# On the trajectories of null and timelike geodesics in different wormhole geometries

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## Abstract

The paper deals with an extensive study of null and timelike geodesics in the background of wormhole geometries. Starting with a spherically symmetric spacetime, null geodesics are analyzed for the Morris–Thorne wormhole (WH) and photon spheres are examined in WH geometries. Both bounded and unbounded orbits are discussed for timelike geodesics. A similar analysis has been done for trajectories in a dynamic spherically symmetric WH and for a rotating WH. Finally, the invariant angle method of Rindler and Ishak has been used to calculate the angle between radial and tangential vectors at any point on the photon’s trajectory.

## 1 Introduction

In general relativity, a wormhole (WH) is considered to be a tunnel through which two distant regions of spacetime can be connected [1]. Long back in 1916, Flamm [2] introduced the idea of wormhole, analyzing at that time the recently discovered Schwarzschild solutions. In 1935, Einstein and Rosen [3] constructed a WH type solution considering an elementary particle model as a bridge connecting two identical sheets. This mathematical representation of space being connected by a WH type solution is known as an “Einstein–Rosen bridge”. Wheeler [4, 5] in the 1950s considered WHs as objects of quantum foam connecting different regions of spacetime and operating at the Planck scale. Subsequently, using this idea, Hawking [6] and collaborators introduced the idea of Euclidean wormholes. But these types of WHs are not traversable and, in principle, would develop some type of singularity [7]. However, these hypothetical shortcut paths, i.e., traversable WHs, have been rekindled by the pioneering work of Morris and Thorne [8] which is considered as the modern renaissance of WH physics. Subsequently, it was claimed that there is no strong ground [9, 10] for the energy conditions and hence one considered a WH, with two mouths and a throat, to be an object of nature, i.e., an astrophysical object.

On the other hand, in general relativity, WH physics is a specific example where the matter stress-energy tensor components are evaluated from the spacetime geometry by solving Einstein’s field equations. But for a traversable WH, the stress-energy tensor components so obtained always violate the null energy condition [1, 8]. As the null energy condition (NEC) is the weakest of all the classical energy conditions, its violation signals that the other energy conditions are also violated. In fact, they violate all the known pointwise energy conditions and averaged energy conditions, which are fundamental to the singularity theorems and theorems of classical black hole thermodynamics. Generally, it is believed that a classical matter obeys energy conditions [11] but, in fact, it is known that they also get violated by some quantum fields (namely as regards the Casimir effect and Hawking evaporation [12]). Further, for a quantum system in classical gravity, it is found that the averaged weak or null energy condition (ANEC), which states that the integral of the energy density as measured by a geodesic observer is non-negative, could also be violated by a small amount [13, 14].

Finally, it is worth to mention a few important dynamical WH solutions. Hochberg and Visser [15] and Hayward [16] independently formulated the dynamical WH solutions, choosing a quasi-local definition of the WH throat in a dynamical spacetime. Accordingly, the WH throat is a trapping horizon [17] of different kind, but again matter in both of them violates the NEC. On the other hand, Maeda et al. [18] have developed another class of dynamical WHs (cosmological WHs) which are asymptotically FRW spacetimes with big bang singularity at the beginning. This class of WHs contain matter which not only obey the NEC but also the dominant energy condition everywhere. These two classes of dynamical WHs are distinct from the geometrical point of view. For the former one, the WH throat is a 2D surface of non-vanishing minimal area of a null hypersurface, while for the latter one, there is no past null infinity due to the initial singularity. Hence, the WH throat is defined only on a space-like hypersurface and the spacetime is trapped everywhere without any trapping horizon [19]. Recently, Lobo et al. [20, 21, 22] formulated wormhole solutions which are dynamically generated using a single charged fluid. Also, dynamical WHs are considered with a two-fluid system [23, 24], for a matter distribution relevant to present day observations [25] and using the mechanism of particle creation [26]. Then for evolving WH,^{1} one may refer to Refs. [27, 28, 29, 30, 31].

