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UPPER AND LOWER BOUNDS FOR THE VOLUME OF A COMPACT SPACELIKE HYPERSURFACE IN A GENERALIZED ROBERTSON–WALKER SPACETIME

JUAN ÁNGEL ALEDO

Departamento de Matemáticas, E. S. I. Informática, Universidad de Castilla-La Mancha, 02071 Albacete, Spain

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ALFONSO ROMERO

Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain

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, and

RAFAEL M. RUBIO

Departamento de Matemáticas, Universidad de Córdoba, 14071 Córdoba, Spain

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https://doi.org/10.1142/S0219887814500066Cited by:9 (Source: Crossref)

Abstract

We provide upper and lower bounds for the volume of a compact spacelike hypersurface in an (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime in terms of the volume of the fiber, the hyperbolic angle function and the warping function. Under several geometrical and physical assumptions, we characterize the spacelike slices as the only spacelike hypersurfaces where these bounds are attained. As a consequence of these results, we get an upper bound for the first eigenvalue of a compact spacelike surface in a three-dimensional GRW spacetime whose fiber is a topological sphere, which includes the case of the three-dimensional De Sitter spacetime, and show that the bound is attained if and only if M is a spacelike slice.

Keywords:

AMSC: 53C50, 53C80

References

J. Á. Aledo and L. J. Alí as, Proc. Amer. Math. Soc. 130, 1145 (2002), DOI: 10.1090/S0002-9939-01-06133-0. Web of Science, Google Scholar
L. J. Alías, A. Romero and M. Sánchez, Gen. Relat. Grav. 27, 71 (1995), DOI: 10.1007/BF02105675. Web of Science, Google Scholar
A. Borde, Phys. Rev. D 50(6), 3692 (1994), DOI: 10.1103/PhysRevD.50.3692. Web of Science, Google Scholar
M. Caballero, A. Romero and R. Rubio, Class. Quantum Grav. 28, 145009 (2011), DOI: 10.1088/0264-9381/28/14/145009. Web of Science, Google Scholar
V. Daftardar and N. Dadhich, Gen. Relat. Grav. 26, 859 (1994), DOI: 10.1007/BF02107144. Web of Science, Google Scholar
D. Earlyet al., Commun. Math. Phys. 106, 137 (1986). Web of Science, Google Scholar
R. P. Geroch, Commun. Math. Phys. 13, 180 (1969), DOI: 10.1007/BF01645486. Web of Science, Google Scholar
J. Hersch, C. R. Acad. Sci. Paris Sér. A–B 270, A1645 (1970). Google Scholar
B. O'Neill , Semi-Riemannian Geometry with Applications to Relativity ( Academic Press , 1983 ) . Google Scholar
M. Rainer and H.-J. Schmidt, Gen. Relat. Grav. 27, 1265 (1995), DOI: 10.1007/BF02153317. Web of Science, Google Scholar
M. J. Reboucas, J. E. F. Skea and R. K. Tavakol, J. Math. Phys. 37, 858 (1996). Web of Science, Google Scholar
R. K. Sachs and H. Wu , General Relativity for Mathematicians , Graduate Texts in Mathematics 48 ( Springer-Verlag , New York , 1977 ) . Google Scholar