Staticity Theorem for Higher Dimensional Generalized Einstein-Maxwell System

0.14 MB·English

by  Marek Rogatko

Staticity Theorem for Higher Dimensional Generalized Einstein-Maxwell System
Marek Rogatko
Institute of Physics
Maria Curie-Sklodowska University
20-031 Lublin, pl.Marii Curie-Sklodowskiej 1, Poland
[email protected]
[email protected]
(February 1, 2008)
Wederive formulas for variations of mass, angular momentum and canonical energy in Einstein
(n−2)-gauge forms field theory by means of the ADM formalism. Considering the initial data
for the manifold with an interior boundary which has the topology of (n−2)-sphere we obtained
the generalized first law of black hole thermodynamics. Supposing that a black hole event horizon
comprisesabifurcationKillinghorizonwithabifurcatesurfacewefindthatthesolutionisstaticin
5
theexteriorworld,whentheKillingtimelikevectorfieldisnormaltothehorizonandhasvanishing
0
electric or magnetic fieldson static slices.
0
2

n
a
J

7 I. INTRODUCTION
2

1
Nowadaystherehasbeenasignificantresurgenceofinterestingravityandblackholesinmorethanfourdimensions.
v
6 Itstemsfromtheattempts ofbuilding aconsistentquantumgravitytheoryinthe realmofM/stringtheoryaswellas
1
2 in the range of TeV gravity, where the large or infinite dimensions are taken into account. Especially mathematical
1
0 aspects of classification of n-dimensional black holes attract recently more attention. As far as the problem of
5
classificationofnon-singularblackholesolutionsinfour-dimensionswasconcernedIsrael[1],Mu¨llerzumHagenet al.
0
/
h [2]andRobinson[3],presentedfirstproofs. ThemostcompleteresultswereprovidedinRefs.[4–8]. Theclassification
t
- of both static vacuum black hole solutions as well as the Einstein-Maxwell (EM) black holes was finished in [9,10].
p
e The problem of the uniqueness black hole theorem for stationary axisymmetric spacetime turned out to be more
h
: complicated. It was elaborated in Refs. [11], but the complete proof was provided by Mazur [12] and Bunting [13]
v
i
X (see for a review of the uniqueness of black hole solutions story see [14] and references therein).
r Attempts of building a consistent quantum gravity theory triggered the researches concerning the mathematical
a
aspects of the low-energy string theory black holes. The uniqueness of the black hole solutions in dilaton gravitywas
provedinworks[15,16],whiletheuniquenessofthestaticdilatonU(1)2 blackholesbeingthesolutionofN =4,d=4
supergravitywasprovidedin[17]. The extensionofthe uniquenessproofto the caseofstatic dilatonblackholes with
U(1)N gauge fields was established in Ref. [18].
On the other hand, n-dimensional black hole uniqueness theorem, both in vacuum and charged cases was given in
Refs. [19–22]. The case of nonlinear self-gravitating σ-model in higher dimensions was treated in [23]. The complete
classification of n-dimensional charged black holes having both degenerate and non-degenerate components of event
horizon was provided in Ref. [24].
In Ref. [25]it was pointed out that a black hole being the sourceof both magnetic and electric components of 2-form
F wasastrikingcoincidence. Hence,inordertotreatthisprobleminn-dimensionalgravityoneshouldconsiderboth
µν
electric and magnetic components of (n 2)-gauge form F . In Ref. [26] the proof of the uniqueness of static
− µ1...µn−2
1
higher dimensional electrically and magnetically charged black hole containing an asymptotically flat hypersurface
with compact interior and non-degenerate components of the event horizon was given.
Proving the uniqueness theorem for stationary n-dimensional black holes is much more complicated. It turned
out that generalization of Kerr metric to arbitrary n-dimensions proposed by Myers-Perry [25] is not unique. The
counterexample showing that a five-dimensional rotating black hole ring solution with the same angular momentum
andmassbutthehorizonofwhichwashomeomorphictoS2 S1 waspresentedin[27](seealsoRef.[28]). InRef.[29]
×
it was shown that Myers-Perry solution is the unique black hole in five-dimensions in the class of spherical topology
andthree commuting Killing vectors[29],while in [30]the problem ofa stationarynonlinear self-gravitatingσ-model
in five-dimensionalspacetime was considered. It was proved that when we assume that the horizon had the topology
of S3 then, the Myers-Perryvacuum Kerr solution is the only one maximally extended, stationary,axisymmetric flat
solution having the regular rotating event horizon with constant mapping.