The particle motion in wormhole spacetimes is an important issue related to traversable WHs. It is interesting to examine whether a timelike or null geodesic can tunnel through the throat in the case Cataldo et al. [32] studied, of the motion of test particles in the background of zero tidal force Schwarzschild-like WH spacetime. They showed that particles moving along the radial geodesics reach the throat with zero tidal velocity in finite time, while the particle velocity reaches maximum at infinity if it travels along a radially outward geodesic. For non-radial geodesics on the other hand, the particles may cross the throat with some restrictions. Olmo et al. [33] carried out a detailed investigation of the geodesic structure for three possible WH configurations, namely: the Reissner–Nordström-like WH, the Schwarzschild-like WH and the Minkowski-like WH. They have shown that it is possible to have geodesically complete paths for all these WH spacetimes. Culetu [34] examined both timelike and null geodesics for a WH belonging to the Planck world (WHs whose throat size is of the order of the Planck length \(l_P\)) where quantum fluctuations are supposed to exist and the spacetime smoothness seems to break down. Muller [35] also studied null and timelike geodesics in the WH configuration using elliptic and Jacobian integral functions. He showed that it is possible to connect two distant events geodesically. Regarding a geodesic study in non-static WHs, recently Chakraborty and Pradhan [36] have studied the geodesic structure of the rotating traversable Teo WH. Also, Nedkova et al. [37] discussed the shadow of a class of rotating traversable WH in the framework of general relativity. They showed that the images depend on the angular momentum of the WH and the inclination angle of the observer. Finally, it is worthy to mention the work of Ellis [38]. He constructed a static, spherically symmetric, geodesically complete, horizonless spacetime manifold with a topological hole (drainhole) at its center by coupling the geometry of Schwarzschild spacetime to a scalar field. It is found that on one side of the drainhole the manifold is asymptotic to a Schwarzschild manifold with positive mass parameter ‘m’, and on the other to a Schwarzschild manifold with negative mass parameter ‘\(\bar{m}\)’, with the condition \(-\bar{m}>m\). As a consequence, there is attraction of particles on one side, while there is repulsion on the other side (with higher strength).

The present work presents a detailed investigation of both timelike and null geodesics both for static and dynamical WHs. The paper is organized as follows: Sect. 2 deals with static spherical WHs in which null and timelike geodesics are studied in great detail. A similar geodesic analysis is presented for a dynamical WH in Sect. 3 and for a rotating WH in Sect. 4. Section 5 uses the invariant angle method of Rindler and Ishak to calculate the angle between radial and tangential vectors at a point on the photon’s trajectory. Finally, the paper ends with a short discussion and concluding remarks in Sect. 6. Throughout our analysis, we have chosen to work with wormholes whose material extends all the way from the throat out to infinity.

## 2 Trajectories in a spherically symmetric and static geometry

*E*is the energy and

*L*is the angular momentum of the photon or a particle as measured by observers at asymptotically flat regions far from the source. Thus, we get

### 2.1 Null geodesics

*r*and thus may not approach \(\lim _{r\rightarrow \infty }(r,\pi /2+\alpha /2)\). In that case, the integral will diverge. Such spheres are called photon spheres; they are discussed in Sect. 2.3.

### 2.2 Morris–Thorne wormhole

*b*(

*r*) is the shape function of the wormhole for which \(b(r)\le r\) and equality holds only at the throat. Both the functions are such that they also satisfy asymptotic flat conditions. Thus, the equation of the trajectory for null geodesics, Eq. (6), becomes

*r*cannot be used for describing the whole spacetime since it accounts for a coordinate singularity at the throat and is therefore valid for describing geometry only at one side of the throat. Thus, for geodesics that actually reach and pass through the throat, one should not use this formula for the trajectory equation. Instead, one can always work with the proper distance (

*l*) which must be valid everywhere and throughout the wormhole. As an example, for the metric given in Eq. (1), we have \({\text {d}}l=\sqrt{B(r)}{\text {d}}r\), thus

*r*in terms of

*l*, which, in principle, could be obtained by inverting Eq. (13) to get \(r\equiv r(l)\). In this paper, however, we will mostly be interested in the behavior of trajectories on one side of a throat and so we will mostly work with

*r*for our convenience. Now, using Eq. (10), the null geodesics coming from infinity and not reaching the throat get deflected by the angle

*b*(

*r*) is,

*n*’ the shape exponent.

