Null electromagnetic fields destroy black holes

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by  Koray Düztaş

Null electromagnetic fields destroy black holes
∗
Koray Du¨zta¸s
Bo˘gazi¸ci University, Department of Physics
Bebek 34342, I˙stanbul, Turkey
We consider two test, null electromagnetic fields aligned with the two repeated principal null
directionsofthetypeDbackground,andatest,non-nullfieldspecialinthesensethattheprincipal
null directions of the electromagnetic field lie along the repeated principal null directions of the
space-time. We show that the interaction of the null field along la with Kerr black holes leads to
violation of two fundamental laws/conjectures of black hole physics: a generic violation of cosmic
censorship conjecture, and violation of the area theorem. These results are totally unexpected
considering the fact that the energy-momentum tensor obeys the weak energy condition. We also
show that the special non-null field does not lead to any perturbation of black hole parameters
6 of mass and angular momentum, and the null field directed along na does not challenge cosmic
1
censorship or thearea theorem.
0
2
PACSnumbers: 04.20.Dw,04.20.Gz,04.40.Nr

p
e
S I. INTRODUCTION infinity. Bynohairtheorem,thespace-timesettlestoan-
8 other stationary condition with new parameters M,Q,a
1 According to singularity theorems of Hawking and at the end of the interaction . If the final configura-
Penrose, the formation of singularities is inevitable as a tion of parameters satisfies (1), the interaction with test
c] resultofgravitationalcollapseinclassicalgeneralrelativ- fields or particles has not destroyed the horizon, so the
q ity, given some very reasonable assumptions [1]. In the CCC remains valid. If the final configuration does not
- model developed by Penrose and Hawking the trapped satisfy (1), the black hole has turned into a naked sin-
r
g surface that arises in the spherically symmetric gravi- gularity and CCC is violated. In the first example of
[ tational collapse of a body, is contained in the black these thought experiments Wald showed that particles

hole region of the space-time, so the singularity is sur- with enough charge or angular momentum to destroy
2
v rounded by an event horizon. This singularity can be the horizoneither miss or are repelled by the black hole.
5 considered harmless for distant observers, as opposed to ManyauthorsfollowedWaldtoconstructsimilarthought
3 a nakedone which intersects a Cauchy surface rendering experimentstotestthevalidityofCCCintheinteraction
7 the initial conditions undefined, thus disabling asymp- of black holes with test particles or fields. [4–26]
1 totic predictability. Penrose proposed the Cosmic Cen- Recently we have constructed a thought experi-
0
sorship Conjecture (CCC) [2], in order to avoid these ment[27] inwhich a free testelectromagneticfield inter-
.
1 pathologiesandpreservethedeterministicnatureofgen- acts with an extremal Kerr black hole, and showed that
0 eral relativity. The weak form (WCCC) asserts that the CCC remains valid in this case. In this work we check
6 singularities that arise in gravitational collapse are al- the validity ofCCC forthe specialelectromagneticfields
1
ways hidden behind event horizons. In this respect, a interacting with Kerr black holes. We consider two test,
:
v distant observer does not encounter singularities or any nullelectromagneticfields alignedwith the two repeated
Xi effects propagating out of them, and the consistency of principalnulldirectionsofthetypeDbackground,anda
the theory of general relativity is reassured at least in test,non-nullfield,specialinthesensethattheprincipal
r
a the space-time excluding the black hole region bounded null directions of the electromagnetic field lie along the
by the event horizon. repeated principal null directions of the space-time. In
It has not been possible to establish a concrete proof Newman Penrose (NP) two spinor formalism [28] these
of CCC. For that reason Wald constructed an alterna- fields correspond to special solutions of Maxwell equa-
tivethoughtproceduretotestthe stabilityofeventhori- tions with a single non-vanishing NP Maxwell scalar. In
zons [3]. Consider a Kerr-Newmanblack hole defined by sectionsIIandIIIwereviewtheexistenceandbehaviour
three parameters (Mass M, charge Q, and angular mo- ofthesefieldsinatypeDvacuumbackground. Insection
mentum per unit mass a), which should satisfy IVwetestthevalidityofCCCintheinteractionofblack
holes with special electromagnetic fields.
