Unfolded Description of $AdS_4$ Kerr Black Hole

0.21 MB·English

by  V. E. Didenko

FIAN/TD/13-08
Unfolded Description of AdS Kerr Black Hole
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V.E. Didenko, A.S. Matveev and M.A. Vasiliev
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I.E. Tamm Department of Theoretical Physics, Lebedev Physical Institute,
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2 Leninsky prospect 53, 119991, Moscow, Russia

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[email protected], [email protected], [email protected]
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] Abstract
c
q
- ItisshownthatAdS4Kerrblackholeisasolutionofsimpleunfoldeddifferential
r
g equations that form a deformation of the zero-curvature description of empty
[ AdS space-time. Our construction uses the Killing symmetries of the Kerr
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2 solution. All known and some new algebraic properties of the Kerr-Schild
v solution result from the obtained black hole unfolded system in the coordinate-
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independent way. Kerr-Schild type solutions of free equations in AdS for
1 4
2 massless fields of any spin associated with the proposed black hole unfolded
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system are found.
.
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8 1 Introduction
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:
v
i Since its discovery in 1963, the celebrated Kerr metric [1] is still a hot research
X
topic1. Being of great physical significance, it exhibits deep mathematical beauty. In
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a particular, the Kerr-Schild form [3] of the Kerr metric illustrates the remarkable fact
that Kerr black hole (BH), being an exact solution of Einstein equations, also verifies
its first order fluctuational part, i.e., the second order terms vanish independently.
In other words, Kerr BH solves free spin-two equations. This property suggests the
relevance of perturbative study of corrections to Kerr-like solutions in such field-
theoretical extensions of gravity as string theory or higher-spin (HS) gauge theory
(see [4, 5, 6, 7, 8] for reviews). The Kerr-Schild form allowed Myers and Perry to
generalize the four-dimensional Kerr metric to any dimension [9].
BHs with non-zero cosmological constant have also been extensively studied in
the context of AdS/CFT correspondence [10, 11, 12, 13] where the (anti)-de Sitter
1For the history of question we refer to the recent publication by Roy Patric Kerr [2].
1
geometry plays a distinguished role. Although the generalization of the Kerr solution
that describes a rotating BH in AdS was discovered by Carter [14] soon after Kerr’s
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paper [1], its higher dimensional generalization has been found only recently. A
generalization of the Myers-Perry solution with non-zero cosmological term in five
dimensions has been given by Hawking, Hunter and Taylor-Robinson in [15] and
then extended to any dimension by Gibbons, Lu, Page and Pope in [16]. These
higher dimensional BHs are shown [17] to possess the hidden symmetry associated
with the Yano-Killing tensor [18] which is the characteristic feature of Petrov type D
metrics.
Despite considerable progress in the construction of different BH metrics, in-
cluding the charged rotating Kerr-Newman solution [19], the generalization of AdS
4
Taub-NUT to any dimension [20] and discovery of various supergravity BHs in the
series of works by Chong, Cvetic, Gibbons, Lu, Pope [21, 22], a number of prob-
lems remain open. In particular, a generalization of the well-known four-dimensional
Kerr-Newman solution to charged rotating BH solution in any dimension is available
neither in flat nor in (A)dS case. This suggests that some more general approaches
can be useful.
Usually an analysis in the BH background uses a particular coordinate system.
TheaimofthispaperistoreformulatetheAdS KerrBHinacoordinate-independent
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way using the formalism of the unfolded dynamics [23, 24] that operates with first-
order covariant field equations. We show how the AdS Kerr BH arises as a solution
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of the BH unfolded system (BHUS) associated with a certain BH Killing vector.
