Liouvillian integrability of gravitating static isothermal fluid spheres
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, autonomous second order ordinary differential equation (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 autonomous LotkaVolterra quadratic polynomial differential system in {R}^2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the LotkaVolterra 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 LotkaVolterra system, and consequently of the second order ODE for the mass function must be nonLiouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 October 2014
 DOI:
 10.1063/1.4897213
 Bibcode:
 2014JMP....55j2501I