Kerr--Newman Memory Effect

Abstract

We bring the Kerr--Newman spacetime into the Bondi--Sachs gauge by means of zero angular momentum, null geodesics. We compute the memory effect produced at the black hole horizon by a transient gravitational shock wave, which from future null infinity is seen as a Bondi-Metzner-Sachs supertranslation. This results in a change of the supertransformation charges at infinity between the spacetime geometries defined by the black hole before, and after, the shockwave scattering. For an extremal Kerr--Newman black hole, we give the complementary description of this process in the near-horizon limit, as seen by an observer hovering over the horizon. In this limit, we compute the supertranformation charges and compare them to those calculated at null infinity. We analyze the effect of these transformations on the electromagnetic gauge field and explore the self-interaction between this and the angular momentum of the black hole.

Full text

arXiv:2407.15289v1 [gr-qc] 21 Jul 2024
Kerr–Newman Memory Effect
Marco Galoppo ID
,aRudeep Gaur ID
,bChristopher Harvey-Hawes ID
a
aSchool of Physical & Chemical Sciences, University of Canterbury, Private Bag 4800, Christchurch
8041, New Zealand.
bSchool of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington
6140, New Zealand.
E-mail: marco.galoppo@pg.canterbury.ac.nz,rudeep.gaur@sms.vuw.ac.nz,
christopher.harvey-hawes@pg.canterbury.ac.nz
Abstract:
We bring the Kerr–Newman spacetime into the Bondi–Sachs gauge by means of zero an-
gular momentum, null geodesics. We compute the memory effect produced at the black
hole horizon by a transient gravitational shock wave, which from future null infinity is seen
as a Bondi-Metzner-Sachs supertranslation. This results in a change of the supertrans-
formation charges at infinity between the spacetime geometries defined by the black hole
before, and after, the shockwave scattering. For an extremal Kerr–Newman black hole, we
give the complementary description of this process in the near-horizon limit, as seen by
an observer hovering over the horizon. In this limit, we compute the supertranformation
charges and compare them to those calculated at null infinity. We analyze the effect of
these transformations on the electromagnetic gauge field and explore the self-interaction
between this and the angular momentum of the black hole.

Contents
1 Introduction 1
1.1 The Bondi–Sachs Metric 2
1.2 Asymptotic Killing Vectors 4
1.3 Associated Charges and Charge Conservation 4
1.4 Memory Effect and Supertranslations 5
2 Kerr–Newman Spacetime in the Bondi–Sachs Gauge 5
3 The Memory Effect at Null Infinity 11
4 Near Horizon Physics: Extremal Kerr–Newman 14
4.1 Near Horizon Metric and Gauge Four-Potential 16
4.2 Near Horizon Supertranslations and Charges 17
5 Conclusions 20
1 Introduction
The Kerr–Newman spacetime [1] describes a rotating, charged Black Hole (BH) and rep-
resents the most general of the asymptotically Minkowskian, stationary BH solutions to
the Einstein–Maxwell equations. It is a direct generalisation of the Kerr solution [2] for
a chargeless, rotating BH. The Kerr solution is widely accepted as providing an accurate
description of the exterior spacetime surrounding realistic BHs. In particular, the match-
ing of its ray tracing predictions with the recent direct observations of Sagittarius A* and
M87* further support its relevance [3–6]. In spite of these successes, a more realistic repre-
sentation of a BH would have to include the effects of its inherent electromagnetic charge,
in principle varying over time due to the in-fall of charged matter. The Kerr–Newman
solution represents a first step in this direction, as it provides the spacetime geometry
for an intrinsically charged, stationary, rotating BH. As such, the study of this solution
is of critical interest for understanding the dynamics and structure of physically realistic
BHs.
In particular, studying the asymptotic structure of Kerr–Newman BHs may provide further
insight into the gravitational memory effect – i.e., the permanent alteration of the system
due to the passage of a gravitational wave [7–14] – and its relation to the set of asymptotic
infinite symmetries at null infinity, dubbed supertransformations [15–19]. This group of
symmetries, composed of supertranslations and superrotations, was introduced by Bondi,
van der Burg, Metzner and Sachs [20–23], thus it is known as the BMS group. A complete
understanding of the BMS group in physically relevant spacetimes is of great interest, as
– 1 –

the initial and final states of two particles left permanently displaced by a gravitational
wave are related by supertranslations at null infinity. Specifically, in light of the next gen-
eration of gravitational wave detectors soon to be operational – e.g., the Einstein Telescope
[24] and the Laser Interferometer Space Antenna (LISA) [25,26] – the memory effect may
soon be observationally detected [27]. Furthermore, recent development in the study of
the BMS algebra and its associated charges [28–30] have led to interesting insight into
the “scattering” problem in general relativity [15,17] and in the information loss problem
[18,19,31].
Moreover, a regime of interest is the Near Horizon (NH) limit of the Kerr–Newman space-
time, specifically the case of an extremal Kerr–Newman BH. Indeed, the study of the NH
limit of classical BHs is of fundamental importance for the investigation of their geome-
try and topology, carrying consequences for any traditional approach to quantum gravity
[32,33]. Here, extremal BHs are highly relevant because even as semi-classical objects, they
remain inert, since they do not emit any Hawking radiation [34,35]. As such, they repre-
sent simple objects for investigating links between quantum physics and general relativity.
Furthermore, the NH limit gives a framework for describing the gravitational shockwave
scattering as seen by an observer hovering over the horizon. Therefore, it provides a com-
plementary analysis to the memory effect study carried out at null infinity. For such cases
as the Reisnner Nordstr¨om and Kaluza–Klein BHs, the NH observer is known to measure a
horizon superrotation after the scattering process has occurred — something absent at null
infinity [36,37]. Moreover, the passage of a gravitational shockwave imparts soft electric
hairs on the horizon of a charged BH, thus showing the interplay between the gravita-
tional and electromagnetic fields. We will reproduce these calculations for the near horizon
extremal Kerr–Newman BH, and further show that the interaction between angular mo-
mentum and the electromagnetic field is present even for the bald extremal Kerr–Newman
solution.
The structure of this paper is as follows. We will begin by summarising the expanded
Bondi–Sachs (BS) metric, asymptotic Killing vectors, their respective symmetries and as-
sociated charges, and the relationship between the memory effect and supertranslations.
In Section 2we put the Kerr–Newman metric in the Bondi-Sachs gauge. In Section 3we
supertranslate the resulting spacetime, electromagnetic gauge field and discuss the physical
implications of this procedure in the presence of charge. In Section 4we explore NH physics
for an extremal Kerr–Newman BH and relate the effect of outgoing gravitational radiation
to null infinity with the respective modifications of the BH horizon. Section 5presents a
brief summary of the results and a discussion regarding future lines of research.
1.1 The Bondi–Sachs Metric
A natural framework for the investigation of the memory effect is given via casting the
spacetime into the Generalised Bondi–Sachs (GBS) form. This metric form was introduced
by Bondi, van der Burg, Metzner, and Sachs [20–23] in an attempt to define a concept
of asymptotic flatness at null infinity. The falloffs needed to be restrictive enough that
– 2 –

