Liouvillian integrability of gravitating static isothermal fluid spheres
Iacono, Roberto (ENEA-C. R. Casaccia(Italy))
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2014
Abstract:	We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, authonomous second order ODE for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent authonomous Lotka-Volterra quadratic polynomial differential system in R2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka-Volterra system possesses Liouvillian first integrals for n = −1, −3, −5, which descend from the existence of invariant algebraic curves of degree one, and for n = −6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka-Volterra system, and consequently of the second order ODE for the mass function must be non Liouvillian. This makes quite unlikely the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410 European Commission 318999 European Commission 316338
Note:	Agraïments: FEDER/UNAB10-4E-378.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Gravitating static isothermal fluid spheres ; Liouvillian integrability ; Lotka-Volterra system
Published in:	Journal of Mathematical Physics, Vol. 55 (2014) , p. 102501, ISSN 1089-7658

DOI: 10.1063/1.4897213

Postprint 25 p, 433.3 KB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles