New identities for linearized gravity on the Kerr spacetime

0.24 MB·

by  Steffen Aksteiner

ON THE STRUCTURE OF LINEARIZED GRAVITY ON VACUUM
SPACETIMES OF PETROV TYPE D
STEFFENAKSTEINER,LARSANDERSSON,ANDTHOMASBA¨CKDAHL
Abstract. Inthispaperweproveanewidentityforlinearizedgravityonvacuumspacetimes
of Petrov type D. The new identity yields a covariant version of the Teukolsky-Starobinsky
identitiesforlinearizedgravity,whichinadditiontothetwoclassicalidentitiesfortheextreme
linearizedWeylscalarsincludesthreeadditionalequations. Byanalogywiththespin-1case,we
expectthenewidentitytoberelevantinderivingnewconservationlawsforlinearizedgravity.
6
1
0 1. Introduction
2
The black hole stability problem, i.e. the problem of proving dynamical stability of the Kerr

n
vacuum black hole solution is one of the most important open problems in general relativity.
a
Proving dispersive estimates for fields, including Maxwell and linearized gravity on the Kerr
J
background is an essential step towards solving this problem. In the paper [4] a new conserved
2
2 energy-momentum tensor for the Maxwell field on the Kerr spacetime was constructed. This
construction relies on an analysis of the Teukolsky Master Equation (TME) and the Teukolsky-

] Starobinsky Identities (TSI) implied by the Maxwell field equations. In this paper we shall,
c
q motivated by the above considerations, study the TSI for linearized gravity on vacuum space-
- times of Petrov type D, which includes in particular the Kerr spacetime. Our main result is a
r
g fundamentalnewidentityforlinearizedgravitywhichleadstoanew,covariantformofthespin-2
[ TSI. This extends the two classically known TSI relations by three additional identities. In the

following, we shall refer to the two classical TSI as the extreme TSI.
1
v In its classical form, the Teukolsky-Press, or Teukolsky-Starobinsky relations [19, 18] relate
4 the solutions of the radial Teukolsky equations for ±s, and is thus valid only for the separated
8 form of the equations. For the case of linearized gravity on the Kerr spacetime, a derivation of
0
the extreme TSI using the Newman-Penrose formalism, which does not require a separation of
6
variables, was given by Torres del Castillo [20], later corrected by Silva-Ortigoza [17], see the
0
. paper by Whiting and Price [22] for discussion and background. We shall refer to this pair of
1
classical identities for the Maxwell and linearized gravity cases as the extreme TSI.
0
Consider a Petrov type D spacetime and let a principal null tetrad be given1. Recall that in
6
1 this case, only one of the Newman-Penrose Weyl scalars,Ψ is non-zero. Let κ ∝Ψ−1/3, let φ ,
2 1 2 i
v: i = 0,1,2 the Maxwell scalars and let Ψ˙0,Ψ˙4 be the gauge invariant linearized Weyl scalars of
i spin weights ±2. In terms of the GHP operators þ,þ′,ð,ð′ the TME take the form
X
(þ−ρ−ρ¯)þ′(κ φ )−(ð−τ −τ¯′)ð′(κ φ )=0, (1.1a)
r 1 0 1 0
a (þ′−ρ′−ρ¯′)þ(κ φ )−(ð′−τ¯−τ′)ð(κ φ )=0. (1.1b)
1 2 1 2
for the spin-1 case, while the spin-2 TME for linearized gravity are given by
(þ−3ρ−ρ¯)þ′(κ Ψ˙ )−(ð−3τ −τ¯′)ð′(κ Ψ˙ )−3Ψ (κ Ψ˙ )=0, (1.2a)
1 0 1 0 2 1 0
(þ′−3ρ′−ρ¯′)þ(κ Ψ˙ )−(ð′−3τ′−τ¯)ð(κ Ψ˙ )−3Ψ (κ Ψ˙ )=0, (1.2b)
1 4 1 4 2 1 4
see [2]. The spin-1 extreme TSI are
ð′ð′(κ2φ )= þþ(κ2φ ) (1.3a)
1 0 1 2
þ′þ′(κ2φ )= ðð(κ2φ ) (1.3b)
1 0 1 2
2010 Mathematics Subject Classification. 83C20,83C60.
Keywords and phrases. linearizedgravity,spingeometry, algebraicallyspecialspacetimes.
1In this paper, we shall use the GHP formalism and 2-spinor formalisms, see [3, 7, 15] for notation and
background.
1
2 S.AKSTEINER,L.ANDERSSON,ANDT.BA¨CKDAHL
and the spin-2 extreme TSI are
ð′ð′ð′ð′(κ4Ψ˙ )= þþþþ(κ4Ψ˙ )+κ3Ψ L Ψ˙ (1.4a)
1 0 1 4 1 2 ξ 0
þ′þ′þ′þ′(κ4Ψ˙ )= ðððð(κ4Ψ˙ )−κ3Ψ L Ψ˙ (1.4b)
1 0 1 4 1 2 ξ 4
where in the Kerr spacetime, κ3Ψ = M/27, with M the ADM mass, and the Killing field
1 2
ξa ∝ (∂ )a, cf. Remark 2.2. A version of the spin-1 TSI in Newman-Penrose notation can be
t
found in [10]. For the extreme spin-2 TSI, see [22]. However, this form does not make the L
ξ
term explicit. In general κ3Ψ and ξa are a complex constant and Killing field, respectively. See
1 2
(2.2) for the definition of ξa. Here we have stated the form of the equations for the case without
sources. ThegeneralversionoftheTME,withsources,isgiveninCorollary3.3,andtheaversion
of TSI with sources can be deduced from Remark 4.3.
