Exact geometric optics in a Morris-Thorne wormhole spacetime

The notion of a wormhole was first introduced in 1962 by John Wheeler [1] who reinterpreted the Einstein-Rosen bridge [2] as a connection between two distant places in spacetime with no mutual interaction. However, he realized together with Robert Fuller [3] that this Schwarzschild wormhole cannot be traversed even by a single particle. In 1988, Michael Morris and Kip Thorne [4] presented the most simple metric which serves as a wormhole that could in principle be traversed by human beings. [5] From that time on, there are a lot of publications which suggest new types of wormholes, see e.g. [7–13]. But all of them have in common that they violate the weak energy condition. For a detailed discussion see, for example, Visser [14]. One of the difficulties in curved spacetimes is to find a geodesic which connects two distant events. The common astrophysical application is the gravitational lensing of a distant object by means of a very massive object like a galaxy or a black hole. Here, the null geodesics connecting the distant object with the observer are searched. In the case of the Schwarzschild spacetime, Frittelli et al [15] construct the exact lens equation. A short discussion of gravitational lensing by wormholes can be found in Cramer et al [16] or Nandi et al [17]. A detailed review of gravitational lensing in curved spacetime with several examples is given by Perlick [18]. In general, there is no mathematical procedure which could find a geodesic in a four-dimensional spacetime connecting two events in a reasonable time. The shooting method [19] might be applicable in a two-dimensional problem. Another possibility would be the precalculation and tabulating of geodesics. But the disadvantage of this method is the extreme amount of data which must be searched. Furthermore, the ambiguity which appears when connecting two events drastically complicates the solution. The only practical method is, so far as it exists, to use the analytic solution of the geodesic equation. In contrast to the Schwarzschild case as shown by Čadež and Kostić [20], the analytic solution of the geodesic equation in the Morris-Thorne (MT) spacetime is quite straightforward. Starting from the Lagrangian equations for the MT metric, one immediately gets the orbital equation for a geodesic as an elliptic integral of the first kind in standard form. Like in the Schwarzschild case, one has to make a distinction where the null geodesic starts and ends. However, in the MT case, we do not have to deal with complex arguments or modules in the elliptic integrals which definitely simplifies the calculations. The aim of this article is to derive the exact analytic solution of the geodesic equation in the MT spacetime and to show its relevance for connecting two events with a lightlike geodesic. The most prominent application is the determination of the exact gravitational lens equation, compare Perlick [21]. In contrast to Perlick, we will formulate the lens equation in terms of elliptic integral functions which makes the orbits of the geodesics more transparent. A second application might be the visualization of the MT spacetime from a first-person’s point of view. By means of objects in motion or at rest some aspects of the topology and the inner geometry of the spacetime become visible. The importance of visualization to get a better insight of special and general relativity is demonstrated, for example, in [22–25]. A visualization of the Morris-Thorne wormhole is given by the author [26]. In general, the ray tracing method is used where a null geodesic is traced back in time from the observer to the point of emission to render the view of an observer. But this method is quite time consuming and is not capable of including a correct illumination of the scenario. This limitation can be bypassed with the exact solution of the geodesic equation. Finally, an interactive visualization by means of today’s fully programmable graphics processing units (GPUs) becomes possible. In Sec. II we give a short introduction to the MorrisThorne spacetime and explain the topological structure by means of an embedding diagram. For the initial conditions of the geodesics we take the perspective of a local observer whose reference frame is represented by a local tetrad. This is a more intuitive approach than the use of angular momentum and longitude of periapsis. The main part of this article concerns with the analytic solution of the geodesic equation which will be discussed in Sec. III. As we will see, the orbits of lightlike, timelike, and spacelike geo*[email protected] PHYSICAL REVIEW D 77, 044043 (2008)