We have presented an analytic theory for the Casimir force on a Bose-Einstein condensate (BEC) which is confined between two parallel plates. We have considered Dirichlet boundary conditions for the condensate wave function as well as for the phonon field. We have shown that, the condensate wave function (which obeys the Gross-Pitaevskii equation) is responsible for the mean field part of Casimir force, which usually dominates over the quantum (fluctuations) part of the Casimir force.

The stationary background flow in the spherically symmetric infall of a compressible fluid, coupled to the space-time defined by the static Schwarzschild metric, has been subjected to linearized perturbations. The perturbative procedure is based on the continuity condition and it shows that the coupling of the flow with the geometry of space-time brings about greater stability for the flow, to the extent that the amplitude of the perturbation, treated as a standing wave, decays in time, as opposed to the amplitude remaining constant in the Newtonian limit. In qualitative terms this situation simulates the effect of a dissipative mechanism in the classical Bondi accretion flow, defined in the Newtonian construct of space and time. As a result of this approach it becomes impossible to define an acoustic metric for a conserved spherically symmetric flow, described within the framework of Schwarzschild geometry. In keeping with this view, the perturbation, considered separately as a high-frequency travelling wave, also has its amplitude reduced.

A general method is described by which exact solutions of Einstein's field equations are obtained for a nonstatic spherically symmetric distribution of a perfect fluid. In addition to the previously known solutions which are systematically derived, a new set of exact solutions is found, and the dynamical behaviour of the corresponding models is briefly discussed.Nonstatic solutions for spherically symmetric systems containing a perfect fluid of inhomogeneous density and pressure have been obtained in isotropic coordinates by several authors previously (McVittie 1967; Nariai 1967;Faulkes 1969;Banerjee and Banerji 1975). Faulkes found a very simple solution that gave a collapsing model by solving the differential equation resulting from isotropy of pressure. Kuchowicz (1972) utilized such an equation to obtain exact solutions corresponding to static fluid spheres. In the present note we have applied a special technique to solve this equation and have systematically derived the previous solutions of Faulkes, Nariai and Banerjee and Banerji as special cases. In addition, under more general conditions, we have obtained a new set of exact solutions satisfying Einstein's field equations for a spherically symmetric distribution of a perfect fluid which can be matched with the exterior Schwarzschild solution at the boundary. It is found that the models may collapse, bounce or oscillate, depending on the boundary conditions. Integration of Field EquationsWe consider the isotropic form of the line element where v and OJ are functions of rand t. Assuming that the fluid is perfect and using comoving coordinates, we find that Einstein's field equations give (Faulkes 1969) (la, b) where the first relation is obtained from isotropy of pressure and the second follows from the condition T14 = O. In equations (1), x = r2, R = e-tro , the subscript 1 denotes differentiation with respect to x and a dot indicates a time derivative. A substitution of the form R = {~(r, t) + 8}v (x)

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SynopsisThe Ambarzumyan theorem connecting the Sturm–Liouville problem and the corresponding problem associated with the Fourier differential equation is extended to a class of second order matrix differential systems.