Metric completeness of C(p, q) in a globally hyperbolic spacetime
B.S. Choudhury, H.S. Mondal1, B.S. Choudhury, H.S. Mondal2
(1) Department of Mathematics, Bengal Engineering and Science University, Shibpur, P.O. - B. Garden, Howrah, 711103, West Bengal, India
(2) Department of Mathematics, Midnapore College, Midnapore, Paschim Medinipur, 721101, West Bengal, India
We consider a Lorentzian manifold M which is globally hyperbolic. We define a metric on C(p, q), the set of all equivalence classes of causal curves connecting two causally related points p and q. We show that C(p, q) is a complete metric space with the metric thus defined. Here, by completeness we mean that every Cauchy sequence (a sequence with a tendency to converge) in C(p, q) finds a point in it to converge. We also give an example to show that the result does not hold in general when the spacetime is not globally hyperbolic. The work is in line with research on causality in relativistic spacetimes.
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- S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973).
- R. Penrose, Techniques of Differential Topology in Relativity (CBMS-NSF Regional conference Series in Applied Mathematics, Philadelphia: SIAM, 1972).
- R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
- R. Geroch, J. Math. Phys. 11, 437 (1970).
- A. N. Bernal and M. Sö?nchez, Commun. Math. Phys. 243, 461 (2003).
- A. N. Bernal and M. Sö?nchez, Commun. Math. Phys. 257, 43 (2005).
- R. J. Low, Lect. Notes Phys. 692, 35 (2006).
- R. J. Low, Class. Quantum Grav. 7, 943 (1990).
- H. J. Seifert, Zs. f. Naturfor. 22a, 1356 (1967).
- Burago et al., A Course in Metric Geometry (Graduate Studies in Mathematics, American Mathematical Society, 33, ISSN 1065-7339, 2001).
- J. K. Beem, P. E. Ehrlich, and K. L. Easley, Global Lorentzian Geometry (Dekker, New York, 1996).
- E. Minguzzi, J. Math. Phys. 49, 092501 (2008).
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