*b*(

*r*) in terms of \(r_o\) and

*n*becomes

*l*depends upon the radial coordinate

*r*and the shape exponent

*n*. If we keep \(b_o=1\), then we have

### 2.3 Photon spheres

*l*gives

### 2.4 Timelike geodesics

*E*/

*m*) and the angular momentum per unit mass (

*L*/

*m*), respectively, then

*l*is the proper length. If we differentiate Eq. (32) with respect to the affine parameter, we obtain for the second derivative of the radial coordinate

#### 2.4.1 Unbounded orbits

#### 2.4.2 Bounded orbits

*V*is smaller than \({\widetilde{E}}\).

#### 2.4.3 Circular orbits

**Case I:**\(n>2\) For this case, it is easy to see that

*f*(

*r*) is such that it will start from a negative value at \(r=0\) and will grow further negative with the increase in

*r*until it hits a turning point at \(r=r_p\), after which it increases monotonically. Thus, it can be inferred that

*f*(

*r*) will have only one positive real root, \(r_c\) say. Then it is clear that \(f(r)>0\ \forall \ r>r_c\). Thus, from Eq. (51), we can say that this root must also satisfy condition III. Since there is only one turning point of

*f*(

*r*), it is obvious from Eq. (48) that this corresponds to the maximum of the potential, in which case it will always lead to an unstable orbit.

Hence, for \(n>2\), there will be only one unstable circular orbit for a particular value of \({\widetilde{E}}\) and \({\widetilde{L}}\).

**Case II:**\(n=2\) For n=2,

*f*(

*r*) becomes

Hence, for \(n=2\), there is a possibility of only one unstable circular orbit depending upon \({\widetilde{L}}\) and \(b_o\) and \({\widetilde{E}}\).

**Case III:**\(0<n<2\) For this case, we have

Meanwhile if \(f(r_p) > 0\), we will have two real roots. Now, from Eq. (48), it is clear that the smaller of these roots will correspond to a local maximum and the larger root will correspond to a local minimum. Using Eq. (51) and the condition in Eq. (59), it can be inferred that both of these roots will also satisfy condition III.

Hence, for \(0<n<2\), there is a possibility for one unstable circular orbit or a combination of one unstable circular orbit and a stable circular orbit. Note that this is the only case where we have the possibility of stable circular orbits. It is also interesting to note that the Schwarzschild geometry, for which \(n=1\), lies in this case.

As mentioned before, any bound orbit, which is not circular, is possible only when it oscillates around the radius of a stable circular orbit. Thus, we can say that we surely have no non-circular bound orbits when \( n\ge 2\) for any \({\widetilde{E}}\) and \({\widetilde{L}}\) of the particle.

*r*satisfying \(r>b_o\). Then the condition for circular orbit, Eq. (59), becomes

#### 2.4.4 Time period of circular orbits

*n*and \(b_o\). Using Eq. (49) and the fact that \(E=V(r)\), we can write

#### 2.4.5 Choice of \(\epsilon (r)\)

where \(\tau \) represents the tidal force. The above expression for the tidal force is obtained by considering the gradient of the potential throughout the spaceship and using the fact that the gradients, except at the two ends of the spaceship, should almost vanish for it to not deviate much from its rigid structure. Pictorially, it is just the absolute difference between the slopes at the two points on \(V^2(r)\) vs. *r* curve in Fig. 4. It can also be noted that without considering the significance of \(\epsilon (r)\), we get a discontinuity in the slopes of the potential at the throat. That would correspond to an impulse of force which will be experienced by a particle at the throat while traversing through the wormhole. It would be like hitting a thin membrane of a tough material. However, it should be noted that even in the presence of infinite tidal forces, causal contact is never lost among the elements making up the observer; this suggests that curvature divergences may not be as pathological as traditionally thought [47].

