Use of Gödel Universe to Construct A New Zollfrei Metric with Topology
Moninder Singh Modgil ^{1}^{1}1 PhD in Physics, from Indian Institute of Technology, Kanpur, India, and
B.Tech.(Hons.) in Aeronautical Engineering, from Indian Institute of Technology, Kharagpur, India.
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Abstract
A new example of dimensional Zollfrei metric, with the topology , is presented. This metric is readily obtained from the celebrated  dimensional rotating Gödel universe . This is because has the interesting property that, the light rays which are confined to move on the plane perpendicular to the rotation axis, return to their origin after a time period where is the angular velocity of the universe. Hence by  the topological identification of pairs of points on the time coordinate, seperated by the time interval . and droping the flat coordinate  which is directed along the rotation axis; one obtains the dimensional Zollfrei metric with the topology.
KEY WORDS: Gödel Universe, Zollfrei Metric, Closed Null Geodesics (CNCs), Closed Timelike Curves (CTCs), Periodic Time
A Lorentzian manifold is said to possess Zollfrei metric, if all its null geodesics are closed [2]. Part of attraction for studying Zollfrei metrics in physical context comes from the Segal’s cosmological model [4] which is based upon the topology of spacetime. Examples of manifolds with Zollfrei metric are  and  with commensurate radii of spatial and temporal factors. To our knowledge, all previously known examples of Zollfrei metric have compact topology for the spatial factor  i.e., or . Here, we construct a new example of Zollfrei metric in dimensions, with the topology. This construction is based upon the Gödel universe [1]. We shall refer to this as the GödelZollfrei metric, and denoted it by .
The D axisymmetric, homogenous, Gödel universe [1] with the topology , has the line element 
(1) 
which satisfies the Einstein equations for a uniform matter density,
(2) 
and a cosmological constant,
(3) 
where, is Newton’s gravitational constant. The universe rotates about the axis, with the angular velocity,
(4) 
Lets denote the curved space defined by the three coordinates as . Now can be regarded as the product of and the flat coordinate (which has the topology ), i.e.,
(5) 
Pfarr [3] worked out geodesic and nongeodesic trajectories for . For particles confined to move on the plane  i.e., the plane perpendicular to the rotation axis , he showed that the geodesically moving particles, move on circles and consequently return to the point of their origin. In particular, the light rays move on the largest circles (null geodesics), and reach out to a maximal distance , from the point of their origin,
(6) 
The circle defined by , may be termed as the Gödel horizon. The circles of radius are Closed Null Curves (CNCs), while circles of radius are Closed Timelike Curves (CTCs). The time taken by the light rays to return to the point of their origin is [3] 
(7) 
Accordingly if in 

One drops the flat coordinate, which is directed along the rotation axis, and

Compactifies the time coordinate , to , and

Chooses the time period of time factor equal to ;
one obtains this new example of dimensional Zollfrei metric with the topology.
Gödel [1] gave a list of nine interesting properties of . Remarkably, in the fifth property, he considered the possibility of both an open and a closed time coordinate.
References
 [1] Gödel, K.: Rev. Mod. Phys. 21, (1949), 447.
 [2] Guillemin, V.: Cosmology in (2+1)dimensions, cyclic models and deformations of , Princeton University Press, (1989).
 [3] Pfarr, J.: Gen Rel. and Grav., 13, 1073, (1981).
 [4] Segal I.E. : Mathematical Cosmology and Extragalactic Astronomy, Cambridge University Press (1976).