Table Of ContentHorizon area-angular momentum-charge-magnetic
fluxes inequalities in 5D Einstein-Maxwell-dilaton
gravity
Stoytcho Yazadjiev∗
3
1
0
2 Department of Theoretical Physics, Faculty of Physics, Sofia University
n
5 J. Bourchier Blvd., Sofia 1164, Bulgaria
a
J
8
]
h
Abstract
t
-
p In the present paper we consider 5D spacetimes satisfying the Einstein-Maxwell-
e
h dilaton gravity equations which areU(1)2 axisymmetric but otherwise highly dynami-
[ cal. We derive inequalities between the area, the angular momenta, the electric charge
1 andthemagneticfluxesforanysmoothstablyoutermarginally trapped surface.
v
8
4
1 Basic notions and setting the problem
5
1
.
1 Thestudyofinequalitiesbetweenthehorizonareaandtheothercharacteristicsofthehorizon
0
has attracted a lot of interest recently. Within the general theory of relativity, lower bounds
3
1 for the area of dynamical horizons in terms of their angular momentum or/and charge were
v: givenin[1]–[8], generalizingthesimilarinequalitiesforthestationaryblackholes[9]–[11].
i Theseremarkableinequalitiesarebasedsolelyongeneralassumptionsandtheyholdforany
X
axisymmetricbutotherwisehighlydynamicalhorizoningeneralrelativity. Foranicereview
r
a
on the subject we refer the reader to [12]. The relationship between the proofs of the area-
angular-momentum-chargeinequalitiesforquasilocalblackholesandstationaryblackholes
isdiscussedin[13]-[15]. Inequalitiesbetweenthehorizonarea,theangularmomentum,and
thecharges werealso studiedin some4D alternativegravitationaltheories[16].
Ageneralizationofthe4Dhorizonarea-angularmomentuminequalitytoD-dimensional
vacuum Einstein gravity with U(1)D−3 group of spatial isometries was given in [17]. The
purpose of the present work is to derive some inequalities between the horizon area, hori-
zon angular momentum, horizon charges and magnetic fluxes in the 5D Einstein-Maxwell-
dilaton gravity including as a particular case the 5D Einstein-Maxwell gravity. It should be
stressedthatthederivationofthementionedinequalities inthehigherdimensionalEinstein-
Maxwelland Einstein-Maxwell-dilatongravityismuchmore difficultand isnotsostraight-
forwardas inthehigherdimensionalvacuumgravityeveninspacetimesadmittingU(1)D−3
∗[email protected]
1
isometry group. The main reason behind this is the lack of nontrivial group of hidden sym-
metries for the dimensionally reduced Einstein-Maxwell-dilaton equations in the general
case 1[18]. In contrast, the dimensionally reduced vacuum Einstein equations (in space-
timeswithU(1)D−3isometrygroup)possessnontrivialgroupofhiddensymmetries,namely
SL(D−2,R)andamatrixsigmamodelpresentationispossible. Someofthedifficultiesdue
tothepresenceofaMaxwellfieldcanbecircumventedbyfollowingamethodsimilartothat
usedin the4D Einstein-Maxwell-dilatongravity[16]as weshowbelow.
Let (M,g ,F ,j ) be a 5-dimensional spacetime satisfying the Einstein-Maxwell-
ab ab
dilatonequations
g
G =2¶ ¶j j −(cid:209) c(cid:209)j j g −2V(j )g +2e−2ja F F c− abF Fcd , (1)
ab a b c ab ab ac b cd
4
(cid:209) e−2ja Fab =0=(cid:209) F , (cid:16) (cid:17) (2)
a [a bc]
(cid:16) dV((cid:17)j ) a
(cid:209) (cid:209) aj = − e−2ja F Fcd, (3)
a dj 2 cd
where g is the spacetime metric, (cid:209) is its Levi-Civita connection, G = R − 1g R is
ab a ab ab 2 ab
the Einstein tensor and F is the Maxwell field. The dilaton field is denoted by j , V(j ) is
ab
its potentialand a is the dilatoncoupling parameter. We assumethat the dilatonpotential is
non-negative,V(j ) ≥ 0. The Einstein-Maxwell gravity is recovered by first putting a = 0
andV(j )=0 and thenj =0.
As an additional technical assumption we require the spacetime to admit U(1)2 group
of spatial isomerties. The commuting Killing fields are denoted by h and h and they are
1 2
normalized to have a period 2p . We also require the Maxwell and the dilaton fields to be
invariantundertheflowoftheKillingfields, i.e. £h F =£h j =0.
I I
LetusfurtherconsideracompactclosedsmoothsubmanifoldB ofdimensiondimB =3
invariantundertheactionofU(1)2. TheinducedmetriconB anditsLevi-Civitaconnection
aredenotedbyg andD ,respectively. ThefuturedirectednullnormalstoB willbedenoted
ab a
by n and l with the normalization condition g(n,l)=−1 and with −l pointing outward. In
whatfollowswerequire B to beastablyoutermarginallytrapped surfacewhich means that
Q =0and £ Q ≤0with Q beingtheexpansionofn onB.
n l n n
As a 3-dimensionalcompact manifoldwith an actionofU(1)2, B is topologicallyeither
S3, S2×S1 oralens space L(p,q) with p and q being co-prime integers[22, 23]. Moreover,
the factor space Bˆ = B/U(1)2 can be identified with the closed interval [−1,+1]. As it
was shown in [22, 23], certain linear combinations of the Killing fields h , with integer
I
coefficients,vanishattheendsofthefactorspace. Inotherwords,thereexistintegervectors
a ∈Z2 such that aI h →0 at x=±1, where x is the coordinate parameterizing the factor
± ± I
space. Equivalently,theGram matrixdefined by
H =g(h ,h ) (4)
IJ I J
isinvertibleintheinterioroftheinterval [−1,1]andhasone-dimensionalkernelattheinter-
valend points,i.e. H aI →0at x=±1.
IJ ±
Infact,theintegervectorsa determinethetopologyofB. ByaglobalSL(2,Z)redefini-
±
tionoftheKillingfields[22,23]wemaypresenta intheform a =(1,0)anda =(p,q)
± + −
1Fortunately,therearesectorsinEinstein-Maxwell-gravitywhicharecompletelyintegrable[19]-[21].
2
with p and q being coprime integers. The topology of B is then S3 when (p=±1,q =0),
S2×S1 when (p=0,q=±1) and thatofalens spaceL(p,q)in theothercases.
Proceeding further we consider a small neighborhood O of B. When the neighborhood
is sufficiently small it can be foliated by two-parametric copies B(u,r) of B =B(0,0) and
parameterized by the so-called null Gauss coordinates defined by a well-known procedure
[24]. In Gauss nullcoordinatesthemetricin O can bewrittenin theform
g=−2du(dr−r2¡ du−rb dya)+g dyadyb, (5)
a ab
where n = ¶ , l = ¶ , and the function ¡ and the metric g are invariantly defined on each
¶ u ¶ r
B(u,r). Usingthesecoordinatesonecan showthaton B it holds
1
Rg −Dab a− b ab a−2Gabnalb =−2£lQ n ≥0, (6)
2
where it has been taken into account that Q n =0 on B. Here Rg and Da are the Ricci scalar
curvatureandLevi-Civitaconnectionwithrespecttothemetricg onB. Takingintoaccount
ab
thatthedilatonpotentialisnonnegativethisinequalitycan berewrittenintheform
1
Rg −Dab a− b ab a−2G˜abnalb =2V(j )−2£lQ n ≥0, (7)
2
where G˜ =G +2V(j )g .