unphyiscal spacetimes — such as those with infinite energy — would be ruled out, yet not
so restrictive that physical spacetimes and gravitational waves would be excluded. In the
canonical BMS framework, employed throughout the paper, the falloffs are [17]
guu =−1 + O(r−1), gur =−1 + O(r−2), guA =O(r0), gAB =r2γAB +O(r),(1.1)
with the additional constraints
grr =grA = 0 .(1.2)
The class of allowed asymptotic line elements for these falloffs can be cast in the form
ds2=−du2−2dudr+r2γAB dΘAdΘB
+2mbondi
rdu2+rCAB dΘAdΘB+DBCAB dudΘA
+1
16r2nCF DCF D odudr
+1
r4
3NA+4u
3∂Ambondi −1
8∂AnCF DCF D odudΘA
+1
4γAB nCF DCF D odΘAdΘB+... (1.3)
where ΘA∈ {θ, φ}and the uppercase Latin indices run over θ, φ.DAis the covariant
derivative on the 2–sphere with respect to the 2–sphere metric, γAB . The function mbondi
is the Bondi mass aspect, which is in general a function of u, θ, and φ. Its integral over
the 2–sphere at null infinity gives the total Bondi mass for the spacetime. In the case
of the Kerr–Newmann spacetime, the Bondi mass coincides with the canonical mass of
the black hole. NAis the angular moment aspect. Contracting NAwith the generator of
rotations and integrating over the entire sphere then gives the total angular momentum of
the spacetime. CAB is another field, symmetric and traceless (γAB CAB = 0). The retarded
time derivative of CAB defines the Bondi news tensor,
NAB := ∂uCAB .(1.4)
The news tensor is the gravitational analogue of the Maxwell field strength and its square
is proportional to the gravitational energy flux across I+(future null infinity) [16,17].
We note that NAis defined slightly differently in various parts of the literature — usually
depending on the asymptotic expansion (1.3). We conform to the convention employed
by Strominger [17] and Comper´e [16], which uses the decomposition (1.3), leading to NA
being defined as 2
3NA−1
16∂ACBC CB C := g(1)uA .(1.5)
Here, g(1)
uA corresponds to the r−1expansion in guA .
– 3 –

1.2 Asymptotic Killing Vectors
The symmetries of a spacetime are directly associated to its Killing vectors. Thus, before
discussing the charges associated with the BMS asymptotic symmetries, we must introduce
its asymptotic Killing vectors. The most general Killing vector, ξα, that preserves the
metric (1.3) to leading order is [17]
ξα∂α:= f∂u+"−1
rDAf+1
2r2CABDBf+Or−3#∂A(1.6)
+"1
2D2f−1
r(1
2DAfDBCAB +1
4CAB DADBf)+Or−2#∂r,(1.7)
where fis a function of that angular coordinates (θ, φ) only and D2is the standard
Laplacian on the 2–sphere. However, since we conduct an analysis to the leading order —
as done in [19], the Killing vector is truncated
ξα∂α=f ∂u+1
2D2f∂r−1
rDAf∂A.(1.8)
1.3 Associated Charges and Charge Conservation
It is well known that the symmetries of spacetime are associated to conserved charges
via Noether’s theorem. It follows that the BMS group, which is an infinite dimensional
group of symmetries at null infinity, also has an infinite number of charges associated to
supertranslations and superrotations. The supertranslation charge and its conservation is
given by [17]
Q+
f=1
4πZI+
d2Θ√γ f m bondi =1
4πZI−
d2Θ√γ f m bondi =Q−
f,(1.9)
where the integration is carried out over the 2–sphere at null infinity, and fis a function
of angular coordinates that can be thought of as the generator of supertranslations. In
general, the supertranslation charges will depend on advanced/retarded time, following the
respective dependence of mbondi . For f= 1 the conserved charge is the spacetime energy
whilst in the case where fis a l= 1 spherical harmonic function we have conservation of
ADM momentum.
The superrotation charge and its conservation is given by [17]
Q+
Y=1
8πZI+
d2Θ√γ Y ANA=1
8πZI−
d2Θ√γ Y ANA=Q−
Y,(1.10)
where YAis an arbitrary vector field on the 2–sphere. If YAis chosen to be one of the 6
global conformal Killing vectors on the 2–sphere, then (1.10) expresses the conservation of
ADM angular momentum and boost charges. Moreover, the notation employed indicates
that the supertransformation charge at future null infinity, I+, will match the supertrans-
formation charge at past null infinity, I−. These matching conditions are a requirement
– 4 –

for a well-posed scattering problem of gravitational waves in general relativity.
1.4 Memory Effect and Supertranslations
The direct correspondence between the gravitational memory effect and the action of su-
pertransformations of the BMS group has been a subject of ongoing debate [16–19,38–41].
For example, in the case of a Schwarzchild BH there is an approximate mapping between
the action of a BMS supertranslation at null infinity and a gravitational shockwave [19].
This mapping has not been shown for more general cases e.g. for the Kerr and Kerr–
Newman BH solutions. However, this identification is believed to be consistent throughout
the classical family of BH solutions. As such, we assume that this physical mapping proven
for non-rotating BHs can also be applied to rotating cases. A supertranslation is given by
taking the Lie derivative of the spacetime metric along the asymptotic Killing vector
ξµ∂µ=f∂v−1
2D2f∂r+1
rDAf∂A.(1.11)
As shown in [19], the action of such a supertranslation on the Schwarzchild metric can only
equate to part of the deformation a gravitational wave would produce when striking the
BH. Indeed, the full memory effect due to the gravitational shockwave is written in this
case as
δgµν =Lξgµν +2µ
rδvµδvν,(1.12)
where δgµν refers to the permanent change in the spacetime due to the passage of a grav-
itational wave. Notice that the second term on the r.h.s. of (1.12) is to be expected, as
gravitational waves carry energy, linear and angular momentum. Therefore, the shockwave
scattering on the BH will bring about changes in these quantities in both. Hence, focusing
on the case discussed in [19], the mass of the hairy Schwarzschild BH would be m=M+µ
after the gravitational shockwave strikes the BH. However, the Bondi mass of the BH —
at linear order — does not change. Thus emphasising that, in general, the memory effect
is not entirely captured by BMS supertransformations, although it is seen as so if studied
from null infinity.
2 Kerr–Newman Spacetime in the Bondi–Sachs Gauge
The Kerr–Newman line element in Boyer–Lindquist coordinates ¯
t, ¯r, ¯
θ, ¯
φis
ds2=−d¯r2
¯
∆+ d¯
θ2¯ρ2+ (d¯
t−asin2¯
θd¯
φ)2¯
∆
¯ρ2−(¯
A2d¯
φ−ad¯
t)2sin2¯
θ
¯ρ2,(2.1)
with
¯
∆(¯r) = ¯r2+a2−Ξ,(2.2)
¯
Ξ(¯r) = 2M¯r−Q2,(2.3)
¯ρ2(¯r, ¯
θ) = ¯r2+a2cos2¯
θ, (2.4)
¯
A2(¯r) = ¯
r2+a2,(2.5)
– 5 –

where M, a and Qare the mass, angular momentum per unit mass, and electric charge of
the Kerr–Newman black hole in geometrised units, respectively. We aim to cast (2.1) into
the BS gauge. To do so, we have to first move from the Boyer–Lindquist coordinates to
the GBS coordinates, in which the metric has to respect the constraints (1.2), and then
impose the falloffs (1.1) through a further coordinate transformation. Only once (2.1) is
put into the BS gauge, then a meaningful analysis of the asymptotic structure can be made
possible.
In analogy to the pioneering work of Fletcher & Lun on the Kerr metric [42] and the
following expansion by Houque & Virmani to the Kerr–de Sitter solution [43], we begin
by considering Zero Angular Momentum Null Geodesics (ZANGs) in the Kerr–Newman
spacetime in Boyer–Lindquist coordinates 1. These are
¯ρ2d¯
t
dλ=¯
Σ2
¯
∆E , (2.6)
¯ρ4d¯r
dλ2
=¯
B2E2,(2.7)
¯ρ4d¯
θ
dλ2
=¯
Ω,(2.8)
¯ρ2d¯
φ
dλ=a¯
Ξ
¯
∆E , (2.9)
where λis an affine parameter along the ZANGs, Eis the constant of motion interpreted as
the energy of the photons, and the remaining functions appearing in equations (2.6)–(2.9)
are
¯
Σ2(¯r, ¯
θ) = ¯
A4−a2¯
∆ sin2¯
θ , (2.10)
¯
B2(¯r) = ¯
A4−a2¯
X2¯
∆,(2.11)
¯
Ω(¯r, ¯
θ) = a2E2(¯
X2−sin2¯
θ).(2.12)
¯
X=¯
X(¯r, ¯
θ) is related to Carter’s separation constant, K, by K=a2E2¯
X2[44]. Hence,
it also results as a constant of geodesic motion
d
dλ ¯
X(¯r, ¯
θ) = 0 .(2.13)
When Q= 0, equations (2.6)–(2.9) reduce to equations (12)–(15) of [42], as it should be
expected. Moreover, we point out that as for the pure Kerr case solution, the r.h.s. of
(2.9) is a function of ¯ralone. We can now proceed to write the Kerr–Newman metric in
GBS coordinates. We start with the coordinate transformation
¯
t= ˇu+J(ˇr, ˇ
θ),(2.14)
¯r= ˇr , (2.15)
1To maintain a consistent nomenclature with the existing literature, we resolve to use the same notation
adopted by Fletcher & Lun and Houque & Virmani [42,43].
– 6 –

¯
θ=¯
θ(ˇr, ˇ
θ),(2.16)
¯
φ=ˇ
φ+L(ˇr, ˇ
θ),(2.17)
where the functions J(ˇr, ˇ
θ), ¯
θ(ˇr, ˇ
θ) and L(ˇr, ˇ
θ) are arbitrarily defined, at this stage. The
coordinate transform is chosen in this manner as to preserve the simple form of the killing
vector fields in the new coordinate system
∂¯
t=∂ˇu,(2.18)
∂¯
φ=∂ˇ
φ.(2.19)
We further impose that the integral curves of the ZANGs, in the new coordinates, are lines
of constant {ˇv, ˇ
θ, ˇ
φ}, i.e.
dˇv
dλ= 0 ,(2.20)
dˇ
θ
dλ= 0 ,(2.21)
dˇ
φ
dλ= 0 .(2.22)
Applying the coordinate transformation (2.14)–(2.17) with conditions (2.20)–(2.22) to
(2.6)–(2.9) gives
∂J
∂ˇr=¯
Σ2
¯
B¯
∆,(2.23)
∂L
∂ˇr=a¯
Ξ
¯
∆¯
B,(2.24)
∂¯
θ
∂ˇr2
=¯
Ω
¯
B2E2,(2.25)
with dˇr
dλ=d¯r
dλ=¯
B E
¯ρ2.(2.26)
The choice of the positive root for ˇ
B, combined with (2.14) and (2.23), indicates that we
are using a retarded time coordinate and that the ZANGs are outgoing rather than ingoing
null geodesics. Furthermore, from (2.13) and (2.21) we deduce,
¯
X=ˇ
X(ˇ
θ).(2.27)
Since we have picked ¯r= ˇrwe can take the square root of (2.25) and integrate to ob-
tain
Z¯
θdθ′
pX2−sin2θ′=±Zˇradr′
ˇ
B(r′)=: ±αX,(2.28)
– 7 –

where
αX(ˇr) = Zˇr
adr′
p(r′2+a2)2−a2X2(r′2+a2−2Mr′+Q2),(2.29)
and dαX
dˇr=a
B(ˇr).(2.30)
Here, when ais positive, αX, is chosen to be a negative, monotonically increasing function.
To integrate (2.28) we notice that the l.h.s. is the Legendre incomplete integral of the first
kind and hence defines the Jacobi elliptic sine (sn) function. Thus, we have
sin ¯
θ=






