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

Weak gravitational lensing by black holes and wormholes in the context of massive gravity (Bebronne and Tinyakov 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 $\lambda$ 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 $\lambda$. Then, by introducing a simple coordinate transformation $r^\lambda=S+v^2$ into the black hole metric, we were able to find a massless wormhole solution of Einstein-Rosen (ER) \cite{Einstein} 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 $\lambda$. 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.


I. 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], anisotropy in the Cosmic Microwave Background radiation [4] and SDSS [5,6]. 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 Ω DE of the order of Ω DE ∼ 0.70. One of the simplest candidates generating the dark energy is the cosmological constant, but its characterization has two well-known 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 [7,8] 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 [9] and has also gone by the name unimodular gravity [10][11][12]. Several alternative models have been suggested to incorporate the cosmological constant problems, namely, quintessence [13], tachyon field [14], phantom model [15] and k-essence [16] 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 This paper is structured as follows. In Sec. II we review the black hole solution in massive gravity. In Sec. III we consider the geodesic equations in massive gravity theory and analyse the deflection angle in four special cases. In Sec. IV we consider the same problem viewed in terms of the Gauss-Bonnet theorem. In Sec. V the time delay problem is considered. In Sec. VI 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 Sec. VII we consider the time delay problem in the context of wormholes. Finally in Sec. VIII we comment on our results.

II. 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: where R is as usual the scalar curvature and F is a function of the scalar fields φ i and φ 0 , 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 Λ. The value of Λ is of the order of mM 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 ij , are defined as where the constant Λ 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: where the metric function with the scalar fields are assumed in the following form with 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".

III. 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] 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 [50]. 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 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 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 identitẏ After some algebraic manipulations one can show that the following relation can be recovered −ṫ 2 (s) On the other hand, from Eqs. (8) and (9) we finḋ t(s) Hence we can recast Eq. (11) in terms of the impact parameter b as follows where b is defined as We proceed by considering four special cases for different values of the parameter λ in the metric (6).
To begin, we shall consider the affine parameter along the light rays to be E = 1, therefore one should find the following condition u max = 1/r 0 , where r 0 gives the distance of the closest approach. Next, we can evaluate the constant l from Eq. (14) in leading order terms as This leads us to the following differential equation where From the above equation we find where It is well known that the solution to the above equation in the weak limit can be written as follows [51] ∆ϕ = π +α, (22) whereα is the deflection angle which should be calculated. Moreover, from the above equation the deflection angle is shown to be calculated as follows [51] Using this relation, from Eq. (19) the deflection angle is found to bê Furthermore if we let S = 0, we find the Schwarzschild deflection angle with second-order correction terms which is in perfect agreement with [48].
Our second case will be λ = 2. Going through the same procedure as in the last example the constant l is found to be We obtain the following differential equation From the above equation we get that where Consequently the deflection angle has the form Now as a special case we can find the charged black hole deflection angle by simply letting S = −Q 2 . In that case we find the RN deflection anglê In a similar way, letting λ = 3 we found The differential equation takes the form du dϕ 2 1 where From the above equation we find where The deflection angle is given bŷ Finally, in our last case we let λ = 4, it follows We find the following differential equation where From the above equation we obtain where Expanding in Taylor series and integrating we derive the expression

A. Gaussian optical curvature
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 ds 2 = 0. In the equatorial plane θ = π/2 we find For the following considerations, it is convenient to introduce a radial Regge-Wheeler tortoise coordinate r , with a new function f (r ) as follows: This prescription allows us to write the line element of the optical metric in the form Using this static coordinates system, it is now clear that the equatorial plane in the optical metric is a surface of revolution when it is embedded, in R 3 . We utilized the following mathematical formulae to calculate the Gaussian curvature K, of the optical surface as With the help of Eq. (54) the optical Gaussian curvature may be expressed as (for further review see [42])