ab ab ab
Making use of (7), for every axisymmetric function f (i.e. every function f invariant
undertheisometrygroup)wehave
1
0≤ −Dab a− b ab a+Rg −2G˜abnalb f2dS
ZB 2
(cid:18) (cid:19)
1
= 2fb aDaf − b ab af2+Rg f2−2G˜abnalbf2 dS, (8)
ZB 2
(cid:18) (cid:19)
where dS is the surface element on B. Now we consider the unit tangent vector Na on B
whichis orthogonalto h . Withits help and takingintoaccount that g ab−NaNb D f =0,
I b
we find b b a = g ab−NaNb b b +(Nab )2 and 2fb aD f = 2(fNab ) NbD f which
a a b a a (cid:0) a (cid:1)b
gives
(cid:0) (cid:1) (cid:0) (cid:1)
1 1
0≤ 2(fNab a) NbDbf − (Nab a)2 f2− g ab−NaNb b ab bf2+Rg f2−2G˜abnalbf2 dS. (9)
ZB 2 2
(cid:20) (cid:21)
(cid:16) (cid:17) (cid:16) (cid:17)
Finally,takingintoaccountthat
1 2
2(fNab ) NbD f − (Nab )2 f2 ≤2 NbD f (10)
a b a b
2
(cid:16) (cid:17) (cid:16) (cid:17)
3
and that NbD f 2 = g ab−NaNb D fD f +NaNbD fD f =g abD fD f =D fDaf we
b a b a b a b a
obtaintheimportantinequality
(cid:0) (cid:1) (cid:0) (cid:1)
1
0≤ 2DafDaf − g ab−NaNb b ab bf2+Rg f2−2G˜abnalbf2 dS. (11)
ZB 2
(cid:20) (cid:21)
(cid:16) (cid:17)
In order to extract the constructive information from this inequality we should perform
a dimensionalreduction and express the inequalityas an inequalityon the factor space Bˆ =
B/U2(1) = [−1,1]. The dimensional reduction can be performed along the lines of [18].
So we shall give here only some basic steps and results without going into detail. As a first
stepitisveryconvenienttopresenttheKillingfieldsh inadaptedcoordinates,i.e. h = ¶
I I ¶f I
wherethecoordinates f I are 2p -periodic. Then theinduced metric g onB takestheform
ab
dx2
g = +H df Idf J, (12)
C2h IJ
whereC>0isaconstantandh=det(H ). Theabsenceofconicalsingularitiesrequiresthe
IJ
followingconditiontobesatisfied
h H aI aJ
lim C2 IJ ± ± =1. (13)
x→±1 1−x2 1−x2
Thearea A ofB can beeasily foundfrom (12)andtheresultis
A =8p 2C−1. (14)
Thereforethecondition(13)can berewrittenin theform
A h H aI aJ 1/4 h H aI aJ 1/4
= lim IJ + + lim IJ − − . (15)
8p 2 x→1 1−x2 1−x2 x→−1 1−x2 1−x2
(cid:18) (cid:19) (cid:18) (cid:19)
Since the factor space O/U(1)2 is simply connected we can introduce electromagnetic
potentialsF and Y invariantundertheisometrygroup anddefined by
I
dF I =ih F, dY =e−2ja ih ih ⋆F. (16)
I 2 1
TheMaxwell2-formcan thenbewrittenin theform
F =HIJh ∧dF +h−1e2ja ⋆(dY ∧h ∧h ). (17)
I J 1 2
Using the field equations one can show that there exist potentials c invariant under the
I
isometrygroup suchthatthetwist w =⋆(h ∧h ∧dh ) satisfies
I 1 2 I
4
w =dc +2F dY −2Y dF . (18)
I I I I
By direct computation of the twist using the metric (5) one finds that on B it holds b =
I
ih b =Ci¶ w I orinexplicitform
I
¶ x
b =¶ c +2F ¶ Y −2¶Y F . (19)
I x I I x x I
Alsoonecan showthaton B wehave
G˜ nalb =D j Daj +2e−2ja HIJD F DaF +h−1e2ja D Y DaY . (20)
ab a a I J a
Usingtheexplicitform(12)ofthemetricinduced on B, by direct computationwefind
Rg =C2h −¶ 2xh+1h−2(¶ xh)2−1Tr H−1¶ xH 2 . (21)
h 4 4
(cid:20) (cid:21)
(cid:0) (cid:1)
Finally,choosing
1−x2 1/2
f = , (22)
h
(cid:18) (cid:19)
substituting(19),(20)and(21)into(11)andtakingintoaccountthatdS=C−1dx(cid:213) df I,we
I
obtain