*r*as shown in Fig. 5.

It is clearly visible that we have removed the discontinuity in the slope at the throat. Also note that there exists a local minimum of \({\dot{r}}^2\) at the throat. Such a local minimum will correspond to an unstable bound orbit. Thus, for our choice of \(\epsilon (r)\), the throat will correspond to a region of unstable circular orbits.

## 3 Trajectories in a dynamic spherically symmetric wormhole

*a*(

*t*).

### 3.1 Null geodesics

*E*is a positive constant of integration.

*r*is itself a comoving coordinate. We should define a new coordinate, \(r'(r,t)=a(t).r\), so that any surface \(r' = const.\), \(t = const.\) is a two-sphere of area \(4\pi r'^2\) and circumference \(2\pi r'\). This coordinate \(r'\) can then be called the ’curvature coordinate’. In this coordinate, the equation of the trajectory becomes

*r*by the following equation:

### 3.2 Timelike geodesics

*E*is a constant of integration. Using it, we get

## 4 Trajectories in a rotating wormhole

*r*that determines the proper radial distance

*R*, i.e., \(R \equiv rK\). We also require this metric to be asymptotically flat, which implies

*J*. Then

*t*and \(\phi \), the corresponding momenta one-forms are conserved. Thus, we can write [50]

Let \(\Delta = AC+B^2=(N^2-r^2K^2\omega ^2)(r^2K^2)+(r^2K^2\omega )^2=N^2r^2K^2\) .

*E*and

*L*in terms of \({\dot{t}}\) and \({\dot{\phi }}\) become

### 4.1 Null geodesics

*r*,

*K*(

*r*),

*N*(

*r*) are all non-negative functions, we get

*J*is the angular momentum of the rotating wormhole. This assumption also implies that \(\omega (r)>0\). Now, we have two possibilities for the conserved angular momentum (

*L*) of the photon: it can be either \(L>0\) or \(L<0\). This will determine whether the light ray is traversing along the direction of frame dragging or opposite to it.

*N*(

*r*) is a smooth decreasing function, then this equation proves that the distance of closest approach is greater when the light ray is moving in the direction of frame dragging than that of light moving opposite to it. Now, from Eqs. (93) and (94), we can write the equation of motion of photon trajectory as

### 4.2 Timelike geodesics

*E*and \({\widetilde{L}}\) as

*L*. Now, we can rewrite the above equation as

## 5 Invariant angle method of Rindler and Ishak

*d*represent the tangential direction at any point on the photon’s trajectory. Let \(\psi \) be the angle between them. Then the invariant formula for \(\cos \psi \) becomes

*d*and \(\delta \) in the \((r,\phi )\) basis can be written as

*b*(

*r*). This independence is true in the case of dynamical and rotating wormhole geometries as well.

## 6 Short discussion and concluding remarks

A detailed study of particle and photon trajectories has been conducted in the background of wormhole geometry. Starting with the Morris–Thorne wormhole, null geodesics and photon spheres have been analyzed, while for particle trajectories both bounded and unbounded orbits are considered. Subsequently, both null and timelike geodesics are analyzed in the geometry of dynamic spherically symmetric WH and rotating WH. Finally, using the invariant angle method of Rindler and Ishak, the angle between radial and tangential vectors on the photon’s trajectory has been evaluated.