Ana¨ıveargumentcountingdegreesoffreedomindicatesthatitshouldbepossibletorepresent
the full dynamicaldegreesof freedomof the Maxwellandlinearizedgravityfields interms of one
complex scalar potential, solving the TME. However, although one may use a Debye potential
construction to produce a new Maxwell of linearized gravity field from a solution of the second
order TME [9, 12, 11, 13], both Maxwell and linearized gravity on a type D background have
conserved charges, which correspond to non-radiative modes, and these cannot be represented
in terms of Debye potentials. For the Maxwell field on the Kerr spacetime, the non-radiative
mode is the well-known Coulomb solution, while for linearized gravity, the non-radiative modes
correspond to variations of the moduli parameters M,a. Thus, it is only the radiative modes
which can be represented by Debye potentials. In particular, it must be emphasized that the
TME is not equivalent to the Maxwell or linearized gravity systems, and in particular, if we
consider a Maxwell or linearized gravity field, it is not possible to reconstruct either of these
fields given only one of the extreme scalars.
For the Maxwell field, the full set of TSI in fact contains a third relation, cf. [10], which can
be written in the form
τ¯′þ(κ2φ )+þð(κ2 φ )=τ¯þ′(κ2φ )+þ′ð′ (κ2φ ). (1.5)
1 2 1 2 1 0 1 0
As mentioned above, the full set of TME/TSI equations implied by the Maxwell field equation,
has the important consequence that the symmetric tensor V introduced in [4] is conserved.
ab
The tensor V is, in contrast to the standard Maxwell stress-energy tensor independent of the
ab
non-radiative modes of the Maxwell field, and is therefore a suitable tool to construct dispersive
estimates for the Maxwell field on the Kerr spacetime.
We now recall the Debye potential construction of solutions of linearized gravity. We first
restrict to Minkowski space. Following Sachs and Bergmann [16], let H be an anti-selfdual
abcd
Weyl field2, i.e. a tensor with the symmetries of the Riemann tensor, H = H = H ,
abcd [ab]cd cdab
H =0, satisfying Ha =0 and 1ǫ efH =−iH , and let
[abc]d bac 2 ab efcd abcd
g =∇c∇dH . (1.6)
ab acbd
Then, if ∇e∇ H =0, it holds that g solves the linearized vacuum Einstein equation.
e abcd ab
Theanalogousconstructionformasslessspin-sfieldsonthe4-dimensionalMinkowskispacewas
discussed by Penrose [14]. In [6] this was used to prove decay estimates for such fields, based on
decay estimates for the waveequation. We shallnow describe the analogof the Sachs-Bergmann
construction in the case of a vacuum type D metric.
Introduce the following complex anti-selfdual tensors with the symmetries of the Weyl tensor
Z0 =4m¯ n m¯ n ,
abcd [a b] [c d]
Z4 =4l m l m ,
abcd [a b] [c d]
where (la,na,ma,m¯a) constitutes a principal null tetrad. These are analogs of the anti-selfdual
bivectors Z0 ,Z2 , see [3, §2]. For a weighted scalar χ , let H be given by
ab ab 0 abcd
H =κ4χ Z0 . (1.7)
abcd 1 0 abcd
2Hereweuseacomplexanti-selfdualWeylfieldforconsistencewiththerestofthepaper,althoughthisisnot
usedin[16].
ON THE STRUCTURE OF LINEARIZED GRAVITY ON VACUUM SPACETIMES OF PETROV TYPE D 3
Define the 1-form U by
a
U =−∇ log(κ ),
a a 1
cf. (2.7), and consider the following analog of (1.6), which sends H to a 2-tensor g ,
abcd ab
g =∇c(∇ +4U )H d .
ab d d (a b)c
A calculation shows that g is a complex solution to the linearized vacuum Einstein equation
ab
provided the scalar χ solves the Teukolsky Master Equation (TME) for spin weight +2 [12].
0
Applying the GHP prime operation la ↔ na, ma ↔ m¯a to the above construction, leads to
the fact that the analogous construction with
H =κ4χ Z4 (1.8)
abcd 1 4 abcd
yields a solution to the linearized Einstein equation in the same way, provided that now the
scalar κ4χ solves the TME for spin weight −2. Note that in general, the linearized metrics g
1 4 ab
constructedfrom (1.7) and (1.8) aredifferent. We are now able to state the tensor versionof our
main result, which describes this difference.
Theorem 1.1 (Tensor version). Let δg be a solution to the linearized Einstein equation on a
ab
vacuum type D background, and let Ψ˙ ,Ψ˙ be the corresponding linearized Weyl scalars of spin
0 4
weight ±2. Let
H =κ4Ψ˙ Z0 −κ4Ψ˙ Z4 , (1.9)
abcd 1 0 abcd 1 4 abcd
and let b
M =∇c(∇ +4U )H d . (1.10)
ab d d (a b)c
Then there is a complex vector field A , depending onbup to three derivatives of the linearized
a
metric δg , and independent of the extreme linearized Weyl scalars Ψ˙ ,Ψ˙ , such that
ab 0 4
M =∇ A + 1Ψ κ3L δg . (1.11)
ab (a b) 2 2 1 ξ ab
Remark 1.2. (1) From (1.11), it is clear that M is a complex solution of the linearized
ab
Einstein equation.
(2) In the Maxwell case, the analogue of (1.11) is that the vector potential arising out of
the Debye potential construction by taking the difference of the extreme Maxwell scalars
from the same Maxwell field is pure gauge, see [1, Proposition 5.2.5]. In the spin-2 case
above, theterminvolvingL δg isanewfeature,whichindicatesanimportantqualitative
ξ ab
difference between the spin-1 and spin-2 cases.
As willbe shownbelow,the identity (1.11) implies the the fullTSI for linearizedgravity,aset
of five scalar equations of fourth order. To see that the extreme TSI follows from this identity, it
is sufficient to recall that the extreme linearized Weyl scalars Ψ˙ ,Ψ˙ are gauge invariant.
0 4
In order to discuss further details of the main result and its proof, we shall use the 2-spinor
formalism,see[15]and[7]forbackground. Weshallfurthermakeuseofthevariationalformalism
for spinors, which was introduced in [8].
The fundamental operators C ,C† ,T ,D acting on symmetric spinors of valence k,l are
k,l k,l k,l k,l
definedas the irreduciblepartsofthe covariantderivative∇AA′φBC···DB′C′···D′ ofthe symmetric
spinor φAB···DA′B′···D′ of valence k,l. They were introducedin the paper [5], which alsocontains
a detailed discussionof their properties. The mainfeatures of the type D geometryis encodedin
the Killing spinor κ =κ o ι , satisfying
AB 1 (A B)
(T2,0κ)ABCA′ =0.
We shall also make use of the extended operators C ,C† ,T ,D , introduced here
k,l,m k,l,m k,l,m k,l,m
for the first time, defined in an analogous manner as the irreducible parts of
(∇AA′ +mUAA′)φBC···DB′C′···D′, (1.12)
see Section 2.2 below for details.
Given a 2-tensor δg solving the linearized Einstein equations, define
ab
G=δgCCC′C′, GABA′B′ =δg(AB)(A′B′),
and let
1
φ = (C C G) . (1.13)
ABCD 3,1 2,2 ABCD
2
4 S.AKSTEINER,L.ANDERSSON,ANDT.BA¨CKDAHL
Then φ is a modified linearized Weyl spinor, see (3.3). The extreme scalars φ ,φ coincide
ABCD 0 4
with Ψ˙ ,Ψ˙ introduced above.
0 4
Theorem 1.1′ (Spinor version). Let δgABA′B′ be a solution to the linearized Einstein equation
on avacuumtypeD background, letφ begiven by (1.13), and letφ ,φ bethecorresponding
ABCD 0 4
linearized Weyl scalars of spin weight ±2. Let o ,ι be a principal dyad. Define
A A
φ =ι ι ι ι κ4φ −o o o o κ4φ , (1.14)
ABCD A B C D 1 0 A B C D 1 4
b
and let
MABA′B′ =(C3†,1C4†,0,4φ)ABA′B′. (1.15)
Then, there is a complex vector field AAA′ depenbding on up to three derivatives of the linearized
metric δgABA′B′, and independent of the extreme linearized Weyl scalars φ0,φ4, such that
MABA′B′ = 12∇AA′ABB′ + 12∇BB′AAA′ + 12Ψ2κ31(Lξδg)ABA′B′. (1.16)
Remark 1.3. Equation (1.14) can be written purely in terms of the Killing spinor κ , without
AB
reference to a specific dyad, see Example 2.8.
Wenowapplytwomorederivativestobothsidesofequation(1.16)(orequivalently,toequation
(1.11)). This gives the following result, which is the covariant form of the spin-2 TSI.
Corollary 1.4. For a vacuum type D spacetime, we have the covariant generalization of the
Teukolsky–Starobinski identities for linearized gravity
(C1†,3C2†,2C3†,1C4†,0,4φ)A′B′C′D′ =κ31Ψ2(Lξφ¯)A′B′C′D′ +(LˆAΨ¯)A′B′C′D′, (1.17)
b
where φ is defined in (1.13), and
ABCD
(LˆAΨ¯)A′B′C′D′ = 12Ψ¯A′B′C′D′(D1,1,0,6A)+2Ψ¯(A′B′C′F′(C1†,1,0,2A)D′)F′. (1.18)
is the Lie derivative on spinors.
Remark 1.5. Given a pair φ ,φ of spin weight ±2 scalars, we say that they solve the spin-2
0 4
TME/TSI system, provided they solve the second order spin-2 TME system (1.2) and also that
they solve the fourth order spin-2 TSI system (1.17), or alternatively the second order identity
(1.16), in a suitable sense. In view of the result by Coll et al. mentioned above, it is natural to
conjecture at this point that the combined spin-2 TME/TSI system (1.2) and (1.17) (or alterna-
tively (1.2) and (1.16)) yields an equivalent system for linearized gravity, in the sense that given
such a pair of scalars, it is possible, modulo gauge conditions and charges, to reconstruct the rest
of the linearized gravitational field.