The uniqueness theorem for black holes is closely related to the problem of staticity for non-rotating black holes
and circularity for rotating ones. For the first time the problem of staticity was tackled by Lichnerowicz [31]. The
next extension to the vacuum spacetime was attributed to Hawking [32], while the extension taking into account
electromagnetic fields was provided by Carter [33]. But only recently the complete proof of staticity theorem [34,35]
by means of the ADM formalism was given. In the case of the low-energy string theory the problem of staticity was
studied in Refs. [36,37].
InourpaperweshallstudytheproblemofstaticitytheoremintheEinstein(n 2)-gaugeformsF(n−2) theory. Sec.II
−
will be devoted to the canonical formalism of the underlying theory. In Sec.III we tackle the problem of canonical
energy and angular momentum and derive the first law of thermodynamics for black holes with (n 2)-gauge forms
−
F(n−2) fields. Our derivation of the first law of black hole thermodynamics relies on the assumption that the event
horizonis a Killingbifurcation (n 2)-dimensionalsphere. Then, we find the conditions for staticity for non-rotating
−
black holes in n-dimensions.
In whatfollows the Greek indices will rangefrom 0 to n. They denote tensors onann-dimensionalmanifold, while
the Latin ones run from 1 to n and denote tensors on a spacelike hypersurfaceΣ. The adequate covariantderivatives
are signed respectively as and .
α i
∇ ∇
II. HIGHER DIMENSIONAL GENERALIZED EINSTEIN-MAXWELL SYSTEM
In this section we shall examine the generalized Maxwell (n 2)-gauge form F in n-dimensional spacetime
− µ1...µn−2
described by the following action:
I = dnx√ g (n)R F2 , (1)
Z − (cid:20) − (n−2)(cid:21)
where gµν is n-dimensional metric tensor, F(n−2) = dA(n−3) is (n 2)-gauge form field. The canonical formalism
−
divides the metric into spatial and temporal parts, as follows:
ds2 = N2dt2+h dxa+Nadt dxb+Nbdt (2)
ab
− (cid:18) (cid:19)(cid:18) (cid:19)
where general covariance implies the great arbitrariness in the choice of lapse and shift functions Nµ (N, Na).
2
A point in the phase space for the underlying theory is related to the specification of the fields
(h , π , A , E ) on (n 1)-dimensional hypersurface Σ. The field momenta are found in the usual
ab ab j1...jn−3 j1...jn−3 −
way by varying the Lagrangianwith respect to h , A , where denotes the derivative with respect to
∇0 ab ∇0 j1...jn−3 ∇0
the time coordinate. Thus, the momentum canonically conjugates to a Riemannian metric is
πab =√h Kab habK , (3)
−
(cid:0) (cid:1)
where K is the extrinsic curvature of the hypersurface Σ. Similarly, the momentum canonically conjugates to
ab
(n 2)-gauge form field F is defined as
− µ1...µn−2
δ
π(F) = L =2(n 2)E . (4)
j1...jn−3 δ( 0Aj1...jn−3) − j1...jn−3
∇
While the electric field E implies
j1...jn−3
E =√hF nα, (5)
j1...jn−3 αj1...jn−3
where nµ is the unit normal timelike vector to the hypersurface Σ. The Hamiltonian is defined by the Legendre
transform may be written as follows:
H=πab ∇0hab+π(jF1.)..jn−3 ∇0Aj1...jn−3 −L(R, F(n−2)) (6)
=NµCµ+A˜0j2...jn−3B˜j2...jn−3 +Hdiv,
whereforbrevityofthe notationwehavedenotedby ˜ =(n 3)!A . Onthehand,thetotalderivative
Aj1...jn−3 − j1...jn−3
part of the Hamiltonian is given by
div
H
N πij
Hdiv =2(n−3)(n−2)∇j1(cid:18)Ej1...jn−3 A˜0j2...jn−3(cid:19)+2√h∇i(cid:18) √jh (cid:19). (7)
The gauge field ˜ has no associated with it kinetic terms. Therefore one can consider it as a Lagrange
A0j2...jn−3
multiplier corresponding to the generalized Gauss law of the form as follows:
0=B˜j2...jn−3 =2(n−3)(n−2)∇j1(cid:18)Ej1...jn−3(cid:19). (8)
In our paper we shall consider the asymptotically flat initial data, i.e., in asymptotic region of hypersurface Σ which
is diffeomorphic to Rn−1 B, where B is compact, one has the following conditions to be satisfied:
−
1
h δ + ( ), (9)
ab ab
≈ O r
1
π ( ), (10)
ab ≈O r2
1
A ( ), (11)
j1...jn−3 ≈O r
1
E ( ). (12)
j1...jn−3 ≈O r
Atinfinitywealsoassumethestandardbehaviourofthelapseandshiftfunctions,i.e.,N 1+ (1)andNa (1).