Based on the above study, we have found that in a Morris–Thorne wormhole and its dynamic and rotating counterparts, the throat itself is a photon sphere. We have also seen that in such geometries, the angle between tangential and radial vectors at any point on a photon’s trajectory is independent of the shape function *b*(*r*). The geodesics in ultra-static wormholes with shape exponents have already been studied in great detail in Ref. [43]. Also, in Ref. [32] Cataldo et al. studied a Schwarzschild-like traversable WH which is obtained by putting \(n=1\) with some slight modification. For geodesics, they showed that a test particle which is radially moving towards the throat always reaches it with zero velocity and at a finite time, while for radially outward geodesics the particle velocity tends to a maximum value, reaching infinity. However, in this paper we have shown that it is true for all possible *n*. Also, general conditions for non-radial geodesics were derived which are required to be satisfied in order for it to cross the throat. These results are in agreement with our study and can, roughly, be obtained by putting \(n=1\) in our general equations for arbitrary *n*. Similarly, in Ref. [38], the Ellis wormhole (\(n=2\)) is studied in great detail including the behavior of geodesics in such geometry. For the Ellis wormhole, the particles are attracted on one side and are repelled on the other and so the throat is of a saddle nature. In our paper, we have mainly stressed the geodesics that remain on one side of the wormhole, unlike the above-mentioned references where geodesics through the throat are studied in detail.

Furthermore, we have analyzed the possibility of bounded timelike orbits for different shape exponents in a different WH geometry, which can be regarded as the generalization of the Schwarzschild geometry far from the throat. There, we used the fact that any bounded timelike orbit in a spherically symmetric geometry is either a circular orbit or an orbit that oscillates around the radius of a stable circular orbit. For this geometry, we found that, for a wormhole with shape exponent \(n>2\), there always exists the possibility of one unstable circular orbit while for \(n=2\), there exists one unstable circular orbit only when \(L>b_o\) and no bound orbits otherwise. That means, no non-circular bound orbits exist when shape exponent \(n\ge 2\) for any value of the impact parameter. For \(0<n<2\), we found that depending upon the value of *L* it can either have the possibility of one unstable circular orbit or a combination of one unstable circular orbit and a stable circular orbit. While studying trajectories in a rotating wormhole geometry, we have seen that the equations of motion of both photon and particle depend upon whether it is traveling in the direction of frame dragging or opposite to it.

## Footnotes

- 1.
These are not as popular as static WHs and also they are not well understood.

## Notes

### Acknowledgements

The authors are thankful to the Inter University Centre for Astronomy and Astrophysics (IUCAA), Pune (India) for their hospitality as the initiation of this work was taken during a visit there. Anuj is also thankful to the library facility at the department of mathematics of Jadavpur University.

