of course $\hat{\nabla}\phi = \mathrm{d}\phi$ :

In [82]:

nab_phi.display()

$\displaystyle \mathrm{d}\phi = \frac{\partial\,\phi}{\partial u} \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \mathrm{d} R + \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}$

In [83]:

nab_phi == diff(phis)

$\displaystyle \mathrm{True}$

In [84]:

T = nab_phi * nab_phi - 1/2*gh.inverse()(nab_phi, nab_phi) * gh
T.set_name('T')
T.display()

$\displaystyle T = \left( \frac{2 \, {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + {\left(R^{8} m^{4} - 4 \, R^{7} m^{3} + 6 \, R^{6} m^{2} - 4 \, R^{5} m + R^{4}\right)} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial R}^{2} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u\otimes \mathrm{d} u + \left( -\frac{{\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} u\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} u\otimes \mathrm{d} {\varphi} + \left( -\frac{{\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} R\otimes \mathrm{d} u + \left(\frac{\partial\,\phi}{\partial R}\right)^{2} \mathrm{d} R\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} R\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} R\otimes \mathrm{d} {\varphi} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta}\otimes \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \mathrm{d} {\theta}\otimes \mathrm{d} R + \left( -\frac{{\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} - \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + \frac{\partial\,\phi}{\partial {\theta}} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\theta}\otimes \mathrm{d} {\varphi} + \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} u + \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} R + \frac{\partial\,\phi}{\partial {\theta}} \frac{\partial\,\phi}{\partial {\varphi}} \mathrm{d} {\varphi}\otimes \mathrm{d} {\theta} + \left( -\frac{1}{2} \, {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial R}\right)^{2} - \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} - \frac{1}{2} \, \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + \frac{1}{2} \, \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} \right) \mathrm{d} {\varphi}\otimes \mathrm{d} {\varphi}$

Energy current

The energy current associated to the Morawez vector field is defined according to Eq. (4.8):

In [85]:

J = T.contract(K)
J.set_name('J')
J.display()

$\displaystyle J = \left( \frac{2 \, {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + 2 \, u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} + {\left({\left(R^{8} m^{4} - 4 \, R^{7} m^{3} + 6 \, R^{6} m^{2} - 4 \, R^{5} m + R^{4}\right)} u^{2} \sin\left({\theta}\right)^{2} + 2 \, {\left(R^{5} m^{2} - 2 \, R^{4} m + R^{3}\right)} u \sin\left({\theta}\right)^{2} + 2 \, {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + {\left({\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} + 2 \, R u \sin\left({\theta}\right)^{2} + 2 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + {\left({\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} + 2 \, R u + 2\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} u + \left( -\frac{u^{2} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + u^{2} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} + {\left({\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} + 4 \, R u \sin\left({\theta}\right)^{2} + 4 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial R}\right)^{2}}{2 \, \sin\left({\theta}\right)^{2}} \right) \mathrm{d} R + \left( u^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} - 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \right) \mathrm{d} {\theta} + \left( u^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} - 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}} \right) \mathrm{d} {\varphi}$

The Hodge dual of $J$ :

In [86]:

star_J = J.hodge_dual(gh)
star_J.display()

$\displaystyle \star J = \left( \frac{u^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\varphi}} - 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\varphi}}}{\sin\left({\theta}\right)} \right) \mathrm{d} u\wedge \mathrm{d} R\wedge \mathrm{d} {\theta} + \left( -u^{2} \sin\left({\theta}\right) \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial {\theta}} + 2 \, {\left(R u \sin\left({\theta}\right) + \sin\left({\theta}\right)\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} \right) \mathrm{d} u\wedge \mathrm{d} R\wedge \mathrm{d} {\varphi} + \left( \frac{{\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u} \frac{\partial\,\phi}{\partial R} + u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial u}^{2} - {\left({\left(R^{5} m^{2} - 2 \, R^{4} m + R^{3}\right)} u \sin\left({\theta}\right)^{2} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + {\left(R u \sin\left({\theta}\right)^{2} + \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + {\left(R u + 1\right)} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2}}{\sin\left({\theta}\right)} \right) \mathrm{d} u\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi} + \left( \frac{u^{2} \sin\left({\theta}\right)^{2} \left(\frac{\partial\,\phi}{\partial {\theta}}\right)^{2} + u^{2} \left(\frac{\partial\,\phi}{\partial {\varphi}}\right)^{2} + {\left({\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} + 4 \, R u \sin\left({\theta}\right)^{2} + 4 \, \sin\left({\theta}\right)^{2}\right)} \left(\frac{\partial\,\phi}{\partial R}\right)^{2}}{2 \, \sin\left({\theta}\right)} \right) \mathrm{d} R\wedge \mathrm{d} {\theta}\wedge \mathrm{d} {\varphi}$

The divergence $\hat{\nabla}^a J_a$ :

In [87]:

divJ = nablah(J).up(gh, 1).trace()
divJ.coord_function()

$\displaystyle \frac{4 \, {\left(R u \sin\left({\theta}\right)^{2} + \sin\left({\theta}\right)^{2}\right)} \frac{\partial^2\,\phi}{\partial u\partial R} \frac{\partial\,\phi}{\partial R} + 2 \, {\left({\left(3 \, R^{4} m^{2} - 5 \, R^{3} m + 2 \, R^{2}\right)} u \sin\left({\theta}\right)^{2} + {\left(2 \, R^{3} m^{2} - 3 \, R^{2} m + 2 \, R\right)} \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R}^{2} + 2 \, {\left({\left(R^{5} m^{2} - 2 \, R^{4} m + R^{3}\right)} u \sin\left({\theta}\right)^{2} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial^2\,\phi}{\partial R ^ 2} + 2 \, {\left(R u \cos\left({\theta}\right) \sin\left({\theta}\right) + \cos\left({\theta}\right) \sin\left({\theta}\right)\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial\,\phi}{\partial {\theta}} + 2 \, {\left(R u \sin\left({\theta}\right)^{2} + \sin\left({\theta}\right)^{2}\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial^2\,\phi}{\partial {\theta} ^ 2} + 2 \, {\left(R u + 1\right)} \frac{\partial\,\phi}{\partial R} \frac{\partial^2\,\phi}{\partial {\varphi} ^ 2} - {\left(2 \, {\left(2 \, R^{3} m^{2} - 3 \, R^{2} m + R\right)} u^{2} \sin\left({\theta}\right)^{2} \frac{\partial\,\phi}{\partial R} + {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u^{2} \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial R ^ 2} + 2 \, u^{2} \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial u\partial R} + u^{2} \cos\left({\theta}\right) \sin\left({\theta}\right) \frac{\partial\,\phi}{\partial {\theta}} + u^{2} \sin\left({\theta}\right)^{2} \frac{\partial^2\,\phi}{\partial {\theta} ^ 2} + u^{2} \frac{\partial^2\,\phi}{\partial {\varphi} ^ 2}\right)} \frac{\partial\,\phi}{\partial u}}{\sin\left({\theta}\right)^{2}}$

This expression can be simplified by taking into account the wave equation (4.4) satisfied by $\phi$ : $\Box_{\hat{g}} \phi + 2 m R (m R - 1) \phi = 0$ To enforce the simplification, we extract $\frac{\partial^2\phi}{\partial\theta^2}$ from this equation and substitute it in $\hat{\nabla}^a J_a$ :

In [88]:

wave_eq = phis.dalembertian(gh) + 2*m*R*(m*R - 1)*phis
d2phidth2 = (wave_eq.expr() + diff(function('phi')(u,R,th,ph), th, th)).simplify_full()
d2phidth2

$\displaystyle \frac{2 \, {\left(R^{2} m^{2} - R m\right)} \phi\left(u, R, {\theta}, {\varphi}\right) \sin\left({\theta}\right)^{2} - 2 \, {\left(2 \, R^{3} m^{2} - 3 \, R^{2} m + R\right)} \sin\left({\theta}\right)^{2} \frac{\partial}{\partial R}\phi\left(u, R, {\theta}, {\varphi}\right) - {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} \sin\left({\theta}\right)^{2} \frac{\partial^{2}}{(\partial R)^{2}}\phi\left(u, R, {\theta}, {\varphi}\right) - 2 \, \sin\left({\theta}\right)^{2} \frac{\partial^{2}}{\partial u\partial R}\phi\left(u, R, {\theta}, {\varphi}\right) - \cos\left({\theta}\right) \sin\left({\theta}\right) \frac{\partial}{\partial {\theta}}\phi\left(u, R, {\theta}, {\varphi}\right) - \frac{\partial^{2}}{(\partial {\varphi})^{2}}\phi\left(u, R, {\theta}, {\varphi}\right)}{\sin\left({\theta}\right)^{2}}$