In general, unfolded equations generalize the Cartan-Maurer equations for Lie
algebras by incorporating p-form gauge potentials. They provide a powerful tool for
the study of partial differential equations. More precisely, consider, following [23], a
set of differential p-forms WA(x) with p 0 and generalized curvatures RA defined
≥
as
RA = dWA +FA(W), (1.1)
where d = dxn ∂ is the space-time exterior differential and some functions FA are
∂xn
built ofwedge productsofthedifferential formsWB. The functionsFA(W)subjected
the generalized Jacobi identity
δFA
FB = 0 (1.2)
δWB
guaranty the generalized Bianchi identity dRA = RB δFA which shows, in particular,
δWB
that the differential equations for WA(x)
RA = 0 (1.3)
are consistent. In this case the equations (1.3) are called unfolded. Note that every
unfolded system is associated with some solution of the generalized Jacobi identity
(1.2) on the functions FA(W) built of exterior products of the differential forms
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WA which, in turn, defines a free differential algebra [25, 26] (more precisely, its
generalization to the case with zero-forms included).
The reformulation of partial differential equations in the unfolded form has a
number of advantages. In particular, the first-order equations formulated in terms of
exterior algebra are manifestly coordinate-independent and reconstruct a local solu-
tionintermsofitsvaluesatanygivenpointofspace-timemodulogaugeinvariancefor
p-forms with p 1. (For the recent summary of the properties of unfolded dynamics
≥
we refer the reader to [27].)
As an example of the unfolded system consider the Killing equation in the AdS
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space written in the local frame
DK = κ hb, Dκ = 2λ2K h , D2 = R = λ2h h , (1.4)
a ba ab [a b] ab a b
∧
whereK issomeKillingvector,κ = κ ,D istheLorentzcovariantdifferential, h
a ab ba a
−
is the AdS vierbein one-form, λ2 is the cosmological constant, and a,b = 0,...,3.
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−
The AdS geometry is described by the zero curvature equations
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dΩ +Ω c Ω λ2h h = 0, (1.5)
ab a cb a b
∧ − ∧
dh +Ω c h = 0, (1.6)
a a c
∧
where Ω is the AdS Lorentz connection one-form. The first of the equations (1.4)
ab 4
is obviously equivalent to the AdS Killing equation as it implies D K +D K = 0,
4 a b b a
while the second one follows as a consequence of Bianchi identities for the space of
constant curvature. The equations (1.4)–(1.6) represent the unfolded system for the
set of forms W = (h ,Ω ,K ,κ ).
A a ab a ab
In this paper we show that there exists a simple deformation of the equations
(1.4)–(1.6) with the same set of forms that describes the AdS Kerr BH in four
4
dimensions withtheBHmassbeingafreedeformationparameter. Sincetheproposed
BH Unfolded System (BHUS) expresses all derivatives of fields in terms of the fields
themselves, all invariant relationships on the derivatives of BH geometric quantities
becomedirectconsequences ofBHUS.Inparticular,itmanifestlyexpresses thePetrov
type D BH Weyl tensor in terms of the Killing two-form.
The AdS Killing two-form that results from the exterior differential of the one-
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form dual to the Killing vector plays the central role in our consideration. The
integrability properties of the AdS BHUS imply, in particular, the existence of the
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sourceless Maxwell tensor and Yano-Killing tensor, both being proportional to the
Killing two-form. Let us stress that, even though the Killing two-form in AdS
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has non-vanishing divergence, it remains related to the BH Weyl tensor and to the
sourceless Maxwell field.
Although the obtained results may have different applications, the original moti-
vation for this study was due to the search of the Kerr BH solution in the nonlinear
HS gauge theory which itself is formulated in the unfolded form and AdS space (see,
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e.g., reviews [5, 8] and references therein). The proposed formulation looks particu-
larly promising in that respect. We show how the proposed BHUS makes it possible
to generate Kerr-Schild like solutions of the free bosonic HS field equations for all
spins s = 0,1... in AdS . Remarkably, for s = 1 and s = 2 the obtained solutions
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satisfy in addition the minimally coupled to gravity equations. For other spins the
covariantization of Fronsdal equations [28] in the Kerr-Schild metric leads to certain
nonlinear terms which are explicitly calculated.
The rest of the paper is organized as follows. First, in Section 2 we recall the basic
facts about the AdS Kerr BH solution, its Kerr-Schild metric decomposition and its
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Killing symmetry. In Section 3 we reconsider the Cartan formalism most appropriate
for our purpose. In Section 4 we present the BH unfolded system, build the null
geodesic congruence using Killing projectors, find relations between the Killing two-
form, Maxwell tensor and Yano-Killing tensor. In Section 5 we reproduce the Kerr
solution from AdS space via a Kerr-Schild type algebraic shift. Finally, in Section
4
6 we show how Kerr-Schild type solutions for free HS equations can be obtained.
Section 7 contains summary and conclusions.
2 AdS Kerr black hole
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A rotating BH of mass M in AdS admits the Kerr-Schild form2 [14]
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2M 2M
g (X) = η (X)+ k (X)k (X), gmn(X) = ηmn(X) km(X)kn(X),
mn mn m n
U(X) −U(X)
(2.1)
where η (X) (m,n = 0,...,3) is the background AdS metric with negative cosmo-
mn 4
logical constant λ2 and km(X) defines the null geodesic congruence with respect to
−
both full metric g (X) and the background one η (X)
mn mn
kmk = 0, km k = kmD k = 0. (2.2)
m m n m n
D
Here and D are full and background covariant differentials, respectively.
D
A useful coordinate system has the background metric of the form [16]
1−a2λ2 0 0 0
 (1+r2λ2)(1−λ2a2r2z2) 
ηmn = 0 1 λ2(x2 a2) λ2xy λ2xz ,
− − − − −
 0 λ2xy 1 λ2(y2 a2) λ2yz 
 − − − − − 
 0 λ2xz λ2yz 1 λ2z2 
 − − − − 
(2.3)
2Throughout this paper we use units with 8πG=1.
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where the radial coordinate r(X) is defined through the ellipsoid of revolution equa-
tion
x2 +y2 z2
+ = 1 (2.4)
r2 +a2 r2
and a is a rotational parameter. The BH has angular momentum J = Ma.
Components of the Kerr-Schild vector km(X) are
1 xr ay yr+ax z
k0 = , k1 = − , k2 = , k3 = , (2.5)
1+r2λ2 −r2 +a2 −r2 +a2 −r
and
a2z2
U(X) = r + . (2.6)
r3
Direct calculation gives
2 1 1
= + = D km = km, (2.7)
U Q Q¯ − m −Dm
where
iaz iaz
Q = r , Q¯ = r + . (2.8)
− r r
Note that the metric (2.1) still provides a Kerr solution after the transformation
τ(a,x,y,z,t) = ( a, x, y, z,t). (2.9)
− − − −
In other words, the Kerr-Schild Ansatz also works for the vector ni = τ(ki).
One can check that the Maxwell tensor
k n
F = dA(1) = dA(2), A(1) = m, A(2) = m (2.10)
m U m U
verifies the sourceless Maxwell equations both in the AdS and in the BH geometry
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D Fm = Fm = 0. (2.11)
m n m n
D
The Kerr BH has two Killing vectors, time translation m and rotation around
Vt
z-axis m
Vφ
∂ ∂
m = = (1,0,0,0), m = = (0,y, x,0). (2.12)
Vt ∂t Vϕ ∂ϕ −
Let us introduce a Killing two-form κ = dK as the exterior derivative of the
one-form dxmK dual to some Killing vector Km. This two-form will also be called
m
Papapetrou field. It was originally introduced in [29], where it was shown to give rise
to the sourceless Maxwell tensor in Ricci flat manifolds with isometries.
A particular linear combination m of the Killing vectors (2.12)
V
m = (1,aλ2y, aλ2x,0) (2.13)
V −
that satisfies the condition k m = 1, will be used later on for the definition of the
m
V
Killing two-form associated with a Kerr BH in AdS .
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5
3 Cartan formalism
Let dxmΩ ab be an antisymmetric one-form Lorentz connection and dxmh a be a
m m
vierbeinone-form. ThesecanbeidentifiedwiththegaugefieldsoftheAdS symmetry
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algebra o(3,2). The corresponding AdS curvatures have the form
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Rab = dΩab +Ωac Ω b λ2ha hb, (3.1)
c
∧ − ∧
Ra = dha +Ωac h , (3.2)
c
∧
where a,b,c = 0,...,3 are Lorentz indices.