## References

- 1.M. Visser, L. Wormholes,
*From Einstein to Hawking*(American Institute of Physics, New York, 1996)Google Scholar - 2.L. Flamm, Beitrage zur Einsteinschen Gravitationstheorie. Phys. Z.
**17**, 448 (1916)MATHGoogle Scholar - 3.A. Einstein, N. Rosen, The particle problem in the general theory of relativity. Phys. Rev.
**48**, 73–77 (1935)ADSCrossRefMATHGoogle Scholar - 4.J.A. Wheeler, Geons. Phys. Rev.
**97**, 511–536 (1955)ADSMathSciNetCrossRefMATHGoogle Scholar - 5.J.A. Wheeler,
*Geometrodynamics*(Academic, New York, 1962)MATHGoogle Scholar - 6.S.W. Hawking, Wormholes in spacetime. Phys. Rev. D
**37**, 904 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar - 7.R.P. Geroch, Topology in general relativity. J. Math. Phys.
**8**, 782 (1967)ADSMathSciNetCrossRefMATHGoogle Scholar - 8.M.S. Morris, K.S. Thorne, Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys.
**56**, 395 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar - 9.B.K. Harrison, K.S. Thorne, M. Wakano, J.A. Wheeler,
*Gravitational theory and gravitational collapse*(University of Chicago Press, Chicago, 1965)Google Scholar - 10.Y.B. Zel’dovich, I.D. Novikov,
*Relativistic astrophysics, Vol. I: stars and relativity*(University of Chicago Press, Chicago, 1971)Google Scholar - 11.S.W. Hawking, G.F.R. Ellis,
*The large scale structure of spacetime*(Cambridge University Press, Cambridge, 1973)CrossRefMATHGoogle Scholar - 12.G. Klinkhammer, Averaged energy conditions for free scalar fields in flat spacetime. Phys. Rev. D
**43**, 2542 (1991)ADSMathSciNetCrossRefGoogle Scholar - 13.F.J. Tipler, Energy conditions and spacetime singularities. Phys. Rev. D
**17**, 2521 (1978)ADSMathSciNetCrossRefGoogle Scholar - 14.T.A. Roman, Quantum stress-energy tensors and the weak energy condition, Phys Rev. D
**33**, 3526 (1986)ADSMathSciNetCrossRefGoogle Scholar - 15.D. Hochberg, M. Visser, Dynamic wormholes, antitrapped surfaces, and energy conditions. Phys. Rev. D
**58**, 044021 (1998)ADSMathSciNetCrossRefGoogle Scholar - 16.S.A. Hayward, Dynamic wormholes. Int. J. Mod. Phys. D
**08**, 373 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar - 17.S.A. Hayward, General laws of black-hole dynamics. Phys. Rev. D
**49**, 6467 (1994)ADSMathSciNetCrossRefGoogle Scholar - 18.H. Maeda, T. Harada, B.J. Carr, Self-similar cosmological solutions with dark energy. II. Black holes, naked singularities, and wormholes. Phys. Rev. D
**77**, 024023 (2008)ADSMathSciNetCrossRefGoogle Scholar - 19.H. Maeda, T. Harada, Cosmological wormholes. Phys. Rev. D
**79**, 044034 (2009)ADSMathSciNetCrossRefGoogle Scholar - 20.F.S.N. Lobo, Dynamical generation of wormholes with charged fluids in quadratic Palatini gravity. Phys. Rev. D
**90**(2), 024033 (2014)ADSCrossRefGoogle Scholar - 21.F.S.N. Lobo, Planck scale physics and topology change through an exactly solvable model. Phys. Lett. B
**731**, 163–167 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar - 22.F.S.N. Lobo et al., Semiclassical geons as solitonic black hole remnants. JCAP
**1307**, 011 (2013)ADSCrossRefGoogle Scholar - 23.M. Cataldo, S. del Campo, Two-fluid evolving Lorentzian wormholes. Phys. Rev. D
**85**, 104010 (2012)ADSCrossRefGoogle Scholar - 24.M. Cataldo, S. del Campo, Phantom evolving wormholes with big rip singularities. Phys. Rev. D
**87**, 064012 (2013)ADSCrossRefGoogle Scholar - 25.S. Pan, S. Chakraborty, Will there be again a transition from acceleration to deceleration in course of the dark energy evolution of the universe? Eur. Phys. J. C
**73**, 2575 (2013)ADSCrossRefGoogle Scholar - 26.S. Pan, S. Chakraborty, Dynamic wormholes with particle creation mechanism. Eur. Phys. J. C
**75**, 21 (2015)ADSCrossRefGoogle Scholar - 27.F.S.N. Lobo,
*Exotic solutions in General Relativity: Traversable wormholes and warp drive spacetimes*. Classical and Quantum Gravity Research, 1–78, Nova Sci. Pub. ISBN:978-1-60456-366-5 (2008). arXiv:0710.4474 [gr-qc] - 28.E. Teo, Rotating traversable wormholes. Phys. Rev. D