M2 Q2+a2. (1)
≥
so that an event horizon exists. Then the black hole A. Electromagnetic and gravitational
is allowed to absorb test particles or fields incident from perturbations in Newman Penrose formalism
Newman Penrose (NP) two spinor formalism [28] has
proved very useful in studying the perturbations and
∗ [email protected] asymptotic structure of space-times. This formalism is
2
based on a spin basis (o,ι) endowed with a symplectic II. EXISTENCE OF SPECIAL FIELDS IN TYPE
structure ǫ = ǫ . In NP formalism the source free D SPACE-TIMES
AB BA
−
Maxwell equations have the form:
AA′φ =0 (2) Algebraically special (null) electromagnetic fields are
∇ AB known to exist in space-times that admit a shear free,
There are four complex equations here by A′,B = 0,1, geodesic null congruence [32]. Existence of the fields
correspondingtoeightrealMaxwellequations. Thesym- refers to the fact that the integrability conditions are
metric valence 2 spinor φ generates 3 complex scalars satisfied by Maxwell equations (also see e.g. [30]). Let
AB
via us consider a null electromagnetic field φ and choose
AB
a spin basis (o,ι) such that φ = φo o . From (4)
φ =φ oAoB φ =φ oAιB φ =φ ιAιB (3) AB A B
0 AB 1 AB 2 AB it follows that the only non-vanishing Maxwell scalar is
TheNPdescriptionofelectromagnetismisgiveninterms φ2 = φ. With our choice of spin basis the null congru-
of these scalars (3). Also note that encehastangentvectorla,andthefactthatitisgeodesic
and shear free implies κ = σ = 0. We also choose ǫ = 0
φ =φ o o 2φ o ι +φ ι ι (4)
AB 2 A B − 1 (A B) 0 A B byaffineparametrizationofthe geodesic. Maxwellequa-
The explicit form of Maxwell’s equations in terms of NP tions (5) reduce to
scalars can be derived.
Dφ=ρφ, δφ=(τ 2β)φ (10)
(D 2ρ)φ (δ¯+π 2α)φ +κφ =0 −
1 0 2
(D−−ρ+2ǫ)−φ2−(δ¯+−2π)φ1+λφ0 =0 (5) A solution for (10) does not necessarily exist. One can
(δ 2τ)φ1 (∆+µ 2γ)φ0 σφ2 =0 show that a solution exists if the backgroundspace-time
− − − −
(δ τ +2β)φ2 (∆+2µ)φ1 νφ0 =0 satisfies (κ = σ = 0); i.e. the space-time admits a
− − −
geodesic, shear-free null congruence. (see e.g. [30])
Any totally symmetric spinor can be decomposed in
There exists another null field on a type D back-
terms of univalent spinors (see e.g. [29, 30]). Hence we
ground,which lies along na. For this field φ = φι ι ,
may decompose the spinor equivalentof Maxwelltensor. AB A B
and the only non-vanishing Maxwell scalar is φ = φ.
0
φAB =α(AβB) (6) The Maxwell equations for this field reduce to
αandβ arecalledthe principalspinorsofφ . Ifα and
AB δ¯φ=(2α π)φ ∆φ=(2γ µ)φ (11)
β are proportional then α is called a repeated principal − −
spinor of φ, and φ is said to be algebraically special or
One can also show that the integrability conditions for
null, or of type N. The corresponding real null vector
(11) are identically satisfied if the space-time that ad-
αa =αAα¯A′ is called a repeated principal null direction. mits a geodesic and shear free null congruence with the
If α and β are not proportional then φ is said to be
tangent vector na (λ=ν =0).
algebraically general or type I, or non-null.
The two principal null directions of Petrov type D
Toformulategravityletusdefinethespinorequivalent
space-times determine two geodesic and shear free null
of the Weyl tensor C .
abcd congruences with tangent vectors la and na. For that
Cabcd+iCa∗bcd =2ΨABCDǫA′B′ǫC′D′ (7) reasontwo nullelectromagneticfields with principalnull
directions la and na exist in type D space-times, such
Ψ istotallysymmetricandsatisfiesthespinorana-
ABCD that the only non-vanishing Maxwell scalars are φ and
2
logue of Bianchi identities in vacuum
φ , respectively. This should not be confused with the
0
DD′Ψ =0 (8) case that a single null field is expressed in terms φ2 or
ABCD
∇ φ by transformation of spin basis. The two null fields
0
TheexplicitformofBianchiidentitiesarederivedin[28]. in type D space-times exist independently and simulta-
(Also see [29, 30]) Since ΨABCD is totally symmetric neously.
there exists univalent spinors αA,βB,γC,δD such that In addition to null fields, we consider an algebraically
general (non-null) test electromagnetic field in type D
Ψ =ψα β γ δ (9)
ABCD (A B C D)
vacuumspace-times,whichisspecialinthesensethatthe
αA,βB,γC,δD arecalledtheprincipalspinorsofΨABCD. principal null directions of the electromagnetic field lie
Thecorrespondingrealnullvectorsdeterminetheprinci- along the repeated principal null directions of the space-
palnulldirectionsofΨABCD. Theclassificationofspace- time. It is known that electrovacuum solutions for type
times according to the principal null directions of the D space-times exist such that the two repeated principal
Weyl tensor is known as Petrov classification [31]. As in null congruences of the Weyl tensor are aligned with the
the caseofelectromagnetismif noneofthe principalnull two principal null congruences of the non-null electro-
directionscoincidethespace-timeisalgebraicallygeneral magneticfield[33]. (Also see[34]andreferencestherein)
oftype I, andif allfour principalnulldirections coincide But a reference for the proofthat type D vacuum space-
the space-time is of type N. If there are two pairs of re- timesadmitsuchspecialnon-nulltestfieldsisnotknown
peated principal null directions the space-time is of type to this author, so we prove it here.
D.
3
Theorem II.1. Type D vacuum space-times admit a space-time. From NP field equations for type D space-
special, test, non-null electromagnetic field, such that times, we have Dρ = ρ2. This leads to a solution for
the two principal null directions of the electromagnetic BianchiidentitiesΨ =ρ3Ψ0,whereΨ0isaconstant[35].
2
field lie along the repeated principal null directions of the In particular Ψ = Mρ3 for a Kerr black hole (see e.g.
2
space-time. [36]). Similarly the solution for Maxwell equations with
φ =φ =0 is given by
0 2
Proof. Naturallywechooseaspinorbasis(o,ι)foratype
D space-time such that the two principal null directions φ =ρ2C (15)
1 1
correspond to lA = oAoA′ and nA = ιAιA′. Consider an
algebraically general test electromagnetic field where C1 is independent of r. In Kerr space-time ρ =
(r iacosθ)−1 so the special field behaves as 1/r2 ev-
− −
φ =φo ι (12) erywhere. In fact, the general expression for ρ in a type
AB (A B)
D vacuum space-time is ρ = (r+iρ0)−1, where ρ0 is
−
which is special in the sense that its principal null direc- independent ofr [35], so 1/r2 behaviourapplies to every
tions are parallelto those of the backgroundspace-time. type D vacuum space-time.
(A test field is one that has a negligible effect on the We are particularly interested in the flux of energy
background geometry.) From (3) and (4) we see that and angular momentum at large distances appropriate
φ = φ = 0. The only non-vanishing Maxwell scalar is for wave propagation,both for the outgoing and ingoing
0 2
φ , which equals φ/2 according to our definition (12). waves. The formal expression for total energy flux per
1
−
Then, Maxwell equations have the form: unit solid angle is (see e.g.[36])
Dφ =2ρφ dE
1 1 = lim r2T1 (16)
∆φ1 =−2µφ1 (13) dtdΩ r→∞ 0
δφ =2τφ
δ¯φ1 = 2π1φ T10 canbesubstitutedbyT13togiveangularmomentum
1 − 1 flux. For the limits to exist, T1 and T1 components of
0 3
the energy momentum tensor must have 1/r2 behaviour
We have to prove that a solution for the system (13),
atlargedistances. Theenergymomentumtensorforelec-
i.e. a solution of Maxwell’s equation such that the only
tromagnetic fields in terms of Maxwell’s scalars,is given
non-vanishing Maxwell scalar is φ , exists in a type D
1
by
background. If a space-time is of type D the only non-
vthaeniBshiainngchsicaidlaenrtoitfietsheinWvaecyulutmensroedruisceΨt2o. In this case 4πTµν ={φ0φ∗0nµnν +2φ1φ∗1[l(µnν)+m(µm∗ν)]+φ2φ∗2lµlν
∗ ∗ ∗
4φ φ n m 4φ φ [l m +2φ φ m m
− 1 0 (µ ν)− 2 1 (µ ν) 2 0 µ ν}
DΨ2 =3ρΨ2 +c.c. (17)
∆Ψ = 3µΨ
δΨ 2=3−τΨ 2 (14) where c.c. denotes complex conjugate. For the special
2 2
δ¯Ψ = 3πΨ non-nullfieldintroducedinsectionIItheenergymomen-
2 2
− tum tensor reduces to
Theintegrabilityconditionsforthesystems(13)and(14)
∗ ∗
4πT =2φ φ [l n +m m ]+c.c. (18)
are identical. In other words the integrability conditions µν 1 1 (µ ν) (µ ν)
for the existence of a special test electromagnetic field
Considering 1/r2 behaviour of the special field, T1 and
in the form (12), in a type D vacuum background, are 0
T1 behave as 1/r4, therefore the flux of energy and an-
identicalwiththeconditionsfortheexistenceofthetype 3
gularmomentumduetothe specialelectromagneticfield
D background itself. Thus, there exists a special non-
vanishesinthelimitr . AlsoconsidertheNPtetrad
null test electromagnetic field in every type D vacuum →∞
for Kerr metric
background.