The zero-torsion condition Ra = 0 expresses algebraically the Lorentz connection
Ω via derivatives of h. Then the λ-independent part of the curvature two-form (3.1)
identifies with the Riemann tensor. Einstein equations imply that the Ricci tensor
vanishes up to a constant trace part proportional to the cosmological constant. In
other words, only those components of the tensor (3.1) may remain non-vanishing
on-shell that belong to the Weyl tensor
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R = hc hdC , (3.3)
ab cdab
2 ∧
where C is the Weyl tensor in the local frame, C = C = C = C .
abcd abcd bacd abdc cdab
− −
The analysis in four dimensions considerably simplifies in spinor notation. Vector
notation are translated to the spinor one and vice versa with the help of Pauli σ-
matrices. For example, for a Lorentz vector U we have
a
1
U = (σa) U , U = (σ )αα˙U . (3.4)
αα˙ αα˙ a a a αα˙
2
Spinor indices are raised and lowered by the sp(2) antisymmetric tensors ε and ε
αβ α˙β˙
ξ = ξβε , ξα = εαβξ , ξ¯ = ξ¯β˙ε , ξ¯α˙ = εα˙β˙ξ¯ , (3.5)
α βα β α˙ β˙α˙ β˙
where ε = ε12 = 1, ε = ε , εαβ = εβα.
12 αβ βα
− −
Lorentz irreducible spinor decompositions of the Maxwell and Weyl tensors F
ab
and C read, respectively, as
abcd
¯ ¯
F = ε F +ε F , C = ε ε C +ε ε C , (3.6)
αα˙ββ˙ αβ α˙β˙ α˙β˙ αβ αα˙ββ˙γγ˙δδ˙ αβ γδ α˙β˙γ˙δ˙ α˙β˙ γ˙δ˙ αβγδ
where3 F and C are totally symmetric multispinors.
αα αααα
To describe Killing symmetries let us rewrite the Cartan equations (3.1), (3.2) in
spinor notation. Lorentz connection one-forms Ω ,Ω¯ and vierbein one-form h
αα α˙α˙ αα˙
3The symmetrization over repeated spinor indices is implied.
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can be identified with the gauge fields of sp(4) o(3,2). Vacuum Einstein equations
∼
with cosmological constant acquire the form
λ2 1
= dΩ +Ω γ Ω = H + HγγC , (3.7)
αα αα α γα αα γγαα
R ∧ 2 8
λ2 1
¯ = dΩ¯ +Ω¯ γ˙ Ω¯ = H¯ + H¯γ˙γ˙C¯ , (3.8)
α˙α˙ α˙α˙ α˙ γ˙α˙ α˙α˙ γ˙γ˙α˙α˙
R ∧ 2 8
1 1
= dh + Ω γ h + Ω¯ γ˙ h = 0, (3.9)
αα˙ αα˙ α γα˙ α˙ αγ˙
R 2 ∧ 2 ∧
¯
where , are the components of the Loretnz curvature two-form
Rαβ Rα˙β˙
1 1
2ξ = βξ + ¯ β˙ξ (3.10)
D αα˙ 2Rα βα˙ 2Rα˙ αβ˙
and
Hαα = hα hαα˙ , H¯α˙α˙ = h α˙ hαα˙ . (3.11)
α˙ α
∧ ∧
The Killing equation is
1 1
= hγ κ + h γ˙κ¯ , (3.12)
αα˙ α˙ γα α γ˙α˙
DV 2 2
where κ and κ¯ just represent non-zero components of first derivatives of the
αα α˙α˙
Killing vector . In vector notation this gives = κ = κ , hence leading
αα˙ m n mn nm
V D V −
to = 0.
(m n)
D V
Differentiation of (3.12) with the help of (3.7), (3.8) and (3.10) yields
1
κ = λ2h γ˙ + hββ˙ β C , (3.13)
D αα α Vαγ˙ 4 V β˙ ββαα
1
κ¯ = λ2hγ + hββ˙ β˙C¯ . (3.14)
D α˙α˙ α˙Vγα˙ 4 Vβ β˙β˙α˙α˙
with compatibility conditions βα˙C = 0, αβ˙C¯ = 0. In case C = 0,
D βααα D β˙α˙α˙α˙ αααα
C¯ = 0 the equations (3.12)–(3.14) describe an isometry of AdS .
