Static electromagnetic fields and charged black holes in general covariant theory of Horava-Lifshitz gravity

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by  Ahmad Borzou

Static electromagnetic fields and charged black holes in general covariant theory of
Hoˇrava-Lifshitz gravity
Ahmad Borzou a,∗ Kai Lin a,b,† and Anzhong Wang a,c‡
a GCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA
b Institute of Theoretical Physics, Chongqing University, Chongqing 400030, China
c Institute for Advanced Physics & Mathematics,
Zhejiang University of Technology, Hangzhou 310032, China
(Dated: January 24, 2012)
Inthispaper,westudyelectromeganeticstaticspacetimesinthenonrelativisitcgeneralcovariant
theoryoftheHoˇrava-Lifshitz(HL)gravity,proposedrecentlybyHoˇravaandMelby-Thompson,and
presentalltheelectricstaticsolutions,whichrepresentthegeneralizationoftheReissner-Nordstr¨om
solutionfoundinEinstein’sgeneralrelativity(GR).Theglobal/localstructuresofspacetimesinthe
2 HL theory in general are different from those given in GR, because the dispersion relations of test
1 particles now contain high-order momentum terms, so the speeds of these particles are unbounded
0 in the ultraviolet (UV). As a result, the conception of light-cones defined in GR becomes invalid
2 and test particles do not follow geodesics. To study black holes in the HL theory, we adopt the

n geometrical optical approximations, and define a horizon as a (two-closed) surface that is free of
a spacetime singularities and on which massless test particles are infinitely redshifted. With such a
J definition,we show that some of oursolutions give rise to(charged) black holes, although theradii
3 of their horizons in general dependon theenergies of thetest particles.
2
PACSnumbers: 04.60.-m;98.80.Cq;98.80.-k;98.80.Bp

]
h
t I. INTRODUCTION differentcases: (a)when only the “darkmatter asanin-
-
p tegration constant” is present [7]; and (b) when a scalar
e field and the “dark matter as an integration constant”
Recently, Hoˇrava proposed a theory of quantum grav-
h
are present [8].
[ ity [1], motivated by the Lifshitz scalar field theory in
solid state physics [2]. Due to several remarkable fea- Another very attractive approach is to eliminate the
2 spin-0 graviton by introducing two auxiliary fields, the
tures, the HL theory has attracted a greatdeal of atten-
v U(1) gauge field A and the Newtonian prepotentail ϕ,
tion (see for example, [3, 4] and references therein). In
6
by extending the Diff(M, ) symmetry (1.1) to include
3 this theory,the generalcovarianceis brokendownto the F
a local U(1) symmetry [11],
6 foliation-preserving diffeomorphisms Diff(M, ),
F
1
. t˜=f(t), x˜i =ζi(t,x), (1.1) U(1)⋉Diff(M, ). (1.2)
0 F
1
Underthisextendedsymmetry,thespecialstatusoftime
1 because of which, in comparison with GR one more de-
remains, so that the anisotropic scaling between space
1 gree of freedom appears in the gravitational sector - the
: spin-0graviton. Thisispotentiallydangerous,andneeds and time,
v
Xi to decouple in the infrared (IR), in order to be consis- x b−1x, t b−3t, (1.3)
tent with observations. Whether this is possible or not → →
r is still an open question [3]. In particular, Mukohyama
a can still be realized, and the theory is UV complete.
studied the spherically symmetric static spacetimes [4],
Meanwhile,becauseoftheeliminationofthespin-0gravi-
andshowedexplicitlythatthespin-0gravitonindeedde-
ton, its IR behavior can be significantly improved. The
couples after nonlinear effects are takeninto account, an
elimination of the spin-0 graviton was done initially in
analogue of the Vainshtein effect [5], initially found in
the case λ= 1 [11, 12], but soon generalized to the case
massive gravity [6]. Similar considerations in cosmology
with any λ [13–15], where λ denotes a coupling constant
were presented in [7, 8] (See also [9] for a class of ex-
thatcharacterizesthedeviationofthekineticpartofthe
actsolutions.), where a fully nonlinear analysisof super-
actionfromthe correspondingone givenin GR. (For the
horizon cosmological perturbations was carried out, by
analysis of the Hamiltonian structure of the theory, see
adopting the so-called gradient expansion method [10].
[11, 16]).
It was found that the relativistic limit of the HL the-
Under the coordinate transformations (1.1), the lapse
ory is continuous, and GR is recovered at least in two
function N, the shift vector N , the 3-metric g , the
i ij
U(1) gauge field A and the Newtonian prepotentail ϕ
transform, reslectively, as
∗Electronicaddress: ahmad˙[email protected] δN = ζk N +N˙f +Nf˙,
†Electronicaddress: k˙[email protected] ∇k
‡Electronicaddress: anzhong˙[email protected] δNi = Nk iζk+ζk kNi+gikζ˙k+N˙if +Nif˙,
∇ ∇
2
δg = ζ + ζ +fg˙ , It should be noted that electromagnetic static space-
ij i j j i ij
∇ ∇
δA = ζi∂ A+f˙A+fA˙, times in other versions of the HL theory were studied in
i
[26],whileanewmechanismforgenerationofprimordial
δϕ = fϕ˙ +ζi∂ ϕ, (1.4)
i magneticseedfieldintheearlyuniversewithoutthelocal
where f˙ df/dt, denotes the covariant derivative U(1) symmetry was considered in [27].
i
≡ ∇
with respect to g , and δg g˜ t,xk g t,xk ,
ij ij ij ij
≡ −
etc. From these expressions one can see that N and N
(cid:0) (cid:1) (cid:0) (cid:1)i II. NONRELATIVISITC GENERAL
play the role of gauge fields of the Diff(M, ). There-
F COVARIANT THEORY
fore, it is natural to assume that N and N inherit the
i
samedependenceonspaceandtimeasthecorresponding
Inthissection,wegiveaverybriefintroductiontothe
generators [1],
nonrelativistic general covariant theory of gravity, pro-
N =N(t), N =N (t,x), (1.5) posed recently by HMT. For details, we refer readers to
i i
[11, 12]. We shall closely follow [12], so that the nota-
whichis often referredto as the projectability condition. tions and conversations will be used directly from there
Ontheotherhand,undertheU(1)gaugetransformation, without further explanations.
