#### Deflection of light by black holes and massless wormholes in massive gravity

Eur. Phys. J. C
Deflection of light by black holes and massless wormholes in massive gravity
Kimet Jusufi 1 2
Nayan Sarkar 0
Farook Rahaman 0
Ayan Banerjee 0 3
Sudan Hansraj 3
0 Department of Mathematics, Jadavpur University , Kolkata 700032 , India
1 Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University , Arhimedova 3, 1000 Skopje , Macedonia
2 Physics Department, State University of Tetovo , Ilinden Street nn, 1200, Tetovo , Macedonia
3 Astrophysics and Cosmology Research Unit, University of KwaZulu Natal , Private Bag X54001, Durban 4000 , South Africa
Weak gravitational lensing by black holes and wormholes in the context of massive gravity (Bebronne and Tinyakov, JHEP 0904:100, 2009) theory is studied. The particular solution examined is characterized by two integration constants, the mass M and an extra parameter S namely 'scalar charge'. These black hole reduce to the standard Schwarzschild black hole solutions when the scalar charge is zero and the mass is positive. In addition, a parameter λ in the metric characterizes so-called 'hair'. The geodesic equations are used to examine the behavior of the deflection angle in four relevant cases of the parameter λ. Then, by introducing a simple coordinate transformation r λ = S + v2 into the black hole metric, we were able to find a massless wormhole solution of Einstein-Rosen (ER) (Einstein and Rosen, Phys Rev 43:73, 1935) type with scalar charge S. The programme is then repeated in terms of the Gauss-Bonnet theorem in the weak field limit after a method is established to deal with the angle of deflection using different domains of integration depending on the parameter λ. In particular, we have found new analytical results corresponding to four special cases which generalize the well known deflection angles reported in the literature. Finally, we have established the time delay problem in the spacetime of black holes and wormholes, respectively.
1 Introduction
At present independent observations have confirmed that
the universe is currently undergoing a phase of accelerated
expansion. The observed late time acceleration has been
confirmed by data from type Ia Supernovae [
3,4
], anisotropy in
the Cosmic Microwave Background radiation [5] and SDSS
[
6,7
]. To describe the present expansion scenario several
models have been proposed so far. Two broad approaches
have emerged to account for the observed accelerated
expansion. The first is the dark energy proposal with the assumption
that nearly 70 % of the total energy-density in the universe
may be in the form of negative pressure fluid with the
associated density parameter DE of the order of DE ∼ 0.70. One
of the simplest candidates generating the dark energy is the
cosmological constant, but its characterization has two
wellknown problems, i.e., fine-tuning and cosmic coincidence.
Moreover, there is a severe discrepancy in the observed value
of the cosmological constant in contrast with the value
predicted by quantum cosmology. Ellis et al. [
8,9
] proposed
the use of the trace-free Einstein equations which effectively
treats the cosmological constant as a mere constant of
integration. This idea was first proposed by Weinberg [
10
] and
has also gone by the name unimodular gravity [
11–13
].
Several alternative models have been suggested to incorporate
the cosmological constant problems, namely, quintessence
[
14
], tachyon field [
15
], phantom model [
16
] and k-essence
[
17
] that also predict cosmic expansion amongst others.
A second approach is that of modified gravity as an
alternative to appealing to exotic matter distributions such as dark
energy or dark matter. Generalizations of general
relativity (GR) appear to avoid introducing matter with
nonstandard physical properties and to solve the singularity problem.
Modified or extended theories of gravity often require higher
dimensional spacetimes. This in itself is no shortcoming as
historically a number of higher dimensional theories have
appeared such as Kaluza–Klein theory and the brane world
concept. It is debatable whether gravitational interactions are
necessarily four dimensional. Indeed if string theory or its
generalization M-theory for quantum effects is to be
consistent with a theory of gravitation then higher dimensions are
necessary. The Einstein–Hilbert action may be modified to
include non-linear geometric terms. One of these proposals
is the f (R) theory [
18–22
], as a simple modification of the
Einstein–Hilbert Lagrangian density by a general function of
the Ricci scalar R. While f (R) theory does have the capacity
to explain the late-time expansion of the universe, the theory
does possess some difficulties in that ghost terms are
manifest in the presence of fourth order derivatives. Of late f (R)
theory has been shown to be equivalent to the Brans–Dicke
scalar tensor theory. A more natural generalization of
general relativity is the Lovelock [
23,24
] lagrangian postulate
in which the action is composed of terms quadratic in the
Ricci scalar, Ricci tensor and the Riemann tensor.