In [89]:

divJ.set_expr(divJ.expr().subs({diff(function('phi')(u,R,th,ph), th, th): d2phidth2}).simplify_full())
divJ.coord_function()

$\displaystyle -2 \, {\left(R^{2} m^{2} - R m\right)} u^{2} \phi\left(u, R, {\theta}, {\varphi}\right) \frac{\partial\,\phi}{\partial u} + 4 \, {\left(R^{2} m^{2} - R m + {\left(R^{3} m^{2} - R^{2} m\right)} u\right)} \phi\left(u, R, {\theta}, {\varphi}\right) \frac{\partial\,\phi}{\partial R} - 2 \, {\left(2 \, R^{3} m^{2} - 3 \, R^{2} m + {\left(R^{4} m^{2} - R^{3} m\right)} u\right)} \frac{\partial\,\phi}{\partial R}^{2}$

This agrees with Eq. (4.16).

Vector field normal to $\mathcal{H}_{s, u_0}$

In [90]:

assume(R*m - 1 < 0)

Expression of $r_*$ in terms of $R$ :

In [91]:

rstar_R = rstar(1/R).simplify_full()
rstar_R

$\displaystyle \frac{2 \, R m + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(-\frac{R m - 1}{R m}\right) - 1}{R^{2} m - R}$

Let us introduce the scalar field $s=-u/r_*$ :

In [92]:

s = Ext.scalar_field(-u/rstar_R, name='s')
s.display(XR)

$\displaystyle \begin{array}{llcl} s:& \mathrm{Ext} & \longrightarrow & \mathbb{R} \\ & \left(u, R, {\theta}, {\varphi}\right) & \longmapsto & -\frac{{\left(R^{2} m - R\right)} u}{2 \, R m + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(-\frac{R m - 1}{R m}\right) - 1} \end{array}$

In [93]:

ds = diff(s)
ds.display()

$\displaystyle \mathrm{d}s = \left( \frac{R^{2} m - R}{2 \, R^{2} m^{2} \log\left(R\right) - 2 \, {\left(R \log\left(R\right) + R\right)} m - 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(-R m + 1\right) + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(m\right) + 1} \right) \mathrm{d} u + \left( -\frac{u}{4 \, R^{4} m^{4} \log\left(R\right)^{2} - 8 \, {\left(R^{3} \log\left(R\right)^{2} + R^{3} \log\left(R\right)\right)} m^{3} + 4 \, {\left(R^{2} \log\left(R\right)^{2} + 3 \, R^{2} \log\left(R\right) + R^{2}\right)} m^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(-R m + 1\right)^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)^{2} - 4 \, {\left(R \log\left(R\right) + R\right)} m - 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m + 2 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)\right)} \log\left(-R m + 1\right) + 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m\right)} \log\left(m\right) + 1} \right) \mathrm{d} R$

This expression can be simplified. Indeed, we have

In [94]:

ds[0] + 1/rstar_R

$\displaystyle 0$

and

In [95]:

F_R = F.subs({r: 1/R})
F_R

$\displaystyle {\left(R m - 1\right)}^{2}$

In [96]:

ds[1] + u/(rstar_R^2 * F_R * R^2)

$\displaystyle 0$

Hence we set

In [97]:

ds[0] = - 1/rstar_R
ds[1] = - u/(rstar_R^2 * F_R * R^2)
ds.display()

$\displaystyle \mathrm{d}s = \left( -\frac{R^{2} m - R}{2 \, R m + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(-\frac{R m - 1}{R m}\right) - 1} \right) \mathrm{d} u -\frac{{\left(R^{2} m - R\right)}^{2} u}{{\left(2 \, R m + 2 \, {\left(R^{2} m^{2} - R m\right)} \log\left(-\frac{R m - 1}{R m}\right) - 1\right)}^{2} {\left(R m - 1\right)}^{2} R^{2}} \mathrm{d} R$