1 (1−x2) 1Tr H−1¶ H 2+1h−2(¶ h)2+
x x
Z 8 8
−1
(cid:26) (cid:20)
1 (cid:0) (cid:1)
h−1HIJ(¶ c +2F ¶ Y −2¶Y F )(¶ c +2F ¶ Y −2¶Y F )
x I I x x I x J J x x J
4
1
+e−2ja HIJ¶ F ¶ F +e2ja h−1(¶ Y )2+(¶ j )2 − dx≤0. (23)
x I x J x x
1−x2
(cid:27)
(cid:3)
Nowwecan introducethestrictlypositivedefinitemetric G givenby
AB
G dXAdXB = 1Tr H−1dH 2+1h−2(dh)2+
AB
8 8
1h−1HIJ(dc +2F (cid:0)dY −2Y (cid:1)dF )(dc +2F dY −2Y dF )+
I I I J J J
4
e−2ja HIJdF dF +e2ja h−1(dY )2+(dj )2 (24)
I J
onthe9-dimensionalmanifoldN ={(H (I≤J),c ,F ,Y ,f )∈R9;h>0}. Intermsofthis
IJ I I
metrictheinequality(23)takes theform
5
1 dXAdXB 1
I [XA]= (1−x2)G − dx≤0. (25)
∗ Z AB dx dx 1−x2
−1
(cid:20) (cid:21)
Inordertotransformthisinequalityintoaninequalityfor thearea,weusecondition(15)
whichcombinedwith(25)gives
A ≥8p 2eI[XA], (26)
where
1 h H aI(x)aJ(x)
I[XA]=I [XA]+ xln IJ |x=1 (27)
∗ 4 1−x2 1−x2 x=−1
(cid:20) (cid:21)
with aI(x) defined by aI(x) = 1(1+x)aI + 1(1−x)aI . We should note that there is an
2 + 2 −
ambiguityindefining thefunctionalI[XA]. Forexample,wecan define itby
1 h H aI(x)aJ(x)
I[XA]=aI [XA]+ xln IJ |x=1 , (28)
∗ 4 1−x2 1−x2 x=−1
(cid:20) (cid:21)
where a is an arbitrary positive number. This ambiguity, however, does not affect the final
resultssinceI [XA]=0 as weshowbelow.
∗
2 Minimizer existence lemma
In orderto put a lowerbound on the area we should find the minimumofthe function I[XA]
withappropriate boundaryconditionsifthe minimumexists. Below we showthat in certain
casestheminimumexists. Thenaturalclassoffunctionsfortheminimizingproblemisgiven
by s = −ln h ∈C¥ [−1,1], ln HIJaIaJ ∈C¥ [−1,1], (c ,F ,Y ,j ) ∈C¥ [−1,1] with
1−x2 1−x2 I I
boundary co(cid:16)ndition(cid:17)s s (±1) = s ±, ((cid:16)c (±1),(cid:17)F (±1),Y (±1),j (±1)) = (c ±,F ±,Y ±,j ±).
I I I I
Since the electromagnetic potentials and the twist potential are defined up to a constant,
withoutlossofgeneralitywecan choose
c + =−c −, F + =−F −, Y + =−Y −. (29)
I I I I
Lemma 1. For dilaton coupling parameter satisfying 0 ≤ g 2 ≤ 8, there exists a unique
3
smoothminimizerofthefunctionalI[XA]with theprescribed boundaryconditions.
Proof. Let usconsiderthetruncated functional
x2 dXAdXB 1
I [XA][x ,x ]= (1−x2)G − dx (30)
∗ 2 1 Z AB dx dx 1−x2
x
1 (cid:20) (cid:21)
6
with−1<x <x <1. Byintroducinganewvariablet = 1ln 1+x thetruncatedfunctional
1 2 2 1−x
takestheform
(cid:0) (cid:1)
t2 dXAdXB
I [XA][t ,t ]= G −1 dt, (31)
∗ 2 1 AB
Z dt dt
t
1 (cid:20) (cid:21)
which is just a modified version of the geodesic functional in the Riemannian space
(N,G ). Consequently the critical points of the functional are geodesics in N. It was
AB
shown in [18] that for 0 ≤ a 2 ≤ 8 the Riemannian space (N,G ) is simply connected,
3 AB
geodesically complete and with negative sectional curvature. Therefore, for fixed points
XA(t ) and Xa(t ) thereexista uniqueminimizinggeodesicconnectingthese points. There-
1 2
fore the global minimizer of I [XA][t ,t ] exists and is unique for 0 ≤ a 2 ≤ 8. Since
∗ 2 1 3
(N,G ) is geodesically complete the global minimizer of I [XA][t ,t ] can be extended
AB ∗ 2 1
to a global minimizer of I [XA]. Indeed, let us take x (e )= −1+e and x (e ) = 1−e (i.e.
