Dilatonic dyon-like black hole solutions in the model with two Abelian gauge fields

0.62 MB·

by  M. E. Abishev

Dilatonic dyon-like black hole solutions in the model
with two Abelian gauge fields
M.E. Abishev1,3, K.A. Boshkayev1,
and V.D. Ivashchuk2,3
7
1
0 1 Institute of Experimental and Theoretical Physics,
2 Al-Farabi Kazakh National University,

r Al-Farabi avenue, 71, Almaty 050040, Kazakhstan,
a
M 2 Center for Gravitation and Fundamental Metrology, VNIIMS,
Ozyornaya St., 46, Moscow 119361, Russia,

4
3 Institute of Gravitation and Cosmology,
2
RUDN University,

]
c Miklukho-Maklaya St.,6, Moscow 117198, Russia
q
-
r Abstract
g
[
Dilatonic black hole dyon-like solutions in the gravitational 4d

3 modelwithascalarfield,two2-forms,twodilatoniccouplingconstants
v
λ (cid:54)= 0, i = 1,2, obeying λ (cid:54)= −λ and the sign parameter ε = ±1 for
9 i 1 2
2 scalar field kinetic term are considered. Here ε = −1 corresponds to a
0 ghost scalar field. These solutions are defined up to solutions of two
2
masterequationsfortwomodulifunctions, whenλ2 (cid:54)= 1/2forε = −1.
0 i
. Some physical parameters of the solutions are obtained: gravitational
1
mass, scalar charge, Hawking temperature, black hole area entropy
0
7 and parametrized post-Newtonian (PPN) parameters β and γ. The
1
PPN parameters do not depend on the couplings λ and ε. A set of
: i
v boundsonthegravitationalmassandscalarchargearefoundbyusing
i
X acertainconjectureontheparametersofsolutions,when1+2λ2ε > 0,
i
r i = 1,2.
a
1
1 Introduction
In this paper we extend our previous work [1] devoted to dilatonic dyon
black hole solutions. We note that at present there exists a certain interest in
spherically symmetric solutions, e.g. black hole and black brane ones, related
toLiealgebrasandTodachains, see[2]-[27]andthereferencestherein. These
solutions appear in gravitational models with scalar fields and antisymmetric
forms.
Here we consider a subclass of dilatonic black hole solutions with electric
and magnetic charges Q and Q , respectively, in the 4d model with metric
1 2
g, scalar field ϕ, two 2-forms F(1) and F(2), corresponding to two dilatonic
coupling constants λ and λ , respectively. All fields are defined on an ori-
1 2
ented manifold M. Here we consider the dyon-like configuration for fields of
2-forms:
F(1) = Q e2λ1ϕ ∗τ, F(2) = Q τ, (1.1)
1 2
where τ = vol[S2] is volume form on 2-dimensional sphere and ∗ = ∗[g]
is the Hodge operator corresponding to the oriented manifold M with the
metric g. We call this noncomposite configuration a dyon-like one in order to
distinguish it from the true dyon configuration which is essentially composite
andmaybechoseninourcaseeitheras: (i)F(1) = Q e2λ1ϕ∗τ+Q τ, F(2) = 0,
1 2
or (ii) F(1) = 0, F(2) = Q e2λ2ϕ∗τ +Q τ. From a physical point of view the
1 2
ansatz(1.1)meansthatwedealherewithachargedblackhole, whichhastwo
color charges: Q and Q . The charge Q is the electric one corresponding
1 2 1
to the form F(1), while the charge Q is the magnetic one corresponding
2
to the form F(2). For coinciding dilatonic couplings λ = λ = λ we get
1 2
a trivial noncomposite generalization of dilatonic dyon black hole solutions
in the model with one 2-form which was considered in ref. [1], see also
[4, 9, 10, 13, 22, 27] and references therein.
The dilatonic scalar field may be either an ordinary one or a phantom
(or ghost) one. The phantom field appears in the action with a kinetic term
of the “wrong sign”, which implies the violation of the null energy condition
p ≥ −ρ. According to ref. [28], at the quantum level, such fields could form
a “ghost condensate”, which may be responsible for modified gravity laws
in the infra-red limit. The observational data do not exclude this possibility
[29].
Here we seek relations for the physical parameters of dyonic-like black
holes, e.g. bounds on the gravitational mass M and the scalar charge Q .
ϕ
2
As in our previous work [1] this problem is solved here up to a conjec-
ture, which states a one-to-one (smooth) correspondence between the pair
(Q2,Q2), where Q is the electric charge and Q is the magnetic charge, and
1 2 1 2
the pair of positive parameters (P ,P ), which appear in decomposition of
1 2
moduli functions at large distances. This conjecture is believed to be valid
for all λ (cid:54)= 0 in the case of an ordinary scalar field and for 0 < λ2 < 1/2 for
i i
the case of a phantom scalar field (in both cases the inequality λ (cid:54)= −λ is
1 2
assumed).
2 Black hole dyon solutions
Let us consider a model governed by the action
(cid:90)
1 (cid:112) (cid:110)
S = d4x |g| R[g]−εgµν∂ ϕ∂ ϕ
µ ν
16πG
1 1 (cid:111)
− e2λ1ϕF(1)F(1)µν − e2λ2ϕF(2)F(2)µν , (2.1)
2 µν 2 µν
where g = g (x)dxµ ⊗ dxν is metric, ϕ is the scalar field, F(i) = dA(i) =
µν
1F(i)dxµ ∧dxν is the 2-form with A(i) = A(i)dxµ, i = 1,2, ε = ±1, G is the
2 µν µ
gravitational constant, λ ,λ (cid:54)= 0 are coupling constants obeying λ (cid:54)= −λ
1 2 1 2
and |g| = |det(g )|. Here we also put λ2 (cid:54)= 1/2, i = 1,2, for ε = −1. For
µν i
λ = λ theLagrangian(2.1)appearsinthegravitationalmodelwithascalar
1 2
field and Yang-Mills field with a gauge group of rank 2 (say SU(3)) when an
Abelian sector of the gauge field is considered.
We consider a family of dyonic-like black hole solutions to the field equa-
tions corresponding to the action (2.1) which are defined on the manifold
M = (2µ,+∞)×S2 ×R, (2.2)
and have the following form
(cid:18) (cid:19)
(cid:110) 2µ
ds2 = g dxµdxν = Hh1Hh2 −H−2h1H−2h2 1− dt2 (2.3)
µν 1 2 1 2 R
dR2 (cid:111)
+ +R2dΩ2 ,
1− 2µ 2
R
exp(ϕ) = Hh1λ1εH−h2λ2ε, (2.4)
1 2
Q
F(1) = 1H−2H−A12dt∧dR, (2.5)
R2 1 2
3
F(2) = Q τ. (2.6)
2
Here Q and Q are (colored) charges – electric and magnetic, respectively,
1 2
µ > 0 is the extremality parameter, dΩ2 = dθ2 + sin2θdφ2 is the canonical
2
metric on the unit sphere S2 (0 < θ < π, 0 < φ < 2π), τ = sinθdθ ∧dφ is
the standard volume form on S2,
1
h = K−1, K = +ελ2, (2.7)
i i i 2 i
i = 1,2, and
A = (1−2λ λ ε)h . (2.8)
12 1 2 2
The functions H > 0 obey the equations
s
d (cid:32) (cid:0)1− 2µ(cid:1)dH (cid:33) (cid:89)
R2 R2 R s = −K Q2 H−Asl, (2.9)
dR H dR s s l
s
l=1,2
with the following boundary conditions imposed:
H → H > 0 (2.10)
s s0
for R → 2µ, and
H → 1 (2.11)
s
for R → +∞, s = 1,2.
In (2.9) we denote
(cid:18) (cid:19)
2 A
(A ) = 12 , (2.12)
ss(cid:48) A 2
21
where A is defined in (2.8) and
12
A = (1−2λ λ ε)h . (2.13)
21 1 2 1
These solutions may be obtained just by using general formulas for non-
extremal (intersecting) black brane solutions from [18, 19, 20] (for a review
see[21]). Thecompositeanalogsofthesolutionswithone2-formandλ = λ
1 2
were presented in ref. [1].
