Distinguishing rotating naked singularities from Kerr-like wormholes by their deflection angles of massive particles

We study the gravitational deflection of relativistic massive particles by Janis-Newman-Winicour (JNW) spacetimes (also known as a rotating source with a surface-like naked singularity), and a rotating Kerr-like wormholes. Based on the recent article [K. Jusufi, Phys. Rev. D 98, 064017 (2018)], we extend some of these results by exploring the effects of naked singularity and Kerr-like objects on the deflection of particles. We start by introducing coordinate transformation leading to an isotropic line element which gives the refraction index of light for the corresponding optical medias. On the other hand, the refraction index for massive particles is found by considering those particles as a de Broglie wave packets. To this end, we apply the Gauss-Bonnet theorem to the isotropic optical metrics to find the deflection angles. Our analysis shows that, in the case of the JNW spacetime the deflection angle is affected by the parameter $0<\gamma<1$, similarly, we find that the deformation parameter $\lambda$ affects the deflection angle in the case of Kerr-like wormholes. In addition to that, we presented a detailed analysis of the deflection angle by means of the Hamilton-Jacobi equation that lead to the same results. As a special case of our results the deflection angle of light is recovered. Finally, we point out that the deflection of particles by Kerr-like wormholes is stronger compared to JNW spacetime, in particular this difference can be used to shed some light from observational point of view in order to distinguish the two spacetimes.

Deflection of light ray in weak gravitational field was the first experimental proof of the theory of general relativity for the Schwarzschild spacetime since 1919 [1], and now it serves as one of the important tool of probing a number of interesting phenomena in astronomy and cosmology.
It is widely used for investigating astrophysical objects e.g., black holes, wormholes, super-dense neutron stars, naked singularity etc. This phenomenon has also been considered to study exoplanet detection from the dark energy, dark matter, and Hubble parameter measurements. Further, gravitational lensing (GL) provides an alternative approach to mass measurement, which leads to measure the mass of the white dwarf Stein 2051 B by using the Hubble space telescope [2]. Early works focussed on the weak field theory of gravitational lensing, which is based on a first order expansion of the small deflection angle [3], but the main disadvantages are when the lens is a very compact object (such as black hole) the weak field approximation is no longer valid.
An interesting method has immersed for computing the angle of light deflection by Gibbons and Werner [14,15]. Considering the source and receiver are located at an asymptotic region, they applied the Gauss-Bonnet theorem (GBT) to the optical metric of a lens that allows us to describe a light ray as a spatial curve. More excitingly, Werner [16] has extended this theory to the stationary metrics and computed the deflection angle by a Kerr black hole. This method allows for computing the deflection of light by global monopole and rotating cosmic string spacetime in the weak limit approximation (see [17,18] and references therein). Nowadays, scientific community has shown their interest in this phenomenon with great interest. Along this line of thinking, for example, Ishihara et al [19] have obtained the bending angle of light using GBT, when finite distance from a lens object to a light source and a receiver. Further this work was extended by considering stationary, axisymmetric and asymptotically flat spacetimes [20]. Jusufi andÖvgün [21] have studied the effect of the cosmological constant on the deflection angle by a rotating cosmic string. On the other hand wormhole [22][23][24], as well as black hole [25][26][27] solutions have been consider using GBT method.
It is worth noticing that in the field of gravity trajectory bending not only happens with light or massless particles, but also to particles with mass. For a review on massive particles whose trajectories can go through a bending process, we refer the reader to [28][29][30]. Thus, the important fact is trajectory bending due to the strong gravity near the black holes and the large velocity of the hypervelocity stars, needs a full relativistic treatment, which will differ from the calculations made using only Newtonian gravity. In recent years, the existence of strong gravitationally lensed supernovae has long been predicted [31][32][33][34]. Thus it is natural to expect that the neutrinos and cosmic rays emitted by these supernova and AGN are also lensed. Neutrino gravitational lensing by astrophysical objects have done in [35]. Besides their positions, they also estimated the depletion of the neutrino flux after crossing a massive object [35]. At the same time, Eiroa and Romero [36] have studied gravitational lensing of neutrinos by Schwarzschild black holes.