Recall that in the case of the Kerr spacetime, κ3Ψ and ξa are real. This means that we can
1 2
take the real and imaginary parts of M , yelding the following result.
ab
Corollary 1.6. On the Kerr spacetime, the imaginary part of (1.16) is gauge invariant and for
the real part we get
ReMABA′B′ = 12∇AA′ReABB′ + 21∇BB′ReAAA′ + 514M(Lξδg)ABA′B′. (1.19)
Remark1.7. Thespinor ReMABA′B′ canbeinterpretedas alinearized metric. Itisindependent
of the coordinate gauge of the original metric and constructed only from the spin-2 part of the
linearized Weyl curvature. Therefore it is gauge invariant. With a particular gauge choice one
could hope to set ReAAA′ = 0. One could then hope to reconstruct δgABA′B′ from the curvature
through integration in time.
The paper is organized as follows. In Section 2 we give some background and preliminary
results. Section 2.1 contains a review of the consequences of the existence of a Killing spinor
on vacuum type D spacetimes. In particular we introduce extended fundamental operators and
projectionoperatorsbasedontheKillingspinor. Example2.8introducesthespin-projected,sign-
reversed linearized Weyl spinor φ which will play a central role in the paper. The section
ABCD
endswithtwotechnicallemmatab. InSection3aspinorialformafthefieldequationsoflinearized
gravity is presented. Corollary 3.2 gives an equation for φ which is a consequence of the
ABCD
linearized Bianchi identity. This equation plays a central rbole in the proof of the main theorem,
ON THE STRUCTURE OF LINEARIZED GRAVITY ON VACUUM SPACETIMES OF PETROV TYPE D 5
given in section 4. In Appendix A we give the GHP form of the vector field AAA′ introduced in
the main theorem, and of the spin-2 TSI.
2. Preliminaries
2.1. Geometric structure of Petrov type D spacetimes. Itis wellknown[21]thatvacuum
spacetimes of Petrov type D admit a non-trivial irreducible symmetric 2-spinor κ solving the
AB
Killing spinor equation
(T2,0κ)ABCA′ =0. (2.1)
Defining the spinors
ξAA′ =(C2†,0κ)AA′, (2.2)
λA′B′ =(C1†,1C2†,0κ)A′B′, (2.3)
the complete table of derivatives reads
∇AA′κBC = − 31ξCA′ǫAB− 31ξBA′ǫAC, (2.4a)
∇AA′ξLL′ = − 12λA′L′ǫAL− 43κBCΨALBCǫ¯A′L′, (2.4b)
∇CC′λA′B′ =2ξCD′Ψ¯A′B′C′D′. (2.4c)
The fact that the system of equations (2.4) for (κAB,ξAA′,λA′B′) is closed, implies in particular
higher derivatives of κ do not give any further information.
AB
Remark 2.1. If we furthermore assume the generalized Kerr-NUT condition that ξAA′ is real,
the middle equation simplifies to λA′L′ = 23κ¯B′C′Ψ¯A′L′B′C′ so the λA′B′ is not an independent
field any more and the system reduces to
∇AA′κBC = − 13ξCA′ǫAB− 13ξBA′ǫAC, (2.5a)
∇AA′ξLL′ = − 34κ¯B′C′Ψ¯A′L′B′C′ǫAL− 34κBCΨALBCǫ¯A′L′. (2.5b)
Using a principal dyad the Killing spinor takes the form
κ = −2κ o ι , (2.6)
AB 1 (A B)
−1/3
with κ ∝ Ψ . Beside the Killing vector field (2.2) another important vector field is defined
1 2
by
κABξBA′
UAA′ = − 3κ2 =−∇AA′log(κ1). (2.7)
1
Because it is completely determined by the Killing spinor (2.6), we have the complete table of
derivatives
(D U)= −2Ψ + ξAA′ξAA′, (2.8a)
1,1 2 9κ2
1
(C U) =0, (2.8b)
1,1 AB
(C1†,1U)A′B′ =0, (2.8c)
(T U) A′B′ = κAB(C1†,1ξ)A′B′ +2U (A′U B′)− ξ(A(A′ξB)B′), (2.8d)
1,1 AB 6κ2 (A B) 9κ2
1 1
in particular UAA′ is closed.
From the integrability condition (L Ψ) =0 it follows that
ξ ABCD
(T4,0Ψ)ABCDFA′ =5Ψ(ABCDUF)A′. (2.9)
The curvature can be expressed in terms of the Killing spinor according to
3Ψ κ κ
2 (AB CD)
Ψ = . (2.10)
ABCD 2κ2
1
Remark2.2. OnKerrwithparameters(M,a)inBoyer-LindquistcoordinatesΨ =− M
2 (r−iacosθ)3
and we can set κ =−1(r−iacosθ). Then ξa =∂ ,Ψ κ3 = 1 M.
