Continuous representation of a globally hyperbolic spacetime with non-compact Cauchy surfaces. (English) Zbl 1327.83019
Summary: In this paper we consider a Lorentzian manifold which is globally hyperbolic with a non-compact Cauchy surface. We show that continuous representation of the spacetime is possible by causally admissible systems of its Cauchy surfaces. For that purpose we use Vietoris topology. One application is also included. The work is in the line with research on causality in relativistic spacetimes.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C10 | Equations of motion in general relativity and gravitational theory |
| 06B35 | Continuous lattices and posets, applications |
| 54F65 | Topological characterizations of particular spaces |
| 53Z05 | Applications of differential geometry to physics |
| 03A05 | Philosophical and critical aspects of logic and foundations |
Keywords:
lorentzian geometry; spacetime; causality; strong causality; global hyperbolicity; Cauchy surface; Vietoris topology; causally admissible systemReferences:
| [1] | Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461 (2003) · Zbl 1085.53060 · doi:10.1007/s00220-003-0982-6 |
| [2] | Bernal, A.N., Sánchez, M.: Smoothness ot time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys. 257, 43 (2005) · Zbl 1081.53059 · doi:10.1007/s00220-005-1346-1 |
| [3] | Esty, \[N.C.: CL(\mathbb{R})\] CL(R) is simply connected under Vietoris topology, applied general. Topology 8(2), 259-265 (2007) · Zbl 1144.54005 |
| [4] | Filippov, V.V.: Basic topological structures of ordinary differential equations. Kluwer Academic Publishers, Netherlands (1998) · Zbl 0905.34001 · doi:10.1007/978-94-017-0841-8 |
| [5] | Geroch, R.: Domain of dependence. J. Math. Phys. 11, 437 (1970) · Zbl 0189.27602 · doi:10.1063/1.1665157 |
| [6] | Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time. Cambridge University Press, Cambridge (1973) · Zbl 0265.53054 · doi:10.1017/CBO9780511524646 |
| [7] | Kim, D.-H.: A note on non-compact Cauchy surfaces. Class. Quantum Gravity 25, 238002 (2008) · Zbl 1155.83006 · doi:10.1088/0264-9381/25/23/238002 |
| [8] | Kim, D.-H.: An Imbedding of Lorentzian manifolds. Class. Quantum Gravity 26, 075004 (2009) · Zbl 1161.83308 |
| [9] | Penrose, R.: Techniques of Differential Topology in Relativity. SIAM, Philadelphia (1972) · Zbl 0321.53001 · doi:10.1137/1.9781611970609 |
| [10] | Vatandoost, M., Bahrampour, Y.: Some necessary and sufficient conditions for admitting a continuous sphere order representation of two-dimensional space-times. J. Math. Phys. 53, 122501 (2012) · Zbl 1278.83037 · doi:10.1063/1.4761822 |
| [11] | Wald, M.: General Relativity. University of Chicago Press, Chicago (1984) · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001 |
| [12] | Minguzzi, E., Sanchez, M.: The causal hierarchy of spacetimes. ESI lect. Math. Phys. 299-358 (2008) · Zbl 1148.83002 |
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