The first boundary condition (2.10) guarantees (up to a possible addi-
tional requirement on the analyticity of H (R) in the vicinity of R = 2µ) the
s
4
existence of a (regular) horizon at R = 2µ for the metric (2.3). The second
condition (2.11) ensures asymptotical (for R → +∞) flatness of the metric.
Remark 1. It should be noted that the main motivation for considering
this and more general 4D models governed by the Lagrangian density L:
m
(cid:112) 1 (cid:88)
L/ |g| = R[g]−h gµν∂ ϕa∂ ϕb − exp(2λ ϕa)F(i)F(i)µν, (2.14)
ab µ ν 2 ia µν
i=1
where ϕ = (ϕa) is a set of l scalar fields, F(i) = dA(i) are 2 forms and
λ = (λ ) are dilatonic coupling vectors, i = 1,...,m, is coming from di-
i ia
mensional reduction of supergravity models; in this case the matrix (h ) is
ab
positive definite. For example, one may consider a part of bosonic sector
of dimensionally reduced 11d supergravity [15] with l dilatonic scalar fields
and m 2-forms (either originating from 11d metric or coming from 4-form)
activated; Chern-Simons terms vanish in this case. Certain uplifts (to higher
dimensions) of 4d black hole solutions corresponding to (2.14) may lead us
to black brane solutions in dimensions D > 4, e.g. to dyonic ones; see
[15, 16, 19, 23, 24] and the references therein. The dimensional reduction
from the 12-dimensional model from ref. [30] with phantom scalar field and
two forms of rank 4 and 5 will lead us to the Lagrangian density (2.14) with
the matrix (h ) of pseudo-Euclidean signature.
ab
Equations (2.9) may be rewritten in the following form:
d (cid:20) dys(cid:21) (cid:88)
(1−z) = −K q2exp(− A yl), (2.15)
dz dz s s sl
l=1,2
s = 1,2. Hereandinthefollowingweusethefollowingnotations: ys = lnH ,
s
z = 2µ/R, q = Q /(2µ) and K = h−1 for s = 1,2, respectively. We are
s s s s
seeking solutions to equations (2.15) for z ∈ (0,1) obeying
ys(0) = 0, (2.16)
ys(1) = ys, (2.17)
0
where ys = lnH are finite (real) numbers, s = 1,2. Here z = 0 (or, more
0 s0
precisely z = +0) corresponds to infinity (R = +∞), while z = 1 (or, more
rigorously, z = 1−0 ) corresponds to the horizon (R = 2µ).
Equations (2.15) with conditions of the finiteness on the horizon (2.17)
imposed imply the following integral of motion:
1 (cid:88) dysdyl (cid:88) dys
(1−z) h A + h (2.18)
s sl s
2 dz dz dz
s,l=1,2 s=1,2
5
(cid:88) (cid:88)
− q2exp(− A yl) = 0.
s sl
s=1,2 l=1,2
Equations (2.15) and (2.17) appear for special solutions to Toda-type equa-
tions [19, 20, 21]
d2zs (cid:88)
= K Q2exp( A zl), (2.19)
du2 s s sl
l=1,2
for functions
zs(u) = −ys −µbsu, (2.20)
s = 1,2, depending on the harmonic radial variable u: exp(−2µu) = 1−z,
with the following asymptotical behavior for u → +∞ (on the horizon)
imposed:
zs(u) = −µbsu+zs +o(1), (2.21)
0
where z are constants, s = 1,2. Here and in the following we denote
s0
(cid:88)
bs = 2 Asl, (2.22)
l=1,2
where the inverse matrix (Asl) = (A )−1 is well defined due to λ (cid:54)= −λ .
sl 1 2
This follows from the relations
1
A = 2B h , B = +εχ χ λ λ , (2.23)
sl sl l sl s l s l
2
where χ = +1, χ = −1 and the invertibility of the matrix (B ) for λ (cid:54)=
1 2 sl 1
−λ , due to the relation det(B ) = 1ε(λ +λ )2.
