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This notebook details some computations in the extremal Reissner-Nordström spacetime.
License: GPL3
Image: ubuntu2004
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.
Spacetime manifold
We declare the spacetime manifold as a 4-dimensional Lorentzian manifold, with the keyword signature='negative'
to indicate that the metric signature is chosen to be :
We consider to be a part of the maximally extended extreme Reissner-Nordström spacetime, namely where and are the open subsets covered by the outgoing and ingoing Eddington-Finkelstein coordinates, respectively:
The Schwarzschild-Droste coordinates on :
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 on :
The black hole exterior is We therefore declare it as follows:
The mass parameter of the extreme Reissner-Nordström spacetime:
The Schwarzschild-Droste coordinates on are defined from the restriction of the chart XSp
to :
The chart XSE
coincides with the restriction of the chart XSm
to :
At this stage, 3 charts have been constructed on 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):
Metric tensor
We define the metric tensor from its components in the charts and :
Eddington-Finkelstein coordinates
Tortoise coordinate [Eq. (2.4)]
Note that the additive constant is chosen so that for :
The chart of the outgoing Eddington-Finkelstein coordinates on :
The transition maps are propagated from to the subset :
The chart of the ingoing Eddington-Finkelstein coordinates on :
The transition maps are propagated from to the subset :
Principal null directions
The tangent vector to the outgoing principal null geodesics is defined according to Eq. (2.7):
is a null vector:
and is a geodesic vector, i.e. it obeys :
Similarly, the tangent vector to the ingoing principal null geodesics is [cf. Eq. (2.7)]:
is a null geodesic vector:
Compactified pictures of , and
We introduce the Euclidean plane to draw some pictures:
and we define a map from to a compact region of by means of the arctangent function:
We use this map to get a compactified view of , with the outgoing null geodesics as solid green lines and the curves depicted in red, via the method plot
of the chart XOEF
= :
Let us superpose the hypersurfaces
(the curvature singularity): orange dotted line
(the past event horizon ): solid black line
(the photon sphere): red dashed line
The above figure is similar to Fig. 1a of the article tilted by .
Similarly, we get a compactified view of via the method plot
of the chart XIEF
=, with the ingoing null geodesics as dashed green lines, the curves depicted in red and the following hypersurfaces:
(the curvature singularity): orange dotted line
(the future event horizon ): solid black line
(the photon sphere): red dashed line
The above figure is similar to Fig. 1b of the article tilted by .
A compactified view of , with the outgoing (resp. ingoing) null geodesics as solid (resp. dashed) green lines and the curves depicted in red:
The above figure is similar to Fig. 2 of the article.
At this stage, 7 charts have been constructed on :
The conformally compactified exterior
Let us introduce the coordinate on via a new chart:
We get the change of coordinates by combining previously defined changes of coordinates via the operator *
:
From now on, we use as the default coordinates on :
The metric in terms of the "compactified" coordinates :
Conformal metric
We use as the conformal factor, defining by
This expression agrees with Eq. (2.9).
Couch-Torrence inversion
We define the Couch-Torrence inversion according to Eq. (3.1):
The Couch-Torrence inversion takes a simple form in terms of the tortoise coordinate :
The inverse of is
is an involution:
The Couch-Torrence inversion as a conformal isometry of
The pullback of by is
We notice that Indeed:
Hence, is a conformal isometry of , with conformal factor .
The Couch-Torrence inversion as an isometry of
Let us check that [Eq. (3.3)]:
Computations for the peeling at infinity
The conformal metric on is
Its inverse is [unumbered equation above Eq. (4.1)]
Its scalar curvature is [Eq. (4.1)]
The volume 4-form of is [Eq. (4.2)]
The d'Alembertian w.r.t. of a generic function is
Check of Eq. (4.5):
The Levi-Civita connection of :
Morawetz vector field
The Morawetz vector field is defined according to Eq. (4.6):
Its Killing form is [Eq. (4.7)]:
Energy-momentum tensor of a generic scalar field
The covariant derivative of :
Since is a scalar field, we have of course :