1 3 t 2 1 27
6 S.AKSTEINER,L.ANDERSSON,ANDT.BA¨CKDAHL
2.2. Extended fundamental spinor operators. Based on the irreducible decomposition of
covariantderivativesonsymmetricspinors,see[5],wedefinetheextendedfundamentaloperators
with additional (extended) indices n,m,
(Dk,l,n,mϕ)A1...Ak−1A′1...A′l−1 ≡h∇BB′ +nUBB′ +mU¯BB′iϕA1...Ak−1BA′1...A′l−1B′, (2.11a)
(Ck,l,n,mϕ)A1...Ak+1A′1...A′l−1 ≡h∇(A1B′ +nU(A1B′ +mU¯(A1B′iϕA2...Ak+1)A′1...A′l−1B′, (2.11b)
(Ck†,l,n,mϕ)A1...Ak−1A′1...A′l+1 ≡h∇B(A′1 +nUB(A′1 +mU¯B(A′1iϕA1...Ak−1BA′2...A′l+1), (2.11c)
(Tk,l,n,mϕ)A1...Ak+1A′1...A′l+1 ≡h∇(A1(A′1 +nU(A1(A′1 +mU¯(A1(A′1iϕA2...Ak+1)A′2...A′l+1). (2.11d)
Forn=m=0itcoincideswiththeusualdefinitionofthe fundamentaloperatorsandtheindices
will be suppressed in that case. Because UAA′ is a logarithmic derivative we have
(Dk,l,n,mϕ)A1...Ak−1A′1...A′l−1 =κn1κ¯m1 (Dk,lκ−1nκ¯−1mϕ)A1...Ak−1A′1...A′l−1, (2.12a)
(Ck,l,n,mϕ)A1...Ak+1A′1...A′l−1 =κn1κ¯m1 (Ck,lκ−1nκ¯−1mϕ)A1...Ak+1A′1...A′l−1, (2.12b)
(Ck†,l,n,mϕ)A1...Ak−1A′1...A′l+1 =κn1κ¯m1 (Ck†,lκ−1nκ¯−1mϕ)A1...Ak−1A′1...A′l+1, (2.12c)
(Tk,l,n,mϕ)A1...Ak+1A′1...A′l+1 =κn1κ¯m1 (Tk,lκ−1nκ¯−1mϕ)A1...Ak+1A′1...A′l+1. (2.12d)
In particular it follows that the commutator of extended fundamental spinor operators with
n = n ,m = m reduces to the commutator of the usual fundamental spinor operators. For
1 2 1 2
commutatorsoftheextendedoperatorswithunequalweightsn ,n ,m ,m onesimplysplitsinto
1 2 1 2
first derivatives and remainder with equal weights.
2.3. Projection operators and the spin decomposition.
Definition 2.3. Given the Killing spinor (2.6), define the operators Ki : S →S ,i =
k,l k,l k−2i+2,l
0,1,2 via
(K0k,lϕ)A1...Ak+2A′1...A′l =2κ−11κ(A1A2ϕA3...Ak+2)A′1...A′l, (2.13a)
(K1k,lϕ)A1...AkA′1...A′l =κ−11κ(A1FϕA2...Ak)FA′1...A′l, (2.13b)
(K2k,lϕ)A1...Ak−2A′1...A′l = − 12κ−11κCDϕA1...Ak−2CDA′1...A′l. (2.13c)
Example 2.4. The “spin raising” operator K0 on a symmetric (2,0) spinor ϕ has compo-
k,l AB
nents
(K0 ϕ) =0, (K0 ϕ) =ϕ , (K0 ϕ) = 4ϕ , (K0 ϕ) =ϕ , (K0 ϕ) =0.
2,0 0 2,0 1 0 2,0 2 3 1 2,0 3 2 2,0 4
The ”sign flip” operator K1 on a symmetric (4,0) spinor ϕ has components
k,l ABCD
(K1 ϕ) =ϕ , (K1 ϕ) = 1ϕ , (K1 ϕ) =0, (K1 ϕ) = − 1ϕ , (K1 ϕ) = −ϕ .
4,0 0 0 4,0 1 2 1 4,0 2 4,0 3 2 3 4,0 4 4
The “spin lowering” operator K2 on a symmetric (4,0) spinor ϕ has components
k,l ABCD
(K2 ϕ) =ϕ , (K2 ϕ) =ϕ , (K2 ϕ) =ϕ .
4,0 0 1 4,0 1 2 4,0 2 3
Definition 2.5 (Spin decomposition). For any symmetric spinor ϕ ,
A1...A2s
• withintegers,defines+1symmetricvalence2sspinors(Pi ϕ) ,i=0...ssolving
2s,0 A1...A2s
s
ϕ = (Pi ϕ) , (2.14)
A1...A2s X 2s,0 A1...A2s
i=0
with (Pi ϕ) depending only on the components ϕ and ϕ .
2s,0 A1...A2s s+i s−i
• with half-integer s, define s+ 1 symmetric valence 2s spinors (Pi ϕ) ,i= 1...s
2 2s,0 A1...A2s 2
solving
s
ϕ = (Pi ϕ) , (2.15)
A1...A2s X 2s,0 A1...A2s
i=1/2
with (Pi ϕ) depending only on the components ϕ and ϕ .
2s,0 A1...A2s s+i s−i
ON THE STRUCTURE OF LINEARIZED GRAVITY ON VACUUM SPACETIMES OF PETROV TYPE D 7
Remark 2.6. The spin decomposition can also be defined for spinors with primed indices and
more generally for mixed valence. In that case the decompositions combine linearly.
