Deflection angle of photon through dark matter by black holes/wormholes using the Gauss-Bonnet theorem

Maxwell's fish eye has been known to be a perfect lens in optics. In this letter, using the Gibbons-Werner method, namely Gauss-Bonnet theorem on optical geometry of black hole, we extend the calculation of the weak gravitational lensing within the Maxwell's fisheye as a perfect lensing in medium composed of an isotropic refractive index that near-field information can be obtained from far-field distances. Moreover, these results provide an excellent tool to observe compact massive object by weak gravitational lensing within the dark matter medium and to understand the nature of the dark matter that may effect the gravitational waves. Moreover, we show that Gauss-Bonnet theorem is a global effect and this method can be used as a new tool on any optical geometry of compact objects in dark matter medium.

Black holes are an essential component of our universe and one of the most important findings in astrophysics is that when stars die they can collapse to extremely small objects. Black holes provide an important tool for probing and testing the fundamental laws of the universe. Recently gravitational waves from black holes and neutron star mergers have been detected [1]. On the other hand, black holes may hint the nature of the quantum gravity at small scales that change the area law of entropy. Observationally, the quantum gravity is far from understood, though theoretically has seen tremendous progress, in few years the event horizon telescope may provide some information about it [2][3][4].
In 1854 Maxwell presented the solution to a mathematical problem relating to the passage of rays through a sphere of variable refractive index and he noted that the potential existence of a medium of this kind retaining exceptional optical properties [5]. This is similar to reflection of the crystalline lens in fish. The absolute tool of physics especially in optics is Maxwell's fisheye (MFE), namely the condition that all light rays form circular trajectories. Nonetheless it was an remarkable effort to visualize that a lens whose refractive index enlarged radially towards a point could form perfect images, that make possible to observe a gradient refractive index material [7].
Then Luneburg discovered that the ray propagation is equivalent to ray propagation on a homogeneous sphere with unit radius and unit refractive index within geometrical optics [6]. This show that the imaging of variants which has been applied to microwave devices and the fisheye lenses in photography that form an extreme wide-angle image, almost hemispherical in coverage. In 2009 Leonhardt showed that MFE is also good for waves and enabling in producing super-resolution imaging with perfect lensing, which requires negative refractive index materials, that begin a hot debate and rich area of research to explore [8][9][10]. The MFE happens when all light rays arising from any point within converge at its conjugate which means that power released from a source can only be fully absorbed at its image point, namely perfect imaging. There has been a rapid increase in the importance of the perfect imaging in theoretical and experimental optics [11][12][13].
Fermat's principle said that light rays always follow extremal optical paths with a path length measured by the refractive index n geodesics. The formula of MFE indicates the interesting possibility that rays generates perfect image in a black hole region. The refractive index depends only on the distance ρ from the origin. The equation for the MFE is [11,12] where the light rays are bounded around spherical surface radius R.
Another useful tool of the astronomy and astrophysics is a gravitational lensing [14], that the light rays from distant stars and galaxies are deflected by a planet, a black hole, or dark matter [15,16] . The detection of the dark matter filaments [17] using the weak gravitational lensing (WGL) is very hot topic because it can help us to understand the large-scale structure in the Universe [18]. Furthermore, to build a sky maps (the index of refraction of the entire visible universe) there are ongoing research on the observation of cosmological weak lensing effect on temperature fluctuations in the Cosmic Microwave Background (CMB) [19]. Theoretical point of view, new methods have been proposed to calculated the deflection angle. One of them is the Gauss-Bonnet theorem (GBT) which is first proposed by Gibbons and Werner (GW), using the optical geometry [20][21][22]. The deflection angle is seen as topologically global effect that can be calculated by integrating the Gaussian curvature of the optical metric outwards from the light ray using the following equation: [20,21] Since a unique perspective of GW's paper on WGL by GBT, this method has been applied in various cases [23][24][25][26][27][28][29][30][31][32][33][34][35].
Main motivation of this letter is to shed light on unexpected features of spacetimes according to MFE and deriving the deflection angle using the Gauss-Bonnet theorem in weak limits. We show that this kind perfect tool can be used in general relativity within the geometry of light by matter of suitable distribution choice, hence creating an interesting light deflector. We also investigate the effect of varying parameters on the imaging resolution, which was not covered in previous studies.
Maxwell Fisheye Effect and Lensing Using Gauss-Bonnet theorem: In this section, first we shall first describe the black hole solution in a static and spherically symmetric spacetime. Then we apply the MFE within the GBT to calculate the WGL.
The Schwarzschild black hole spacetime reads with the metric functions Analysis of the geodesics equation, the ray equation is calculated by where b is the impact parameter of the unperturbed photon. Our universe is homogeneous and isotropic on large scales. Now we consider isotropic coordinates which are non-singular at the horizon and the time direction is a Killing vector. Moreover, time slices become Euclidean with a conformal factor and one can calculate the index of refraction, n, of light rays around the black hole. Other important feature of the isotropic coordinates is that they satisfy Landau's condition of the coordinate clock synchronization Using the following transformation the Schwarzschild black hole rewritten in isotropic coordinates (where ρ is isotropic radial coordinate) [35] with The metric becomes nonsingular at the horizon r = 2M . It can be also written in Fermat form of metric: and it can be approximated for large ρ M n ≈ 1 + 4M ρ .
Now the ray equation becomes ϕ = bdρ To discuss the gravitational lensing and extract information of MFE, the GBT will be used instead of the null geodesics method. The GBT is calculated using the negative Gauss curvature of the optical metric. For the Schwarzschild black hole spacetime, the optical metric in an equatorial plane reads using the transformation of radial optical distance dr = n(ρ)dρ, and f (r ) = ρn(ρ) in tortoise coordinate: Case 1: Let us start from the full MFE profile as refractive index [5,6,11,12]: and now we apply the MFE which is an ideal optical instrument on the black hole using the reflective index in Eq.11 to derive: where R is a constant. Then we can obtain the Gaussian optical curvature using where is everywhere negative that gives a universal property of black hole metrics [21]. We can approximate the result of the Gaussian optical curvature in leading orders: This result will be used to evaluate the deflection angle using a non-singular domain outside the light ray ( D r , with boundary ∂D r = γg ∪ C r ) [20]: where κ stands for the geodesic curvature κ = g (∇γγ,γ) and K is Gaussian optical curvature, with the exterior angles θ i =(θ O , θ S ) and the Euler characteristic number χ(D r ) = 1. At weak limits, ( ρ → ∞), θ O + θ S → π. Then the GBT reduces to (22) For the geodesics property of γg, geodesic curvature vanishes κ(γg) = 0, and we have with C r := ρ(ϕ) = r =const. The radial part is calculated: It is noted that first term is zero. The geodesic curvature in weak limit approximation is For very large radial distance we have also so that κ(C R )dt = d ϕ. Conveniently we choose the deflection line as r = b/ sin ϕ, to obtain the deflection angle by GBT:α where (30) After nontrivial calculation, we calculate the deflection angle of the Schwarzschild black hole in the leading order terms isα where R = (9 Mb 2 ) 1/3 . It means that Maxwell fisheye effect can be applied by the GBT which provide a global and even topological effect. This method as a quantitative tool can be used in any metrics. Case 2: Now we apply the different model of MFE that is the archetype of the absolute optical instrument with the profile [9] n = n 0 1 + z 2 .
As wave propagation in two-dimensional space described by the complex coordinate. One shows that it can be defined by stereographic projection [9]: After using above equations with the refractive index of black hole in Eq. 11, we derive theθ for our spacetime: with and it is plotted in Fig.1 To calculate the deflection angle using GBT method, we rewrite our refractive index that reduces to in this form We can approximate the above result in leading orders to find The Gaussian curvature of the optical metric approximating in leading orders is everywhere negative. The geodesics curvature in weak limit approximation is Next, for very large radial distance, we have Then using the same method we have and after a simple calculation we derive the deflection angle as follows:α At the z = √ 2 − 1, it reduces to exact Schwarzschild case. We plot theθ using the calculated z and Eq. 33 respect to α × b in Fig. 2 to see how it changes with deflection angle and impact parameter. It means that MFE is global effect similar to GBT and can be applied to any metrics.