lµ =[(r2+a2)/∆,1,0,a/∆]
The integrability conditions for the existence of type
nµ =[(r2+a2, ∆,0,a]/(2Σ)
D vacuum space-times were derived by Kinnersley [35].
−
The same conditions assure the existence of a solution mµ =[iasinθ,0,1,i/sinθ]/[√2(r+iacosθ)] (19)
for the system (13), on a type D vacuum background.
where Σ=r2+a2cos2θ and ∆=r2 2Mr+a2. Using
− ∗
thistetradweseethattheexpression[l n +m m ]=
(1 0) (1 0)
III. BEHAVIOUR OF SPECIAL FIELDS IN [l n +m m∗ ]=0. ThisimpliesthatT1 andT1 van-
(1 3) (1 3) 0 3
KERR SPACE-TIME ish everywhere if φ is the only non-vanishing Maxwell
1
scalar for an electromagnetic field in Kerr space-time,
Insection(II)weprovedtheexistenceofaspecialelec- regardless of the solution for the field. Thus, the contri-
tromagnetic field in type D space-times, such that the bution of the special electromagnetic field introduced in
only non-vanishing Maxwell scalar is φ . We are par- section (II) to the radial energy and angular momentum
1
ticularly interested in its asymptotic behaviour in Kerr fluxes vanishes everywhere, in Kerr background.
4
Let us evaluate the behaviour of null fields in Kerr where the label b.h. refers to the black hole and S∞ is
background. The nonvanishing spin coefficients are the spherical surface as r . Next we follow [6] to
→ ∞
define an indicator for CCC
ρ= 1/(r iacosθ), β = ρ∗cotθ/2√2
τ =−iaρρ∗s−inθ/√2, γ =µ+−ρρ∗(r M)/2 C =M2−a2 (28)
−
π =iaρ2sinθ/√2, µ=ρ2ρ∗∆/2, α=π β∗(20) Then, using a=L/M
−
Forthenullfieldwhichliesalongla,(usingDρ=ρ2)the δ = dCdt= 2 (M2+a2)dM adL dt
solution for (D ρ)φ2 =0 is C Z dt Z M (cid:26) dt − dt(cid:27)
−
(29)
φ2 =ρC2 (21) implying
so φ behavesas 1/r. T1 andT1 behave as 1/r2, as re- d 2
2 0 3 C = √ g[(M2+a2)( T1 ) aT1 ]dθ dφ (30)
quiredby(16)sothatthecontributionofthenullfieldto dt M Z − − 0 − 3
S
energy and angular momentum fluxes can be calculated.
Forthe nullfieldalongna,we have∆φ =(2γ µ)φ . Anextremalblackholewouldsaturatethemaincriterion
0 0
∗ ∗ ∗ − (1). Inthatcaseδ shouldalwaysremainpositivesothat
As r , ρ ρ , µ µ ρ/2, and γ γ 0. So C
→ ∞ ∼ ∼ ∼ ∼ ∼ the event horizon is preserved. If the initial state is an
the Maxwell equation is reduced to
extremalblack hole andδ turns out to be negative,the
C
lim ∆φ ( 1/2)ρφ (22) final state describes a naked singularity.
r→∞ 0 ∼ − 0 Thevalidity ofCCCinthe caseofthe specialnon-null
field is trivial. For this field we have T1 = T1 = 0
Consider the following NP field equation in a vacuum 0 3 ⇒
δ = 0. The non-null field does not lead to any per-
type D space-time C
turbation of black hole parameters of mass and angular
∆ρ δ¯τ = ρµ¯+(β¯ α τ¯)τ +(γ+γ¯)ρ Ψ (23) momentum, hence does not challenge CCC whether we
2
− − − − − start with an extremal or a nearly extremal black hole.