sn±αXX + H(ˇ
θ),1
X2X2>1
tanh±αX+ H(ˇ
θ)X2= 1
Xsn(±αX+ H(ˇ
θ),X2) sin2¯
θ≤X2<1
,(2.31)
where H(ˇ
θ) is an arbitrary function of ˇ
θ. We now require ¯
θ→ˇ
θfor ˇr→ ∞, that is, the
two angular coordinates must match at large distances. Therefore, we obtain
H = 






sn−1(sin ˇ
θ, 1
X2)X2>1
tanh−1(sin ˇ
θ)X2= 1
sn−1sin ˇ
θ
X,X2sin2¯
θ≤X2<1
.(2.32)
Finally, by requiring a fixed equatorial plane under the transformation of coordinates –
¯
θ=±π/2←→ ˇ
θ=±π/2 – the case X2= 1 is selected2. This corresponds to choosing
the simplest possible class of ZANGs with non-zero energy. Indeed, it forces both Carter’s
constant (Q=K−a2E2) and the total angular momentum about the axis of symmetry
to be zero.
Here, we must also stress an interesting difference between the coordinate systems built
following this procedure for the Kerr, Kerr–de Sitter and Kerr–Newman spacetimes. For
the latter, due to the presence of the charge term in the denominator of (2.29), the coor-
dinate system is not well-defined over the whole spacetime. Indeed, by analysing (2.29)
for X= 1, we see that the coordinate chart develops a singularity at the real positive root
of
P(r;M;a;Q) = (r2+a2)2−a2(r2+a2−2Mr +Q2).(2.33)
Therefore, in the Kerr–Newman case, the coordinate system will not be global, unlike in
Kerr and Kerr–de Sitter. However, the coordinate singularity appears only below the outer
horizon of the charged BH. Thus, the coordinate chart built using ZANGs can still be used
in studying the asymptotic structure of the spacetime. Then, by using equations (2.31)
and (2.32) we obtain
tanh−1(sin ¯
θ) = tanh−1(sin ˇ
θ)±α , (2.34)
2Henceforth, the subscript Xis dropped from αX.
– 8 –

with
α(ˇr) = −Z∞
ˇr
adr′
pr′4+a2(r′2+r′2M−Q2).(2.35)
From (2.34), and choosing the plus side in front of α, we directly deduce
sin ¯
θ=D
C,(2.36)
cos ¯
θ=cos ˇ
θ
Ccosh α,(2.37)
where
C= 1 + tanh αsin ˇ
θ , (2.38)
D= tanh α+ sin ˇ
θ . (2.39)
From (2.36), (2.37), (2.38) and (2.39) we obtain
∂¯
θ
∂ˇr=cos ˇ
θ
Ccosh α
dα
dˇr=cos ˇ
θ
Ccosh α
a
B(ˇr),(2.40)
∂¯
θ
∂ˇ
θ=1
Ccosh α.(2.41)
Therefore, we have
d¯
t= dˇu+ˇ
Σ2
ˇ
Bˇ
∆dˇr+g(ˇr, ˇ
θ) dˇ
θ , (2.42)
d¯
φ= d ˇ
φ+aˇ
Ξ
ˇ
Bˇ
∆dˇr+h(ˇr, ˇ
θ) dˇ
θ , (2.43)
d¯
θ=cos ˇ
θ
Ccosh α
a
Bdˇr+1
Ccosh αdˇ
θ , (2.44)
where g(ˇr, ˇ
θ) = ∂J (ˇr, ˇ
θ)/∂ ˇ
θand h(ˇr, ˇ
θ) = ∂L(ˇr, ˇ
θ)/∂ ˇ
θ. To complete the coordinate trans-
formation, we need to establish the function form of g(ˇr, ˇ
θ) and h(ˇr, ˇ
θ). From the condi-
tion
gˇrˇ
θ= 0 ,(2.45)
we deduce the form of g(ˇr, ˇ
θ) as
g(ˇr, ˇ
θ) = acos ˇ
θ
C2cosh2α,(2.46)
whilst from the integrability condition
∂ˇ
θ∂ˇrL(ˇr, ˇ
θ) = ∂ˇr∂ˇ
θL(ˇr, ˇ
θ),(2.47)
– 9 –

we get
h(ˇr, ˇ
θ) = h(ˇ
θ).(2.48)
Without losing any generality we are then free to choose
h(ˇ
θ) = 0.(2.49)
Therefore, (2.23), (2.24), (2.40), (2.41), (2.46) and (2.49) completely define the correct
coordinate transform – (2.14)-(2.17) – to cast the line element (2.1) into the GBS form
ds2=−1−ˇ
Ξ
ˇρ2dˇu2−2ˇρ2
ˇ
Bdˇudˇr−21−ˇ
Ξ
ˇρ2acos ˇ
θ
C2cosh2αdˇudˇ
θ−2aˇ
Ξ
ˇρ2D
C2
dˇudˇ
φ
+ˇρ2
C2cosh2α−1−ˇ
Ξ
ˇρ2a2cos2ˇ
θ
C4cosh4αdˇ
θ2−2a2cos ˇ
θ
C2cosh2αD
C2ˇ
∆
ˇρ2dˇ
θdˇ
φ
+D
C2ˇ
Σ2
ˇρ2dˇ
φ2.(2.50)
Finally, to put the line element (2.50) into the BS gauge we apply the following coordinate
transformation [45]
ˇu=u , (2.51)
ˇ
θ=θ , (2.52)
ˇ
φ=φ , (2.53)
ˇr=r+a
2
cos 2θ
sin θ+a2
8 4 cos 2θ+1
sin2θ!1
r.(2.54)
At the expansion order of interest in r, we find the metric components to be3
guu =−1 + 2M
r−aM csc θcos 2θ+Q2
r2+Or−3,(2.55)
gur =−1 + a2csc2θ
8r2+a2(2M+acos 4θcsc θ)
2r3+Or−4,(2.56)
guφ =−2aM sin2θ
r+asin θ3aM cos 2θ+ 2aM +Q2sin θ
r2+Or−3,(2.57)
guθ =1
2acot θcsc θ+acos θacsc3θ+ 8M
4r−acot θcsc θ4a2+Q2cos 2θ
2r2(2.58)
−acot θcsc θ3a2cos 4θ+ 2 2a2−7aM sin θ+ 3aM sin 3θ+Q2
4r2+Or−3,
gθθ =r2+ar csc θ+1
2a2csc2θ+a2acsc θcos 4θ+ 8Mcos2θ
4r+Or−2,(2.59)
3The calculations put forward in this paper have been checked with the Mathemathica codes described
in the Appendix.
– 10 –

gθφ =−2a2Msin2θcos θ
r+a2sin θcos θ5aM cos 2θ+Q2sin θ
r2+Or−3,(2.60)
gφφ =r2sin2θ−ar sin θ+a2
2+a3sin θcot2θ(cos 4θ−2 cos 2θ)
4r
+a2sin θ4asin4θ−5asin2θ+a+ 2Msin3θ
r+Or−2.(2.61)
Furthermore, we can compute the electromagnetic four-potential in the selected gauge. We
start by considering the four-potential in Boyer–Lindquist coordinates
Aµd¯xµ=¯rQ
¯ρ2d¯
t+a¯rQ
¯ρ2sin2¯
θd¯
φ . (2.62)
We then find for the GBS coordinates
Aµdˇxµ=ˇrQ
ˇρ2dˇu+ˇrQ
ˇ
Bˇ
∆(ˇr2+a2)dˇr+ˇr Q
ˇρ2
acos ˇ
θ
C2cosh2αdˇ
θ+aˇrQ
ˇρ2D
C2
dˇ
φ , (2.63)
where ˇρ= ˇr2+a2−a2(D/C)2. Given that Aˇris solely a function of ˇr, it can be set to
zero via a classical U(1) gauge transformation. Moving into the BS gauge we find
Aµdxµ=Q
r−aQ cos 2θcsc θ
2r2du+aQ cos θ
r+a2Qcsc θ(cos θ−3 cos 3θ)
4r2dθ
+aQ sin2θ
r−a2Qsin θ(3 cos 2θ+ 2)
2r2dφ+O(r−3).(2.64)
Thus, we can now move to the evaluation of the asymptotic structure of the space-
time.
3 The Memory Effect at Null Infinity
The gravitational memory effect as seen by an observer at null infinity has been shown to
be equivalent to a BMS supertranslation. Following the investigation of the Kerr mem-
ory effect at null infinity [45], we now focus on the Kerr–Newman memory effect. The
supertranslated metric functions are calculated via
δgµν =Lξgµν ,(3.1)
where ξαis the asymptotic Killing vector, (1.8). We find4
δguu =1
r3(−Mr +Q2+aM (1 −2 sin2θ)
sin θ)D2f
+aM
r3((−2 + cos 2θ) cot θcsc θ)∂θf+Or−4,(3.2)
4By comparison to the Kerr spacetime [45], there are extra orders in the expansion. This is so one can
observe the change in the charge.
– 11 –

δgur =1
r2(acos θ
2 sin2θ)D2f , (3.3)
δgu θ =−(∂θf+1
2∂θD2f)+1
r(2M∂θf−∂θ a
2
cos θ
sin2θ∂θf!)−1
r2(Q2∂θf)
+1
r2(aM csc θcos 2θ)∂θf+O(r−3) ; (3.4)
δgu φ =−(∂φf+1
2∂φD2f)+1
r(2M∂φf−a
2
cos θ
sin2θ∂φ∂θf)−1
r2(Q2∂φf)
+1
r2(aM csc θcos 2θ)∂φf+O(r−3) ; (3.5)
δgθθ =(2∂θ2f−D2f)r−a
sin θ(+1
2D2f+ 2 cos θ
sin θ∂θf−2∂θ2f)+Or−1; (3.6)
δgφφ =(2∂φ2f+ 2 sin θcos θ ∂φf−sin2θD2f)r
−a
sin θ(1
2sin2θD2f+ cos θ ∂φf+ 2∂φ2f)+Or−1.(3.7)
Additionally, the supertranslated gauge field components at null infinity are:
δAu=−1
2
Q
r2D2f−1
r2csc θcos 2θcot θ+ 2 sin 2θ∂θf+O(r−3) ; (3.8)
δAB=1
2D2f∂rAB−∂θf ∂θAB+AC∂BDCf(3.9)
+ Q
r−aQ cos 2θcsc θ
2r2!∂θf+O(r−3).
As can be seen, the supertranslated gauge field has components which match the leading
order parts of the original gauge field. Therefore, when an observer at null infinity measures
the Maxwell field through Fµν , they will observe a difference in a bald Kerr–Newman
spacetime and hairy Kerr–Newman spacetime.
Comparing the new supertranslated metric components with (1.3) we find CAB CAB ,CAB ,
NA, and mbondi, after the impact of the gravitational wave5.
CAB CAB =2a2
sin2θ+16acos θ
sin2θD2f , (3.10)
5There have been developments in the BMS group where authors have started investigating higher order
terms in the expansion. For instance, in [46,47] there are higher order terms, such as ‘EAB ’ and ‘FAB ’ which
appear in the gAB expansion. However, these modifications are made for the inclusion of a cosmological
constant. The relevance of these terms in our analysis and the effect these may have on charges that we
observe at null infinity remains unclear and is perhaps an avenue for further research. Furthermore, the
incorporation of these terms would likely also require tweaking of the transformation (2.51), similar to what
is done in [47].
– 12 –

CAB dxAdxB= a
sin θ+ 2∂θ2f−D2f!dθ2
− asin θ−2∂φ2f−2cos θ
sin θ∂φf−sin2θD2f!dφ2,(3.11)
Nθ= 3M{acos θ+∂θf}+3
2a∂θ(cos θ
sin2θhD2f−1
2∂θfi),(3.12)
Nφ= 3M{−asin2θ+∂φf}+3
2a∂φ(cos θ
sin2θhD2f−1
2∂θfi),(3.13)
mbondi =M . (3.14)
We are now in a position to discuss the supertranslation and superrotation charges that
are implanted on the BH horizon, as seen by an observer at null infinity. As expected
[18,45], the scattering of a gravitational wave by the BH will not excite supertranslation
charge. However, this process, equivalent to a supertranslation at null infinity, will modify
the superrotation charge. The superrotation charge that is measured at future null infinity
is given by
QY=1
8πZI+
d2Θ√γ Y ANA.(3.15)
Using (3.10) we get
QY=Yθ=1
8πZI+
√γd2ΘYθ3Ma cos θ+
1
8πZI+
√γd2ΘYθ"∂θf+3
2a∂θ(cos θ
sin2θhD2f−1
2∂θfi)#,
(3.16)
and
QY=Yφ=1
8πZI+−√γd2ΘYφ3M a sin2θ+
1
8πZI+
√γd2ΘYφ"∂φf+3
2a∂φ(cos θ
sin2θhD2f−1
2∂φfi)#.
(3.17)
The first terms in (3.16) and (3.17) correspond to the bald Kerr–Newman BH superro-
tation charges and can easily be recovered if the supertranslation function fvanishes.
Furthermore, when a= 0 we recover the superrotation charges of the hairy Schwarzschild
BH [19,37,48]. Moreover, as shown by Barnich and Troessaert in [29], when Yφis the
Killing vector, ∂φ, (3.17) corresponds to conservation of angular momentum for both the
bald Schwarzschild and Kerr BH. For the hairy Kerr–Newman BH, one may see that the
zero-mode superrotation charge (when f= 0 and Yφ= 1) given by (3.17), does not change
and will still correspond to the conservation of angular momentum.
We note that the calculated charges are no different from those obtained for the Kerr BH
– 13 –

[45]. Therefore, within the current framework, the expected memory effect at null infinity
in these two spacetimes is indistinguishable at the level of the metric. This follows from the
electric charge, Q, appearing only at a higher order than r−1in the expansion of the metric
in the BS gauge. In our opinion, this result represents a clear drawback of the current,
first-order framework. A higher-order approach is needed to distinguish fundamentally
different spacetimes, such as the Kerr and Kerr–Newman solutions, and should therefore
be pursued as an important milestone for the field [49].
Nonetheless, we point out that the presence of an electromagnetic field in the Kerr–Newman
spacetimes gives a novel method to measure the scattering of a gravitational wave from
the BH, via the change in the field. In particular, if such a change were to be detected and
agree with our calculations, it could be considered as an indirect test for the presence of
supertransformation charges. However, such a measurement clearly presents observational
challenges not easily met.
4 Near Horizon Physics: Extremal Kerr–Newman
We now shift our attention to the NH form of the Kerr–Newman spacetime. In particular,
contrary to the null infinity analysis, we show that the two spacetimes differ in their
response to the scattering of a gravitational shockwave. Indeed, in the Kerr–Newman
case, the gravitational wave excites supertransformation charges and implants soft, electric
hair on the horizon, due to its interaction with the electromagnetic four-potential. To
determine the charges that are implanted on the horizon, we must first find the NH metric
components, and secondly, derive an expression for the electromagnetic gauge field in the
NH limit. Chrusciel [50] shows that the general form of a NH metric is given by
ds2=−2Rκdv2+ 2dvdR+ 2R θAdvdxA+ ΩAB dxAdxB+... , (4.1)
where vis the advanced time, xAare angular coordinates, θA, ΩAB ≡ΩaγaAB 6are in
principle arbitrary metric functions of vand xA, and κis the surface gravity. Note, that
when dealing with an extremal horizon — as is the case in this paper — the surface
gravity vanishes, i.e., κ= 0. In the coordinate used in (4.1), the horizon is now located
at R= 0 and the ellipsis are to denote terms that are O(R2). Furthermore, we have the
constraints
gRR = 0 , gvR = 1 , gAR = 0 .(4.2)
Additionally, in analogy to [36,39], we use the boundary conditions
gvv =−2κR +O(R2), gvA =θAR+O(R2), gAB = ΩAB +O(R).(4.3)
We can then find a set of asymptotic Killing vectors that preserve (4.2) and (4.3), generating
an algebra consisting of both supertranslations and superrotations. The resulting Killing
6Note the use of the internal index, a, here. This is required in the case of the NH Kerr–Newman metric
as we can not use only one scaling factor for ΩΘΘ and ΩΦΦ .
– 14 –

vectors are
ξµ∂µ=f∂v+YA−∂BfZρ
dρ′gAB ∂A+Z(v, xA)−ρ∂vf+∂AfZρ
dρ′gAB gvB ∂ρ.(4.4)
In contrast with [36,39], we find a vector, YA, that is a “constant” of integration which
represents the horizon superrotations7. Then, using the NH asymptotic Killing vector, we
compute the general supertranslated metric functions; κ,θA, and ΩAB subject to (4.3)
8.
δξκ=Lξκ= 0 ,(4.5)
δξθA=LYθA+f∂vθA−2κ∂Af−2∂v∂Af+ ΩBC ∂vΩAB DCf , (4.6)
δξΩAB =f∂vΩAB +LYΩAB .(4.7)
To properly study the NH physics of a charged BH, we must also discuss the NH expansion
of the gauge field. The Taylor expansion of the U(1) electromagnetic gauge field near R= 0
is given by [36]
Av=A(0)
v+RA(1)
v(v, xA) + O(R2),(4.8)
AB=A(0)
B(xA) + RA(1)
B(v, xA) + O(R2),(4.9)
AR= 0.(4.10)
Here A(0)
vis the Coulomb potential. In particular, we find that the supertranslated gauge
field components take the form:
δξAv= 0,(4.11)
δξAB=YC∂CA(0)
B+A(0)
C∂BYC+∂BU. (4.12)
Where Uis an arbitrary function of angular coordinates and is referred to as the electromag-
netic charge generator, just as fis referred to as the generator of supertranslations.
We now discuss the NH extremal Kerr-Newman spacetime and provide the supertranslated
metric functions. This will allow us to examine the effect of a gravitational shockwave –
under the identification of supertranslations with the scattering of such waves by the BH
– on the extremal horizon as seen by an observer near the horizon. This leads to a horizon
superrotation that is absent at null infinity9, similarly to the Schwarzschild and Kaluza–
7It is important to point out that one does not need a gravitational shockwave here to have “a superro-
tation/supertranslation charge”. These aspects exist as a property of the asymptotic structure of the NH
metric.
8We correct a small mistake here that is present in ref [39]. This third equation now correctly states
that the Lie derivative of ΩAB is along Yαand not ξα.
9While not discussed in this paper, various parts of the literature make explicit the fact that super-
translations turn on superroation charge and superrotations turn on supertranslation charges. We stated
that there is a change in the superrotation charges at null infinity due to the supertranslation. However,
in the near horizon case, (mathematically due to the boundary conditions) we note that there is also a
superrotation that has associated supertranslation charges discussed in subsection 4.2.
– 15 –