In a recent argument Crisnejo and Gallo [37] showed that how one can successfully study the GBT theorem in a plasma medium in a static and spherically symmetric gravitational field. Even in the GBT methode, they computed the deflection of massive particles with the same symmetries. On the other hand, Jusufi [38] introduced a method based on GBT to compute the deflection angle formassive particles in a rotating spacetime geometry such as the Kerr black hole [39] and Teo wormhole [40]. In doing so, he used an isotropic type metrics for a linearized rotating gravitational field, where the refractive index of the corresponding optical media can be obtained. Motivated by the works we study the gravitational deflection of massive particles by Janis-Newman-Winicour (JNW) spacetimes [41] (also known as a rotating source with a surface-like naked singularity), and a rotating Kerr-like wormholes [46]. Along this line of thought, recently different aspects of naked singularities and Kerr-like wormholes have been studied, including the iron line spectroscopy [42], accretion disc properties [43], spin precession of a test gyroscope [44,45], echoes of Kerr-like wormholes [46], shadow images [47] the strong and the weak deflection of light by Kerr-like wormholes [48,49]. In the present article, we study the deflection of massive particles by assuming the propagating particles as a de Broglie wave packets introduced in [50].
The paper is organized as follows. In Sect. II, we use a coordinate transformation to recover an isotropic metric form for the rotating naked singularities, after which we derive the refractive index of the corresponding optical media. In Sect. III, we consider the problem of the gravitational deflection of massive particles by Kerr-like wormholes. In Sect. IV, we present a detailed analyses of the deflection of massive particles in terms of the Hamilton-Jacobi equation. In Sect. V, we comment on our results. We use natural units, G = c =h = 1.

II. DEFLECTION OF MASSIVE PARTICLES BY ROTATING NAKED SINGULARITIES
In this section, we calculate the deflection angle of rotating naked singularities thereby we consider Janis-Newman-Winicour (JNW) spacetime. It is a solution of the Einstein equation described with the assumption of spherical symmetry, asymptotic flatness with the static massless scalar field Φ. The action associated with this spacetime [41,42] is the variation of action turns out the following field equations where the scalar field is given by The corresponding rotating spacetime metric [42] has the form where m = M/γ, M is the Arnowitt-Deser-Misner (ADM) mass, and Σ = r 2 + a 2 cos 2 θ, ω = a sin 2 θ, ∆ = r 2 + a 2 − 2mr, γ = M Note that a = J/M, where J is the spin angular momentum of the source. For an instant if we set γ = 1 and q = 0, one can recover the Kerr black hole solution and the larger root of ∆ = 0 provides the radius of the event horizon, It can be easily checked that for 0 < γ < 1, and q = 0, the metric (4) describes rotating naked singularities with a scalar curvature and diverges on the surface r is the surface-like singularity, where curvature invariants diverge and the spacetime is geodetically incomplete as well. In addition, the scalar curvature diverges where g tt = 0 and represent the presence of a curvature singularity at as the domain of variation of θ depends on the ratio γ 2 a 2 /m 2 . Also, it follows that in the transition from a black hole (γ = 1) to a naked singularity (γ < 1), the singularity is indeed naked, not the event horizon.

A. Deflection angle
Since, we are mainly interested in the weak limit, we shall simplify the problem by considering the deflection of particles in the equatorial plane (θ = π/2) with a linearized metric in a, which results in Now introducing a coordinate transformation via we deduce an isotropic metric with the line element given by where F 2 (ρ) is The velocity of light in this particular optical media can be found by setting ds 2 = 0, and using the definition v = |d ρ|/dt, where |d ρ| 2 = dρ 2 + ρ 2 dϕ 2 . With this information in hand, we can find the effective refractive index of the optical media by using n(ρ) = c/v(ρ) and c = 1, yielding We can, therefore, write down the optical metric in terms of the refractive index as follows By inverting the coordinate transformation (10), we obtain which shows leading terms in ρ r − m. The above relations holds for the deflection of light.