∗ 1 2
t (e )=−t (e )= 1ln e )withe beingasmallpositivenumberandconsiderthetruncated
1 2 2 2−e
functional
(cid:0) (cid:1)
Ie [XA]= x2(e ) (1−x2)G dXAdXB − 1 dx (32)
∗ Zx1(e ) (cid:20) AB dx dx 1−x2(cid:21)
with boundary conditions XA(x (e )) and XA(x (e )). Consider now the unique minimizing
1 2
geodesicG e in N between thepointsXA(x1(e ))and XA(x2(e )). Then wehave
Ie [XA]≥Ie [XA]|G (33)
∗ ∗ e
where the right hand side of the above inequality is evaluated on the geodesic G e . Taking
intoaccountthat l 2e =GABddXtAddXtB isconstanton thegeodesicG e , weobtain
I∗e [XA]|G e =Zt1t(2e()e )(cid:20)GABddXtAddXtB −1(cid:21)dt = l 2e −1 (t2(e )−t1(e )). (34)
(cid:0) (cid:1)
Our next step is to evaluate l 2 and this can be done by evaluating G dXAdXB at the
e AB dt dt
boundary points which are in a small neighborhood of the poles x =±1. For this purpose
wefirst write l 2 in theform
e
(1−x2)2 dH 2 (1−x2)2 dh 2
l 2 = Tr H−1 + h−2 (35)
e
8 dx 8 dx
(cid:18) (cid:19) (cid:18) (cid:19)
(1−x2)2 dc dY dF dc dY dF
+ h−1HIJ I +2F −2Y I J +2F −2Y J
I J
4 dx dx dx dx dx dx
(cid:18) (cid:19)(cid:18) (cid:19)
dF dF dY 2 dj 2
+(1−x2)2e−2ja HIJ I J +(1−x2)2e2ja h−1 +(1−x2)2 .
dx dx dx dx
(cid:18) (cid:19) (cid:18) (cid:19)
7
Withintheclass offunctionsweconsider,wehave
(1−x2)2 dh 2 1
h−2 = +O(e ) (36)
8 dx 2
(cid:18) (cid:19)
inasmallneighborhoodofthepoles.
InordertoestimatethetermassociatedwithH wetakeintoaccountthatH−1dH satisfies
dx
itsowncharacteristicequation,namely Tr H−1dH 2 =h−2 dh 2−2h−1detdH. Hence we
dx dx dx
find
(cid:0) (cid:1) (cid:0) (cid:1)
(1−x2)2 dH 2 1
Tr H−1 = +O(e ). (37)
8 dx 2
(cid:18) (cid:19)
Proceeding further we notice that ¶ /c¶ are Killing fields for the metric G and conse-
I AB
quentlywe havethefollowingconstantsofmotionon thegeodesicsG e
1 dc dY dF 1−x2 dc dY dF
h−1HIJ I +2F I −2Y I = h−1HIJ I +2F I −2Y I =cIe .
2 dt dt dt 2 dx dx dx
(cid:18) (cid:19) (cid:18) (cid:19)
(38)
Henceweobtain
(1−x2)2 dc dY dF dc dY dF
h−1HIJ I +2F −2Y I J +2F −2Y J =
I J
4 dx dx dx dx dx dx
(cid:18) (cid:19)(cid:18) (cid:19)
hH cIcJ =O(e ). (39)
IJ e e
Fortheremainingterms,itis easy toseethat theybehaveas
dF dF
(1−x2)2e−2ja HIJ I J =O(e ), (40)
dx dx
dY 2
(1−x2)2e2ja h−1 =O(e ), (41)
dx
(cid:18) (cid:19)
dj 2
(1−x2)2 =O(e 2). (42)
dx
(cid:18) (cid:19)
Summarizingtheresultsso far, weconcludethatthebehaviorof l 2 forsmalle is
e
l 2 =1+O(e ). (43)
e
Thereforewehave
8
limIe [XA]|G =0 (44)
e →0 ∗ e
which,inviewof(33), gives
I [XA]= limIe [XA]≥0. (45)
∗ e →0 ∗
Therefore, there exists a unique global minimizer of the functional I [XA]. Since the
∗
functionals I[XA] and I [XA] differ in boundary terms the global minimizerof I [XA] is also
∗ ∗
aglobalminimizerof I[XA]. Thiscompletestheproof.