**58**, 024014 (1998)ADSMathSciNetCrossRefGoogle Scholar - 29.M. Cataldo, P. Mella, P. Minning, J. Saavedra, Phys. Rev. Lett.
**662**, 314 (2008)CrossRefGoogle Scholar - 30.A. Banijamali, B. Fazlpour, Phys. Rev. Lett.
**703**, 366 (2011)CrossRefGoogle Scholar - 31.Y.-F. Cai, J. Wang, Class. Quant. Grav.
**25**, 165014 (2008)ADSCrossRefGoogle Scholar - 32.M. Cataldo, L. Liempi, P. Rodriguez, Traversable Schwarzschild-like wormholes. Eur. Phys. J. C
**77**(11), 748 (2017)ADSCrossRefGoogle Scholar - 33.G.J. Olmo, D. Rubiera-Garcia, A. Sanchez-Puente, Geodesic completeness in a wormhole spacetime with horizons. Phys. Rev. D
**92**(4), 044047 (2015)ADSMathSciNetCrossRefGoogle Scholar - 34.H. Culetu, On a particular Morris–Thorne wormhole. Phys. Scr.
**90**(8), 085001 (2015)ADSMathSciNetCrossRefGoogle Scholar - 35.T. Muller, Exact geometric optics in a Morris–Thorne wormhole spacetime. Phys. Rev. D
**77**, 044043 (2008)ADSMathSciNetCrossRefGoogle Scholar - 36.C. Chakraborty, P. Pradhan, Behavior of a test gyroscope moving towards a rotating traversable wormhole. JCAP
**1703**, 035 (2017)ADSMathSciNetCrossRefGoogle Scholar - 37.P.G. Nedkova, V.K. Tinchev, S.S. Yazadjiev, Shadow of a rotating traversable wormhole. Phys. Rev. D
**88**(12), 124019 (2013)ADSCrossRefGoogle Scholar - 38.H.G. Ellis, Ether flow through a drainhole: a particle model in general relativity. J. Math. Phys.
**14**, 104 (1973)ADSMathSciNetCrossRefGoogle Scholar - 39.B.F. Schutz,
*A first course in general relativity*(Cambridge University Press, Cambridge, 2014)Google Scholar - 40.C.W. Misner, K.S. Thorne, J.A. Wheeler,
*Gravitation*(W.H. Freeman and Company, San Francisco, 1973)Google Scholar - 41.A. Bhattacharya, A.A. Potapov, Bending of light in Ellis wormhole geometry. Mod. Phys. Lett. A
**25**, 2399 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar - 42.N. Tsukamoto, T. Harada, K. Yajima, Can we distinguish between black holes and wormholes by their Einstein-ring systems? Phys. Rev. D
**86**, 104062 (2012)ADSCrossRefGoogle Scholar - 43.P. Taylor, Propagation of test particles and scalar fields on a class of wormhole space-times. Phys. Rev. D
**90**, 024057 (2014)ADSCrossRefGoogle Scholar - 44.S.M. Carroll,
*Spacetime and geometry: An introduction to general relativity*(Addison-Wesley, San Francisco, 2004)MATHGoogle Scholar - 45.R.M. Wald,
*General relativity*(University of Chicago Press, Chicago, 1984)CrossRefMATHGoogle Scholar - 46.V. Balakrishnan,
*Lecture series on classical physics, lecture recording, dynamics in phase space*(IIT Madras, Chennai, 2009)Google Scholar - 47.G.J. Olmo, D. Rubiera-Garcia, Impact of curvature divergences on physical observers in a wormhole space-time with horizons. Class. Quant. Grav.
**33**(11), 115007 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar - 48.J.B. Hartle, Slowly rotating relativistic stars. I. Equations of structure. Astrophys. J.
**150**, 1005 (1967)ADSCrossRefGoogle Scholar - 49.M. Azreg-Aïnou, Wormhole solutions sourced by fluids, I: Two-fluid charged sources. Eur. Phys. J. C
**76**, 3 (2016)ADSCrossRefGoogle Scholar - 50.V. Ferrari, L. Gualtieri,
*General relativity*, chapter 20. INFN, Rome (2014–2015)Google Scholar - 51.S.V. Iyer, E.C. Hansen, Light’s bending angle in the equatorial plane of a Kerr black hole. Phys. Rev. D
**80**, 124023 (2009)ADSCrossRefGoogle Scholar - 52.W. Rindler, M. Ishak, Contribution of the cosmological constant to the relativistic bending of light revisited. Phys. Rev. D
**76**, 043006 (2007)ADSCrossRefGoogle Scholar - 53.O.F. Piattella, On the effect of the cosmological expansion on the gravitational lensing by a point mass. Universe
**2**, 25 (2016)ADSCrossRefGoogle Scholar

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