α˙α˙α˙α˙ 4
4 Black hole unfolded equations
4.1 Consistency condition for Killing unfolded system
Let us investigate solutions of Einstein equations with negative cosmological constant
and the Weyl tensor of the form
C = f(X)κ κ , (4.1)
αααα αα αα
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where f(X) is some function of space-time coordinates X, and κ is a Killing two-
αα
form corresponding to some Killing vector . So, we assume at least one isometry.
αα˙
V
The Weyl tensor (4.1) is of the type D by Petrov classification [30].
From the Killing equations (3.12)–(3.14) we derive the following unfolded equa-
tions
1 1
= hγ κ + h γ˙κ¯ , (4.2)
αα˙ α˙ γα α γ˙α˙
DV 2 2
f
κ = λ2h γ˙ + hββ˙ β κ , (4.3)
D αα α Vαγ˙ 4 V β˙ ββαα
¯
f
κ¯ = λ2hγ + hββ˙ β˙κ¯ , (4.4)
D α˙α˙ α˙Vγα˙ 4 Vβ β˙β˙α˙α˙
λ2 f
= H + Hββκ , (4.5)
αα αα ββαα
R 2 8
λ2 f¯
¯ = H¯ + H¯β˙β˙κ¯ , (4.6)
Rα˙α˙ 2 α˙α˙ 8 β˙β˙α˙α˙
h = 0, (4.7)
αα˙
D
1 5fλ2
df = ( f2 + )hαγ˙ α κ , (4.8)
− 12 2κ2 V γ˙ αα
1 5f¯λ2
df¯ = ( f¯2 + )hγα˙ α˙κ¯ , (4.9)
− 12 2κ¯2 Vγ α˙α˙
where Hαα and H¯α˙α˙ are defined in (3.11), and ¯ are the curvatures (3.7) and
αα α˙α˙
R R
(3.8), and we use notations κ κβ = κ2ε ,κ = κ κ .
αβ γ αγ αααα αα αα
The system (4.2)–(4.9) has the unfolded form (1.3) and is formally consistent. We
call it BHunfolded system (BHUS). It is invariant under thefollowing transformation
τ : ( ,κ ,κ¯ ,f,f¯) (µ ,µκ ,µκ¯ ,µ−2f,µ−2f¯), (4.10)
µ αα˙ αα α˙α˙ αα˙ αα α˙α˙
V → V
where µ is a real parameter. As we show below, the transformation (4.10) with
µ = 1, that does not affect f, describes the τ-symmetry transformation (2.9).
−
The equations (4.8) and (4.9) result from the Bianchi identities for the curvatures
(4.5), (4.6). It can be shown that f(X) = f(κ2) as a consequence of (4.3). Indeed,
we have
1
dκ2 = (λ2 + fκ2)hαα˙ α κ . (4.11)
α˙ αα
3 V
Comparing (4.8) and (4.11), we obtain
dκ2 df
+ = 0. (4.12)
2λ2 + 2fκ2 1f2 +5λ2 f
3 6 κ2
Its general solution is
3
f = 6 G , (4.13)
Mκ2
8
where the complex parameter appears as an integration constant and is defined
M G
implicitly by
3 √ κ2 = λ2 (4.14)
MG −G −
and satisfies
2
d = G hαα˙ α κ . (4.15)
α˙ αα
G −2√ κ2 V
−
Analogously,
¯3
f¯= 6 ¯ G , ¯ ¯3 ¯√ κ¯2 = λ2. (4.16)
Mκ¯2 MG −G −
4.2 Killing projectors
Let two pairs of mutually conjugated projectors Π± and Π¯± have the form
αβ α˙β˙
1 1 1 1
Π± = (ǫ κ ), Π¯± = (ǫ κ¯ ). (4.17)
αβ 2 αβ ± √ κ2 αβ α˙β˙ 2 α˙β˙ ± √ κ¯2 α˙β˙
− −
They satisfy
Π±βΠ± = Π± , Π±βΠ∓ = 0, Π¯±β˙Π¯± = Π¯± , Π¯±β˙Π¯∓ = 0. (4.18)
α βγ αγ α βγ α˙ β˙γ˙ α˙γ˙ α˙ β˙γ˙
From the definition (4.17) it follows that
Π± = Π∓ , Π¯± = Π¯∓ . (4.19)
αβ − βα α˙β˙ − β˙α˙
Hereinafter we will focus on the holomorphic (i.e., undotted) sector of the BHUS. All
relations in the antiholomorphic sector result by conjugation.