Remarkably this higher curvature theory generates up to second order
derivative terms in the equations of motion and is accordingly
ghost-free. To zeroth order the Lovelock polynomial is
identical to the cosmological constant, to first order the Einstein
action is regained while to second order the action is known
as the Gauss–Bonnet action.
In this paper, we consider massive gravity as a
modification of GR. These include massive gravitons and have
attracted much attention recently. In addition the theory
incorporates massive spin-2 particles which have two degrees
of freedom. This theory has a rich phenomenology, such as
explaining the accelerated expansion of the universe without
invoking dark energy. Additionally, the resolution of the
hierarchy problem and brane-world gravity scenarios also
generate arguments for the existence of massive modes; hence
massive gravity as in the Refs. [
25,26
] emerged. In this
direction the pioneer work was done by Fierz and Pauli [27]
in the context of linear theory. It is worthwhile to
mention that the original theory suffered from the existence of
vDVZ (van Dam–Veltman–Zakharov) discontinuity. Later,
Vainshtein introduced a well known mechanism [
30–32
] to
resolve the long standing problem of the vDVZ discontinuity
by considering a nonlinear framework but this raised another
problem of the Fierz and Pauli theory which is known as
the Boulware–Deser (BD) [
28,29
] ghost instability at the
non-linear level. In order to avoid such instability, de Rham,
Gabadadze and Tolley (dRGT) [
33,34
] have proposed a new
massive gravity theory with an extension of the Fierz-Pauli
theory. Recently other versions of massive gravity have been
proposed, namely, new massive gravity [
35
] and bi-gravity
[
36
].
Massive gravity theories are also studied in the
astrophysical context. Black hole solutions and their thermodynamical
properties have been analyzed in dRGT massive gravity [
37–
40
]. Katsuragawa et al. [41] devised a neutron star model that
demonstrated that massive gravity dynamics deviates only
slightly from GR. It was recently proposed by Bebronne and
Tinyakov [
2
] that vacuum spherically-symmetric solutions
do exist in massive gravity. The black hole solution depends
on the mass M and an extra parameter S which is referred
to as the ‘scalar charge’. Additionally, in Ref. [
42
] the
validity of the laws of thermodynamics in massive gravity have
been checked for the same black hole solutions. A number
of articles on black holes in massive gravity have appeared
recently; some solutions have been reported in [
43–47
].
It is important to understand the deflection of light in the
presence of a mass distribution. This becomes an important
and effective tool for probing a number of interesting
phenomena. As early as 1919 Eddington [
48
] studied the weak
gravitational lensing of the Schwarzschild spacetime. This
seminal work initiated the study of gravitational lensing (GL)
theory [
49–52
]. It is also known that in the vicinity of
massive compact objects (such as neutron stars or black holes)
electromagnetic radiation is generated. The importance of
examining light deflection in the weak field limit lies in the
ability to probe large-scale structures, as well as exotic
matter, wormholes, naked singularity, etc (The reader is referred
to the more detailed review in [
53–57
]). It is thus imperative
to investigate the GL effect of black holes in massive
gravity and to search for their possible observational signatures
in the weak field limit. In contrast to the lensing situation
already studied in the literature, we apply the higher
curvature Gauss–Bonnet theorem (GTB) [86] to calculate the
deflection angle.
It is well known that the deflection of light (i.e.