As r , τ ρ2 0, and Ψ ρ3 0. The NP field Let us now consider a Kerr black hole interacting
2
equat→ion∞(23)∼is red∼uced to ∼ ∼ with the null field that lies along na. For this field the
only non-vanishing Maxwell scalar is φ and the energy-
0
lim ∆ρ ( 1/2)ρ2 (24) momentum tensor (17) reduces to
r→∞ ∼ −
1
Using (22) and (24) we see that T = φ 2n n (31)
µν 0 µ ν
2π| |
rl→im∞φ0 ∼ρC0 (25) By direct substitution from the NP tetrad (19)
φ0 also behaves as 1/r and therefore satisfies the limit 2πT1 = φ 2 ∆2 2πT1 = φ 2asin2θ ∆2 (32)
condition(16)sothatitscontributiontofluxesofenergy 0 −| 0| 4Σ2 3 | 0| 4Σ2
and angular momentum can be calculated.
Note that as r , (∆2/4Σ2) 1. Then
→∞ ∼
2 √ g φ 2
IV. SPECIAL ELECTROMAGNETIC FIELDS δ = − | 0| (M2+a2cos2θ)dθ dφdt (33)
C M Z 2π 4
AND COSMIC CENSORSHIP S∞
The expression (33) is positive definite. The result is
AKerrblackholeisuniquelyparametrisedbyitsmass independent from the specific form of φ provided that
0
M and angular momentum a. The changes in the black the limit condition (16) is satisfied. Thus, whether we
holeparameterscanbeexpressedasfluxesintotheblack start with an extremal or a nearly extremal Kerr black
hole. Let K be a Killing vector. The Killing equation, hole, CCC always remains valid in the interaction of the
K = 0, combined with the local conservation of black hole with the null field along na.
(a b)
∇energy-momentum Tac =0 in generalrelativity,leads Now let us consider the interaction of the black hole
c
to the current cons∇ervation equation (TacK ) = 0. with the null field along la. For this field the only non-
a c
∇
For K =∂/∂t we have vanishing Maxwell scalar is φ , and the energy momen-
2
tum tensor (17) reduces to
dM
= √ gT1 dθdφ (26) 1
(cid:18) dt (cid:19)b.h −ZS∞ − 0 Tµν = 2π|φ2|2lµlν (34)
and for K =∂/∂φ Againwe make a direct substitution from the NP tetrad
(19)
dL
= √ gT1 dθdφ (27)
(cid:18)dt(cid:19)b.h ZS∞ − 3 2πT10 =|φ2|2 2πT13 =−(asin2θ)|φ0|2 (35)
5
CCC indicator takes the form the formation of a naked singularity, since the area of
theeventhorizoncannotdecrease. Recentlyitwasshown
2 √ g
δ = − ( φ 2)(M2+a2cos2θ)dθ dφdt thattestfieldsthatsatisfythenullenergycondition,can-
2
C M Z 2π −| |
S∞ notdestroyextremalblackholesviaaderivationthatre-
(36)
lies on black hole thermodynamics [26]. In that case it
The expression (36) is negative definite, independent
may not be clear how to explain the generic violation of
fromthespecificformofφ . Theinteractionwiththenull
2 CCC by null electromagnetic fields. Let us check if the
field, decreases the values of both mass and angular mo-
area theorem actually holds for the null field along la.