A normal vector field to $\mathcal{H}_{s, u_0}$ can be obtained as minus the dual of $\mathrm{d}s$ with respect to $\hat{g}$ :

In [98]:

n = - ds.up(gh)
n.display()

$\displaystyle \left( -\frac{u}{4 \, R^{4} m^{4} \log\left(R\right)^{2} - 8 \, {\left(R^{3} \log\left(R\right)^{2} + R^{3} \log\left(R\right)\right)} m^{3} + 4 \, {\left(R^{2} \log\left(R\right)^{2} + 3 \, R^{2} \log\left(R\right) + R^{2}\right)} m^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(-R m + 1\right)^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)^{2} - 4 \, {\left(R \log\left(R\right) + R\right)} m - 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m + 2 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)\right)} \log\left(-R m + 1\right) + 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m\right)} \log\left(m\right) + 1} \right) \frac{\partial}{\partial u } + \left( \frac{2 \, R^{4} m^{3} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{2} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m - {\left(R^{4} m^{2} - 2 \, R^{3} m + R^{2}\right)} u - 2 \, {\left(R^{4} m^{3} - 2 \, R^{3} m^{2} + R^{2} m\right)} \log\left(-R m + 1\right) + 2 \, {\left(R^{4} m^{3} - 2 \, R^{3} m^{2} + R^{2} m\right)} \log\left(m\right) - R}{4 \, R^{4} m^{4} \log\left(R\right)^{2} - 8 \, {\left(R^{3} \log\left(R\right)^{2} + R^{3} \log\left(R\right)\right)} m^{3} + 4 \, {\left(R^{2} \log\left(R\right)^{2} + 3 \, R^{2} \log\left(R\right) + R^{2}\right)} m^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(-R m + 1\right)^{2} + 4 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)^{2} - 4 \, {\left(R \log\left(R\right) + R\right)} m - 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m + 2 \, {\left(R^{4} m^{4} - 2 \, R^{3} m^{3} + R^{2} m^{2}\right)} \log\left(m\right)\right)} \log\left(-R m + 1\right) + 4 \, {\left(2 \, R^{4} m^{4} \log\left(R\right) - 2 \, {\left(2 \, R^{3} \log\left(R\right) + R^{3}\right)} m^{3} + {\left(2 \, R^{2} \log\left(R\right) + 3 \, R^{2}\right)} m^{2} - R m\right)} \log\left(m\right) + 1} \right) \frac{\partial}{\partial R }$

Check of the expression of $n$ given in Remark 4.1:

In [99]:

n[0] + u/((1 - m*R)^2 * (rstar_R*R)^2)

$\displaystyle 0$

In [100]:

n[1] + (1 + u/rstar_R)/rstar_R

$\displaystyle 0$

Extremal Reissner-Nordström spacetime

Spacetime manifold

Metric tensor

Eddington-Finkelstein coordinates

Principal null directions

Compactified pictures of E+E_+E+​, E−E_-E−​ and M=E+∪E−\mathscr{M} = E_+ \cup E_-M=E+​∪E−​

The conformally compactified exterior

Conformal metric g^\hat{g}g^​

Couch-Torrence inversion

The Couch-Torrence inversion as a conformal isometry of ggg

The Couch-Torrence inversion as an isometry of g^\hat{g}g^​

Computations for the peeling at infinity

Morawetz vector field

Energy-momentum tensor of a generic scalar field ϕ\phiϕ

Energy current

Vector field normal to Hs,u0\mathcal{H}_{s, u_0}Hs,u0​​

Compactified pictures of $E_+$ , $E_-$ and $\mathscr{M} = E_+ \cup E_-$

Conformal metric $\hat{g}$

The Couch-Torrence inversion as a conformal isometry of $g$

The Couch-Torrence inversion as an isometry of $\hat{g}$

Energy-momentum tensor of a generic scalar field $\phi$

Vector field normal to $\mathcal{H}_{s, u_0}$