Gravitational lensing) is now one of useful tools to search not
only for dark and massive objects, but also wormholes. In
recently past, several attempts have been made to calculate
the elliptical integral by Virbhadra and Ellis [
58,59
]. Soon
after the Eiroa et al. have studited Riessner–Nordstrom black
hole lensing in strong gravitational region [60]. The black
hole gravitational lenses have been widely demonstrated in
[
61–72
]. In addition, after the pioneer works by Kim and
Cho [79], the gravitational lensing by a negative Arnowitt–
Deser–Misner (ADM) mass was studied in [
80–85
]. As a
consequence, several forms of the deflection angle by the
Ellis wormhole (particular example of the Morris–Thorne
traversable wormhole) have been studied in the strong field
limit [
73–78
]. The computation of the deflection angle in
the weak field limit for spherically symmetric static
spacetimes may be accomplished through a simple algorithm. Very
recently, Werner [
87
] extended and applied the optical
geometry to the case of stationary black holes. Further, under some
physically realistic assumptions GBT was used in studies of
various astrophysical objects, such as Ellis wormholes by
Jusufi [
88
], wormholes in Einstein–Maxwell-dilaton theory
[
89–91
], black holes with topological defects and deflection
angle for finite distance by Ishihara et al. [
88,98–103
]. In Ref.
[
105
], the authors have studied the strong deflection limit
from black holes and explored the role of the scalar charge
in massive gravity. In the present work, we aim to investigate
the deflection angle by black holes and charged wormholes
in massive gravity in the weak limit approximation using the
optical geometry as well as the geodesic method.
This paper is structured as follows. In Sect. 2 we review the
black hole solution in massive gravity. In Sect. 3 we consider
the geodesic equations in massive gravity theory and analyse
the deflection angle in four special cases. In Sect. 4 we
consider the same problem viewed in terms of the Gauss–Bonnet
theorem. In Sect. 5 the time delay problem is considered. In
Sect. 6 we shall consider deflection of light by wormholes.
By applying the GBT of gravitational lensing theory to the
optical geometry, we calculate the deflection angle produced
by charged and massless wormhole in massive gravity. In
Sect. 7 we consider the time delay problem in the context of
wormholes. Finally in Sect. 8 we comment on our results.
2 Black hole solution in massive gravity
We commence with a brief discussion about black holes in
massive gravity. An action of a four-dimensional massive
gravity model which is used in this paper, is given by:
S =
d4x √−g
R
16π +
4F (X, W i j ) ,
where R is as usual the scalar curvature and F is a function
of the scalar fields φi and φ0, which are minimally coupled
to gravity. These scalar fields play the crucial role for
spontaneously breaking Lorentz symmetry. Actually, this action
in massive gravity can be treated as the low-energy effective
theory below the ultraviolet cutoff . The value of is of
the order of m M pl , where m is the graviton mass and M pl
is the Plank mass. The function F which depends on two
particular combinations of the derivatives of the Goldstone
fields, X and W i j , are defined as
X =
W i j =
∂0φi ∂0φi
4
,
∂μφi ∂μφi
4
−
∂μφi ∂μφ0∂ν φ j ∂ν φ0
,
where the constant has the dimension of mass. From this,
one can arrive at the new type of black hole solution, namely,
massive gravity black hole (for detailed derivation can be
found in [
2
]). The ansatz for the static spherically symmetric
black hole solutions can be written in the following form:
dr 2
ds2 = − f (r )dt 2 + f (r ) + r 2 dθ 2 + sin2 θ dϕ2 ,
2xi ,
(5)
,
−1
dr 2
(6)
where the metric function with the scalar fields are assumed
in the following form
2M
f (r ) = 1− r
− rSλ , φ0 =
2(t +h(r )) and φi =
with
h(r ) = ±
dr
f (r )
1 − f (r )
Sλ(λ − 1) 1
12m2 r λ+2 + 1
where M accounts for the gravitational mass of the body
and λ is a parameter of the model which depends on the
scalar charge S. The presence of the scalar charge represents
a modification of the Einstein’s gravitational theory. When
S = 0 the usual Schwarzschild potential is regained.
However, at large distances with positive M the solution (2) has
an attractive behavior, whereas with negative M the Newton
potential is repulsive at large distances and attractive near the
horizon. Our goal is to study the when M > 0 and S > 0,
so that black hole has attractive gravitational potential at all
distances and the size of the event horizon is larger than 2M .
Another reason for considering such a solution is that the
asymptotic behaviour of the gravitational potential is
Newtonian with finite total energy, featuring an asymptotic behavior
slower than 1/r and generically of the form 1/r λ. Therefore,
the attraction the modified black hole solution exhibits is
stronger than that of the usual Schwarzschild black hole due
to the presence of “hair λ”.
(1)
(2)
(3)
(4)
3 Geodesic equations
Let us turn our attention to the problem of the deflection angle
in massive gravity theory in the framework of the geodesic
equations. Recently a new black hole solution in the context
of the massive gravity theory was found to be [
2
]
ds2 = −
This solution does not describe asymptotically flat space
in the case λ < 0. For λ = −2 the metric coincides with
the familiar Schwarzschild de-Sitter spacetime consisting of
a constant stress energy tensor in the form of the (positive)
cosmological constant [
106
]. In the present paper we shall
focus on the case λ ≥ 1. Immediately it may be recognized
that the case λ = 2 corresponds with the Reissner–Nordström
solution for the exterior of a charged perfect fluid sphere.
Applying the variational principle to the metric (6) we find
the Lagrangian
2 L =
2M S
1 − r (s) − r λ(s)
closest approach. Next, we can evaluate the constant l from
Eq. (14) in leading order terms as
l =
4M S
r02
dϕ
1
1
2M
,
where
A1 = 2M 2 S + 4M 2r0 + 4M S2
+ 4M Sr0 + 2Mr02 + S2r0 + Sr02,
C1 = 2Mr02 + 4M S + Sr0 + r0 .
2
It is well known that the solution to the above equation in
the weak limit can be written as follows [
107
]
It is worth noting that L is +1, 0, and −1, for timelike,
null, and spacelike geodesics, respectively. Taking the
equatorial plane θ = π/2, the spacetime symmetries implies two
constants of motion, namely l and E , given as follows
∂L
pϕ = ∂ϕ˙ = r (s)2ϕ˙ = l,
∂L
pt = − ∂t˙ =
2M S
1 − r (s) − r λ(s)
t˙ = E .
To proceed further we need to introduce a new variable,
say u(ϕ), which is can be given in terms of the radial
coordinate as r = 1/u(ϕ) which yields the identity
r˙ dr 1 du
ϕ = dϕ = − u2 dϕ
˙
After some algebraic manipulations one can show that the
following relation can be recovered
t˙2(s) t˙2(s) M u + Suλ t˙2(s)
− ϕ˙2(s) + 2 ϕ˙2(s) ϕ˙2(s)
du 2
where αˆ is the deflection angle which should be calculated.
Moreover, from the above equation the deflection angle is
shown to be calculated as follows [
107
]
αˆ = 2|ϕu=1/b − ϕu=0| − π.
Using this relation, from Eq. (19) the deflection angle is
found to be
We proceed by considering four special cases for different
values of the parameter λ in the metric (6).
αˆ λ=1
To begin, we shall consider the affine parameter along the
light rays to be E = 1, therefore one should find the following
condition umax = 1/r0, where r0 gives the distance of the
Furthermore if we let S = 0, we find the Schwarzschild
deflection angle with second-order correction terms which is
in perfect agreement with [
104
].
3.2 Case λ = 2
Our second case will be λ = 2. Going through the same
procedure as in the last example the constant l is found to be
l =
4M S
r03
du 2
,
where
The differential equation takes the form
1
where
Finally, in our last case we let λ = 4, it follows
l =
4M S
r05
S 2M
+ r 4 + r0 + 1 r0.
0
We find the following differential equation
(36)
(37)
(38)
(39)
4M
M2
r0 + r02
M Q2
− r03
Fig. 1 We plot the deflection angle as a function of x = r0/2M. In the
first plot we have chosen M = 1 and S = 0.8. One can observe that
with the increase of λ the deflection angle decreases for fixed valued of
M and the scalar charge being positive i.e. S > 0
where
A4 = 2M Sr04 + Sr05 + 4M S2 + S2r0,
B4 = 4M 2r 4
0 + 2Mr05 + 8M 2 S + 2M Sr0,
C4 = 2Mr04 + 4M S + Sr0 + r0 .