the above quantities transform as The basic variables are A, ϕ, N, N and g , in terms
i ij
of which the spacetime is given by,
δ N = 0, δ N =N α, δ g =0,
α α i i α ij
∇
δ A = α˙ Ni α, δ ϕ= α, (1.6) ds2 = N2c2dt2+g (dxi+Nidt)(dxj +Njdt). (2.1)
α i α ij
− ∇ − −
whereα[=α(t,x)]isthegeneratorofthelocalU(1)gauge The total action takes the form,
symmetry, and Ni = gikN . For the detail, we refer
k
readers to [11, 12]. S = ζ2 dtd3xN√g + +
K V ϕ A
It should be noted that all the above hold only in the L −L L L
case with projectability condition [1, 3]. However, the Z (cid:16) 1
elimination of the spin-0 graviton can be also realized +ζ2LM , (2.2)
in the non-projectability case with the extended sym- (cid:19)
metry (1.2) [17, 18]. In addition, the number of inde- where g =detg , and
ij
pendent coupling constants can be significantly reduced
(from more than 70 [19, 20] to 15), by simply impos- = K Kij K2,
K ij
L −
ing the softly breaking detailed balance condition, while
= ϕ ij 2K + ϕ ,
the theory still remains UV complete and has a healthy Lϕ G ij ∇i∇j
IR limit. When applying it to cosmology, a remarkable A (cid:16) (cid:17)
= 2Λ R . (2.3)
resultisobtained: theFriedmann-Robertson-Walkeruni- A g
L N −
verse is necessarily flat. (cid:16) (cid:17)
In this paper, we study electromagnetic static space- Here the coupling constant Λg, acting like a 3-
timesintheHoˇrava-Melby-Thompson(HMT)setup[11], dimensional cosmologicalconstant, has the dimension of
in which λ = 1 and the projectability condition (1.5) is (length)−2. Kij is the extrinsic curvature of the hyper-
adopted. Specifically, after giving a brief introduction to surfacest=Constant,and ij isthe3-dimensional“gen-
G
the HMT theory in Sec. II, we consider its coupling to a eralized” Einstein tensor, defined, respectively, by
vectorfieldinSec. III,whileinSec. IVwestudy electric
1
static spacetimes, and find all the static solutions. In K = ( g˙ + N + N ),
ij ij i j j i
2N − ∇ ∇
Sec. V we study the existence of horizons andshow that
1
some of these solutions found in Sec. IV have black hole = R g R+Λ g , (2.4)
ij ij ij g ij
structuresinthegeometricalopticalapproximations[21]. G − 2
It should be noted that because of the breaking of the
where the Ricci tensor R refers to the three-metric g ,
general covariance, x˜µ =ζµ(t,x),(µ=0,1,2,3), the dis- and R[= gijR ] denotesitjhe 3D Ricci scalar. is thije
ij M
persion relations of test particles, including the massless L
matterLagrangianand anarbitraryDiff(Σ)-invariant
V
ones,containgenericallyhigh-ordermomentumterms,so L
local scalar functional built out of the spatial metric
the speeds of these particles are unbounded in the ultra-
g ,its Riemanntensorandspatialcovariantderivatives,
ij
violet (UV), and the causal structures of spacetimes are
without the use of time derivatives. In [28], by assuming
quite different from that presented in GR. In particular,
that the highest order derivatives are six and that the
intheHLtheorytheconceptionoflight-conesisnotvalid
theory respects the parity and time-reflection symmetry,
anylonger,andthemotionsoftestparticlesdonotfollow
the most general form of is given by [28] (See also
V
geodesics [21]. As a result, the definitions of black holes L
[29]),
giveninGR [22–25] cannotbe appliedto the HL gravity
without modifications. In Sec. VI we present our main 1
= ζ2g +g R+ g R2+g R Rij
conclusions. LV 0 1 ζ2 2 3 ij
(cid:0) (cid:1)
3
1
+ζ4 g4R3+g5RRijRij +g6RjiRkjRik where f(ij) ≡(fij +fji)/2, and
(cid:16) (cid:17)
−ζ14 g7R∇2R+g8(∇iRjk) ∇iRjk , (2.5) K2 ij ≡ KilKlj,
1
(cid:2) (cid:0) (cid:1)(cid:3) (cid:0) F(cid:1)ij ARij i j gij 2 A ,
where the coupling constants g (s = 0,1,2,...8) are all A ≡ N − ∇ ∇ − ∇
s
dimensionless, and Fij h1 δ √gL(cid:16)V = 8 g ζns(cid:17)(Fi)ij,
s s
Λ= 1ζ2g0, (2.6) ≡ −√g (cid:0)δgij (cid:1) Xs=0
2 3
Fij Fij , (2.16)
denotes the cosmologicalconstant. The relativistic limit ϕ ≡ (ϕ,n)
in the IR requires, nX=1
g1 =−1, ζ2 = 161πG, (2.7) tweinthsornss(F=s)ij(2a,n0d,−F2(i,ϕj−,n2),a−re4,g−iv4e,n−i4n,−A4p,p−e4n)d.ix TAh.eTh3e-
stress 3-tensor τij is defined as
where G denotes the Newtonian constant.
Variation of the total action (2.2) with respect to the
lapse function N(t) yields the Hamiltonian constraint, τij = 2 δ √gLM . (2.17)
√g (cid:0) δgij (cid:1)
d3x√g + ϕ ij ϕ =8πG d3x√gJt,
LK LV − G ∇i∇j The matter, on the other hand, satisfies the conserva-
Z Z
(cid:0) (cid:1) (2.8) tion laws,
where
1 2N
δ(N ) d3x√g g˙ τkl √gJt + k √gJk
Jt =2 LM . (2.9) kl − √g ,t N√g ,t
δN Z (cid:20)
(cid:0) (cid:1) (cid:0) (cid:1)
A
Variationofthe actionwith respectto the shift vector −2ϕ˙Jϕ− N√g (√gJA),t =0, (2.18)
N yields the supermomentum constraint, (cid:21)
i
1 Jk
kτ (√gJ ) ( N N )
j πij ϕ ij =8πGJi, (2.10) ∇ ik− N√g i ,t− N ∇k i−∇i k
∇ − G
wherethesupermo(cid:16)mentumπij(cid:17)andmattercurrentJiare Ni kJk+Jϕ iϕ JA iA=0. (2.19)
−N ∇ ∇ − 2N∇
defined as
πij Kij +Kgij,
≡ − III. COUPLING OF A VECTOR FIELD
δ
Ji N LM. (2.11)
≡ − δN
i To couple the gravitational sector (N,N ,g ,A,ϕ)
i ij
Similarly, variations of the action with respect to ϕ and withavectorfield,weborrowtherecipeof[13],inwhich
A yield, respectively, itwasshownthatforanygivenmatterfield,say,ψn,that
is invariant under Diff(M, ), its motion is described by
F
ij K + ϕ =8πGJ , (2.12) the action, Sˆ (N,N ,g ;ψ ). Then, the action,
ij i j ϕ M i jk n
G ∇ ∇
R (cid:16)2Λ =8πGJ (cid:17), (2.13)
− g A S (N,N ,g ,A,ϕ;ψ )=Sˆ N,Nˆ ,g ;ψ
M i jk n M i jk n
where
(cid:16) (cid:17)
+ dtd3xN√gZ(g ,ψ ) A ), (3.1)
δ M δ(N M) ij n −A
Jϕ L , JA 2 L . (2.14) Z
≡− δϕ ≡ δA (cid:0)
has the enlarged symmetry (1.2), where Z(g ,ψ ) de-
ij n
On the other hand, variationwith respect to g leads to
ij notes the mostgeneralscalaroperatorof dimensiontwo,
the dynamical equations,
[Z]=2, of the enlarged symmetry, and
1 √g πij ϕ ij = 2 K2 ij +2KKij Nˆ N +N ϕ,
N√g − G ,t − i ≡ i ∇i
1
(cid:2)1 (cid:0) (cid:1)(cid:3) (cid:0) (cid:1) ϕ˙ +Nk ϕ+ N ϕ kϕ . (3.2)
+ k Nkπij 2πk(iNj) A ≡ − ∇k 2 ∇k ∇
N∇ −
1 h i (cid:0) (cid:1)(cid:0) (cid:1)
+ ( + + )gij In [29], the general action of a massive vector field
K ϕ A
2 L L L (A ,A ) with the Diff(M, ) symmetry (1.1) was con-
0 i
+Fij +Fij +Fij +8πGτij, (2.15) structed. Applying the aboFve recipe to this vector field,
ϕ A
4
we obtain the action of a massive vector field that is in- ε li
+ k a m jBk
variant under the enlarged symmetry (1.2), 2√g∇l∇j∇m 7∇ ∇
(cid:16) (cid:17)
2g2
S = 1 dtd3x√g 2 gij( Nˆk ) − NeK′AiBjBj(A−A)
EM 4ge2 Z "N F0i− Fki g2ε ik
+ e j Bj(A )
( Nˆl )+m2e(A NˆiA )2 N√g ∇k K −A
× F0j − Fjl N 0− i g2ε ji h i
e k Bk(A ) , (3.8)
j
N +4g2 B Bi(A ) , (3.3) −N√g ∇ K −A
− G eK i −A # h i
where a prime denotes the ordinary derivative with re-
spect to the indicated argument.
where m denotes the mass of the vector field, is an
e
arbitrary function of AkA , and K On the other hand, when the electromagnetic field is
k the only source, we find that Jt,J ,J ,J and τ are
i ϕ A ij
= ∂ A ∂ A , =∂ A ∂ A , given by Eq.(B.1) in Appendix B.
0i i 0 t i ij j i i j
F − F −
1ε jk
B = i , iB =0,
i jk i
2 √g F ∇ IV. SPHERICAL STATIC SPACETIMES
= a +a ζ +a ζ2+a ζ3+a ζ +a ζ ζ FILLED WITH AN ELECTROMAGNETIC FIELD
G 0 1 1 2 1 3 1 4 2 5 1 2
+a ζ +a ζ , (3.4)
6 3 7 4
Spherically symmetric static vacuum spacetimes with
with a being arbitrary functions of AkA only, and projectability condition in the HMT setup were studied
n k
systematically in [21, 30–32]. In particular, the metric
ζ = B Bi, ζ =( B ) iBj , can be cast in the form,
1 i 2 i j
∇ ∇
ζ3 = ( iBj) iBk jB(cid:0)k , (cid:1) ds2 = c2dt2+e2ν dr+eµ−νdt 2+r2d2Ω, (4.1)
∇ ∇ ∇ −
ζ4 = (∇i∇jB(cid:0)k) ∇i(cid:1)∇(cid:0)jBk .(cid:1) (3.5) in the spherical coordin(cid:0)ates xi = (r(cid:1),θ,φ), where d2Ω =
In terms of the magnetic fi(cid:0)eld B , th(cid:1)e 3-tensor can dθ2+sin2θdφ2, and
i ij
F
be written as µ=µ(r), ν =ν(r), Ni =eµ−νδi. (4.2)
r
εk ThecorrespondingtimelikeKillingvectorisξ =∂ . With
ij t
B . (3.6)
ij k thegaugefreedomoftheNewtonianprepotentialandthe
F ≡ √g
electromagnetic field, without loss of the generality, we
