Effect of the Hair on Deflection Angle by Asymptotically Flat Black Holes in
Einstein-Maxwell-Dilaton Theory
W. Javed,1,∗ J. Abbas,1,† and A. ¨Ovg ¨un2, 3,‡ 1Department of Mathematics, University of Education,
Township, Lahore-54590, Pakistan.
2Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4950, Valpara´ıso, Chile
3Physics Department, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, North Cyprus, via Mersin 10, Turkey (Dated: June 12, 2019)
In this paper, we are interested in a model of exact asymptotically flat charged hairy black holes in the background of dilaton potential. We study the weak gravitational lensing in the spacetime of hairy black hole in Einstein-Maxwell theory with a non-minimally coupled dilaton and its non-trivial potential. In doing so, we use the optical geometry of the flat charged hairy black hole for some range of parameterγ. For this purpose, by using Gauss-Bonnet theorem, we obtain the deflection angle of photon in a spherically symmetric and asymptotically flat spacetime. Moreover, we also in-vestigate the impact of plasma medium on weak gravitational lensing by asymptotically flat charged hairy black hole with a dilaton potential. Our analytically analyses show the effect of the hair on the deflection angle in weak field limits.
PACS numbers: 04.70.Dy; 04.70.Bw; 11.25.-w
Keywords: relativity and gravitation; classical black hole; deflection angle; Gauss-Bonnet; non-linear electrody-namics.
I. INTRODUCTION
At the darkest points in the universe, their boundaries perilous and invisible, space warps. The singularity consti-tutes the centre of a black hole and is hidden by the object’s surface, the event horizon. A black hole is a location in space that possesses so much gravity, nothing can escape its pull, even light. It is said that fact is sometimes stranger than fiction, and nowhere is this more true than in the case of black holes. Since the first image of a black hole by Event Horizon Telescope [1], physicist now try to take even sharper images so that Einstein’s theory of general rela-tivity can be tested and also see the properties of the black holes [2–10] because there are many theoretically obtained black hole solutions with different properties.
In this paper, we try to understand the effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton (EMD) theory, where is derived from a string theory at low energy limits [11]. The EMD black holes have a scalar hair, in addition to mass, rotation and charge so that they offers an fascinating theoretical black hole models to check experimentally in black hole experiments, and investigate possible differences between Einstein gravity and modified gravity theories [11–13].
Further, let us now briefly review the GBT which connects the topologically surfaces. First, using Euler charac-teristic χ and a Riemannian metric g, one can choose the subset oriented surface domain as (D,χ,g) to find the Gaussian curvatureK. Then the Gauss-Bonnet theorem is defined as follow [14]
Z Z
DKdS+
I
∂D
κdt+
∑
i
θi=2πχ(D) (1)
where κ is the geodesic curvature for ∂D : {t} → Dand θi is the exterior angle withith vertex. Following this approach, global symmetric lenses are considered to be Riemannian metric manifolds, which are geodesic spatial light rays. In optical geometry, we calculate the Gaussian optical curvatureKto find the asymptotic bending angle which can be calculated as follows [14]:
ˆ α=−
Z Z
D∞
KdS. (2)
∗Electronic address:[email protected]; [email protected]
†Electronic address:[email protected]
Note that this equation is an exact result for the deflection angle. In this equation, we integrate over an infinite region of the surfaceD∞which is bounded by the light ray. By assumtion, one can use the above relation only for asymptotically Euclidian optical metrics. Therefore it will be interesting to see the form of the deflection angle in the case of non-asymptotically Euclidian metrics. This method has been applied in various papers for different types of spacetimes [15–43].
This paper is organized as follows: In section2, we review some basic concepts about asymptotically flat hairy BH. In section3, we compute the Gaussian optical curvature for deflection angle and calculate the deflection angle by using GBT forγ=1 also find the deflection angle in a plasma medium. In section4, we calculate the deflection angle forγ =
√
3 also find the deflection angle in a plasma medium. The last section comprises of concluding remarks and results obtained from graphical analysis.