B. Deflection angle
Theorem: Let S R be a non-singular region with boundary ∂S R = γ g op ∪ γ R , and let K and κ be the Gaussian optical curvature and the geodesic curvature, respectively. Then GBT reads [42] in which θ i are the exterior angles at the i th vertex. In our setup, however, the Euler characteristic is χ(S R ) = 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(S R ) = 0.
Furthermore, for computing the deflection angle of light, we need first to compute the geodesic curvature in terms of the following relation In doing so we should take into account the unit speed condition which is stated as follows g op (γ,γ) = 1, withγ being the unit acceleration vector. Next, if we simply allow R → ∞, one can show that our two jump angles (θ O , θ S ) yield π/2. Put it differently, if we take the total sum of our jump angles at S and O, we find θ O + θ S → π [42]. It follows from the simple geometry that κ(γ g op ) = 0 due to the simple fact that γ g op is a geodesic. Hence we are left with the following relation in which γ R := r(ϕ) = R = constant. In this way, one is left with the following non-zero radial part note thatΓ r ϕϕ 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 via the conditiong ϕϕγ But for very large radial distance Eq. (53), suggest that provided that λ > 0. From GBT we find where the surface element is given by dA = det g op 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 bê 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): 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 Substituting into Eq. (66) generates the value of the deflection angle in terms of the integral In order to evaluate the above integral note that det g op dr = r dr 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 r 0 in terms of the impact parameter Consequently the deflection angle takes the form 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.

λ = 2
Let us substitute this equation into Eq. (66) then we find that the deflection angle is given in terms of the following integralα where det g op dr = r dr 1 + 3M The deflection angle in terms of the impact factor is found to bê 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 = −Q 2 , if we expand (25) in series form the last result we recover Eq. (34) up to the second order terms in M and Q. 3. λ = 3 Let us substitute this equation into Eq. (66) then we find that the deflection angle is given in terms of the following integralα The deflection angle has the form Hence in a similar way using Eq. (35) we recover Eq. (43) up to the second order in M, but in leading order in S. 4. λ = 4 Let us substitute this equation into Eq. (66) then we find that the deflection angle is given in terms of the following integralα where det g op dr = r dr 1 + 3M The deflection angle is given bŷ Or, after we use Eq. (44) the deflection angle in terms of the distance of the closest approach readŝ (84)

V. TIME DELAY
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 H 0 and was first pointed out by Refsdal [52].
The time delay of a light signal passing through the gravitational field of this configuration is express as where r 1 and r 2 are distances of the observer and the source from the configuration and r 0 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 r e and r s be distances of the observer (Earth) and the source from the black hole respectively. Further r 0 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 [51].
where t(r e , r 0 ) = and t(r s , r 0 ) = r s r 0 for our considered metric, given in the Eq. (6). Considering the approximations (as r e ,r s , r 0 >> 2M) the integrand of these expressions assume the form So, we can express the Eq. (85) as In the absence of gravitational field (M = S = 0) the time is Now, the delay in time is express as the following equation Finally, we can estimate the time delay due to the gravitational field of the black hole as 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.
Therefore, the required delay in time corresponding to λ = 1 is Therefore, the required delay in time corresponding to λ = 4 is

VI. 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 + v 2 into the metric (6), in that case we find the wormhole solution given by the Einstein-Rosen (ER) bridge form The throat of the wormhole is located v = 0, with radius R thro. = S 1 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 = −Q 2 the 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 Going through same procedure and introducing a new variable r = 1/u as in the black hole case, we find the following equation On the other hand the wormhole optical metric reads with dr = 2(S + r 2 ) 1/λ dr λr , f (r ) = (S + r 2 ) (2+λ)/2λ r .
The Gaussian optical curvature is found to be We shall consider the deflection angle by the spacetime metric (104) in terms of the GB method.
The Gaussian optical curvature from Eq. (109) in the case when λ = 1 reads Substituting this result into Eq. (66) generates the value of the deflection angle in terms of the integral In order to evaluate the above integral we need to find the equation for the light ray which can be found from Eq. (106) which yields 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 Solving this differential equation and using the condition u(0) = 0 and u(π/2) = 1/b we find Finally the light ray equation in terms of the old coordinate gives The deflection angle is found to bê In this case when λ = 2 the Gaussian optical curvature yields We Substitute this equation in the deflection angle led to the following integral Considering a series expansion around S in Eq. (106) and then take only the straight line approximation led to the following differential equation Solving this equation we find the light ray equation Using the above result for the deflection angle we find (122)

FIG. 2:
We plot the deflection angle as a function of the impact factor b. We have chosen S = 0.5. We see that with the increase of λ the deflection angle actually increases.

VII. 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 as above, we calculate the delay in time for the cases corresponding to the values of λ = 1, 2, 3 and 4 respectively.
Here, we find the time delay as Here S + v 2 = r 2 , hence S + v 2 e = r 2 e and S + v 2 0 = r 2 0 . In this case, we obtain the time delay as C. Case λ = 3 Corresponding the value of λ = 3, time delay is found as In this case we calculate the time delay as

VIII. 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 + v 2 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.