VariationsofS withrespecttoA andA yieldthe choose the gauges,
EM 0 i
generalized Maxwell equations, given respectively, by
ϕ=0, A (r)=A (r)δr, (4.3)
i 1 i
1
k = m2 A NˆiA , (3.7) that is, we consider only the electric field and set the
∇ F0k 2 e 0− i magnetic field to zero. Then, we find that
(cid:16) (cid:17)
N1√g∂t(cid:20)√Nggij(cid:16)F0j −NˆlFjl(cid:17)(cid:21)=−N12(gkj∇khNˆi πLϕ == 0,eµ+Fνϕijµ=′δ0r,δrB+ir=e−02,νΩ
ij − i j ij
Nˆl gij Nˆk Nˆl (cid:16) 2 (cid:17)
× F0j − Fjl − ∇k F0j − Fjl ) +eµ−νgij µ′+ ,
(cid:0) (cid:1)i h (cid:0) (cid:1)i r
+2mN2e2 A0−NˆjAj Nˆi+ 12Ai a′0+a′1ζ1+a′2ζ12 Kij = eµ+ν µ′δir(cid:16)δjr+re(cid:17)−2νΩij ,
+a′ζ3(cid:0)+a′ζ +a′ζ(cid:1)ζ +a′ζ (cid:16)+a′ζ R = 2ν′δr(cid:16)δr +e−2ν rν′ 1(cid:17) e2ν Ω ,
3 1 4 2 5 1 2 6 3 7 4 ij r i j − − ij
+2ε√kjgi∇j a1+2a2ζ1+3a3ζ12+a5ζ2(cid:17)Bk LK = −r22e2(µ−ν)(2rµh′+1)(cid:0), (cid:1)i
+εjik h(cid:0) 2(a +a ζ ) lBj (cid:1) i LA = 2rA2 e−2ν(1−2rν′)+ Λgr2−1 , (4.4)
4√g∇k∇l( 4 5 1 ∇ where Ω δθδhθ +sin2θδφδφ, and(cid:0) is to(cid:1)oicompli-
ij ≡ i j i j LV
+a lBm jB + lB mBj catedto be givenexplicitly here. Then, the Hamiltonian
6 m m
∇ ∇ ∇ ∇ constraint (2.8) reads,
h(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
+ mBl mBj + 8πGJt eνr2dr =0, (4.5)
∇ ∇ ) LK LV −
(cid:0) (cid:1)(cid:0) (cid:1)i Z
(cid:0) (cid:1)
5
where A. Massless Electromagnetic Field with Nr 6=0
Jt =−21g2 2A′02e−2ν +m2e A0−A1eµ−ν 2 . (4.6) When Nr = 0 (or µ = ), for a massless electro-
e 6 6 −∞
h (cid:0) (cid:1) i magnetic field, Eq.(4.7) immediately givesν =ν0, where
The momentum constraint (2.10) reduces to, ν is an integration constant. Then, Eq.(4.10) yields,
0
2πGm2 ν =0=Λ . (4.17)
ν′eµ = e A eν A eµ rA , (4.7) g
− g2 0 − 1 1
e (cid:16) (cid:17) When me =0, Eqs.(4.15) and (4.16) yield,
where
Q
m2 A0 = +Q0, A1 =0, (4.18)
Ji = e(A A eµ−ν)A e−2νδi. (4.8) r
2g2 0− 1 1 r
e where Q and Q are two integration constant. Without
0
Eqs.(2.12) and (2.13), on the other hand, now read, re- loss of generality, we can always set Q = 0. It is re-
0
spectively, markablethatthe abovesolutionforthe electromagnetic
fieldisthesameasthatgiveninGR.Ontheotherhand,
e2ν Λ r2 1 +1 eµ+νµ′ 2 ν′ Λ re2ν eµ+ν when ν =0, the spatial part is flat, R =0, and all the
g g ij
− − −
high order spatial derivative terms vanish, so we have
h=8π(cid:0)Gr2e4νJϕ(cid:1), i (cid:16) (cid:17) (4.9) Fij = Λg . Then, Eqs.(4.12) and (4.13) reduce to,
ij
−
2rν′ e2ν Λ r2 1 +1 =0, (4.10)
− g − 1 4πGQ2
2µ′+ e2µ =Λr2 2rA′+ , (4.19)
where JA =h0, a(cid:0)nd (cid:1) i (cid:18) r(cid:19) − ge2r2
1 1
m2e−2ν µ′′+2µ′ µ′+ e2µ = Λr (rA′)′
Jϕ = e2ge2 "(A1A0)′−2A1A′1eµ−ν (cid:20) (cid:18) r(cid:19)(cid:21) r" −
4πGQ2
2
−A21eµ−ν µ′−2ν′+ r − ge2r3 #. (4.20)
(cid:18) (cid:19)
2 Note that these two equations are not independent. In
A A ν′ . (4.11)
− 0 1(cid:18) − r(cid:19)# fact,onecanobtainEq.(4.20)fromEq.(4.19). Therefore,
wehaveone equationfortwounknowns, µ andA. Thus,
The dynamical equations (2.15) yield, similartothevacuumcase[31],foranychosengaugefield
A, the metric coefficient µ is given by
1 1
e2µ 2 µ′+ν′ + + re2ν = r F
r 2 LA − rr 1 2m 1 4πGQ2
(cid:20) (cid:21) (cid:16) µ = ln + Λr2
+FA +(cid:0) 8πGτ(cid:1) , (4.12) 2 r 3 − ge2r2
rr rr
e2µ µ′′+ 2µ′ (cid:17)ν′ µ′+ 1 + 1e2ν 2A+ 2 rA(r′)dr′ . (4.21)
− r 2 LA − r !
(cid:20) (cid:18) (cid:19)(cid:21) Z
(cid:0) (cid:1)
= e2ν F +FA +8πGτ , (4.13) Then, the Hamiltonian constraint (4.5) reduces to
− r2 θθ θθ θθ! ∞
A′rdr =0. (4.22)
where
Z0
1
τ = e2ν +2A′2 2a′A2 Therefore, for any given A, the solutions of Eqs.(4.17),
rr −4ge2 G 0 − 0 1 (4.18) and (4.21) representsolutions of the HL theory in
h
m2e2ν A A eµ−ν 2 , the HMT setup, coupled with an electromagnetic field,
− e 0− 1 provided that Eq.(4.22) is satisfied.
r2e−2ν (cid:0) (cid:1) i When A = Constant, the above solutions reduce ex-
τ = e2ν 2A′2
θθ − 4g2 G − 0 actly to the Reissner-Nordstro¨m solution found in GR
e
h but written in the Painleve-Gullstrandcoordinates [33].