II. ASYMPTOTICALLY FLAT BLACK HOLES IN EINSTEIN-MAXWELL-DILATON THEORY
In this section, we briefly review the asymptotically flat hairy BH in EMD theory in the background of dilaton potential. We consider the following action of Einstein-Maxwell-dilaton theory [11,12]:
I[gµν,Aµ,φ] = 1 2k
Z
Md
4xp
−g[R−eγϕF2−1 2(∂ϕ)
2−V(ϕ)], (3)
where
F2=FµνFµν,(∂ϕ)2=∂µϕ∂µϕ, (4)
V(ϕ)is the dilaton potential, andc=GN =4πe0=1 by using the conventionk=8π. The equations of motion for gauge field, dilaton and metric are following:
Rµν− 1
2gµνR = T ϕ µν+T
EM (5)
∂µ( p
−geγϕFµν) = 0 (6)
1 √
−g∂µ(
p
−ggµν∂ νϕ) =
dV(ϕ)
dϕ +γe
γφF2 (7)
where the stress tensors of matter fields are defined as
Tµνϕ ≡ 1
2∂µφ∂νϕ− 1 2gµν[
1 2(∂φ)
2+V(φ)], (8)
TµνEM ≡2eγϕ(F µαFνα−
1 4gµνF
2). (9)
For exact regular hairy BH solutions, for a general scalar potential there is a new method in a number of papers [13], by using a specific ansatz. For flat spacetime, we should apply the similar special ansatz for the metric and dilaton. For the sake of simplicity, we are going to focus on two particular cases for which the exponent coefficient of the dilaton coupling with the gauge field takes the valuesγ=1 andγ=
√
3, but for more general solution there exist a literature [12].
III. WEAK DEFLECTION ANGLE OF CALCULATION OF PHOTON LENSING FORγ=1BY GAUSS-BONNET THEOREM
For this solution we consider the following scalar field potential:
V(ϕ) =2α(2ϕ+ϕcoshϕ−3 sinhϕ), (10)
hereαis an arbitrary parameter. Forγ=1 the static hairy BH metric and gauge field, yields that:
ds2 = Ω(x)
−f(x)dt2+ η 2dx2
x2f(x)+dΣ
2 (11)
F = 1
2Fµνdxµ∧dxν=
qe−φ
whereηandqare defined as independent parameters of the solution and correspond to the mass and charge of this BH.dΣ2=dθ2+sin2θdφ2is spherical line element, and the coordinatexis restricted to be positive,x ∈[0,∞). We can suppose thatη>0. One can use the conformal factor as follows:
Ω(x) = x
η2(x−1)2 (12)
and then check that the equations of motion are satisfied for the following spacetime metric function:
f(x) =α
x2−1
2x −ln(x)
+η
2(x−1)2
x
1−2q
2(x−1)
x
. (13)
It is appropriate to first find the black hole optical metric by imposing the null conditionds2=0, and solving the space-time metric fordtand also set the metric into equatorial plane withθ= π2, which yield as:
dt2= η 2
x2f(x)2dx 2+ 1
f(x)dϕ
2. (14)
The optical geometry is in two dimensions, and is obtained for thermodynamically stable asymptotically flat hairy BH with a dilaton potential as follows [14]. By using Gauss-Bonnet theorem, initially we find the Gaussian curvature K of the optical spacetime, as
K = RicciScalar
2 (15)
K ≈ −η2x−η2
x +1/4
η2
x2 +3/2η
2+1/4x2
η2−1/2xαln(x)−1/2αln(x)
x +1/4αx
2−1/4 α
x2 −x 2
η2q2
+5η2q2x−11η 2q2
x2 +16 η2q2
x +3
η2q2
x3 −12η
2q2+5αq2
x −3/2
αq2
x3
−4αq 2ln(x)
x2 +3
αq2ln(x)
x +xαln(x)q
2−3
αq2−1/2x2αq2+1/2xαq2−1/2αq 2
x2 (16)
For multiple images, we use the global theory (Gauss-Bonnet theorem) to relate with the local feature of the space-time such that Gaussian optical curvature.
