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

Eur. Phys. J. C (2018) 78:374 https://doi.org/10.1140/epjc/s10052-018-5854-5 Regular Article - Theoretical Physics On the trajectories of null and timelike geodesics in different wormhole geometries Anuj Mishra1,2,a , Subenoy Chakraborty2,b 1 National Institute of Technology, Rourkela, Odisha 769008, India 2 Department of Mathematics, Jadavpur University, Kolkata 700032, India Received: 31 October 2017 / Accepted: 30 April 2018 / Published online: 11 May 2018 © The Author(s) 2018 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 a e-mail: b e-mail: 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. 123 374 Page 2 of 16 For the former one, the WH throat is a 2D surface of nonvanishing 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. [20–22] 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. [27–31]. 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 (...truncated)