From (4.17) and (4.2) and (4.3) it follows that
Π± = (Π+ Π+ +Π− Π− ) β hβγ˙. (4.20)
D αα ±G αβ αβ αβ αβ V γ˙
The projectors (4.17) split the two-dimensional (anti)holomorphic spinor space into
the direct sum of two one-dimensional subspaces. For any ξ we set
α
ξ± = Π±βξ , ξ+ +ξ− = ξ , (4.21)
α α β α α α
sothatΠ∓βξ± = 0.Thisallowsustobuildlight-likevectorswiththeaidofprojectors.
α β
Indeed, consider an arbitrary vector U . Using (4.17) define U± and U±∓ as
αα˙ αα˙ αα˙
U± = Π±βΠ¯±β˙U , U+− = Π+βΠ¯−β˙U , U−+ = Π−βΠ¯+β˙U . (4.22)
αα˙ α α˙ ββ˙ αα˙ α α˙ ββ˙ αα˙ α α˙ ββ˙
Obviously, U± U±αγ˙ = 0 and U± U±βα˙ = 0. Then U− can be cast into the form
αβ˙ αα˙ αα˙
U− = ψ ζ¯ , (4.23)
αα˙ α α˙
From this it follows that
(U−+U+−)
U± U± = U± U± , U−+U+− = U− U+ , (4.24)
αβ˙ βα˙ αα˙ ββ˙ αβ˙ βα˙ − (U−U+) αα˙ ββ˙
where (AB) = A Bαα˙.
αα˙
9
4.3 Kerr-Schild vector in BH unfolded system
Let us identify κ in (4.17) with κ in the BHUS and introduce two null vectors
αα αα
2 2
k = − , n = + (4.25)
αα˙ ( − +)Vαα˙ αα˙ ( − +)Vαα˙
V V V V
with the evident property 1k αα˙ = 1n αα˙ = 1. It is a matter of definition
2 αα˙V 2 αα˙V
which of two vectors k or n to identify with a Kerr-Schild vector. Indeed, the
αα˙ αα˙
equations (4.2)–(4.9) are invariant under the symmetry τ (4.10) which acts on k
−1 αα˙
as τ (k ) = n . Let us choose k as a Kerr-Schild vector.
−1 αα˙ αα˙ αα˙
−
Obviously,
k kβα˙ = 0. (4.26)
αα˙
From BHUS and projector properties the geodesity condition follows by straightfor-
ward calculation
kαα˙ k = 0. (4.27)
Dαα˙ ββ˙
In addition, k is the eigenvector of the Papapetrou field κ
αα˙ αα
κ kβ = √ κ2k (4.28)
αβ α˙ αα˙
−
and has the following properties as a consequence of (4.2), (4.3) and (4.15)
kαα˙ = 2( + ¯), kαα˙ = 2 2, (4.29)
αα˙ αα˙
D − G G D G G
(( + ¯)kαα˙) = 4 ¯, ( ¯kαα˙) = 0, (4.30)
αα˙ αα˙
D G G − GG D GG
k α˙ k = k α˙ k . (4.31)
α αα˙ γγ˙ α γα˙ αγ˙
D G V
The BHUS (4.2)–(4.9) admits a sourceless Maxwell tensor. Indeed, consider F
αα
of the form
2
F = G κ . (4.32)
αα αα
√ κ2
−
That (4.32) solves Maxwell equations can be easily verified as follows. Using (4.14),
(4.25) and (4.2), (4.3) one can make sure that
1
F = (( + ¯)k α˙) (4.33)
αα αα˙ α
2D G G
and F is τ -invariant. In other words, the vector-potential A = 1( + ¯)k gives
αα −1 m 2 G G m
the Maxwell tensor field F = dA.
Due to (4.32), the Weyl tensor (4.1) can be rewritten in the form
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C = MF F . (4.34)
α(4) αα αα
−
G
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