Conclusions:
We have constructed a MFE on the black hole using the GBT. This has been achieved by constructing a isotropic optical metrics. In summary, we have investigate that the MFE is global similarly GBT on the WGL. We have showed it by using two different cases. In the first case, we have used the full MFE profile as refractive index. Then constructing the optical geometry on the isotropic coordinates, we have been able to use the GBT to obtain the deflection angle in weak limits. The deflection angle of the Schwarzschild black hole has been calculated correctly with the MFE embedded in the constant R. In the second case, we have used the different model of MFE and repeat the calculation for this model and showed that it gives similar effect. Here all the emitted light rays travel along the great circles, then meet at a antipodal point. Clearly, the agreement has been shown to arise from optics to general relativity where we exploit conformal-mapping to transfer the full Maxwell fisheye into a finite circle. Moreover, we have showed that the MFE can be applied by GBT which provides a global and even topological effect. This method as a quantitative tool can be used in any metrics.
The main message of this article is that, these results provide an excellent tool to observe compact massive object by WGL within the MFE, namely perfect imaging and to understand the nature of the medium that may effect the gravitational waves.
Needless to say, since the discovery of gravitational waves by LIGO, new discoveries are waiting to happen in future, we wish to be part of this cutting-edge research or become pioneer of a new field of physics.