2 sl 2 1 2
The energy integral of motion for (2.19), which is compatible with the
asymptotic conditions (2.21),
1 (cid:88) dzsdzl
E = h A (2.24)
s sl
4 du du
s,l=1,2
1 (cid:88) (cid:88) 1 (cid:88)
− Q2exp( A zl) = µ2 h bs,
2 s sl 2 s
s=1,2 l=1,2 s=1,2
leads to eq. (2.18).
Remark 2. The derivation of the solutions (2.3)-(2.6), (2.9)-(2.11) may
be extracted from the relations of [18, 19, 20], where the solutions with a
horizon were obtained from general spherically symmetric solutions governed
6
by Toda-like equations. These Toda-like equations contain a non-trivial part
corresponding to a non-degenerate (quasi-Cartan) matrix A. In our case
these equations are given by (2.19) with the matrix A from (2.23) and the
condition detA (cid:54)= 0 implies λ (cid:54)= −λ . The master equations (2.9) are
1 2
equivalent to these Toda-like equations. Fortunately, for λ = −λ and ε =
1 2
+1 the solution does exist. It obeys eqs. (2.3)-(2.6) and (2.9)-(2.11) with
Q2i
Hi = HQ21+Q22, i = 1,2, where H = 1+ PR and P > 0 satisfies P(P +2µ) =
K (Q2 +Q2), K > 0. For λ = −λ the solution reads:
1 1 2 1 1 2
(cid:110) (cid:18) 2µ(cid:19) dR2 (cid:111)
ds2 = Hh1 −H−2h1 1− dt2 + +R2dΩ2 ,
R 1− 2µ 2
R
Q
exp(ϕ) = Hh1λ1ε, F(1) = 1H−2dt∧dR, F(2) = Q τ.
R2 2
We have verified this solution by using MATHEMATICA. It is also valid for
ε = −1 and λ2 < 1.
1 2
3 Some integrable cases
Explicit analytical solutions to eqs. (2.9), (2.10), (2.11) do not exist. One
may try to seek the solutions in the form
(cid:88)∞ (cid:18)1(cid:19)k
H = 1+ P(k) , (3.1)
s s R
k=1
whereP(k) areconstants,k = 1,2,...,ands = 1,2,butonlyinfewintegrable
s
cases the chain of equations for P(k) is dropped.
s
For ε = +1, there exist at least four integrable configurations related to
the Lie algebras A +A , A , B = C and G .
1 1 2 2 2 2
3.1 (A + A )-case
1 1
Let us consider the case ε = 1 and
(cid:18) (cid:19)
2 0
(A ) = . (3.2)
ss(cid:48) 0 2
7
We obtain
1
λ λ = . (3.3)
1 2
2
For λ = λ we get a dilatonic coupling corresponding to string induced
1 2
model. The matrix (3.2) is the Cartan matrix for the Lie algebra A + A
1 1
(A = sl(2)). In this case
1
P
s
H = 1+ , (3.4)
s
R
where
P (P +2µ) = K Q2, (3.5)
s s s s
s = 1,2. For positive roots of (3.5)
(cid:112)
P = P = −µ+ µ2 +K Q2, (3.6)
s s+ s s
we are led to a well-defined solution for R > 2µ with asymptotically flat
metric and horizon at R = 2µ. We note that in the case λ = λ the
1 2
(A +A )-dyon solution has a composite analog which was considered earlier
1 1
in [7, 9]; see also [14] for certain generalizations.
3.2 A -case
2
Now we put ε = 1 and
(cid:18) (cid:19)
2 −1
(A ) = . (3.7)
ss(cid:48) −1 2
We get
λ = λ = λ, λ2 = 3/2. (3.8)
1 2
This value of dilatonic coupling constant appears after reduction to four
dimensions of the 5d Kaluza-Klein model. We get h = 1/2 and (3.7) is the
s
Cartan matrix for the Lie algebra A = sl(3). In this case we obtain [19]
2
P P(2)
s s
H = 1+ + , (3.9)
s R R2
where
P (P +2µ)(P +4µ)
2Q2 = s s s , (3.10)
s P +P +4µ
1 2
P (P +2µ)P
P(2) = s s s¯ , (3.11)
s 2(P +P +4µ)
1 2
8
s = 1,2 (s¯= 2,1).