Klein cases discussed in [36,37] respectively.
4.1 Near Horizon Metric and Gauge Four-Potential
To derive the extremal NH Kerr–Newman metric, we begin by defining [50]
¯
t=ǫ−1ˆ
t , (4.13)
¯r=M+ǫˆr , (4.14)
¯
θ=ˆ
θ , (4.15)
¯
φ=ˆ
φ+ǫ−1a
r2
0
ˆ
t , (4.16)
where r2
0=M2+a2. After taking the limit ǫ→0, the metric becomes
ds2=1−a2
r2
0
sin2ˆ
θ−ˆr2
r2
0
dˆ
t2+r2
0
ˆr2dˆr2+r2
0dˆ
θ2
+r2
0sin2ˆ
θ1−a2
r2
0
sin2ˆ
θ−1dˆ
φ+2a M
r4
0
r dˆ
t2
.(4.17)
This metric is clearly singular on the horizon. Hence, we apply the following coordinate
transform
ˆ
t=V−r2
0
r,(4.18)
ˆr=R , (4.19)
ˆ
θ= Θ ,(4.20)
ˆ
φ= Φ −2M a
r2
0
log ˆr
r0,(4.21)
leading to the line element
ds2=r2
0−a2sin2Θ
r2
0−R2
r2
0
dV2−2dVdR+r2
0dΘ2
+r4
0sin2Θ
r2
0−a2sin2ΘdΦ + 2aM
r4
0
RdV2
,(4.22)
which is regular for R= 0. We may now read off the metric functions in (4.1):
κ= 0; (4.23)
θΘ= 0; (4.24)
θΦ=2a M sin2Θ
r2
0−a2sin2Θ; (4.25)
ΩΘΘ =r2
0−a2sin2Θ; (4.26)
ΩΦΦ =r4
0sin2Θ
r2
0−a2sin2Θ.(4.27)
– 16 –

We must now bring the Kerr–Newman gauge potential into the form (4.8). Performing the
same coordinate transformations as we did for the metric, we first find:
Aµdˆxµ=
QM2−a2cos2ˆ
θ
r2
0M2+a2cos2ˆ
θˆrdˆ
t+QaM sin2ˆ
θ
M2+a2cos2ˆ
θdˆ
φ , (4.28)
where we have eliminated the constant term ǫ−1(M Q/r2
0) in Aˆ
tthrough a U(1) gauge
transformation before taking the limit for small ǫ. With the final coordinate transformation
(4.18), we obtain
AµdXµ=QM2−a2cos2Θ
r2
0(M2+a2cos2Θ) RdV+Q aM sin2Θ
M2+a2cos2ΘdΦ ,(4.29)
where we have renormalised AR=−(M2−a2)(Q/r2
0)R−1with a further U(1) gauge
transform. Note, that the Coulomb potential does not appear here. However, because it
is coordinate-independent, this can be added back in at any point without changing the
Maxwell field. Moreover, as expected, we will see that the Coulomb potential will not
appear in the expressions for surface charges.
4.2 Near Horizon Supertranslations and Charges
Bringing the asymptotic Killing vector, (1.8) to the NH limit for the extremal case – i.e.,
M2=a2+Q2– and supertranslating the NH Kerr–Newman spacetime (4.22) we find the
following metric components
gV V =−R2
r2
0r2
0−a2sin2Θ
r2
0
+r4
0sin2Θ
r2
0−a2sin2Θ2aMR
r4
02
,(4.30)
gΘΘ =nr2
0−a2sin2Θon1−2
r+
∂2
Θfo+1
r+
a2sin 2Θ ∂Θf , (4.31)
gΦΦ =n1
r+
r4
0
r2
0−a2sin2Θonr+sin2Θ−r2
0sin 2Θ
r2
0−a2sin2Θ∂Θf−2∂2
φfo,(4.32)
gV R =−r2
0−a2sin2Θ
r2
0
,(4.33)
gVΦ=(1
r+
2a M
r2
0−a2sin2Θ)(sin2Θ−sin 2Θ r2
0
r2
0−a2sin2Θ∂Θf−∂2
Φf)R , (4.34)
gVΘ=(1
r+
2a M
r2
0−a2sin2Θ)(2 cot Θ ∂Φf−∂Θ∂Φf)R , (4.35)
gRR = 0 ,(4.36)
gRΘ= 0 ,(4.37)
gRΦ= 0 ,(4.38)
gΘΦ = 0 .(4.39)
– 17 –

We can now compare our results with the supertranslated extremal NH Kerr–Newman
spacetime, see (4.17), and the general NH supertranslated metric functions, see (4.5)-
(4.7). Since ΩAB does not depend on retarded/advanced time, from (4.7) we find the
corresponding horizon superrotation to be
YA=1
MDAf . (4.40)
Surprisingly, this does not differ from the horizon superrotation of a Schwarzschild BH
found in [36]. Indeed, this is perhaps not expected as the Kerr class of solutions are already
rotating. However, this could be intuitively understood by noticing that the correlation
between the memory effect and supertranslations — in both regimes, null infinity, and NH
— is only examined at linear order. In fact, one can show that if ΩAB does not depend on
advanced or retarded time, we will always have a horizon superrotation of this form — up
to a factor which depends on the horizon radius.
The associated charges with the diffeomorphisms generated by asymptotic Killing vectors
have associated horizon charges. The derivation of these charges stem from [51] and are
also discussed in [36,52]. The NH charges take the form:
Q[X, Y A, U ] = 1
16πZdΘ dΦ sin Θ r2
0 2X−YAθA−4UA(1)
V−4A(0)
BYBA(1)
V!.(4.41)
In the extremal case, it is apparent that the surface gravity vanishes and so too does
the Hawking temperature [34,35]. This leads to an interesting scenario in which the
Hawking temperature is no longer the associated zero-mode for the first charge. In this case,
this zero-mode (the supertranslation charge) is associated with the product of Bekenstein-
Hawking entropy and the geometric temperature [36,53]. The second term is analogous to
the superrotation charge found at null infinity. The third term is due to the electromagnetic
generator and the last term mixes the superrotation vector field with the gauge field.
Let the associated charges to X,YA, and Ube X,YA, and Urespectively. The associated
zero-mode (bald) horizon charges are
X=Q[1,0,0] = r2
0
2,(4.42)
YΘ=Q[0, Y Θ= 1, Y Φ= 0,0] = 0 ,(4.43)
YΦ=Q[0, Y Φ= 1,0] = 1
16πZdΘ dΦ sin Θ r2
0 θΦ−4AΦA(1)
V!,(4.44)
U=Q[0,0,1] = −1
4πZdΘdΦ sin Θ r2
0A(1)
V
=Q1−2M
aarctan a
M.(4.45)
The zero mode of YΦgives the angular momentum of the BH as measured by the hovering
– 18 –

