On the Area of the Symmetry Orbits of the Einstein-Vlasov-Scalar Field System with Plane and Hyperbolic Symmetry

D. Tegankong

Abstract


We prove in the case of cosmological models for the Einstein-Vlasov-scalar field system, that the area radius of compact hypersurfaces tends to a constant value as the past boundary of the maximal Cauchy development is approached. In other case, there is at least one Cauchy hypersurface of constant areal time coordinate in plane and hyperbolic symmetric spacetimes. Moreover, we show that the areal time coordinate $R=t$ which covers these spacetimes runs from zero at infinity with the singularity occuring at $R=0$. The sources of the equations are generated by a distribution function and a massless scalar field, subject to the Vlasov and wave equations respectively.

Keywords


Einstein; Vlasov; Scalar field; Areal coordinates; Surface symmetry; Hyperbolic differential equations; Global existence

Full Text:

PDF

References


H. Andréasson, Global foliations of matter spacetimes with Gowdy symmetry, Commun. Math. Phys. 206 (1999), 337-366.

H. Andréasson, G. Rein and A.D. Rendall, On the Einstein-Vlasov system with hyperbolic symmetry, Math. Proc. Camb. Phil. Soc. 124 (2003), 529-549.

J. Ehlers, A survey of General Relativity Theory Astrophysics and cosmology, W. Israel (ed.), 1-125, Reidel, Dordrecht, 1973.

A.D. Rendall, Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry, Class. Quantum Grav. 12 (1995), 1517-1533.

J. Smulevici, On the area of the symmetry orbits of cosmological spacetimes with toroidal or hyperbolic symmetry, Analysis & PDE 4 (2) (2011), 191-245.

D. Tegankong, N. Noutchegueme and A.D. Rendall, Local existence and continuation criteria for solutions of the Einstein-Vlasov-scalar field system with surface symmetry, Journ. Hyperb. Diff. Eq. 1(4) (2004), 691-724.

D. Tegankong, Global existence and asymptotic behaviour for solutions of the Einstein-Vlasov-scalar field system with surface symmetry, Class. Quantum Grav. 22 (2005), 2381–-2391.

D. Tegankong and A.D. Rendall, On the nature of initial singularities for solutions of the Einstein-Vlasov-scalar field system with surface symmetry, Math. Proc. Camb. Phil. Soc. 141 (2006), 547-562.

M. Weaver, On the area of the symmetry orbits in $T^2$ symmetric spacetimes with Vlasov matter, Class. Quantum Grav. 21 (2004), 1079.




DOI: http://dx.doi.org/10.26713%2Fjims.v5i3.219

eISSN 0975-5748; pISSN 0974-875X