5
Expanding in Taylor series and integrating we derive the
expression
αˆ λ=4
4M
r0
M S
+ r05
In Figs. (1 and 2) the deflection angle for different
parametric values are plotted as a function of x = r0/2M . From
Fig. 1, one may observe that deflection angle is monotonic
decreasing for fixed valued of M with the increase of λ when
the scalar charge being positive i.e. S > 0. In Fig. 2, we show
that deflection angle is monotonic increasing when scalar
charge being negative i.e. S < 0.
In this subsection we consider null geodesics deflected by a
black hole in massive gravity models. We start by considering
the optical metric from spacetime metric (6), by choosing
the null geodesic equations ds2 = 0. In the equatorial plane
θ = π/2 we find
With the help of Eq. (54) the optical Gaussian curvature may
be expressed as (for further review see [
86
])
lim
lim κ(γR ) = R→∞
R→∞
∇γ˙R γ˙R ,
K =
Theorem Let SR be a non-singular region with boundary
∂SR = γgop ∪ γR , and let K and κ be the Gaussian
optical curvature and the geodesic curvature, respectively. Then
GBT reads [
86
]
in which θi are the exterior angles at the i th vertex. In our
setup, however, the Euler characteristic is χ (SR ) = 1 due to
the fact that we consider a non-singular domain outside of
the light ray. It is worth noting that for a singular domain we
have χ (SR ) = 0.
Furthermore, for computing the deflection angle of light,
we need first to compute the geodesic curvature in terms of
the following relation
κ = gop
∇γ˙ γ˙ , γ¨ .
In doing so we should take into account the unit speed
condition which is stated as follows gop(γ˙ , γ˙ ) = 1, with γ¨
being the unit acceleration vector. Next, if we simply allow
R → ∞, one can show that our two jump angles (θO, θS )
yield π/2. Put it differently, if we take the total sum of our
jump angles at S and O, we find θO + θS → π [
86
]. It
follows from the simple geometry that κ(γgop ) = 0 due to
the simple fact that γgop is a geodesic. Hence we are left with
the following relation
κ(γR ) = |∇γ˙R γ˙R |,
in which γR := r (ϕ) = R = constant. In this way, one is
left with the following non-zero radial part
∇γ˙R γ˙R
r
= γ˙Rϕ ∂ϕ γ˙ Rr + ˜ ϕrϕ γ˙Rϕ 2 ,
note that ˜ ϕrϕ is the Christoffel symbol associated with the
optical metric geometry. While is clear that the first term in
this equation must vanish, we can calculate the second term
ϕ ϕ
via the condition g˜ϕϕ γ˙R γ˙R = 1. Finally we find
(58)
(60)
(61)
(62)
But for very large radial distance Eq. (53), suggest that
R
2M S
1 − R − Rλ
⎤
1/2 ⎥⎦ dϕ
(63)
(64)
(66)
(67)
(68)
π ∞
0 rγ
where the surface element is given by d A = √det gop dr dϕ.
It is clear now that we should integrate over the domain S∞
to find the deflection angle. This the deflection angle is found
to be
αˆ G B = −
K
det gop dr dϕ.
One can now compute the deflection angle by choosing
the light ray as r (ϕ) = b/ sin ϕ. However, this equation
corresponds to the straight-line approximation and gives the
correct result only for the linear terms in the deflection angle. In
this paper, we will make use of the following choice for the
light ray which is a solution of our geodesic equation (13):
1
rγ =
sin (ϕ)
b
1 M (3 + cos (2 ϕ))
+ 2 b2
1 M2 (37 sin (ϕ) + 30 (π − 2 ϕ) cos (ϕ)−3 sin (3 ϕ))
+ 16 b
Let us now elaborate on the following special cases:
Let us first calculate the Gaussian optical curvature from Eq.
(58) in the case when λ = 1. One can easily find that
αˆ λG=B1 = −
π ∞
0 rγ
and expanding in a Taylor series the previous equation results
in the expression
Using the above result for the deflection angle we find
On the other hand we can use the relation (15) to express
the last result in terms of the minimal distance r0 in terms of
the impact parameter
1 1
b = r0
M S
1 − r0 − 2r0 + · · ·
Consequently the deflection angle takes the form
3π
1 − 16
Thus we have shown that by modifying the integration
domain our result is in perfect agreement up to the second
order in M , and agrees only in the linear term in S. In order to
find the exact result including the second order terms in S we
have to modify the equation for the light ray (65). However
this goes beyond the scope of this paper.
4.2.2 λ = 2
αˆ λ=2 = −
π ∞
0 rγ
S2
+2 r 6
Let us substitute this equation into Eq. (66) then we find that
the deflection angle is given in terms of the following integral
Let us substitute this equation into Eq. (66) then we find that
the deflection angle is given in terms of the following integral
π ∞
0 rγ
M S S2
+ 18 r 7 + 6 r 10
M M 2 S
−2 r 3 + 3 r 4 − 10 r 6
M
−2 r 3 +
3 M 2 − 3 S
r 4
M S
+ 6 r 5
where
3M
= r dr 1 + r +
As already noted, the disagreement in the last two terms is
to be expected due to the integration domain. Finally,
neglecting these terms and letting S = −Q2, if we expand (25) in
series form the last result we recover Eq. (34) up to the second
order terms in M and Q.
Let us substitute this equation into Eq. (66) then we find that
the deflection angle is given in terms of the following integral
π ∞
0 rγ
S 15 S2
− 6 r 5 + 4 r 8
M 2 M M S
3 r 4 − 2 r 3 + 11 r 6
det gop dr dϕ.
3M
= r dr 1 + r +
+
where
where
Or, after we use Eq. (44) the deflection angle in terms of
the distance of the closest approach reads
4M
r0
15π
4
.
We analyze here the time delay due to the massive
gravitational field of the black hole solution. Suppose that two
photons emitted at the same time but follow different paths to
reach the observer. They will take two different times to reach
the observer and this time difference is called the time delay.
It is important to discuss the time delays between lensed
multiple images which is directly related to determining the
Hubble constant H0 and was first pointed out by Refsdal
[
108
].
We consider light propagation in a static spherically
symmetric spacetime given by the line element
ds2 = − A(r )dt 2 + B(r )dr 2 + C (r )(dθ 2 + sin2 θ dφ2).
(85)
The time delay of a light signal passing through the
gravitational field of this configuration is express as
T = 2
r1 ⎢
⎢
r0 ⎢⎢
⎣
⎡
r2 ⎢
⎢
r0 ⎢⎢
⎣
⎡
A(r) A2(r) C(r0)
B(r) − B(r)C(r) A(r0)
1
1
A(r) A2(r) C(r0)
B(r) − B(r)C(r) A(r0)
−
−
⎤
⎤
where r1 and r2 are distances of the observer and the source
from the configuration and r0 is the closest approach to the
configuration. With help of this algorithm we will calculate
the time delay due to the massive gravitational field of the
black hole. Let re and rs be distances of the observer (Earth)
and the source from the black hole respectively. Further r0 is
the closest approach to the black hole.
Therefore, the total time required for a light signal passing
through the gravitational field of the black hole to go from
the observer (Earth) to the source and back after reflection
from the source is given by the following equation [
107
].
Te,s = 2 [t (re, r0) + t (rs , r0)] ,
(87)
where
t (re, r0) =
and
t (rs , r0) =
re
r0
⎛
rs
r0
⎛
× ⎜⎜ 1 −
⎝
× ⎜⎜ 1 −
⎝
S 1 + rr0 + ( rr0 )2 + · · · ( rr0 )λ−1
rs
r0
2r (r + r0)r0λ−2
1
r02 − 2
1 − r 2
for our considered metric, given in the Eq. (6).
Considering the approximations (as re,rs , r0 >> 2M ) the
integrand of these expressions
−1
⎛
⎜⎜ 1 −
⎝
1 − Mr − rSλ r02 ⎟
r 2 ⎟⎠
M S
1 − r0 − rλ
0
2M
1 + r
is
⎡
Te ,s = 2 ⎣
and we may proceed to calculate the delay in time for the
cases corresponding to the values of λ = 1, 2, 3, and 4
respectively.
