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

Data: 2014
Resum: 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.
Nota: Número d'acord de subvenció MINECO/MTM2008-03437
Nota: Número d'acord de subvenció AGAUR/2009/SGR-410
Nota: Número d'acord de subvenció EC/FP7/2012/318999
Nota: Número d'acord de subvenció EC/FP7/2012/316338
Nota: Agraïments: FEDER/UNAB10-4E-378.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: Gravitating static isothermal fluid spheres ; Liouvillian integrability ; Lotka-Volterra system
Publicat a: Journal of Mathematical Physics, Vol. 55 (2014) , p. 102501, ISSN 0022-2488

DOI: 10.1063/1.4897213


Preprint
25 p, 433.3 KB

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