m2e2ν A A eµ−ν 2 . (4.14)
− e 0− 1
(cid:0) (cid:1) i
The Maxwell equations (3.7) and (3.8) now become, B. Massless Electromagnetic Field with Nr =0
a′A +m2eµ+ν A A eµ−ν =0, (4.15)
0 1 e 0− 1 In the diagonal case, we have
2 1
A′′ A′ν′+ A(cid:0)′ m2e2ν A(cid:1) eµ−νA =0. (4.16)
0 − 0 r 0− 2 e 0− 1 Nr =0, or µ= . (4.23)
−∞
(cid:0) (cid:1)
6
Then, K = 0 = π , for which the momentum con- 2M Λ
ij ij = 1 gr2
straint is satisfied identically, while Eq.(4.10) becomes, − r − 3
r
1 r
ν′ = 1 e2ν(Λgr2 1)+1 , (4.24) × CA− 6 D(r′)dr′ , (4.32)
2r − (cid:18) Z (cid:19)
h i
which has the general solution, where C is an integration constant, and
A
1 2M Λ
ν =−2ln 1− r − 3gr2 , (4.25) (r) 1 α1r9+α2r6
(cid:18) (cid:19) D ≡ r16 1 2M Λgr2 3
where M is a constant. Then, the Maxwell equations − r − 3
r
(4.15) and (4.16) reduce to, (cid:16) (cid:17)
4πGD2
+α r3+α r+α 1r5 . (4.33)
A1 =0, (4.26) 3 4 5− ge2 !
2
A′0′+A′0 r −ν′ =0. (4.27) For the special case Λg =0, A is given explicitly by
(cid:16) (cid:17)
Eq.(4.27) has the general solution, D2 M 2M
A(r) = 1+ 1 1 +C 1
12g2M2 r − A − r
dr e (cid:18) (cid:19) r
A =D +D , (4.28)
0 1Z r4 1− 2Mr − Λ3gr2 2 +g0ζ2 r2+5Mr 30M2 1
(cid:16) (cid:17) 8 " −
where D and D are two integrationconstants. For the
1 2
2M
solutions µ and ν, given by Eqs.(4.23) and (4.25), it can 1 ln √r+√r 2M
be shown that only one of the two dynamical equations −r − r − !#
(cid:16) (cid:17)
(4.12) and (4.13) is independent, and can be cast in the g
+ 3 M3+M2r+2Mr2 2r3
form, 10M2r3ζ2 −
(cid:16) (cid:17)
α
4 7M5+5M4r+4M3r2
A′+P(r)A=Q(r), (4.29) −378M6r5
where
+4M2r3+8Mr4 8r5
Λ r3 3M − !
g
P(r) = − ,
r[3(r 2M) Λ r3]
Q(r) = − 1 − g 4πGD12 −1386αM5 7r6 21M6+14M5r+10M4r2
2[3(r 2M) Λ r3] g2r2
− − g (cid:18) e
α r2 α2 α3 α4 α5 , (4.30) +8M3r3+8M2r4+16Mr5 16r6 . (4.34)
− 1 − r − r4 − r6 − r7 − !
(cid:17)
with
3 2 V. CHARGED BLACK HOLES
α = g ζ2+Λ + 2g + g Λ2ζ−2
1 −2 0 g 2 3 3 g
(cid:18) (cid:19)
4 The causalstructureofspacetimesinthe HL theoryis
+ 12g4+4g5+ 3g6 Λ3gζ−4, differentfromthatinGR,becauseofthe breakingofthe
(cid:18) (cid:19) Lorentzsymmetry. Inparticular,thedispersionrelations
α2 = 6M 24g2+10g3 MΛgζ−2 ofparticlescontainhigh-ordermomentumterms[1,9,34,
−
− 72g4(cid:0)+28g5+12(cid:1)g6−2g8 MΛ2gζ−4, 35],
α3 = (cid:0)3g3M2ζ−2+ 78g5+90g6(cid:1) 75g8 M2Λgζ−4, k2 k4
− − ω2 =m2+k2 1+ + , (5.1)
α4 = 27M2ζ−4 7g8−(cid:0)9g6−8g5 , (cid:1) k (cid:18) MA2 MB4(cid:19)
α5 = −18M3ζ−(cid:0)4 20g8−25g6−(cid:1)22g5 . (4.31) where MA and MB are the suppression energy scales of
The general solution(cid:0)of Eq.(4.29) is give(cid:1)n by, thefourthandsixthorderderivativeterms. Asanresult,
thespeedsofpartilcesv ( dω /dk)becomeunbounded
p k
A(r) = e−RrP(r′)dr′ in the UV, and their mo≡tions do not follow geodesics.
This immediately makesallthe definitions ofblackholes
rQ(r′)eRr′P(r′′)dr′′dr′+C given in GR invalid [22–25]. To provide a proper defi-
A
× ! nition of black holes, anisotropic conformal boundaries
Z
7
[36] and kinematics of particles [37] have been studied whereE isanintegrationconstant,representingthetotal
in the HL theory. In particular, in [21] black holes and energy of the test particle. eliminating e from Eqs.(5.8)
globalstructureofspacetimeswerestudiedby defining a and (5.9) we find that
horizonastheinfinitelyredshifted2-dimensional(closed)
Xn p(r)X q(r,E)=0, (5.10)
surface of massless test particles [29]. Such a definition
− −
reducestothatgiveninGRwhenthe dispersionrelation
where
is relativistic, where M ,M k, as one can see from
A B
Eq.(5.1). ≫ √ 1/(n−1) r′+Nr 1/(n−1)
To study the black hole structure of the solutions pre- X D = | | ,
sented in the last section, following [21] let us consider ≡ t˙ ! (cid:18) n√f (cid:19)
a scalar field with a given dispersion relation F(ζ). In p(r) Nr, q(r,E) EN1/(n−1), (5.11)
the geometrical optical approximations, ζ is given by ≡ √f ≡
ζ = g kikj, where k denotes the 3-momentum of the
ij i with r′ r˙/t˙ = dr/dt. Once X is found by solving
corresponding massless particle. With this approxima-
≡
Eq.(5.11), from it we obtain
tion, the trajectory of a test particle is given by
dr
1 t=t + , (5.12)
0
S dτ H(r,E)
p ≡ Lp Z
Z0
where
1 1 c2N2
= dτ t˙2+e F(ζ) 2ζF′(ζ) , (5.2)
2Z0 ( e h − i) H(r,E) = ǫne−νXn−1−Nr
N1/(n−1)
where e is a one-dimensional einbein, and ζ is now con- = (ǫn 1)Nr+ǫnE e−ν,(5.13)
− X
sidered as a functional of t,xi,t˙,x˙i and e, given by the
relation, with ǫ=sign(r˙+Nrt˙). In the laststep ofthe aboveex-
pressions,weusedEq.(5.10)toreplaceXn−1. Fordetail,
ζ [F′(ζ)]2 = 1 g x˙i+Nit˙ x˙j +Njt˙ , (5.3) we refer readers to [21].