≈ O r ≈O r
On the hypersurface Σ the initial data are restricted to the constraint manifold on which at each point x Σ the
∈
following quantities vanish
3
1 1 (n 2)
0=C0 =√h(cid:20)−(n−1)R+ h(cid:18)πijπij − 2π2(cid:19)(cid:21)+ √−h Ej1...jn−3 Ej1...jn−3 +√hFj1...jn−2 Fj1...jn−2, (13)
π
0=Ca =2(n−2)Faj1...jn−3 Ej1...jn−3 −2√h ∇i(cid:18)√iha(cid:19),
0=B˜j2...jn−3 =2(n−3)(n−2)∇j1(cid:18)Ej1...jn−3(cid:19),
where is the derivative operator on Σ, while (n−1)R denotes the scalar curvature with respect to the metric h
j ab
∇
on the considered hypersurface.
The equations of motion for this theory can be formally derived from the volume integral contribution to the
V
H
Hamiltonian and subject to the pure constraint form as follows:
H
HV =Z dΣ(cid:18)NµCµ+NµA˜µj2...jn−3B˜j2...jn−3(cid:19). (14)
Σ
One can verify that the change caused by arbitrary infinitisemal variations (δhab, δπab, δA˜j1...jn−3, δEj1...jn−3) of
compact support, after integration by parts leads us to the expression
δHV =Z dΣ(cid:18)Pab δhab+Qab δπab+Rj1...jn−3 δA˜j1...jn−3 +Sj1...jn−3 δEj1...jn−3(cid:19), (15)
Σ
where ab , ab, j1...jn−3, j1...jn−3 are written as
P Q R S
ab =√h N aab+√h hab j N a bN πab, (16)
j Ni
P (cid:18) ∇ ∇ −∇ ∇ (cid:19)−L
N
= 2π πh +2 N , (17)
ab ab ab a b
Q √h(cid:18) − (cid:19) ∇
j1...jn−3 = 2(n 2) a Faj1...jn−3 + NiEj1...jn−3 , (18)
R − − (cid:20) ∇ (cid:18) (cid:19) L (cid:21)
2(n 2)
Sj1...jn−3 = √−h N Ej1...jn−3 +2(n−2) LNiA˜j1...jn−3 +2(n−2)(n−3)∇j1(cid:18)NA˜0j2...jn−3(cid:19). (19)
While aab takes the form as
1
aab = F Fj1...jn−2 (n 2)Faj2...jn−2Fb (20)
2 j1...jn−2 − − j2...jn−2
(n 2) (n 2)(n 3)
− 2√−h habEj1...jn−3Ej1...jn−3 + − √h − Eaj2...jn−3Ebj2...jn−3
1 1 1 1
+ 2πa πbj πhab hab π πij π2 +(n−1)Rab hab (n−1)Rab.
j ij
h(cid:20) − − 2 (cid:18) − 2 (cid:19)(cid:21) − 2
Intheaboverelations Ni denotestheLiederivativecalculatedonthehypersurfaceΣ. TheLiederivativeofEj1...jn−3
L
is understood as the Lie derivative of the adequate tensor density.
On using the Hamiltonians principle and evaluating variationsof the compactsupport of Σ, we finally reachto the
evolutions equations which can be written as follows:
π˙ab = ab, (21)
−P
h˙ = , (22)
ab ab
Q
4
E˙j1...jn−3 = j1...jn−3, (23)
−R
˙˜ = . (24)
Aj1...jn−3 Sj1...jn−3
As was mentioned in Ref. [34] expression (14) depicts rather the volume integral contribution to the Hamiltonian.
The non-vanishing surface terms arise when we take into account integration by parts. In order to get rid of these
surface contribution terms one can add the surface terms given by
H=Hv+ZS∞dSj1(cid:20)2(n−2) NaA˜aj2...jn−3Ej1...jn−3 +2(n−2)(n−3) N A˜0j2...jn−3Ej1...jn−3(cid:21) (25)
2Nbπ
+ dSa N bh h b + ab .
ab a b
ZS∞ (cid:20) (cid:18)∇ −∇ (cid:19) √h (cid:21)
By the direct calculations it can be seen that not only for asymptotically flat perturbations of a compact support of
the hypersurface Σ but also for Nµ and ˜ satisfying the asymptotic conditions at infinity, we get
Aj1...jn−3
δH=Z dΣ(cid:18)Pab δhab+Qab δπab+Rj1...jn−3 δA˜j1...jn−3 +Sj1...jn−3 δEj1...jn−3(cid:19). (26)
Σ
III. FIRST LAW OF BLACK HOLE MECHANICS
We can define the canonical energy as the Hamiltonian function corresponding to the case when Nµ is an asymp-
totical translation at infinity. Thus, one has that N 1, Na 0. We multiply Hamiltonian function by 1/2 and
→ →
reach to the expression
=α M +(n 2)(n 3) dS N ˜ Ej1...jn−3, (27)
E − − ZS∞ j1 A0j2...jn−3
where M is the ADM mass defined as follows:
1
αM = dSa N bh h j , (28)
ab a j
2ZS∞ (cid:20) (cid:0)∇ −∇ (cid:1)(cid:21)
and α= n−3. The remaining term is highly gauge dependent because of the arbitrary choice of ˜ . It yields
n−2 A0j2...jn−3
=(n 2)(n 3) dS N ˜ Ej1...jn−3. (29)
EF − − ZS∞ j1 A0j2...jn−3
We shall call canonical energy of (n 2)-gauge forms fields.
F
E −
We define also the canonicalangular momentum on the constraint submanifold of the phase space as the Hamil-
(i)
J
tonian multiplied by the factor1/2,whenN 0 and the shift vector tends to the appropriateKilling vectorfields
H →
responsible for rotation in the adequate directions. Thus, it reduces to
J((i∞) ) =−21ZS∞dSa(cid:18)2φ(i)bπba+2(n−2)(n−3) φ(i)mA˜mj2...jn−3Eaj2...jn−3(cid:19). (30)
IfoneconsidersthecaseofhypersurfaceΣhavinganasymptoticregionandsmoothinteriorboundaryS andtakeinto
accountthe linearcombinationsofthe translationandrotationsatinfinity, thenwe reachto the followingexpression:
5
n−1
2 δ Ω (i)(∞) (31)
(cid:18) E − (i)J (cid:19)
Xi=1
=Z dΣ(cid:18)Pab δhab+Qab δπab+Rj1...jn−3 δA˜j1...jn−3 +Sj1...jn−3 δEj1...jn−3(cid:19)
Σ
+δ(surface terms).
As in Ref. [34] one can take an asymptotically flat hypersurface Σ which intersects the sphere S of a stationary
n-dimensional black hole. We assume also that (n 2)-sphere S is a bifurcation Killing horizon and set that
n−1 −
Nµ = χµ = tµ + Ω φµ(i), where Ω describe angular velocities of the direction established by φµ(i). We
(i) (i)
iP=1
also choose ˜ so that A˙ = E˙ = 0. Using Eqs.(21)-(24) one can draw a conclusion that the
A0j2...jn−3 j1...jn−3 j1...jn−3
integral over Σ vanishes while, the only one surface term survives because of the fact that on sphere S we have
Nµ = 0. The non-zero term is equal to 2πκδA, where κ is the surface gravity constant on S, while A is the area of
the (n 2)-dimensional sphere S. Thus we reach to the following:
−
Theorem:
Let (hij, πij, A˜j1...jn−3, Ej1...jn−3) be smooth asymptotically flat initial data for a stationary black hole
with (n 2)-gauge forms field on a hypersurface Σ with (n 2)-dimensional bifurcation sphere S. If
− −
(δhij, δπij, δA˜j1...jn−3, δEj1...jn−3) are arbitrary smooth asymptotically flat solutions of the linearized constraints
on a hypersurface Σ, then the following is fulfilled:
n−1
α δM +δ Ω (i)(∞) =κ δA. (32)
F (i)
E − J
Xi=1
Taking into account (32) one can see that any stationary black hole with a bifurcate Killing horizon is an extremum
of mass M at fixed canonical energy of (n 2)-gauge forms fields, canonical momentum and horizon area.