The above equation will be apply to calculate the deflection angle by taking a non singular domainSRoutside of the light ray (along with boundaries ∂SR = γg∪CR) with Euler characteristic χ(SR), Gaussian curvatureK, geodesic curvaturekand exterior jump anglesαi = (αO,αS)at vertices.
Z Z
SR
KdS+ I
∂SR
kdt+
∑
j
αj=2πχ(SR), (17)
at weak limit approximation(ρ→∞),αO+αS→π. Then GBT becomes
Z Z
SR
KdS+ I
Cr
kdt=ρ→∞ Z Z
S∞kdS+
Z π+Θ
0 dϕ=π. (18)
Now, by geodesic property, the geodesic curvature vanishesk(γg) =0, and we get
k(CR) =|∇C˙rC˙r|, (19)
withCr :=ρ(ϕ) =r=constant. Then GBT reduces
lim R→∞
Z π+Θ
0
kgd
σ
dϕ
|CRdϕ=π−Rlim→∞ Z Z
SR
KdS. (20)
Now, for radial distance
Therefore,
lim R→∞kg
dσ
dϕ|CR =1. (22)
In the weak field regions, the light ray follows a straight line approximation, so that we can use the condition of
r=b/ sinϕat zero order.
Θ=− lim R→∞
Z π
0 Z R
b/ sinϕ
KdS, (23)
where
KdS≈1/4η 2
x2+1/4 α
x2 − η2q2
x2 −1/2 αq2
x2 . (24)
After simplification, we find the deflection angle of photon for asymptotically flat hairy BH in leading order terms as
˜
α' −1/2η 2
b −1/2
α
b +2
η2q2
b +
αq2
b (25)
Therefore, we can say that GBT provides a globally as well as topologically effect, this method is very useful for quantitative tool and can be apply in any asymptotically flat metrics.
A. Photon lensing in a plasma medium
In this section, we analyze the effect of plasma medium on the photon lensing by asymptotically hairy black hole. The refractive index for hairy black hole is as follows [15],
n(x) =
s 1− ω2e
ω2∞
x f(x) η2(x−1)2
, (26)
then, the corresponding optical metric yields that
dσ˜2=goptjk dxjdxk= n 2(x)
f(x)
η2
x2f(x)dx 2+d
ϕ2
. (27)
The determinant of above optical metric is:
detgoptxϕ =
x f(x)ω4
e −2η2ω2∞ω2e(x−1)2
η2ω4∞x f(x)2(x−1)4 (28)
where the metric function is given as
f(x) =α
x2−1
2x −ln(x)
+η
2(x−1)2
x
1−2q
2(x−1)
x
(29)
Now, we have
dσ˜
dϕ =n(x)
α2x2
f(x)
1/2
, (30)
hence we get differently which goes toα:
lim x→∞kg
dσ˜
We use straight line approximationr=b/ sinϕ, for the limitx→∞, then GBT stated as
lim x→∞
Z π+Θ
0
kgdσ˜
dϕ
|CRdϕ=π− lim
x→∞ Z π
0 Z x
b/ sinϕ
KdS, (32)
where
KdS = −3/2ωe 2η2q2
x2ω∞2 −3/2
ωe2αq2
x2ω∞2 +1/4 ωe2η2
x2ω∞2+3/8 ω∞2α
x2ω∞2+1/4 α
x2 +1/4 η2
x2−1/2 αq2
x2 − η2q2
x2 . (33)
Note that we only consider the first order terms. After simplification, we obtain the deflection angle in weak field limits as follows:
˜ α'3ωe
2η2q2
bω∞2 +2 η2q2
b +3
ωe2αq2
bω∞2 + αq2
b −1/2
ωe2η2
bω∞2 −1/2 η2
b −3/4
ωe2α
bω∞2−1/2 α
b (34)
The above results shows that the photon rays are moving in a medium of homogeneous plasma.
