224 S.H. Hendi, M. Momennia / Physics Letters B 777 (2018) 222–234
T
M
μν
=−
1
2
g
μν
F
ρσ
F
ρσ
+2F
μλ
F
λ
ν
, (7)
T
YM
μν
=−
1
2
g
μν
F
(a)
ρσ
F
(a)ρσ
+2F
(a)
μλ
F
(a)λ
ν
. (8)
Also, the Faraday tensor F
μν
and the YM tensor F
(
a
)
μν
can be calculated via their related potentials with the following explicit forms
F
μν
=∇
μ
A
ν
−∇
ν
A
μ
, (9)
F
(a)
μν
=∇
μ
A
(a)
ν
−∇
ν
A
(a)
μ
+ f
(a)
(
b)(c)
A
(b)
μ
A
(c)
ν
. (10)
In order to obtain the static spherically symmetric black hole solutions in gravity’s rainbow, we take the following energy dependent
metric into account
ds
2
=−
k(r)
f
2
(ε)
dt
2
+
dr
2
g
2
(ε)k(r)
+
r
2
g
2
(ε)
dθ
2
+sin
2
θdϕ
2
,
(11)
where k(r) is an arbitrary function of radial coordinate which should be determined.
In
order to find the electromagnetic field, we use the following radial gauge potential ansatz
A
μ
=
h
(
r
)
f (ε)
δ
0
μ
, (12)
which satisfies the Maxwell field equations (5) with the following solution
dh(r)
dr
= E(r) =
q
r
2
, (13)
where q is an integration constant related to electric charge of the solutions.
To
solve the YM field, Eq. (6), we use the magnetic Wu–Yang ansatz of the gauge potential [12,16]. In addition, we consider the position
dependent generators t
(r)
, t
(θ)
, and t
(ϕ)
of the gauge group instead of the standard generators t
(1)
, t
(2)
, and t
(3)
. The relation between the
basis of SU(2) group and the standard basis are
t
(r)
= sin θ cos νϕt
(1)
+sin θ sin νϕt
(2)
+cos θ t
(3)
,
t
(θ)
= cos θ cos νϕt
(1)
+cos θ sin νϕt
(2)
−sin θ t
(3)
,
t
(ϕ)
=−sin νϕt
(1)
+cos νϕt
(2)
, (14)
and it is straightforward to show that the generators satisfy the following commutation relations
t
(r)
, t
(θ)
=
t
(ϕ)
,
t
(ϕ)
, t
(r)
=
t
(θ)
,
t
(θ)
, t
(ϕ)
=
t
(r)
. (15)
Since we are looking for the black hole solutions coupled to Wu–Yang monopole, we take the following gauge field characterized by
Wu–Yang ansatz [12,16]
A
(a)
t
= 0, A
(a)
r
= 0, A
(a)
θ
= δ
(a)
(
ϕ)
, A
(a)
ϕ
=−ν sin θδ
(a)
(θ)
, (16)
where the magnetic parameter ν is a non-vanishing integer. One can show that the YM field equations (6) are satisfied under the choice
of (16). Using the YM tensor field (10) and Wu–Yang ansatz (16), one finds that the only non-vanishing component of the YM field is
F
(r)
θ
ϕ
= ν sin θ. (17)
In order to obtain the metric function k(r), one may use the nonzero components of Eq. (4). Our calculations show that the nonzero
components of Eq. (4) can be written as
tt (rr) −component : Eq
tt
=r
2
rk
(r) +k(r) − 1
+
r
4
g
2
(ε)
+
ν
2
g
2
(ε) +q
2
f
2
(ε) = 0, (18)
θθ (φφ) −component : Eq
θθ
=r
3
r
2
k
(r) +k
(r)
+
r
4
g
2
(ε)
−
ν
2
g
2
(ε) −q
2
f
2
(ε) = 0, (19)
where prime denotes d/dr. It is worthwhile to mention that both Eq
tt
and Eq
θθ
equations are not independent because the metric function
k(r) is the only unknown function in these field equations. After some simplifications, one can find the following relation between Eqs. (18)
and
(19)
Eq
θθ
=
r
2
d
dr
−1
Eq
tt
.
Solving Eq. (18), one can find that the metric function k(r) has the following form
k(r) = 1 −
m
r
−
r
2
3g
2
(ε)
+
q
2
f
2
(ε)
r
2
+
ν
2
g
2
(ε)
r
2
, (20)
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