However, we are interested in studying the deflection of massive particles. One way to incorporate this is to consider the propagating massive particles as the de Broglie wave packets. In particular, following [38] and [50], we can argue that by using p =hk and H = E =hω, (note that we have temporarily introducedh) results where µ is the rest mass of the particle. We rearrange this equation to obtain a constant quantity of the given optical media In other words, this equation is a generalization of well-known result in wave-optics. For massive particles, in a given isotropic metric, the expression should be constant everywhere in the optical medium, i.e., λN = const. Thus the refractive index relation for massive particles is given by modifying the optical metric Furthermore, the energy of the particles measured at infinity is given by where w represents the relativistic velocity. The angular momentum can be written as where b is the impact parameter. Following the definition of the impact parameter, we can write which reduces to b in the case of light w = c = 1. Therefore, the quantity dϕ/dt, in case of massive particles is modified as follow yielding This result clearly indicates that the refractive index is modified due to the angular momentum parameter a.
Let us continue to calculate the Gaussian optical curvature using metric (19). Rewriting the optical metric (19), in terms of a new coordinates, thus the optical metric reads In terms of these coordinates, the Gaussian optical curvature K can be expressed as: modified refractive index as In particular, by using (15) we can rewrite the last equation, in terms of the old coordinate r, which in leading order terms results with With these results in hand, we can chose a non-singular region D R with boundary ∂D R = γg ∪ C R .
The GBT provides a connection between geometry (in a sense of optical curvature) and topology (in a sense of Euler characteristic number) stated as follow with κ is the geodesic curvature, and K is the Gaussian optical curvature. Now we can choose a non-singular domain with Euler characteristic number χ(D R ) = 1. The geodesic curvature κ, in the case of massive particles is defined by the relation but also keeping in mind an additional unit speed conditiong(γ,γ) = 1, withγ being the unit acceleration vector. The two corresponding jump angles in the limit Therefore the GBT now is simplified as follow By construction, there is a zero contribution from the geodesics i.e. κ(γg) = 0, therefore, we end up with a contribution only for the the curve C R , thus we can write To evaluate the above expression one can express the curve in terms of the radial coordinate C R := r(ϕ) = R = const, resulting with a nonzero contribution for the radial component the final result We see that the final result of the geodesic curvature is slightly modified, letting w = 1, we recover For an observer located at the coordinate R, we find Putting together these results, from the GBT, we find the following expression for the deflection where the surface element is given by dS = √g dr dϕ N 2 (r)rdrdϕ. In addition to that, from Eq.
(24) the following approximation holds N(r) 2 w 2 . This integral can easily be evaluated, Now we can consider three special cases: If γ = 1 we find the deflection angle of massive particles in Kerr black hole geometry. Furthermore letting w = 1 the deflection angle of light is recovered.