IV. NEW ASYMPTOTICALLY FLAT BLACK HOLES IN EINSTEIN-MAXWELL-DILATON THEORY
In this section, we analyzed the exact asymptotically flat charged hairy BH’s in the background of dilaton potential. There exist a literature related to this BH recently discussed by Astefanesei et al. [12]. We are interested in a action of of Einstein-Maxwell-dilaton theory (κ=8πGN):
I[gµν,Aµ,φ] = 1 2κ
Z
d4xp
−g
R−1 4e
γφF2−1 2∂µφ∂
µφ−V(φ)
(35)
where the gauge coupling and potential are functions of the dilaton. The corresponding equations of motion are
∇µ eγφFµν
=0 (36)
1 √
−g∂µ
p
−ggµν∂ νφ
−∂V ∂φ −
1 4γe
γφF2=0 (37)
Rµν− 1
2gµνR= 1 2 h
Tµνφ +T EM µν
i
(38)
whereTµνφ andTµνEMare the stress tensors of the matter fields and represented as follows
Tµνφ =∂µφ∂νφ−gµν
1 2(∂φ)
2+V(φ) , TEM µν =e
γφ
FµαF
·α ν −
1 4gµνF2
(39)
Now, we consider only asymptotically flat solutions. To implement this condition, we require
lim
x→1Ω(x)f(x) =1 (40)
on this account, we fix f0to obtain [12]
f(x) = η 2
ν
x2+2x 2−ν
ν−2 − ν
ν−2
+f1
xν+2
ν+2 −x
2+ x2−ν 2−ν+
ν2
ν2−4
+ Q
2η2
(1−p)ν2
x3−p+ν 3−p+ν +
x3−p−ν 3−p−ν −2
x3−p 3−p −
2ν2
(3−p) (3−p+ν) (3−p−ν)
+ P
2η4
(1+p)ν2
x3+p+ν 3+p+ν +
x3+p−ν 3+p−ν −2
x3+p
3+p −
2ν2
(3+p) (3+p+ν) (3+p−ν)
A. Solutions with a non-trivial dilaton potential (γ=1)
For this solution we consider the following scalar field potential [12]:
Ω(x) = x η2(x−1)2
, φ(x) =ln(x) (42)
ds2=Ω(x)
−f(x)dt2+ η 2dx2
x2f(x)+dθ 2+sin2
θdϕ2
(43)
Here we study only theγ = 1 case, which is smoothly connected with a solution ofN =4 supergravity which gives the solutions of metric function as follows:
A= Q
xdt+Pcosθdϕ (44)
V(φ) =α[2φ+φcosh(φ)−3 sinh(φ)] (45)
f(x) = η
2(x−1)2
x +
x 4−
1 4x −
1 2ln(x)
α+η
2(x−1)3 2x
η2P2−x−1Q2
(46)
Note that the dilaton field is vanishing at the boundary,x=1 and also the dilaton potential is vanishing whenα=0. At the boundaryx=1, one can find a asymptotically flat spacetime. After make a change of coordinates using:
Ω(x) =r2+O(r−4), (47)
which is given for, thex<1 black holes, by
x=1− 1 ηr+
1 2η2r2−
1 8η3r3+
1
27η5r5 (48)
The spacetime metric becomes asymptotically flat [12]
gtt=Ω(x)f(x) =1−
α+6η2 η2P2−Q2 12η3r +O(r
−2) (49)
grr−1=
x2f(x)
η2Ω(x)
dr
dx
2
=1− α+6η
2 η2P2−Q2
12η3r +O(r
−2). (50)
It is noted that the scalar field potential is regular everywhere, except at the spacetime singularities.
V. DEFLECTION ANGLE OF PHOTONS BY ASYMPTOTICALLY FLAT HAIRY BLACK HOLES IN EINSTEIN-MAXWELL-DILATON THEORY
Initially we find the Gaussian curvatureK of the optical spacetime, as
K = RicciScalar
2 , (51)
and
K = −6η
4p2+6Q2
η2−α −6η4p2+6Q2η2+16η3r−α
In weak field limits,
K =− α
12η3r3
+ α
2
192η6r4 +
−8η3r+αp2 16η2r4
− −8η 3r+
αQ2 16η4r4
+O(Q3,p3). (53)
For multiple images, we use the global theory (Gauss-Bonnet theorem) to relate with the local feature of the space-time such that Gaussian optical curvature.