−
Wegettheextensionofthefirstlawofblackholemechanicswhichistrueforarbitraryasymptoticallyflatperturbations
of a stationary n-dimensional black hole (in four-dimensions the similar result was obtained by Sudarsky and Wald
[34]inEinsteinYang-MillstheoryandinthecaseofEinstein-Maxwellaxion-dilatonblackholesinRef.[36]). Contrary
to the firstlaw of black holes mechanics derivedin Ref. [38]valid for perturbations to a nearbystationary blackhole.
IV. STATICITY CONDITIONS
Now we proceed to find the staticity theorem for non-rotating n-dimensional black holes with (n 2)-gauge field
−
F . To begin with let us suppose that a stationary black hole is regular on and outside a Killing horizon of a
µ1...µn−2
Killing vector field of the form
n−1
χµ =tµ+ Ω φµ(i), (33)
(i)
Xi=1
is normal. The mass of a black hole implies [25] the following:
1
M = ǫ atb. (34)
−αZ j1...jn−2ab∇
S
Furthermore,wedefinethe angularmomentumofblackholeassociatedwitharotationalKillingvectorφ expressed
(i)
as a covariant surface integral
6
1
I = ǫ aφb . (35)
(i)BH 2Z j1...jn−2ab∇ (i)
H
The same procedure as in Ref. [38] leads us to the mass formula
n−1
2 g T 2 2
M = dΣ T + µν tµnν + κA+ Ω I(i) . (36)
αZ (cid:18) µν 2 n(cid:19) α α (i) BH
Σ − Xi=1
Rewriting the latter expression (36) in terms of the considered matter energy momentum tensor yields
n−1
2 2
M κA Ω I(i) (37)
− α − α (i) BH
Xi=1
(n 2) λ n 3
=Z dΣ(cid:20) √−h tmFmj1...jn−3Ej1...jn−3 + hEj1...jn−3Ej1...jn−3 +λ(cid:18)n−2(cid:19)Fj1...jn−2Fj1...jn−2(cid:21),
Σ −
where we defined by λ= n tβ.
β
−
Takingaccountofconstraintequations andchangingthe surface integralinto a volume onewe candeduce that (∞)
J(i)
has the form as
J((i∞) ) =−21Z dΣ(cid:18)πab LNihab+2(n−2) LNiA˜j1...jn−3Ej1...jn−3(cid:19)+J(i)H, (38)
Σ
where we define by the following expression:
(i)H
J
1
J(i)H =−2Z dSa(cid:20)2φ(i)bπba+2(n−2)(n−3)φ(i)mA˜mj2...jn−3Eaj2...jn−3(cid:21). (39)
S
Using the fact that Killing vector fields φ(i) are equal to their tangential projection φ(i) one can readily find that
µ m
(∞) = (i). The first term in relation (39) is equal to I(i) . Then, from (39) it follows immediately the result
J(i) JH BH
n−1
Ω(i)(cid:18)IB(iH) −J(i) (∞)(cid:19)=(n−2)Z dΣ(cid:20)Ej1...jn−3LNiA˜j1...jn−3 −tmFmj1...jn−3Ej1...jn−3(cid:21). (40)
Xi=1 Σ
By virtue of the above equation and the constraint relation (24) we find the expression of the form
n−1
2 2 2
M κA+ Ω (i) (∞) (41)
− α αEF − α (i)J
Xi=1
λ(n 2)
=Z dΣ (cid:20)2λ Fj1...jn−2Fj1...jn−2 −2 h− Ej1...jn−3Ej1...jn−3(cid:21).
Σ
From this stage on we shall restrict our attention to the case of the maximal hypersurface, i.e., for which π a =0.
a
Having this in mind we consider the initial data induced on hypersurface Σ and choose the lapse and shift function
coincidingwithKillingvectorfieldsinthespacetimeunderconsiderations. ItmaybeverifiedthatcontractingEq.(21)
we get
mN =ρ N, (42)
m
∇ ∇
where ρ is given by
n 3 n 1
ρ=(cid:18)n−2(cid:19) Fj1...jn−2Fj1...jn−2 + 2h(n 2)πijπij − 2h(cid:20)(n−1)(n−5)+(3−n)(cid:21) Ej1...jn−3Ej1...jn−3. (43)
− −
7
We remark that ρ will be non-negative for n 4. Consistently with this remark the maximum principle can be
≥
applied to the relation (42) provided that solutions of it can be uniquely determined by their boundary value at S
and their asymptotic value at infinity.
To proceed further, we use as the lapse function λ with the boundary conditions λS = 0, λ∞ = 1. Integrating
| |
Eq.(42) we obtain black hole mass formula as
2 2
M κA= dΣ λ ρ. (44)
− α αZ
Σ
Using the scaling transformationwe cantransforma solutionof Einstein (n 2)-formgauge theory into a new one
−
with the following changes:
M βn−3M, (45)
→
βn−3 , (46)
F F
E → E
Ω β−1Ω , (47)
(i) (i)
→
(i) (∞) βn−2 (i) (∞), (48)
J → J
κ β−1κ, (49)
→
A βn−2A, (50)
→
where β is a constant. Inserting the linearized perturbation connected with the above scaling transformation into
equation (32) one finally left with the second mass formula of the form
n−1
α M 2κA 2 Ω (i) (∞)+ =0. (51)
(i) F
− − J E
Xi=1
n−1
Then, using (41), (44), (51) one solves them for and Ω (i) (∞). The results become as
F (i)
E J
iP=1
n 3 n 3
EF =Z dΣ(cid:20)4λ(cid:18)n−2(cid:19) Fj1...jn−2Fj1...jn−2 +4λ −h Ej1...jn−3Ej1...jn−3(cid:21) (52)
Σ −
while the formula for the angular momenta can be written as
n−1
n 3 λn
Ω(i)J(i) (∞) =Z dΣ(cid:20)3λ(cid:18)n−2(cid:19) Fj1...jn−2Fj1...jn−2 + 2h(n 2) πijπij (53)
Xi=1 Σ − −
λ 3(3 n) (n 1)(n 5)
+ h(cid:18) − 2−(2 −n) − (cid:19) Ej1...jn−3Ej1...jn−3(cid:21)
−
Inthecaseoffour-dimensionalspacetimethecoefficientforE2 inEq.(53)isequaltozero. Thuswehavethesame
(n−3)
result for n=4 as was obtained in Ref. [35].
In Ref. [39]was pointed out that the exterior regionof a black hole can be foliated by maximal hypersurfaces with
boundary S, asymptotically orthogonal to the timelike Killing vector field t , when the strong energy condition for
µ
every timelike vector is satisfied. As one can check this is the case in the considered theory. In the light what has
been shown above, we can establish the following:
Theorem:
8
Let us consider an asymptotically flat solution to Einstein (n 2)-gauge theory possessing a stationary Killing vec-
−
tor field and describing a stationary black hole comprising a bifurcate Killing horizon. Suppose, moreover that
n−1
Ω(i)J(i) (∞) = 0, then the solution is static and has vanishing electric Ej1...jn−3 or magnetic Fj1...jn−2 fields on
iP=1
static hypersurfaces.
One can readily verify the above by applying Eq.(53) to the maximal hypersurfaces. It will be noticed that on
the considered hypersurfaces Σ one has the condition π = 0. Let N denotes the lapse function for the maximal
t ij
hypersurface and nµ depicts unit normalto this hypersurface. We chooseNµ =Nnµ as the evolutionvector field for
these slices. This is sufficient to establish that
Nµπij =π˙ij =0. (54)
L
From Eqs.(17) and (22) , since πab =0 and Na =0, we obtain that h =h˙ =0.
Nµ ab ab
L
We shall first consider the case when Ej1...jn−3 =0. Consequently it yields the result as follows:
NµEj1...jn−3 =E˙j1...jn−3 =0. (55)
L
It can be verified that considering Eqs.(19) and (24) and choosing A = 0 one gets that ˙˜ = 0. By
0j2...jn−3 Aj1...jn−3
virtue of this we can conclude that the solution is static.
Now we take into account the case when F = 0. To begin with let us consider relation (18) from which
j1...jn−2
because of the fact that Na = 0, one has that NµEj1...jn−3 = 0. Thus, we see that E˙j1...jn−3 = 0. Now consider
L
Eq.(24) and Eq.(19) and choose A0j2...jn−3 = 0 as well as Ej1...jn−3 = 0. Then one can draw a conclusion that
˙˜ =0 and the solution is static.
Aj1...jn−3
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