e2 ij A horizon is defined as a surface that is free of space-
(cid:0) (cid:1)(cid:0) (cid:1) time singularities and on which massless test particles
with t˙ dt/dτ, etc. Considering Eq.(5.1), we assume
≡ are infinitely redshifted. Note that the nature of singu-
that
larities in the HL theory was studied in [38], and was
shown that they can be classified into two classes: the
F(ζ)=ζn, (n=1,2,...). (5.4)
coordinatesingularitiesandspacetimesingularities. The
coordinatesingularitiesaretheonesthatcanberemoved
Then, Eq.(5.3) yields,
by the general coordinate transformations (1.1), while
r˙+Nrt˙ 2/(2n−1) 1/(2n−1) the spacetime singularities are ones that cannot be re-
ζ = D , (5.5) moved by (1.1). It should be noted that, although these
ne√f ≡ e2
(cid:18) (cid:19) (cid:18) (cid:19) definitions are the same as those given in GR, there are
fundamental differences, because of the symmetry (1.1)
where
of the HL theory. In [38], some examples are given in
Nr = eµ−ν, f =e−2ν. (5.6) which coordinate singularities in GR become spacetime
− singularities in the HL theory. Spacetime singularities
Note that there is a sign difference between Nr defined canbe further divided into twokinds: the curvatureand
here and the one defined in Eq.(4.2). This corresponds non-curvatureones. Acurvaturesingularityisdefinedas
to the coordinate transformation t t, that is, in the the one in which at least one of the scalars of the sym-
static case, if (N,Nr,ν) is a solutio→n−of the HL theory, metry (1.2) is singular. A non-curvature singularity is
so is the one (N, Nr,ν). Keeping this in mind, and in- defined as the one that does not have curvature singu-
serting the above−into Eq.(5.2), we find that, for radially larity, but some other physical quantities, such as tidal
moving massless particles, is given by forces and/or distortions experienced by a test particle,
p
L become unbounded.
= N2t˙2+ 1 1 2n e1/(1−2n) n/(2n−1). (5.7) Assuming that at a surface, say, rH, H defined by
Lp 2e 2 − D Eq.(5.13) behaves as
(cid:0) (cid:1)
Then, the variations of = 0 with respect to e and t H(r,E)=H (r ,E)(r r )δ+..., (5.14)
p 0 H H
L −
yield, respectively,
as r r , where H (r ,E)=0. Then,
H 0 H
→ 6
N2t˙2 e2(n−1)/(2n−1) n/(2n−1) =0, (5.8)
− D 0, δ >1,
N2t˙ e2(n−1)/(2n−1)Nr 1/[2(2n−1)] =eE, (5.9) H′(r,E) = H0(rH,E), δ =1, (5.15)
− √fD (cid:12)r=rH ( , 0<δ <1.
(cid:12) ±∞
(cid:12)
(cid:12)
8
Now t as r r+ if and only if To see the above explicitly, let us consider the case
→∞ → H
A= Constantand Λ=0, for whichEq.(4.21) reduces to
δ 1, (5.16) Reissner-Nordstro¨msolutionfound in GRbut written in
≥
the Painleve-Gullstrandcoordinates [33],
for which we have
1 r r2
dH(r,E) µ (r)= ln g Q , (5.24)
dr = finite. (5.17) RN 2 r − r2!
(cid:12)r=rH
(cid:12)
This provides the necessa(cid:12)(cid:12)ry condition on f,N,Nr,E,n where rg ≡ 2m, rQ2 ≡ 4πGQ2/ge2. Inserting it into
Eq.(5.20) we find that
forthehypersurfacer tobeahorizon,asdefinedabove.
H
Itshouldbenotedthatr usuallydependsontheenergy
H 1
ETofotshteudtyestthpeabrtlaicclkesh,oalsefistrrsutcntoutreedoifnth[2e1s].olutions pre- rH± = 2e−2µ(n,E) rg ± rg2−4e2µ(n,E)rQ2!. (5.25)
q
sentedinthelastsection,letusconsiderthecasesNr =0
and Nr =0 separately. It canbe shownthat the solutions at rH± is free ofspace-
6 time singularities and Eq.(5.23) is satisfied. Therefore,
providedthatr2 4r2e2µ(n,E) 0,thesolutionshavetwo
g− Q ≥
A. Nr 6=0 horizons at r =r±. When n=1, we have µ =0,
H (n=1,E)
and the above expressions reduce exactly to those given
In this case, the solutions are given by Eqs.(4.17) and inGR[39]. However,whenn>1,fromEq.(5.21)wefind
that
(4.21). InsertingthemintoEq.(5.14)andconsideringthe
Eq.(5.6), we find that 1+n
µ lnE, (5.26)
(n,E) ≃− n
nE
H(r,E)=(1+n)eµ− X . (5.18) as E . Then, Eq.(5.25) show that r± 0, that is,
→ ∞ H ≃
as long as the test particles have enough energy (E
Notethatinwritingtheaboveexpression,wehadchosen 1), the horizons can be as closed to the singularity ≫at
ǫ= 1 [21]. Thus, at r =rH, we have r =0 as desired. A similar situation also happens to the
−
Schwarzschildsolution[21]. Again,this is because of the
nE
X(r ,E)= e−µ(rH). (5.19) violation of the Lorentz symmetry in the UV regime.