Example 2.7. (1) For s=2 the decomposition is given by
ϕ =(P0 ϕ) +(P1 ϕ) +(P2 ϕ) (2.16)
ABCD 4,0 ABCD 4,0 ABCD 4,0 ABCD
and the components, written as vectors, are
ϕ 0 0 ϕ
0 0
       
ϕ 0 ϕ 0
1 1
ϕ2=ϕ2+ 0+ 0 . (2.17)
       
ϕ3 0  ϕ3  0 
       
ϕ4 0   0 ϕ4
In terms of the operators (2.13) they read
(P0 ϕ) = 3(K0 K0 K2 K2 ϕ) , (2.18a)
4,0 ABCD 8 2,0 0,0 2,0 4,0 ABCD
(P1 ϕ) =(K0 K1 K1 K2 ϕ) , (2.18b)
4,0 ABCD 2,0 2,0 2,0 4,0 ABCD
(P2 ϕ) =(K1 K1 K1 K1 ϕ) − 1 (K0 K1 K1 K2 ϕ) . (2.18c)
4,0 ABCD 4,0 4,0 4,0 4,0 ABCD 16 2,0 2,0 2,0 4,0 ABCD
(2) For s=3/2 on a (3,1) spinor the decomposition is given by
ϕABCA′ =(P13/,12ϕ)ABCA′ +(P33/,12ϕ)ABCA′ (2.19)
and the components, written as vectors, are
ϕ0A′ 0 ϕ0A′
     
ϕ1A′ = ϕ1A′ + 0 . (2.20)
ϕ2A′ ϕ2A′  0 
     
ϕ3A′  0  ϕ3A′
In terms of the operators (2.13) they read
(P13/,12ϕ)ABCA′ = 43(K01,1K23,1ϕ)ABCA′, (2.21a)
(P33/,12ϕ)ABCA′ = − 112(K01,1K23,1ϕ)ABCA′ +(K13,1K13,1ϕ)ABCA′. (2.21b)
Example 2.8. Given a symmetric spinor ϕ , let
ABCD
ϕ =κ4(K1 P2 ϕ) .
ABCD 1 4,0 4,0 ABCD
b
Then the components of ϕ are
ABCD
b ϕ κ4ϕ
 0  1 0 
ϕ 0
b1
ϕb2= 0 .
   
ϕb3  0 
   
ϕb4 −κ41ϕ4
b
Lemma 2.9. For any symmetric spinors ϕABCD, ϕAB, ϕ, ϕAA′, ϕABCA′ and an integer w, we
have the algebraic identities
(K1 P1 ϕ) = 1(K0 K1 K2 ϕ) , (2.22a)
4,0 4,0 ABCD 2 2,0 2,0 4,0 ABCD
(K11,1K11,1ϕ)AA′ =ϕAA′, (2.22b)
(K2 K1 ϕ)=0, (2.22c)
2,0 2,0
(P33/,12K01,1φ)ABCA′ =0, (2.22d)
(K1 K1 K1 φ) =(K1 φ) , (2.22e)
2,0 2,0 2,0 AB 2,0 AB
(K13,1K13,1K13,1φ)ABCA′ = − 29(K01,1K11,1K23,1φ)ABCA′ +(K13,1φ)ABCA′, (2.22f)
8 S.AKSTEINER,L.ANDERSSON,ANDT.BA¨CKDAHL
and the first order differential identities
(C4†,0,wK14,0ϕ)ABCA′ =(K13,1C4†,0,wϕ)ABCA′ + 12(T2,0,−4+wK24,0ϕ)ABCA′, (2.23a)
(C4†,0,wK02,0ϕ)ABCA′ =(K01,1C2†,0,−1+wϕ)ABCA′ −(T2,0,−4+wK12,0ϕ)ABCA′, (2.23b)
(C3†,1,wK01,1ϕ)ABA′B′ =(K00,2C1†,1,−1+wϕ)ABA′B′ − 34(T1,1,−3+wK11,1ϕ)ABA′B′, (2.23c)
(C2†,0,wK12,0ϕ)AA′ =(K11,1C2†,0,wϕ)AA′ +(T0,0,−2+wK22,0ϕ)AA′, (2.23d)
(C2†,0,wK24,0ϕ)AA′ =(K23,1C4†,0,1+wϕ)AA′, (2.23e)
(C2†,0,wK00,0ϕ)AA′ = −2(K11,1T0,0,−2+wϕ)AA′, (2.23f)
(D K1 φ)=2(K2 C φ), (2.23g)
1,1,w 1,1 2,0 1,1,−2+w
(D K2 φ)=(K2 D φ), (2.23h)
1,1,w 3,1 2,0 3,1,1+w
(K13,1P33/,12T2,0,wφ)ABCA′ = − 14(C4†,0,4+wK02,0K12,0K12,0φ)ABCA′ + 43(T2,0,wK12,0φ)ABCA′. (2.23i)
Proof. For (2.22a) we calculate
(K1 P1 ϕ) =(K1 K0 K1 K1 K2 ϕ)
4,0 4,0 ABCD 4,0 2,0 2,0 2,0 4,0 ABCD
= 1(K0 K1 K1 K1 K2 ϕ)
2 2,0 2,0 2,0 2,0 4,0 ABCD
= 1(K0 K1 K2 ϕ) .
2 2,0 2,0 4,0 ABCD
In the first step uses (2.18b), the second one is a commutator of K1 and K0 and the third step
makes use of the fact that three sign-flips are equalto one sign-flip. For (2.22b) we note that K1
on a (1,1) spinor changes sign in two of the four components,
(K11,1ϕ)00′ =ϕ00′, (K11,1ϕ)01′ =ϕ01′, (K11,1ϕ)10′ = −ϕ10′, (K11,1ϕ)11′ = −ϕ11′,
so K1 K1 = Id. Equation (2.22c) is true because K1 cancels the middle component of ϕ
1,1 1,1 2,0 AB
andK2 singlesoutthatmiddle component. The restofthe algebraicidentities areprovedanal-
2,0
ogously. The proof of the differential identities relies on a straightforwardbut tedious expansion
of projectors (2.13) and fundamental operators (2.11). We only calculate (2.23e).