observer. As one may note, there is a contribution to this zero-mode from the gauge field
which does not vanish. Therefore, we see a strong interaction between the electromagnetic
gauge potential and the angular momentum of the BH, with the former influencing the
latter for the chosen observer. As the gauge field vanishes, we retrieve the extremal NH
Kerr solution, and the angular momentum depends solely on θΦ. The final charge, U,
whose zero-mode charge corresponds to the electromagnetic charge generator, gives the to-
tal electric charge of the BH as measured in the NH limit. Here, the complementary effect
is observed and the angular momentum of the BH effectively shields the intrinsic charge for
the NH observers. Remarkably, these unexpected effects of the self-interactions between
angular momentum and electric charge are not found via a null infinity analysis. Thus,
they further indicate the importance in general relativity of studying the same phenomena
using a plurality of observers. Lastly, one may verify that these charges do indeed agree
with the extremal Reissner–Nordstr¨om horizon when a→0 as seen in [36].
We may also use (4.41) to determine the zero-mode of the NH charges of the supertranslated
horizon. To do so, we compute
θΘ=(1
M
2a M
r2
0−a2sin2Θ)(2 cot Θ ∂Φf−∂Θ∂Φf); (4.46)
θΦ=(1
M
2a M
r2
0−a2sin2Θ)(sin2Θ−sin 2Θ r2
0
r2
0−a2sin2Θ∂Θf−∂2
Φf); (4.47)
A(1)
V=Q
r2
0(M2−a2cos2Θ
(M2+a2cos2Θ) −1
M∂Θ
(M2−a2cos2Θ)
(M2+a2cos2Θ)∂Θf); (4.48)
A(0)
B=QaM sin2Θ
M2+a2cos2Θ+δAB,(4.49)
(4.50)
where δABis given below. Interestingly, we see that once the NH spacetime is super-
translated by the passage of a gravitational wave, then θΘis no longer zero. However,
even though the NH geometry is transformed, the zero-mode horizon charges remain un-
changed. This is because we are setting the supertranslation generator, f, to zero10 in all
cases.
In the bald and supertranslated metric analysis, we already see an interplay between the
electromagnetic field and angular momentum. However, a further interaction between the
gravitational and electromagnetic fields can be investigated by determining the change in
the electromagnetic field generator due to the memory effect in the NH limit
U=ZYC∂CA(0)
B+A(0)
C∂BYC−δξABdxB.(4.51)
10In fact, f, can be expanded in Fourier modes which relate it linearly to Xin the extremal case. Hence,
when setting Xto zero, we are also setting fto zero, and the zero-modes now correspond solely to the bald
near-horizon geometry.
– 19 –

Here, YAis the horizon superrotation,
YA=1
MDAf , (4.52)
and
δξAB=−Qa (∂Θ sin2Θ
M2+a2cos2Θ!∂Θf− 1
M2+a2cos2Θ!∂Θ∂Φf).(4.53)
This illustrates that the gravitational memory effect due to the passing of a gravitational
wave is not only seen as a supertranslation from null infinity, but in the NH limit implants
soft electric hair on the extremal Kerr–Newman horizon.
5 Conclusions
Motivated by the rising relevance of the gravitational memory effect, in this paper we have
investigated the connection between the scattering of a gravitational shockwave by the
Kerr–Newman black hole, as seen in the near-horizon region and in the far asymptotic
region.
In [18] the authors showed that a transient gravitational shockwave modifies the black
hole geometry in a way that can be interpreted as a BMS supertranslation at null infinity.
Here, we have brought for the first time the Kerr–Newman black hole in the Bondi–Sachs
gauge and computed the action of a BMS supertranslation on its asymptotic structure. In
particular, we discussed the change in the supertransformation charges due to the super-
translation hair implanted on the Kerr–Newman black hole by the gravitational wave. We
have shown that the supertranslation charge was absent at null infinity, whilst a superrota-
tion charge is instead detectable. Furthermore, the zero-mode of the superrotation charge
remains unchanged, as any change in mass cannot be captured by the action of pure BMS
supertranslations.
Following the pioneering work of Donnay et al [52], we studied the gravitational memory
effect in the near horizon limit of an extremal Kerr–Newman black hole. Surprisingly,
we found that the corresponding horizon superrotation matches the one computed for
non-rotating black holes. Moreover, we find that no non-trivial supertranslation charge is
turned on at the horizon, due to the extremality of the black hole. Finally, we show that
the scattering of the gravitational shockwave by the black hole implants soft electric hair
on the horizon, via its interaction with the electromagnetic gauge field.
Some questions remain open and require further study. Indeed, we showed that a higher-
order formalism is needed to properly capture the full properties of the spacetime when
dealing with the memory effect. Consequently, a rigorous definition, and interpretation, of
higher-order charges would be required. Furthermore, we have found a series of previously
unexplored interactions between the gravitational and electromagnetic fields. To wit, the
presence of electric charge invalidates the construction of the Bondi–Sachs coordinates as
– 20 –

a global coordinate patch for the spacetime, failing below the horizon. Moreover, we have
showed that the electric charge, and angular momentum, inferred for the black hole by a
near horizon observer differ from what would be measured in the asymptotic region, on
account of the interplay between these two quantities. This unexpected interaction between
spin and charge requires further clarification, with a possible avenue of research leading to
the study of a similar effect in higher dimensional, charged, rotating black holes.
Acknowledgments
MG and CHH were supported by the University of Canterbury Doctoral Scholarship. RG
was supported by a Victoria University of Wellington PhD Doctoral Scholarship. MG and
CHH would like to warmly thank Matt Visser for his hospitality during the preparation of
this paper. The authors would like to acknowledge Chris Stevens, Matt Visser and David
Wiltshire for their numerous comments and insights on this paper.
Appendix: Mathematica Codes
We have submitted three ancillary Mathematica files with this submission to the arXiv. A
brief explanation on these files is as follows.
1. KerrNewmanGeneralisedBondiSachsForm.nb: this file writes the Kerr–Newmann
metric into the generalised Bondi–Sachs form.
2. KerrNewmanBondiGaugeComponentsExpansions.nb: this file computes the
Kerr–Newmann metric components expansions at null infinity in the generalised
Bondi–Sachs form; transforms the metric into the Bondi–Sachs gauge and computes
their asymptotic expansion.
3. KerrNewman4PotentialNullInfinity.nb: this file computes the electromagnetic
four-potential in the Bondi–Sachs gauge.
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