In the weak field regions, the light ray follows a straight line approximation, so that we can use the condition of
r=b/ sinφat zero order.
˜
α=− lim
R→∞ Z π
0 Z R
b/ sinϕ
KdS. (54)
Now by using Eq. (18), the deflection angle of photon by exact asymptotically flat charged hairy black hole with a dilaton potential in weak field limit is found as:
˜
α = 3Q 2p2π
32b2 + ηp2
b − Q2 bη +
πQ2α 64b2η4+
α
6bη3+O(Q
3,p3). (55)
A. Deflection angle of photons in plasma medium by asymptotically flat hairy black holes in Einstein-Maxwell-dilaton theory
In this section, we analyze the effect of plasma medium on the photon lensing by asymptotically hairy black hole. The refractive index for hairy black hole is as follows [15],
n(x) =
s
1− ωe2 ω∞2
x f(x) η2(x−1)2
, (56)
then, the corresponding optical metric yields that
dσ˜2=goptjk dxjdxk= n 2(x)
f(x)
η2
x2f(x)dx 2+d
ϕ2
. (57)
The determinant of above optical metric is:
detgoptxϕ =
x f(x)ωe4−2η2ω∞2ωe2(x−1)2
η2ω∞4x f(x)2(x−1)4 . (58)
Now, by using Eq. 17 and 29, in weak field limit the Gaussian optical curvature stated as follows:
K = −ω∞
2 ω
e2−2ω∞2Q2 4r3η(ωe2−ω∞2)2 +
ω∞2 3Q2ωe2ω∞2+3Q2ω∞4+2ηrωe4−6ηrωe2ω∞2+4ηrω∞4p2 8r4(ωe2−ω∞2)3
+ α (Ψ)ω∞ 2
48η4r5(ωe2−ω∞2)4
, (59)
where
Ψ=9Q2ηp2ωe4ω∞2+9Q2ηp2ωe2ω∞4−3η2p2rωe4ω∞2+3η2p2rω∞6+3Q2rωe4ω∞2−3Q2rω∞6+2ηr2ωe6
−8ηr2ωe4ω∞2+10ηr2ωe2ω∞4−4ηr2ω∞6.(60)
Now, we have
dσ˜
dϕ =n(x)
α2x2
f(x)
1/2
, (61)
hence we get differently which goes toα:
lim x→∞kg
dσ˜
We use straight line approximationr=b/ sinϕ, for the limitx→∞, then GBT stated as
lim x→∞
Z π+˜α
0
kgdσ˜
dϕ
|CRdϕ=π−xlim→∞ Z π
0 Z x
b/ sinϕ
KdS. (63)
After simplification, we obtain
˜
α = 21πQ 2p2ω
e2 64ω∞2b2 +
3Q2p2π 32b2 +
ηp2ωe2
ω∞2b +
ηp2
b −
7π αp2ωe2 128ω∞2b2η2−
π αp2 64b2η2−
Q2ωe2
ω∞2bη
− Q
2
bη +
7πQ2α ωe2 128ω∞2η4b2+
πQ2α 64b2η4+1/6
α ωe2
ω∞2η3b+1/6 α
bη3 (64)
The proposed deflection angle shows that the photon rays are moving in a medium of homogeneous plasma.
VI. CONCLUSION
In this paper, we obtain the deflection angle of photon to the spherically symmetric and asymptotically flat space-time of hairy BH with Einstein-Maxwell-dilaton system in weak field limit. To this end, we set the photon rays on the equatorial plane in the black hole spacetime. For this purpose, we have used the GBT and obtain the deflection angle of photon by integrating a domain outside the impact parameter. Moreover, we also found the deflection angle of photon by asymptotically flat hairy BH in plasma medium. We examined that the proposed deflection angle shows that gravitational lensing can be affected from the hair of the black hole and it is a global effect as well as a valuable tool to study the nature of singularities of black holes.
Acknowledgments
This work is supported by Comisi ´on Nacional de Ciencias y Tecnolog´ıa of Chile (CONICYT) through FONDECYT Grant No 3170035 (A. ¨O.).
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