(C2†,0,−1K24,0φ)AA′ −(K23,1C4†,0φ)AA′ = UBA′κC2κDφABCD + κBC(C4†2,0κφ)ABCA′ +(C2†,0K24,0φ)AA′
1 1
UBA′κCDφABCD φABCD(T2,0κ)BCDA′
= −
2κ 2κ
1 1
κCDφABCD(T0,0κ1)BA′
+
2κ2
1
=0.
The rest are proved along similar lines. (cid:3)
Lemma 2.10. The following identities hold
0=(K00,2C1†,1,−5T0,0,−3ϕ)ABA′B′ + 43(T1,1,−4K11,1T0,0,−6ϕ)ABA′B′
− 34(T1,1,−7K11,1T0,0,−3ϕ)ABA′B′, (2.24a)
0=(C3†,1,−1K01,1T0,0ϕ)ABA′B′ +2(K00,2C1†,1T0,0ϕ)ABA′B′ +12(K12,2T1,1T0,0ϕ)ABA′B′
− 332(T1,1,−1K11,1T0,0,3/2ϕ)ABA′B′, (2.24b)
0= − 112(C3†,1,−1K01,1D2,2,4ϕ)ABA′B′ +(C3†,1,−1K13,1C2,2,1ϕ)ABA′B′ −Ψ2(K12,2ϕ)ABA′B′
−(K12,2C3†,1C2,2ϕ)ABA′B′ − 31(K12,2T1,1D2,2ϕ)ABA′B′ + (Lξϕ3)κABA′B′
1
+ 29(T1,1,−1K11,1D2,2,1ϕ)ABA′B′ − 32(T1,1,−1K23,1C2,2,−2ϕ)ABA′B′. (2.24c)
Proof. This can be verified by expanding in components. (cid:3)
ON THE STRUCTURE OF LINEARIZED GRAVITY ON VACUUM SPACETIMES OF PETROV TYPE D 9
3. Spinorial formulation of linearized gravity
In this section we review the field equations of linearized gravity for a general vacuum back-
ground and allow for sources of the linearized field. The spinor variational operator ϑ developed
in [8] will be used. Let δgABA′B′ be a solution of the linearized Einstein equations
12ǫABǫ¯A′B′ϑΛ−4ϑΦABA′B′ =(cid:3)δgABA′B′ −∇AA′FBB′ −∇BB′FAA′
−2Ψ¯A′B′C′D′δgABC′D′ −2ΨABCDδgCDA′B′, (3.1)
with Λ the Ricci scalar, ΦABA′B′ the trace-free Ricci spinor, ΨABCD the Weyl spinor and
FAA′ = −21∇AA′δgBBB′B′ +∇BB′δgABA′B′ the gauge source function. Observe that we make
the variation with the indices down, and raise them and take traces afterwards. We define the
irreducible parts of the linearized metric as
GABA′B′ =δg(AB)(A′B′), G =δgCCC′C′, (3.2)
and introduce
φ = 1GΨ +ϑΨ . (3.3)
ABCD 4 ABCD ABCD
asamodificationofthe variedWeylspinorϑΨ 3. Thenweget,see[8], forageneralvacuum
ABCD
background
φ = 1(C C G) , (3.4a)
ABCD 2 3,1 2,2 ABCD
ϑΦABA′B′ = 21GCDA′B′ΨABCD+ 21(C3†,1C2,2G)ABA′B′ + 61(T1,1D2,2G)ABA′B′
− 18(T1,1T0,0G)ABA′B′, (3.4b)
ϑΛ= − 1 (D D G)+ 1 (D T G). (3.4c)
24 1,1 2,2 32 1,1 0,0
Inparticular(3.4b)and(3.4c)togetherareequivalentto(3.1). NotealsothatϑΛ=0=ϑΦABA′B′
in the source-free case. As a consequence of (3.4) we derive the linearized Binachi identity.
Lemma 3.1. For a general vacuum background the modified Weyl spinor (3.4a) satisfies
(C4†,0φ)ABCA′ =(C2,2ϑΦ)ABCA′ + 12ΨABCD(D2,2G)DA′ − 23Ψ(ABDF(C2,2G)C)DFA′
− 18ΨABCD(T0,0G)DA′ + 21GDFA′B′(T4,0Ψ)ABCDFB′. (3.5)
Restricting to a type D background this simplifies to
(C4†,0φ)ABCA′ =(C2,2ϑΦ)ABCA′ − 32Ψ2(C2,2,1G)ABCA′ − 83Ψ2(K01,1K11,1D2,2,4G)ABCA′
+ 332Ψ2(K01,1K11,1T0,0G)ABCA′ + 49Ψ2(K01,1K23,1C2,2,1G)ABCA′. (3.6)
Proof. We apply C on (3.4b), commute C C† , use (3.4a) to get
2,2 2,2 3,1
(C2,2ϑΦ)ABCA′ = 12(C2,2C3†,1C2,2G)ABCA′ + 16(C2,2T1,1D2,2G)ABCA′ − 81(C2,2T1,1T0,0G)ABCA′
− 13ΨABCD(D2,2G)DA′ + 21Ψ(ABDF(C2,2G)C)DFA′
− 21GDFA′B′(T4,0Ψ)ABCDFB′ (3.7)
= 16(C2,2T1,1D2,2G)ABCA′ − 81(C2,2T1,1T0,0G)ABCA′ +(C4†,0φ)ABCA′
− 13ΨABCD(D2,2G)DA′ + 23Ψ(ABDF(C2,2G)C)DFA′
− 21GDFA′B′(T4,0Ψ)ABCDFB′ − 18(T2,0D3,1C2,2G)ABCA′. (3.8)
3InatypeDprincipalframethismodificationonlyaffects themiddlecomponent.