H
1+n
When A = Constant, the local and global structures
6
of the corresponding spacetimes depend on the choice of
Inserting it into Eq.(5.10), we obtain
A(r) (as well as Λ). It is not difficult to see that the
spacetimes have very rich structures, and some of them
µ(r) µ =0, (5.20)
(n,E)
− willquitesimilartotheReissner-Nordstro¨msolution,and
at r =rH, where ingeneraltheradiusrH willdependontheenergyofthe
observers,i.e., r =r (E).
H H
1 1 1+n n−1
µ ln . (5.21)
(n,E) ≡ n "n(cid:18) nE (cid:19) # B. Nr =0
On the other hand, from Eqs.(5.10) and (5.18), we also
When Nr =0, we have X =E1/n and
have
H(r,E) = ǫnE(n−1)/ne−ν(r)
1
H′(rH,E)= 2(1+n)eµ(n,E)µ′(rH). (5.22) = ǫnE(n−1)/n 1 2M Λgr2. (5.27)
− r − 3
Thus, the condition (5.17) requires r
Tohaveahorizon,thenecessarycondition(5.16)requires
µ′(r) = finite. (5.23) that
|r=rH
2M Λ
Insertingthegeneralsolution(4.21)intoEq.(5.20),one 1 gr2 =0, (5.28)
− r − 3
canfindalltheroots,r =r . Providedthatthesolutions
H
have no spacetime singularities and the condition (5.23) has at least two equal and positive roots. It is easy to
holds, the 2-sphere r = r represents a horizon. As show that this is not possible for any choice of Λ and
H g
mentioned above, r in general depends on E, that is, M. Therefore,this class ofsolutionsdoes nothaveblack
H
the radius of the horizon is observer-dependent. Such hole structures. The global structures of the spacetimes
a dependence is the reflection of the fact that the HL for various choices of the free parameters Λ and M are
g
theory breaks the Lorentz symmetry. givenin[21],andweshallnotrepeattheseanalyseshere.
9
VI. CONCLUSIONS solutions is consistent with observations only when the
gauge field A is considered as part of the lapse function
In this paper, we have studied electromagnetic static in the IR, = N A. In [31] a different point of view
N −
spacetimes in the Hoˇrava and Melby-Thompson (HMT) wasadopted,inwhichthegaugefieldaswellastheNew-
setup [11], in which λ = 1. After writing down specif- tonianprepotentialwasconsideredasindependentofthe
ically the coupling of the theory with a massive vector spacetime metric, although they are part of the gravita-
field in Sec. III, we have applied the general formu- tionalfieldandinteractwithspacetime throughthe field
las to electric static spacetimes with sphericalsymmetry equations, roles quite similar to the Brans-Dicke scalar
in Sec. IV, and found all the solutions of the massless field in the Brans-Dicke theory of gravity [40]. This is
vector field under the gauge (4.3). In particular, when seemingly supported by the results presented recently in
Nr = 0, the metric coefficients are given by Eqs.(4.17) [41]. Moreover,somepreliminaryresultsofstabilityanal-
and6(4.21) for any given gauge field A, subjected to the ysis show that this class of solutions might not be stable
constraint (4.22). The corresponding electromagnetic [42]. Clearly, to understand these solutions better, fur-
field(A ,A ,0,0)isgivenbyEq.(4.18). WhenA=Con- ther investigations are highly demanded, including the
0 1
stant, the solutions reduce to the Reissner-Nordstro¨m possibilities of considering them as describing the space-
one,found inGR butwritten inthe Painleve-Gullstrand timeoutsideofachargedstar. Thegeneraljunctioncon-
coordinates [33]. ditions acrossthe surface ofa star havebeen workedout
InthediagonalcaseNr =0,themetriccoefficientsare in [31], although internal solutions of charged stars have
given by Eqs.(4.23) and (4.25), while the corresponding not been found, yet.
electromagnetic field (A ,A ,0,0) and the gauge field A
0 1
are given, respectively, by Eqs.(4.26), (4.28) and (4.32). Acknowledgements: The work of AW was supported
When Λ = 0, the gauge field A is explicitly given by in part by DOE Grant, DE-FG02-10ER41692. KL was
g
Eq.(4.34). partially supported by NSFC No. 11178018 and No.
In Sec. V, using the geometrical optical approxima- 11075224.
tions [21], we have studied the existence of horizons and
shown explicitly why the definitions of black holes given
in GR [22–25] cannot be applied to the HL gravity, and
Appendix A: (F ) and Fij
how to generalize those definitions to the HL theory, be- s ij (ϕ,n)
causeofthe breakingofthe Lorentz-invariance,inwhich
theconceptionoflight-conesisnolongervalid. Applying (F ) and Fij appearing in Eq.(2.16) are given, re-
s ij (ϕ,n)
the new definition of horizons to our solutions presented spectively,
inSec. IV,wehavefoundthatsomeofthesolutionswith
Nr =0giverisetoblackholes,althoughthe locationsof 1
6 (F ) = g ,
the horizonsdepend onthe energies ofthe test particles. 0 ij ij
−2
With sufficient high energy, the horizoncan be as closed 1
to the central singularity at origin as desired. On the (F1)ij = gijR+Rij,
−2
other hand, the solutions with Nr =0 do nothaveblack
1
hole structures. It shouldbe notedthat one mightargue (F2)ij = −2gijR2+2RRij −2∇(i∇j)R
that by the coordinate transformations,
+2g 2R,
ij
∇
dr 1
dr˜= +dt, (6.1) (F ) = g R Rmn+2R Rk 2 k R
Nr(r) 3 ij −2 ij mn ik j − ∇ ∇(i j)k
+ 2R +g Rmn,
one can bring the metric with Nr = 0 into the diagonal ∇ ij ij∇m∇n
6 1
form, (F ) = g R3+3R2R 3 R2
4 ij −2 ij ij − ∇(i∇j)
ds2 = dt2+e2ν˜d2r˜+r2d2Ω, (6.2) +3g 2R2,
ij
− ∇
1
where ν˜ = ν(r) +ln Nr(r). However, now r is time- (F5)ij = gijRRmnRmn+RijRmnRmn
| | −2
dependent, r = r(r˜,t) (and so is ν˜), and then the tra-
+2RR Rk (RmnR )
jectories of massless particles are no longer described by ki j −∇(i∇j) mn
Eqs.(5.8) and (5.9), so the discussions presented in Sec. 2 n RR +g 2(RmnR )
(i j)n ij mn
− ∇ ∇ ∇
V cannot be applied to the “dynamical” case.
