INTRODUCTION

Discovering dark energy as source of accelerating expansion of our universe [3], many efforts have gone into understanding its nature. One of the prime candidates is the cosmological constant $\mathrm{\Lambda }$ [4] which its effects on local phenomena such as null geodesics, time delay of light [5], and the perihelion precession [6] are studied. In these circumstance, local cases, many authors have investigated the effects of cosmological constant on the bending of light.

The argument for the non-influence of $\mathrm{\Lambda }$ was first discussed by Islam through investigating the null geodesic equation in a spherically symmetric spacetime [7] and has been re-affirmed by other authors, see e.g. [8]. However, in the last decade, Rindler and Ishak [1], by considering the intrinsic properties of the Schwarzschild-de Sitter space-time proposed a new method for calculating the deflection angle of light. Also, different aspects of their method such as integration of the gravitational potentials and Fermats principle have been studied [9]. Sultana in [10] and Heydari-Fard et al [11] have investigated light bending in Kerr-de Sitter and Reissner-Nordstrom-de Sitter space-time through Rindler-Ishak method. Also, this method has been applied to investigate Mannheim-Kazanas solution of conformal Weyl gravity [12].

Dyonic black holes enjoy the duality of electric/magnetic charges and possibly mass/dual mass [13]. In [14] is shown that two constants of a Taub-NUT system can be interpreted as a gravitating dyon with both ordinary mass and its dual where role of Nut charge is the mass duality such as the duality between electric and magnetic charges in the U(1) Maxwell theory [15]. Dyonic black hole and its properties have been investigated in literature (see [16]). In this letter, we take into account bending of light around dyonic black holes in massive gravity theory to examine massive gravity effects on light deflection.

For static spherically symmetric space-time, the dyonic black hole in massive gravity, massive dyonic black hole, is given by [2].

where

where $m_0$ is the total mass of dyonic black holes, $m$ denotes the massive parameter, $c$, $c_1$ and $c_2$ are constants of model, $l^2=-3\mathrm{\Lambda }$ and $q_E$ and $q_M$ identified as electric and magnetic charges, respectively.

The standard approach for calculating the bending angle is [17]

where $r_0$ denotes the closest distance to the black hole. However, the space-time here is not asymptotically flat, and so we cannot use usual way to calculate the deflection angle of light around massive dyonic black hole. Surprisingly, the Rindler-Ishak method proposed in [1] gives new approach to calculate the deflection angle in an asymptotically non-flat space-time. Rindler and Ishak have shown that by considering the effects of cosmological constant on the geometry of space-time, one can obtain the contribution of $\mathrm{\Lambda }$ to the bending angle near massive celestial objects (for example see [20]). Using the Euler-Lagrange equations for null geodesics in equatorial plan, $\theta =\pi /2$, we obtain the following equation

where $u\equiv 1/r$. For $c=0$, and in the absence of electric and magnetic charges, we find the standard orbital equation for light bending in Schwarzschild-de Sitter space-time (see e.g. [18]).

The orbit that is usually considered as small perturbation of the undeflected straight line in flat space

So according to the standard orbital equation of deflection of light, we have two different approaches to consider differential equation (4): first approximation case (see [1], [18]), where the small perturbation of the undeflected straight line (5) substituted into the right-hand terms of (4) or solving it by using a perturbation method up to the third order and consider a solution as

where ${\mathsf{u}}_0=\frac{\sin \phi }{R}$ and corrections $\delta {\mathfrak{u}}_1,\delta {\mathfrak{u}}_2$ and $\delta {\mathfrak{u}}_3$ satisfy the following equations

here we use perturbation method for small effects of electric-magnetic charge and massive parameter $m$ on the deflection of light with respect to standard one.

Applying these approaches on equation (4) gives:

where ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are solutions of equation (4) for first approximation and perturbation method, respectively.

To obtain the one-sided deflection angle at the point where

$\phi \ll 1$, we obtain

According to the Rindler-Ishak method, one able to compute angle $\psi$ between photon orbit direction d and direction of $\phi =$ const. line, $\delta ,$ by the invariant formula (see Figure 1)

Fig. 1: The orbital map, light bending in the space-time of a black hole. The one-sided deflection angle is $\psi -\phi \equiv \epsilon$.

where $g_{ij}$ are the coefficients of the 2-metric on $\theta =\pi /2$ and $t=$ const. surface. Substituting $d=(dr,d\phi )$ and $\delta =(\delta r,0)$ in Eq. (9), gives

or equivalently

using equations (12) or (13) for $m\ll 1$

finally, by substituting in equations (12) and (13), we find their corresponding expressions for the total deflection angle

The deflection angle is modified by new terms containing the massive parameter and cosmological constant in both equations (18) and (19).

Canceling out massive parameter and cosmological constant effects gives the same results for both approaches in equations (18) and (19) as

which equals to deflection light equation for charged Schwarzschild black holes.

The effect of the cosmological constant, electric charge and magnetic charge on deflection angle at small scales such as the solar system is expected to be negligible. So by using $1\to \mathrm{\infty }$ or $\mathrm{\Lambda }\approx 0$ and canceling out $q_{\mathrm{E}}$ and $q_{\mathrm{M}}$ from equations (18) and (19), we have

To find constraint on constant $m$, we use the observational data on light deflection by the sun, from long baseline radio interferometry [19]. According to this observational data, $\delta {\varphi }_{LD}=\delta {\varphi }_{LD}^{(GR)}(1+{\mathrm{\Delta }}_{LD})$ with ${\mathrm{\Delta }}_{LD}\le \le 0.0002\pm 0.0008$, where $\delta {\varphi }_{LD}^{(GR)}=1.7510$ arc sec. Assuming ${\mathrm{\Delta }}_{LD}$ as the geometric effects of the conformal terms, the observational results constrain the two last equations as follows

Assuming

where we set $c=1$. This selection leads us to find massive parameter $m$ as function of constants $c_1$ and $c_2$. Taking for $R$ and $m0$ values of the radius and mass of the sun, $R_{\odot }\approx 6.95\times {10}^8\text{m}$ and $M_{\odot }\approx 1.99\times 1030\text{kg}$, we find following constraints on $m$ according to our approaches, first approximation and perturbative method

In conclusion, in this letter, we have investigated the bending of light in the dyonic black holes in massive gravity.

Following Rindler-Ishak method, we have shown that when the geometry of the Schwarzschild-de Sitter space-time is taken into account, $\mathrm{\Lambda }$ indeed contributes to the light-bending in massive dyonic black holes. Also, using the observational data on bending of light by the sun and constant $c=1$ leads us to find strong constraint on massive constant $m$.

Generally, if we assume that both approaches give same result, one need to set $c_1=0$.

ACKNOWLEDGEMENT

The author thanks A. H. Fazlollahi for his helpful cooperation and comments.