10 S.AKSTEINER,L.ANDERSSON,ANDT.BA¨CKDAHL
We then commute the C T operators, and in the last step we commute D C and use
2,2 1,1 3,1 2,2
C T =0 to get
1,1 0,0
(C2,2ϑΦ)ABCA′ =(C4†,0φ)ABCA′ − 12ΨABCD(D2,2G)DA′ + 23Ψ(ABDF(C2,2G)C)DFA′
+ 81ΨABCD(T0,0G)DA′ − 21GDFA′B′(T4,0Ψ)ABCDFB′
+ 112(T2,0C1,1D2,2G)ABCA′ − 116(T2,0C1,1T0,0G)ABCA′
− 81(T2,0D3,1C2,2G)ABCA′ (3.9)
=(C4†,0φ)ABCA′ − 21ΨABCD(D2,2G)DA′ + 23Ψ(ABDF(C2,2G)C)DFA′
+ 81ΨABCD(T0,0G)DA′ − 21GDFA′B′(T4,0Ψ)ABCDFB′. (3.10)
This gives(3.5). Onatype Dspacetime,wecanuse(2.9)and(2.10). The resultingUAA′ spinors
canbe incorporatedas extendedindices, and the κ spinorscanthen be rewrittenin terms the
AB
Ki operators to get (3.6). (cid:3)
NotethatonaMinkowskibackgroundandwithoutsourcestherighthandsideof (3.5)vanishes
andthelinearizedBianchiidentityreducestothespin-2equation. ThelinearizedBianchiidentity
(3.6)isoffundamentalimportanceandnextwederivesomedifferentialidentities foritwhichare
needed for the main theorem.
Corollary 3.2. The rescaled spin-projected Weyl spinor φ =κ4(K1 P2 φ) satisfies
ABCD 1 4,0 4,0 ABCD
b
(C4†,0,4φ)ABCA′ = −(cid:0)κ41K13,1P33/,12C4†,0P14,0φ(cid:1)ABCA′ − 32(cid:0)κ41Ψ2K13,1P33/,12C2,2,1G(cid:1)ABCA′
b + κ4K1 P3/2C ϑΦ . (3.11)
(cid:0) 1 3,1 3,1 2,2 (cid:1)ABCA′
Proof. Applying the operator K1 P3/2κ4C† to (2.16) and using (2.23a) gives the identity
3,1 3,1 1 4,0
(C4†,0,4φ)ABCA′ = −(cid:0)K13,1P33/,12C4†,0,4(κ41P14,0φ)(cid:1)ABCA′ +(cid:0)K13,1P33/,12(κ41C4†,0φ)(cid:1)ABCA′. (3.12)
b
The result follows from (P33/,12K01,1φ)ABCA′ =0 together with (3.6). (cid:3)
Corollary 3.3 (Covariant TME). The covariant form of the spin-2 Teukolsky Master equation
with source is given by
(C C† φ) = −3Ψ φ +κ4(P2 K1 C C ϑΦ) . (3.13)
3,1 4,0,4 ABCD 2 ABCD 1 4,0 4,0 3,1,−4 2,2 ABCD
Proof. Apply the operabtor P2 C to (3.b11) and use
4,0 3,1
P2 C K1 P3/2C† (κ4P1 φ) =0, (3.14a)
(cid:0) 4,0 3,1 3,1 3,1 4,0,4 1 4,0 (cid:1)ABCD
P2 C K1 P3/2(κ4C ϑΦ) =κ4(P2 K1 C C ϑΦ) , (3.14b)
(cid:0) 4,0 3,1 3,1 3,1 1 2,2 (cid:1)ABCD 1 4,0 4,0 3,1,−4 2,2 ABCD
P2 C K1 P3/2(κ4Ψ C G) =Ψ κ4(P2 K1 C C G) . (3.14c)
(cid:0) 4,0 3,1 3,1 3,1 1 2 2,2,1 (cid:1)ABCD 2 1 4,0 4,0 3,1 2,2 ABCD
(cid:3)
4. Main theorem
Weshallnowproveourmaintheorem. ThefollowingisthedetailedstatementofTheorem1.1′.
Theorem 4.1. Let φ be the modified linearized Weyl spinor given in (3.3) and let
ABCD
φ =κ4(K1 P2 φ) (4.1)
ABCD 1 4,0 4,0 ABCD
b
be the rescaled spin-projected field as in example 2.8, and let
MABA′B′ =(C3†,1C4†,0,4φ)ABA′B′. (4.2)
b
Then we have
MABA′B′ = 12∇AA′ABB′ + 12∇BB′AAA′ + 12Ψ2κ31(Lξδg)ABA′B′, (4.3)
where the complex vector field AAA′ is given by
AAA′ = − 21Ψ2κ31ξBB′(K00,2K22,2G)ABA′B′ + 32κ31ξBA′(K12,0K24,0φ)AB
+(K11,1T0,0K22,0K24,0(κ41φ))AA′ − 41Ψ2κ41(K11,1T0,0,2G)AA′. (4.4)