In the composite case [1] the Kaluza-Klein uplift to D = 5 gives us the
well-known Gibbons-Wiltshire solution [5], which follows from the general
spherically symmetric dyon solution (related to A Toda chain) from ref. [4].
2
3.3 C and G cases
2 2
If we put ε = 1 and
(cid:18) (cid:19) (cid:18) (cid:19)
2 −1 2 −k
(A ) = or (A ) = , (3.12)
ss(cid:48) −k 2 ss(cid:48) −1 2
we also get integrable configurations for k = 2,3, corresponding to the Lie
algebras B = C and G , respectively, with the degrees of polynomials (3,4)
2 2 2
and (6,10). From (2.8), (2.13) and (3.12) we get the following relations for
the dilatonic couplings:
(cid:18) (cid:19)
1 1 1
+λ2 = k +λ2 , 1−2λ λ = − −λ2, (3.13)
2 2 2 1 1 2 2 2
or
(cid:18) (cid:19)
1 1 1
+λ2 = k +λ2 , 1−2λ λ = − −λ2. (3.14)
2 1 2 2 1 2 2 1
√
Solving eqs. (3.13) we get (λ ,λ ) = ±( 2, √3 ) for k = 2 and (λ ,λ ) =
1 2 1 2
2
(cid:16) (cid:113) (cid:17)
± √5 ,3 3 for k = 3. The solution to eqs. (3.14) is given by permutation
6 2
of λ and λ .
1 2
The exact black hole (dyonic-like) solutions for Lie algebras B = C and
2 2
G will be analyzed in detail in separate publications. They do not exist for
2
the case λ = λ . We note that for the B = C case (k = 2) the polynomials
1 2 2 2
H , i = 1,2, were calculated in [31].
i
3.4 Special solution with two dependent charges
There exists also a special solution
(cid:18) P(cid:19)bs
H = 1+ , (3.15)
s
R
9
with P > 0 obeying
K
sQ2 = P(P +2µ), (3.16)
b s
s
s = 1,2. Here bs (cid:54)= 0 is defined in (2.22). This solution is a special case of
more general “block orthogonal” black brane solutions [32, 33, 34].
The calculations give us the following relations:
2λ
bs = s¯ K , (3.17)
s
λ +λ
1 2
(λ +λ ) 1
Q2 1 2 = P(P +2µ) = Q2, (3.18)
s 2λ 2
s¯
where s = 1,2 and s¯ = 2,1, respectively. Our solution is well defined if
λ λ > 0, i.e. the two coupling constants have the same sign.
1 2
For positive roots of (3.18)
(cid:114)
1
P = P = −µ+ µ2 + Q2 (3.19)
+
2
we get for R > 2µ a well-defined solution with asymptotically flat metric and
horizon at R = 2µ. It should be noted that this special solution is valid for
both signs ε = ±1. We have
(cid:110) (cid:18) 2µ(cid:19) dR2 (cid:111)
ds2 = H2 −H−4 1− dt2 + +R2dΩ2 , (3.20)
R 1− 2µ 2
R
ϕ = 0, (3.21)
Q
F(1) = 1 dt∧dR, F(2) = Q τ, (3.22)
H2R2 2
where H = 1+ P with P from (3.19) and
R
λ λ
Q2 = 2 Q2, Q2 = 1 Q2. (3.23)
1 λ +λ 2 λ +λ
1 2 1 2
By changing the radial variable, r = R+P, we get
ds2 = −f(r)dt2 +f(r)−1dr2 +r2dΩ2, (3.24)
2
Q
F(1) = 1dt∧dr, F(2) = Q τ, ϕ = 0, (3.25)
r2 2
10