mentum parameters of the black hole; but mass loss ex-
The area of the horizon is given by
ceeds angularmomentumloss. (substituting (35)in (26)
and (27), we obtain negative values for both dM/dt and
dL/dt)AsaresultthevalueofM2 a2 decreases. Thus, A= √gθθgφφdθdφ=8πM M + M2 a2
if we start with an extremal black−hole (M2 a2 = 0), Zr=r+ (cid:16) p − (cid:17)
the null field along la turns it into a naked s−ingularity, (37)
Using (26) and (35)
and CCC is violated. So far, the only context where a
thoughtexperimentcanresultinthedestructionofanex-
tremalblackholewithoutfine-tuningwasduetoneutrino (dM) = √ g φ 2dθdφdt <0 (38)
b.h −Z − | 2|
fields [21]. Also, there exists claims of destruction of ex- S∞
tremal black holes with finely tuned particles/fields [19],
Note that d(M2 a2) is δ in our notation and we have
but these are expected to be challenged by backreaction − C
shown that δ < 0 in the previous section (36). Having
or self-force effects. both dM < 0Cand d(M2 a2) < 0 in eq. (37), directly
Since δ is always negative, if we start with a nearly −
C leads to dA < 0. Thus, the area theorem does not hold
extremal black hole the interaction with the null field
in the interaction of the null field that lies along la with
along la drives it to extremality and beyond. The fact
theblackhole,althoughtheenergy-momentumtensorfor
that δ is always negative also implies that the initial
C thefieldsatisfiesthenullenergycondition. Nullfieldsnot
data giving rise to violation is not confined to a set of
only violate CCC but also the second law of black hole
measure zero. In this sense the violationof CCC by null
mechanics. For that reason the derivation in [26] does
fields along la is generic. Such a generic violationoccurs
not apply to null fields.
in the case of neutrino fields where the the black hole is
For the sake of completeness, we should also evaluate
overspun if the incoming field is in the frequency range
the validity of the area theorem in the interaction of the
0 < ω < mΩ. However, the most generic violation of
blackhole with the null fieldalongna. Substituting (32)
CCCamongallthethoughtexperimentsinvolvingparti-
in (26) gives dM > 0, and (33) implies d(M2 a2) > 0.
clesandfieldsturnsouttobethecaseofnullelectromag- −
Thus, dA>0 and the area theorem holds.
netic fields, since no initial conditions has to be imposed
onthe thoughtexperimentto ensurethat δ is negative.
C
VI. SUMMARY AND CONCLUSIONS
V. NULL FIELDS AND BLACK HOLE
MECHANICS In this paper, we have shown that the interaction of a
Kerr black hole with the null electromagnetic field such
The violation of CCC by null electromagnetic fields that the only non-vanishing Maxwell scalar is φ , leads
2
and neutrino fields have common aspects such as be- toviolationoftwofundamentallaws/conjecturesofblack
ing generic and applying to extremal black holes. How- hole physics. First, we have proved that a generic viola-
ever,theenergymomentumtensorforneutrinofields(or tionofcosmiccensorshipconjectureoccurs. If the initial
Dirac fields in general) fundamentally differs from that stateisanextremalblackholethehorizonisdestroyed,if
of bosonic fields in the sense that it does not satisfy the itisasub-extremalblackholetheinteractiondrivesitto
weak energy condition. For that reason the area the- extremality and beyond. Such a generic violation which
orem is not expected to hold for Dirac fields. Since, appliestoextremalblackholeswasshowntooccurinthe
the change in the area of the black hole is given by interaction with neutrino fields [21, 24]. However, this
dA = (8π/κ)(dM ΩdJ), the absorption of the modes violation can be expected since the energy-momentum
−
0 < ω < mΩ decreases black hole’s area, violating the tensor for the neutrino fields does not satisfy the weak
second law of mechanics. The absorption of the same energy condition. On the contrary bosonic fields satisfy
modesleadstoviolationofCCC[21,24]. Thisresultcan the weak energy condition, and the area theorem is ex-
also be considered as a consequence of the fact that the pected hold for these fields which would forbid such a
energymomentumtensordoesnotsatisfythenullenergy genericdestructionofthe eventhorizon. In this workwe
condition. have shown that the area theorem does not hold either
On the other hand the energy-momentum tensor for forthenullelectromagneticfieldthatliesalongthe prin-
bosonic fields satisfies the null energy condition, and the cipalnulldirectionla. Theseviolationsareevidentlyper-
area theorem is expected to hold [37]. This will forbid plexing considering the fact that the energy-momentum
6
tensor for the field satisfies the weak/null energy condi- alongna doesnotchallengecosmiccensorshipconjecture
tion. or the area theorem.
We have also considered the null field that lies along
theprincipalnulldirectionna,andaspecialnon-nullfield
suchthattheprincipalnulldirectionsofthefieldliealong
the repeated principal null directions of the space-time.
ACKNOWLEDGMENTS
We proved the existence of this field on type vacuum
background. Weshowedthatthecontributionofthenon-
null field to mass and angular momentum parameters of TheauthorthanksI˙.Semizforhelpfulsuggestionsand
a Kerr black hole identically vanishes, and the null field comments.
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