×

Recovering the geometry of a flat spacetime from background radiation. (English) Zbl 1298.83125

Summary: We consider globally hyperbolic flat spacetimes in 2 + 1 and 3 + 1 dimensions, in which a uniform light signal is emitted on the \(r\)-level surface of the cosmological time for \(r \to 0\). We show that the frequency shift of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2 + 1, we show that this observed frequency shift function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this frequency shift function for a few simple examples.

MSC:

83F05 Relativistic cosmology
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C80 Analogues of general relativity in lower dimensions
54F65 Topological characterizations of particular spaces
83C35 Gravitational waves
83C75 Space-time singularities, cosmic censorship, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andersson, L., Barbot, T., Benedetti, R., Bonsante, F., Goldman William, M., Labourie, François, Scannell, K.P., Schlenker, J.-M.: Notes on: “Lorentz spacetimes of constant curvature” [Geom. Dedicata 126, 3-45 (2007); mr2328921] by G. Mess. Geom. Dedicata 126, 47-70 (2007) (MR MR2328922) · Zbl 1126.53042
[2] Apanasov, B.N.: Deformations of conformal structures on hyperbolic manifolds. J. Differ. Geom. 35(1), 1-20 (1992) (MR 1152224 (92k:57042)) · Zbl 0770.53025
[3] Barbot, T.: Globally hyperbolic flat space-times. J. Geom. Phys. 53(2), 123-165 (2005) (MR 2110829 (2006d:53085)) · Zbl 1087.53065
[4] Benedetti, R., Bonsante, F.: Canonical Wick rotations in 3-dimensional gravity. Memoirs of the American Mathematical Society, vol. 198, 164pp (2009) (math.DG/0508485) · Zbl 1165.53047
[5] Benedetti, R., Guadagnini, E.: Geometric cone surfaces and (2 + 1)-gravity coupled to particles. Nuclear Phys. B 588(1-2), 436-450 (2000) (MR MR1787158 (2001g:83094)) · Zbl 1009.83033
[6] Bonahon, F.: Geodesic laminations on surfaces. In: Laminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998). Contemporary Mathematics, vol. 269, pp. 1-37. American Mathematical Society, Providence (MR 1810534 (2001m:57023)) · Zbl 0996.53029
[7] Bonsante, F.: Flat spacetimes with compact hyperbolic Cauchy surfaces. J. Differ. Geom. 69(3), 441-521 (2005) (MR MR2170277 (2006h:53068)) · Zbl 1094.53063
[8] Carlip, S.: Quantum gravity in 2 + 1 dimensions. In: Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1998) (MR 1637718 (99e:83026)) · Zbl 0919.53024
[9] Cornish, N.J., Spergel, D.N., Starkman, G.D.: Measuring the topology of the universe. Proc. Natl. Acad. Sci. 95(1), 82-84 (1998)
[10] Davis, M.W.: The geometry and topology of Coxeter groups. London Mathematical Society Monographs Series, vol. 32. Princeton University Press, Princeton, NJ (2008) (MR 2360474 (2008k:20091)) · Zbl 1142.20020
[11] Epstein, D.B.A., Marden, A.: Convex hulls in hyperbolic spaces, a theorem of Sullivan, and measured pleated surfaces. In: Epstein, D.B.A. (ed.) Analytical and Geometric Aspects of Hyperbolic Space. L.M.S. Lecture Note Series, vol. 111. Cambridge University Press, Cambridge (1986) · Zbl 0612.57010
[12] Goldman, W.M., Margulis, G.A.: Flat Lorentz 3-manifolds and cocompact Fuchsian groups. In: Crystallographic Groups and their Generalizations (Kortrijk, 1999). Contemporary Mathematics, vol. 262, pp. 135-145. American Mathematical Society, Providence, RI (2000) (MR 1796129 (2001m:53124)) · Zbl 1039.53074
[13] Richard Gott, J. III: Topology and the universe. In: Classical Quantum Gravity 15(9), 2719-2731 (1998). Topology of the Universe Conference (Cleveland, OH, 1997). MR 1649670 (99g:83084) · Zbl 0939.83072
[14] Richard Gott, J. III, Melott, A.L., Dickinson, M.: The sponge-like topology of large-scale structure in the universe. Astrophys. J. 306(2), part 1, 341-357 (1986) (MR 861880 (87k:85005))
[15] Johnson, D., Millson, J.J.: Deformation spaces associated to compact hyperbolic manifolds. In: Discrete Groups in Geometry and Analysis (New Haven, Conn., 1984). Progress in Mathematics, vol. 67, pp. 48-106. Birkhäuser, Boston (1987) (MR 900823 (88j:22010))
[16] Kapovich, M.: Deformations of representations of discrete subgroups of SO(3,1). Math. Ann. 299(2), 341-354 (1994) (MR 1275772 (95d:57010)) · Zbl 0828.57009
[17] Kourouniotis, C.: Deformations of hyperbolic structures. Math. Proc. Cambridge Philos. Soc. 98(2), 247-261 (1985) (MR 795891 (87g:32022)) · Zbl 0577.53041
[18] Luminet, J.P., Weeks, J.R., Riazuelo, A., Lehoucq, R., Uzan, J.P.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background. Nature 425(6958), 593-595 (2003)
[19] Mess, G.: Lorentz spacetimes of constant curvature. Geom. Dedicata 126, 3-45 (2007) (MR MR2328921) · Zbl 1206.83117
[20] Meusburger, C.: Cosmological measurements, time and observables in (2 + 1)-dimensional gravity. Class. Quantum Gravity 26(5), 055006, 32 (2009) (MR 2486312 (2010f:83078)) · Zbl 1160.83341
[21] Savaré G., Tomarelli F.: Superposition and chain rule for bounded Hessian functions. Adv. Math. 140, 237-281 (1998) · Zbl 0919.49001 · doi:10.1006/aima.1998.1770
[22] Scannell, K.P.: Infinitesimal deformations of some SO(3,1) lattices. Pacific J. Math. 194(2), 455-464 (2000) (MR 1760793 (2001c:57018)) · Zbl 1019.57007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.