Axially Symmetric Type N Space-Time with Causality Violating Curves and the von Zeipel Cylinder
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Axially Symmetric Type N Space-Time with Causality Violating Curves and the von Zeipel Cylinder Faizuddin Ahmed* Ajmal College of Arts and Science, Dhubri-783324, Assam, India Received February 13, 2020; revised May 10, 2020; accepted May 14, 2020
Abstract—An axially symmetric nonvacuum solution of the Einstein field equations, regular everywhere and free from curvature divergence is presented. The matter-energy content is a the pure radiation field satisfying the energy conditions, and the metric is of type N in the Petrov classification scheme. The spacetime develops circular closed timelike curves everywhere outside a finite region of space i.e., beyond a null curve. Furthermore, the physical interpretation of the solution based on the study of the equations of geodesic deviation is presented. Finally, the von Zeipel cylinders with respect to the Zero Angular Momentum Observers (ZAMOs) is discussed. In addition, circular null and timelike geodesic of pacetime are also presented. DOI: 10.1134/S0202289320030020
1. INTRODUCTION It is very hard to find exact solutions of the Einstein field equations without imposing symmetry conditions. The exact solutions with different symmetry, such as the Schwarzschild, de Sitter, anti-de Sitter, and Einstein static solution are known after A. Einstein in 1915 proposed the General Theory of Relativity (GTR). The first rotating cylindrically symmetric space-time was found by Lewis [1]. van Stockum [2] in 1937 presented a rotating dust cylinder containing closed timelike curves (CTC), the solution of the field equations obtained long before by Lanczos (in 1924), and subsequently analyzed by Tipler [3] ¨ to have CTC. In 1949, Godel presented a rotating cosmological model with a pressureless dust solution of the field equations which admits CTC and closed null geodesics (CNG) [4]. It was noted that the Kerr metric (rotating uncharged black holes) contain CTC [5]. Lorentzian wormholes models, discussed, e.g., by Morris et al., have CTC [6–8]. Other example of CTC space-times would be [9–29]. Of these, the space-time where CTC form everywhere beyond a null curve for some values of the radial coordinate r, is known as eternal time-machine space-time. In some others, CTC appear after a certain instant of time in a causally well-behaved manner, and are known as time machine space-times. The present work comprises the following: in Section 2, an axially symmetric non-vacuum solution of the field equations, admitting closed timelike curves, *
is analyzed; in Section 3, the physical interpretation of the solution is presented; in Section 4, the von Zeipel cylinders with respect to ZAMO are discussed; in Section 5, circular null and timelike geodesics of the metric are presented, and final conclusions are formulated in Section 6. Our conventions are as follows. Greek indices take values 0, 1, 2, 3, and Einstein’s summation convention is used. Our choice of signature is (−, +, +, +). 2. A 4D CURVED SPACE-TIME AND ITS ANALYSIS Consider the following axially symmetric metric in the coordinates (x0 = v, x1
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