Now, the delay in time is express as the following equation
Te,s = Te,s −Te ,s .
(94)
Finally, we can estimate the time delay due to the
gravitational field of the black hole as
1
r02 − 2
1 − r 2
Therefore, the required delay in time corresponding to λ = 1
is
Te,s |λ=1 = 2 (2M + S)
⎛
× ln ⎝⎜⎜
+ (2M + S)
⎟⎠⎟
re
r0
r0
1
r02 − 2
1 − r 2
S 1 + rr0 + ( rr0 )2
2r (r + r0)r0
rs
r0
1
r02 − 2
1 − r 2
S 1 + rr0 + ( rr0 )2
2r (r + r0)r0
dr
2M
r
dr.
⎠
+ tan−1 ⎝
⎛ #rs2 − r02 ⎞ ⎤
r0 ⎠ ⎦ .
2M
r
S Mr0
+ r 3 + r (r + r0)
S Mr0
+ r 3 + r (r + r0)
dr
dr,
(98)
(99)
(100)
⎟⎟
⎠
⎟⎟
⎠
dr.
(96)
Therefore, the required delay in time corresponding to λ = 3
is
Therefore, the required delay in time corresponding to λ = 2
is
Te,s |λ=2
.
1
r02 − 2 ⎡ 2M
1 − r 2 ⎣ r
S Mr0
+ r 4 + r (r + r0)
S 1 + rr0 + ( rr0 )2 + ( rr0 )3 ⎤
⎦ dr
2r (r + r0)r02
rs
r0
Therefore, the required delay in time corresponding to
λ = 4 is
3S ⎡ #re2 − r0
2
+ r02 ⎣ re2
+
#rs2 − r02 ⎤
⎦ .
rs2
6 Light deflection by charged and massless Wormholes
in massive gravity
Let us set the mass to zero i.e. M = 0 and introduce the
following coordinate transformation r λ = S + v2 into the
metric (6), in that case we find the wormhole solution given
by the Einstein–Rosen (ER) bridge form
ds2 = − v2v+2 S dt 2 + λ2(S4+dvv22) λ−λ2 + (S + v2)2/λd 22.
The throat of the wormhole is located v = 0, with radius
1
Rthro. = S λ . This metric represents a massless wormhole
with scalar charge S, and as far as we know this is a new
metric. One can check by setting λ = 2 and S = −Q2 the
S
+ r 2
0
S
+ r 2
0
re − r0
re + r0
above metric takes the form of usual charged ER wormhole.
From now on, we shall consider v = r , in this way from the
metric (104) the Lagrangian yields
2 L =
.
On the other hand the wormhole optical metric reads
4(S + r 2)2/λdr 2
λ2r 2
+
(S + r 2)(2+λ)/λ
r 2
dϕ2,
2(S + r 2)1/λdr
, f (r ) =
λr r
The Gaussian optical curvature is found to be
(S + r 2)(2+λ)/2λ
6.1 Case λ = 1
Kλ=1
4
u2(S + u)2
du 2
dϕ
If we linearize Eq. (113) around S, and then consider the
equation which corresponds to straight line approximation
we are left with the following equation
du 2
dϕ
In this case when λ = 2 the Gaussian optical curvature yields
We Substitute this equation in the deflection angle led to
the following integral
αˆ λG=B2 = −
π ∞
0 rγ
3S
− r 4
det gop dr dϕ.
Considering a series expansion around S in Eq. (106) and
then take only the straight line approximation led to the
following differential equation
The Gaussian optical curvature in the case when λ = 3 is
found to be
Solving this equation we find the light ray equation
Using the above result for the deflection angle we find
3S
− r 4
3π S
det gop dr dϕ = 4b2 . (122)
with the following equation for the light ray
(121)
Kλ=4
αˆ λG=B4 = −
10S
− r 3 .
π ∞
0 rγ
d3u du
dϕ3 + 4 dϕ = 0,
2b
rγ = 1 − cos(2ϕ) .
b
rγ = √sin ϕ .
π ∞
0 sinbϕ
(123)
(124)
(125)
(126)
(127)
(128)
(129)
(130)
(131)
π ∞
0 rγ
On the other hand the light ray equation in this case reduces
to a nonlinear differential equation. However we can
approximate this equation from Eq. (106) as follows
αˆ λG=B3 = −
as above, we calculate the delay in time for the cases
corresponding to the values of λ = 1, 2, 3 and 4 respectively.
Thus we have shown that the deflection angle increases
with the increase of the parameter λ for a constant value of
the scalar charge S, which is shown in Fig. 3. From Fig. 3
we can see that for a fixed value of S = 0.5, the deflection
angle increases when increase of λ. It is a straightforward
calculation to show and check these results in terms of the
geodesic approach (Fig. 3).
7 Time delay due to massless wormhole in massive gravity
Here, we focus to estimate the time delay due to the massless
wormholes in the massive gravity. Using the same technique
Corresponding the value of λ = 3, time delay is found as
(133)
(134)
+
×
×
T |λ=3 = 2S ⎣
⎡ #(S + ve2) 23 − (S + v02) 23
(S + ve2) 31 (S + v02) 23
#(S + vs2) 23 − (S + v02) 23 ⎤
(S + vs2) 31 (S + v02) 23
⎦
+ (S +Sv02) 23 ⎡⎣ ⎛⎝ &''( ((SS ++ vvee22)) 3311 −+ ((SS ++ vv0022)) 3311 ⎞⎠
(S + v02) 31 + 2(S + ve2) 31
(S + ve2) 31
+ (S +Sv02) 23 ⎡⎣ ⎛⎝ ''&( ((SS ++ vvss22)) 3311 −+ ((SS ++ vv0022)) 3311 ⎞⎠
(S + v02) 31 + 2(S + vs2) 31
(S + vs2) 31
⎝
⎛ #
(S + v02) 41
(S + ve2) 21 − (S + v02) 21 ⎞
(S + v02) 41
⎠ ⎦
(S + vs2) 21 − (S + v02) 21 ⎞ ⎤
⎡ ⎛ #
(S + ve2) 21 − (S + v02) 21 ⎞
(S + ve2) 21
⎠
⎠
(136)
7.4 Case λ = 4
In this case we calculate the time delay as
8 Conclusions
In this paper we have studied the weak gravitational lensing
for a black hole and wormhole in massive gravity. The black
hole solution is governed by a parameter λ dependent further
on the mass M and scalar charge S. In the case of
vanishing S, the results of the standard Schwarzschild geometry
are recovered. By deforming the black hole solution in terms
of the following coordinate transformation r λ = S + v2 we
constructed a wormhole solution of ER type bridge which is
regular in the interval −∞ < v < ∞. The deflection angle
is then computed for four different values of the parameter
λ. The extension of this work via Gauss–Bonnet theorem is
nontrivial. First we derive a result showing how the Gaussian
optical curvature and deflection angle is to be computed. The
analysis is aided through the use Taylor series expansions.
The time delay function is also established and computed for
each of the four cases of λ of interest in this investigation.
Graphical plots indicate that for a fixed value of the mass and
positive scalar charge, the deflection angle decreases with
increasing λ, while for negative scalar charge, the deflection
angle increases with an increase in λ. Whereas in the
wormhole case we found that the deflection angle increases with
the increase of the parameter λ for a constant value of the
scalar charge S, provided S > 0.
Acknowledgements AB and FR are thankful to the authority of
Inter-University Centre for Astronomy and Astrophysics, Pune, India
for providing research facilities. FR and NS are also grateful to
DST-SERB (Grant No.: EMR/2016/000193) and CSIR (Grant No.:
09/096(0863)/2016-EMR-I), Govt. of India for financial support
respectively. AB wishes to thank the University of KwaZulu-Natal (ACRU)
for financial support.
T |λ=4
=
5S
2(S + v02) 43 ⎣
+ ⎝
⎛ #
(S + vs2) 21 − (S + v02) 21 ⎞ ⎤
(S + vs2) 21
⎠ ⎦
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
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