+ 2(RR )+g (RRmn),
Inaddition,thegaugefieldAforthenon-diagonalcase ∇ ij ij∇m∇n
1
(See Sec. IV.A) is undetermined. While its physics is (F ) = g RmRnRp +3RmnR R
not clear (even in more general case) [11, 12], the solar 6 ij −2 ij n p m ni mj
3 3
system tests generically require it vanish [31] (See also + 2 R Rn + g RkRln
[32]). However, in the diagonal case it is uniquely de- 2∇ in j 2 ij∇k∇l n
termined by Eq.(4.32). HMT showed that this class of 3 (cid:0) R (cid:1)Rnk , (cid:0) (cid:1)
k (i j)n
− ∇ ∇
(cid:0) (cid:1)
10
1 m2
(F7)ij = −2gij(∇R)2+(∇iR)(∇jR)−2Rij∇2R +2g2eN(A0−NˆkAk)Ai,
e
+2∇(i∇j)∇2R−2gij∇4R, J = 1 ∂ √g B Bi
(F ) = 1g ( R )( pRmn) 4R ϕ −√gN t K i
8 ij −2 ij ∇p mn ∇ −∇ ij m2 h i
+(∇iRmn)(∇jRmn)+2(∇pRin) ∇pRjn +2ge2eN∇l (A0−NˆiAi)Al
h i
+2 n 2R +2 Rn R(cid:0)m (cid:1) 1
∇ ∇(i∇ j)n ∇n m∇(i j) j BiBi(N jϕ Nj)
−N∇ K ∇ −
(cid:16) (cid:17)
2 R Rmn 2 R nRm 1 h i
− ∇n m(j∇i) − ∇n m(i∇ j) + gij l ( Nˆk )
g n(cid:0) m 2R , (cid:1) (cid:16) (A(cid:17).1) 2ge2N ∇ F0i− Fki Fjl
ij mn h
− ∇ ∇ ∇
( Nˆk ) ,
0j jk li
and [12] F − F F
J = 2 B Bi, i
1 A K i
F(iϕj,1) = 2ϕ 2K+∇2ϕ Rij −2 2Kkj+∇j∇kϕ Rik τmn = gmnLEM
n−(cid:16)2(cid:16)2Kki +∇(cid:17)i∇kϕ(cid:17)R(cid:16)jk (cid:17) −2ge21N2"(F0m−NˆkFkm)(F0n−NˆlFnl)
2Λ R 2Kij+ i jϕ ,
Fij = 1 −(cid:16)ϕ igk−2N(cid:17)j(cid:16)+ jϕ ∇ ∇ (cid:17)o ++N(F(0n0−j NˆNkˆFlknjl))((F0mmj−nNˆϕlF+ml)nj mϕ)
(ϕ,2) 2∇k G N ∇ F − F F ∇ F ∇
+ϕ j(cid:26)k 2N(cid:16)i + iϕ ϕ(cid:17) ij 2Nk + kϕ , +N(F0i−NˆlFli)(Fim∇nϕ+Fin∇mϕ)#
G N ∇ − G N ∇
(cid:16) (cid:17) (cid:16) (cid:17)(cid:27) m2
F(iϕj,3) = 21 2∇k∇(ifϕj)k−∇2fϕij − ∇k∇lfϕkl gij , −2ge2eN(A0−NˆiAi)(Am∇nϕ+An∇mϕ)
n (cid:0) (cid:1) o(A.2) 1
a (BmBn B Bigmn)
where −2ge2( 1 − i
fij = ϕ 2Kij + i jϕ 1 2K+ 2ϕ gij . −a′0AmAn−a′1AmAnBjBj
ϕ (cid:26)(cid:16) ∇ ∇ (cid:17)− 2(cid:16) ∇ (cid:17) (cid:27) −a′2AmAn(BjBj)2
(A.3) +2a B Bl(BmBn B Bigmn)
2 l i
−
a′AmAn(B Bj)3+3a (B Bl)2(BmBn
− 3 j 3 l
Appendix B: Jt,Ji,Jϕ,JA and τij for a vector field −BiBigmn)−a′4AmAn(∇lBj)∇lBj
a ( mB ) nBj +( Bm) iBn
4 j i
When a vector field is the only source, the quantities − ∇ ∇ ∇ ∇
h i
Jt,Ji,Jϕ,JA and τij are given by + i(a4 iBj) Bjgmn 2 i(a4 iB(n) Bm)
∇ ∇ − ∇ ∇
h i h i
1 1
Jt = gij( Nˆk )( Nˆl ) + a B(m iBn)+B(m n)Bi
ge2"− N2 F0i− Fki F0j − Fjl ∇i" 4 ∇ ∇
(cid:16)
1
+ gij( 0i Nˆk ki)( lϕ jl) Bi (mBn) a′AmAnB Bk( B ) lBj
N F − F ∇ F − ∇ #− 5 k ∇l j ∇
+ 1 gij( Nˆl )( kϕ ) (cid:17)
N F0j − Fjl ∇ Fki +a5( lBk) lBk (BmBn BiBigmn)
∇ ∇ −
m2
+ Ne(A0−NˆiAi)(∇jϕAj) −a5BkBk ((cid:0)∇mBj)(cid:1)∇nBj +(∇jBm)∇jBn
h i
m2e (A NˆiA )2 , + ∇i(a5BkBk∇iBj) Bjgmn
−2N2 0− i # 2h (a B Bk iB(ni) Bm)
i 5 k
− ∇ ∇
J = 1 gjk ( Nˆl ) h i
i −2ge2N " F0j − Fjl Fki + i a5BkBk B(m iBn)+B(m n)Bi
∇ " ∇ ∇
(cid:16)
+( Nˆl )
0k lk ij
F − F F #