It shouldbenoted thatfrom (25)and (45)immediatelyfollowsthat I [XA]=0.
∗
The extremalstationary nearhorizon geometry is in fact defined by thesamevariational
problem with the same boundary conditions and by the same class of functions. Therefore,
as an directconsequenceoftheprovenlemmaweobtainthefollowing
Corollary. Foreverydilatoncouplingparametera intherange0≤a 2 ≤ 8 theareaA ofB
3
satisfiestheinequality
A ≥A , (46)
ENHG
where A is the area associated with the extremal stationary near horizon geometry of
ENHG
Einstein-Maxwell-dilaton gravity with V(j ) = 0, for the corresponding a . The equality is
saturated only for the area associated with extremal stationary near horizon geometry with
V(j )=0.
3 Horizon area-angular momenta-charge-magnetic fluxes
inequality for critical dilaton coupling parameter
For the critical coupling a 2 = 8 the Riemannian space (N,G ) is an SL(4,R)/O(4) sym-
3 AB
metric space [16] and therefore, there exists a matrix M such that the metric G can be
AB
writtenintheform
G dXAdXB = 1Tr M−1dM 2, (47)
AB
8
where M is positivedefinite and M ∈SL(4,R). (cid:0)Finding th(cid:1)e explicitform of the matrix M is
atedioustaskand herewepresent onlythefinal result. Thematrix M is givenby
E 0 N 0 E S N NS
M = 2×2 2×2 = , (48)
ST E 0 Y 0 E STN STNS+Y
2×2 2×2
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
where E is the unit 2×2 matrix and S, N and Y are 2×2 matrices which have the
2×2
followingexplicitform:
2F 2F
1 2
S= , (49)
c +2F Y c +2F Y
1 1 2 2
(cid:18) (cid:19)
9
N =e 23j h−1 e−4 23j h+4Y 2 −2Y , (50)
q q−2Y 1 !
2j
Y =e 3 H. (51)
q
In termsofthematrix M, theEuler-Lagrangeequationsare
d dM
(1−x2)M−1 =0. (52)
dx dx
(cid:20) (cid:21)
Henceweobtain
dM
(1−x2)M−1 =2A, (53)
dx
where A isaconstantmatrixwith TrA=0, sincedetM =1. Integratingfurtherwefind
1+x
M =M exp ln A (54)
0
1−x
(cid:18) (cid:19)
withM beingaconstantmatrixwiththesamepropertiesas M andsatisfyingATM =M A.
0 0 0
As a positive definite matrix, M can be written in the form M = BBT for some constant
0 0
matrix B with |detB| = 1 and this presentation is up to an orthogonal matrix O, i.e it is
invariant under the transformation B −→ BO. This freedom can be used to diagonalize the
symmetricmatrix BTABT−1. So wecan takeBTABT−1 =diag(l ,l ,l ,l )and weobtain
1 2 3 4
l
1+x 1 0 0 0
1−x
l
M =B (cid:0) 0(cid:1) 11+−xx 2 0 0 BT. (55)
l
0 0 1+x 3 0
(cid:0) (cid:1) 1−x
l
0 0 0 1+x 4
(cid:0) (cid:1) 1−x
The eigenvalues l can be found by comparing the s(cid:0)ingul(cid:1)ar behavior of the left and the
i
righthandsideof(55)atx→±1. TakingintoaccountthatonlythematrixN inM issingular
at x→±1, we find that l =1,l =−1,l =l =0. Even more, if we write the matrix B
1 2 3 4
inblockform
B R
1
B= , (56)
L B
2
(cid:18) (cid:19)
where B , B ,R and L are2×2 matrices,from thesingularbehaviorat x→±1 wefind
1 2
10