III. DEFLECTION OF MASSIVE PARTICLES BY KERR-LIKE WORMHOLES
We consider a Kerr-like wormhole spacetime, recently proposed in [46]. The spacetime metric can be written, in Boyer-Lindquist coordinates (t, r, θ, φ) as where Σ and∆ are expressed by It contains a family of parameters where a and M corresponds to the spin and the mass of wormhole, and λ 2 is the deformation parameter. A non vanishing λ 2 differs this metric from Kerr metric, we can recover the Kerr metric when λ 2 = 0. Although the throat of the wormhole can be easily obtained by equating the∆ to zero, which represents a special region that connects two different asymptotically flat regions. To simplify the problem, we can consider the deflection in the equatorial plane in a linearized Kerr-like wormhole metric in spin a. Such a metric can also describe the gravitational field around the rotating star or planet given by

A. Deflection angle
We follow the same GBT approach to calculate the deflection angle of Kerr-like wormhole as discussed in previous section. Here we use the following coordinate transformation after substitution the metric takes the following form The isotropic coordinate speed of light v(ρ) can be found from the relation Using n(ρ) = c/v(ρ) and c = 1, the last equation yields the effective refractive index for light in the Kerr gravitational field In this way the optical metric reads By inverting the coordinate relation (43), we find which suggests that in leading order terms ρ r − M(1 + λ 2 ). The refractive index relation for massive particles in that case reads We can apply this expression to our Kerr optical media which results with in which Eq. (43) has been used. Therefore, the quantity dϕ/dt, in the case of massive particles is modified as follow yielding This result clearly indicates that the refractive index is modified due to the angular momentum parameter a. In particular, in leading order terms, we find Going through the same procedure we find From the optical metric is quite easy to see that at a very large distance Now from the GBT we find the following expression for the deflection anglê This integral can easily be evaluated, yieldinĝ Setting w = 1, in the last equation we recover the deflection angle of light. In this section, we will derive the equations of motion of an uncharged test particle of mass µ and then we will calculate the deflection angle of the particle moving in the rotating naked singularity or the Kerr-like wormhole spacetime, given by (4) and (39), respectively. The geodesic equations may be derived using Hamilton-Jacobi approach, as we look for a function of the coordinates and the geodesic affine parameter σ, which is a solution of the Hamilton-Jacobi equation In general, the solution S of the Hamiltonian-Jacobi equation depends on four integration constants and satisfy the relations for the conjugate momenta which can be defined by the Lagrangian of the system L = 1 2 g µνẋ µẋν through the equations p µ − ∂L/∂ẋ µ = 0. Here, the dot represents differentiation with respect to the geodesic affine parameter σ. In order to involve the mass of the test particle, σ must be related to the proper time by τ = µσ, which turns the conjugate momenta into where u µ ≡ dx µ /dσ stands for the four-velocity of the particle with normalization condition g σρ u σ u ρ = 1. Therefore, one can obtain the Hamiltonian which does not depend explicitly on geodesic affine parameter σ and is a constant of motion. By taking zero or positive values of µ 2 , the Hamiltonian (63) can be used to give null or timelike geodesics, respectively. The corresponding Hamilton-Jacobi equation is where S is the Jacobi function (59). From the symmetries one can obtain two constants of motion corresponding to conservation of energy, E, and angular momentum with respect to the rotational symmetry axis of the naked singularity or Kerr-like wormhole, J. Therefore, we have p t = g ttṫ + g tφφ = E, It follows that for the time and azimuthal coordinates we obtain the equationṡ In addition, we have the constant of motion given by Eq. (63), corresponding to the conservation of the rest mass of the particle. The energy E and the angular momentum of the particle J can be expressed by relativistic velocity w and impact parameter b at infinity, where the spacetime is asymptotically flat, as it is shown by Eqs. (20) and (21).
Hereafter, we will consider only a light ray laying on θ = π/2 hypersurface of the considered rotating naked singularity or wormhole geometry expecting the largest ray deviation. Therefore, we shall seek a solution of the Hamilton-Jacobi equation of the form Taking into account the relationship between a metric tensor and its inverse, thus the nonzero contravariant metric coefficients are g tφ = g φt = − g tφ g tt g φφ − g 2 tφ . (68) The Hamilton-Jacobi equation than takes the form Using the relation p r = g rrṙ = ∂S/∂r = dS r /dr, it may be related directly to the radial equation of motion of a particle with rest mass µ dr dσ 2 = g φφ E 2 + 2g tφ EJ + g tt J 2 g rr (g 2 tφ − g tt g φφ ) where the solutions of the equation in E g φφ E 2 + 2g tφ EJ + g tt J 2 = 0 (71) represent the effective potentials Evaluating the Hamiltonian at the minimum distance of approach of the particle, whereṙ = 0, we find the precise relation between the impact parameter ξ = J/E = wb and the closest approach which reduces to expected photon impact parameter b for stationary axially symmetric metrics, when w = c = 1 [51]. The sign in front of the square root is chosen to be positive when the orbital motion of the particle and rotation of the compact object are in the same direction.
The orbital equation of the particle in terms of azimuthal φ and radial r coordinates can be obtained by using the fact that the partial derivative of the Jacobi function with respect to the constant of motion J is a constant Solving Eq. (69) or taking into account Eq. (70), we obtain the azimuthal shift function of the particle by Eq. (74) where the sign in front of the integral is changing when the particle passes thought a turning Having the azimuthal shift of the photon as a function of the coordinate distance r, we can calculate the deflection angle of the test-particle as we assume that the source r S and observer r O are placed in the asymptotically flat region of the spacetime, so that r S → ∞ and r O → ∞. Then, after calculating the total azimuthal angle according to the orbital equation (75), the deflection angle isα

A. Deflection angle by Rotating Naked Singularities
In order to calculate the deflection angle of a test-particle in the spacetime of rotating naked singularities, firstly we will calculate the impact parameter of the test-particle ξ = wb as a function of the distance of closest approach r 0 . According to Eq. (73) the equatorial impact parameter of a particle with relativistic velocity w is given by The impact parameter (78) reduces to that for photons, obtained in [52], letting w = 1. For a > 0 the naked singularity rotates counterclockwise, while for a < 0 the naked singularity and the particles rotate in converse direction.
Expecting small deviations in the orbital motion of the test-particles, we will assume that the distance of closest approach r 0 is of the same order as the impact parameter b due to the validity of inequalities, b m and b a. Therefore, considering Eq. (70) in the framework of the assumptions adopted, we suggest the following solution where we are introducing two independent expansion parameters, as the former is in term of metric parameter m related to the ADM mass M and scalar charge q, while the latter is in term of specific angular momentum a of the rotating naked singularity Here the coefficients c m,a are real numbers, and the summation is over all possible combinations of epsilon powers i, j up to and including second order terms.
Applying the procedure to the equationṙ(r 0 ) = 0 as discussed above, we acquire the following The calculation of the deflection angle of massive particles in the spacetime of rotating naked singularities requires integration of the first derivative of the azimuthal shift function dφ/dr, as follows from Eq. (75). Since, we are expecting small deviation in the orbital motion of the particle, we will consider only terms with the most significant contribution, for simplicity. Introducing new small enough parameters ε m = m/r 0 and ε a = a/r 0 , we will perform power series expansion of dφ/dr, in the spacetime under consideration (3), up to and including second order terms of ε m and ε a . After the convenient change of variables r = r 0 /u, we obtain dφ du where we are introducing the quartic polynomial in powers of u Completing the integration of (82) from the minimal distance of closest approach r 0 to observer and source positioned at infinite distance with respect to the center of the compact object, we obtain the following approximate expression for the deflection angle of a massive particle, according to Eq. (76) The most significant contribution to the deflection angle of massive particles is given by massive After performing the calculations, we get the following invariant representation of the deflection for a given relativistic velocity w of the massive particlê By comparison of the calculated approximate expression for the deflection angle of massive particles (84) via geodesic approach with the result (38) found by GBT it appears that up to the second formal order in the expansion parameters the both angles coincide. Letting w = 1, we recover the light deflection angleα presented in the static case in [5,52,53].

B. Deflection angle by Kerr-like wormholes
In the case of the Kerr-like wormhole (39) the impact parameter of the massive particles ξ = wb with relativistic velocity w and the minimal distance of closest approach r 0 obey the following Since, we are considering propagation of massive particles in the asymptotically flat spacetime regions with geometrical curvature, causing small deflection angles of trajectories, then r 0 is expected to be of the same order as the impact parameter b. Therefore, under assumptions b M and b a, we are taking advantage of the two small enough expansion parameters where M and a are the mass and spin of the Kerr-like wormhole, respectively. Hence, we could find an approximate solution of the radial Eq. (70), as for itself we suggest the ansatz Here the coefficients c M,a are real numbers.
Solving the equationṙ(r 0 ) = 0, for the closest approach distance we obtain the following power series expansion Now, we can proceed ahead toward finding of the perturbative expansion of the deflection angle of a massive particle moving in the spacetime of the Kerr-Like wormhole. Since, we are interested in the weak deflection limit, we will perform power series expansion of the derivative dφ/dr, adopting the new small parameters ε M = M/r 0 and ε a = a/r 0 up to and including second order terms. In order to perform the integration afterwards more easily, we introduce a new variable u = r 0 /r in terms of which we obtain dφ du where it is introduced the fourth degree polynomial in variable u Therefore, the bending angle of the light ray, defined by Eq. (76), up to and including the second orders of the expansion parameters ε M and ε a is given bŷ Taking advantage of the power series (89) we can represent the deflection angle of the relativistic particles in terms of the impact parameter b, up to and including second order terms of the small parameters M and a . Therefore, the deviation of trajectories of the massive particles by the Kerr-like wormhole leads to the following components of the deflection anglê As in the previous case of the rotating naked singularities, here as well the most significant contribution to the deflection angle is given by the massive term proportional to O( M ). The second order terms proportional to the mass O( 2 M ) and to the mass and angular momentum of the wormhole O( M a ) give a contribution to the deflection angle of the relativistic massive particles.
The first and second order terms based on the wormhole's angular momentum, O( a ) and O( 2 a ) respectively, do not contribute in the perturbative analysis. In the special case w = c = 1, from expression (92) the deflection angle of light is recovered [49]

V. CONCLUSIONS
We studied the gravitational deflection of massive/massless particles in a rotating naked singularities and Kerr-Like wormholes backgrounds. In the first part of this paper, we used a recent geometric approach to compute the gravitational deflection angle using the refractive index of the optical media and by utilizing the GBT applied to the optical metric. In order to incorporate the effect of relativistic speed on the deflection angle, we have assumed the propagating massive particles as a de Broglie wave packets yielding a modification of the refractive index N, and a constant quantity of a given optical media, λN = const. In particular, we have shown that the refractive index and the Gaussian optical curvature are affected by the mass M, angular momentum parameter a, naked singularity parameter γ, the deformation parameter λ, and the relativistic velocity of the particle w. Although the geodesic curvature κ(C R ) is slightly modified due the velocity of the particle, i.e. κ(C R ) → wR −1 , we have shown that the expression for the deflection angle readŝ In other words we have found exactly the same expression as in the case of deflection of light, which suggests that the above result can be viewed as a form-invariant quantity for asymptotically flat spacetimes. Again, the total deflection angle is found by integrating over a domain outside the light ray described by r γ , hence the global effects are important. Note that in our setup we have simplified the problem by considering a linearized spacetime metrics and we have also assumed that the source and the observer are located at infinity. Applying then the GBT to the isotropic metrics we have found two interesting results for the deflection angles, namelŷ α Naked sing. 2mγ bw 2 (1 + w 2 ) ± 4maγ b 2 w in the case of rotating naked singularity, and in the case of Kerr-like wormholes. From the last two equations we see that an apparent singularity appears when w → 0, therefore an additional constraint should be imposed, namely 0 < w ≤ 1. Now we specialize further the case of λ = 0 and γ = 1, in that case both results coincide with the Kerr deflection angle for massive particles reported recently in [38], but also in [54,55]. Note that our approach works perfectly well for relativistic particles, in this sense the apparent singularity is not problematic. Furthermore, we did a careful analyses by studying the problem of deflection of particles using the Hamilton-Jacobi equation and we find agreement of results in leading order terms. Our results show that, while the deflection angle decreases with γ in the case of JNW spacetime, contrary to this, the deflection angle increases with the deformation parameter λ 2 in the case of Kerr-like wormholes as can be seen from figure (3). Our results are not important only from conceptual point of view, namely the difference between deflection angels may shed light from observational point of view in order to distinguish the two spacetimes.
Finally, we plan in the near future to extend these results by including finite distance corrections on the deflection angle, in other words by assuming that the distance between the source and the observer is finite.