Use of Gödel Universe to Construct A New Zollfrei Metric with R 2 × S 1 Topology
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Use of G ¨odel Universe to Construct A NewZollfrei Metric with R × S Topology
Moninder Singh Modgil Abstract
A new example of (2 + 1) -dimensional Zollfrei metric, with the topol-ogy R × S , is presented. This metric is readily obtained from the cele-brated (3 + 1) - dimensional rotating G ¨odel universe G , . This is because G , has the interesting property that, the light rays which are confined tomove on the plane perpendicular to the rotation axis, return to their originafter a time period T = πω [ √ − -where ω is the angular velocity of theuniverse. Hence by - the topological identification of pairs of points onthe time coordinate, seperated by the time interval T . and droping the flat x coordinate - which is directed along the rotation axis; one obtains the (2 + 1) -dimensional Zollfrei metric with the R × S topology. KEY WORDS:
G ¨odel Universe, Zollfrei Metric, Closed Null Geodesics(CNCs), Closed Timelike Curves (CTCs), Periodic Time PhD in Physics, from Indian Institute of Technology, Kanpur, India, andB.Tech.(Hons.) in Aeronautical Engineering, from Indian Institute of Technology, Kharag-pur, India.email: moni [email protected] S × S topology of spacetime. Examples of manifoldswith Zollfrei metric are - S n × S and P n × S - with commensurate radiiof spatial and temporal factors. To our knowledge, all previously knownexamples of Zollfrei metric have compact topology for the spatial factor- i.e., S n or P n . Here, we construct a new example of Zollfrei metric in (2 + 1) -dimensions, with the R × S topology. This construction is basedupon the G ¨odel universe [1]. We shall refer to this as the G ¨odel-Zollfreimetric, and denoted it by G S , .The (3 + 1) -D axisymmetric, homogenous, G ¨odel universe G , [1] withthe topology R × R , has the line element - ds = a ( dx − dx − dx + e x dx + 2 e x dx dx ) (1)which satisfies the Einstein equations for a uniform matter density, ρ = (8 πGa ) − , (2)and a cosmological constant, Λ = − πGρ, (3)where, G is Newton’s gravitational constant. The universe rotates aboutthe x axis, with the angular velocity, ω = 2( πGρ ) / = ( √ a ) − (4)Lets denote the curved space defined by the three coordinates ( x , x , x ) as G , . Now G , can be regarded as the product of G , and the flat x coordinate (which has the topology R ), i.e., G , = G , × R (5)Pfarr [3] worked out geodesic and non-geodesic trajectories for G , .For particles confined to move on the ( x x ) plane - i.e., the plane perpen-dicular to the rotation axis x , he showed that the geodesically moving2articles, move on circles and consequently return to the point of their ori-gin. In particular, the light rays move on the largest circles (null geodesics),and reach out to a maximal distance r H , from the point of their origin, r H = 2 ω ln( √ (6)The circle defined by r H , may be termed as the G ¨odel horizon. The circlesof radius r = r H are Closed Null Curves (CNCs), while circles of radius r > r H are Closed Timelike Curves (CTCs). The time taken by the lightrays to return to the point of their origin is [3] - T = 2 πω [ √ − (7)Accordingly if in G , -1. One drops the flat x coordinate, which is directed along the rotationaxis, and2. Compactifies the time coordinate x , to S , and3. Chooses the time period of S time factor equal to T = πω [ √ − ;one obtains this new example of (2 + 1) -dimensional Zollfrei metric withthe R × S topology.G ¨odel [1] gave a list of nine interesting properties of G , . Remarkably,in the fifth property, he considered the possibility of both an open R anda closed S time coordinate. References [1] G ¨odel, K.: