Effective geometry of a white dwarf D. Bini Istituto per le Applicazioni del Calcolo ‘‘M. Picone,’’ CNR, I-00185 Rome, Italy and ICRA, University of Rome ‘‘La Sapienza,’’ I-00185 Rome, Italy C. Cherubini and S. Filippi Nonlinear Physics and Mathematical Modeling Lab, Engineering Faculty, University Campus Bio-Medico, I-00128 Rome, Italy and ICRA, University of Rome ‘‘La Sapienza,’’ I-00185 Rome, Italy (Received 4 November 2010; published 29 March 2011) The ‘‘effective geometry’’ formalism is used to study the perturbations of a white dwarf described as a self-gravitating fermion gas with a completely degenerate relativistic equation of state of barotropic type. The quantum nature of the system causes an absence of homological properties, manifested instead by polytropic stars, and requires a parametric study of the solutions both at the numerical and analytical level. We have explicitly derived a compact analytical parametric approximate solution of Pade ´ type, which gives density curves and stellar radii in good accordance with already existing numerical results. After validation of this new type of approximate solutions, we use them to construct the effective acoustic metric governing general perturbations following Chebsch’s formalism. Even in this quantum case, the stellar surface exhibits a curvature singularity due to the vanishing of density, as already evidenced in past studies on nonquantum self-gravitating polytropic stars. The equations of the theory are finally numeri- cally integrated in the simpler case of irrotational spherical pulsating perturbations, including the effect of backreaction, in order to have a dynamical picture of the process occurring in the acoustic metric. DOI: 10.1103/PhysRevD.83.064039 PACS numbers: 04.20.Cv, 51.40.+p I. INTRODUCTION General relativity introduced a century ago the concept of curved space-time as the natural arena for any physical phenomena. Only recently, however, scientists not neces- sarily involved in relativistic astrophysics have been at- tracted by the mathematical structure of this theory because of the emergence of analog geometries in non- relativistic contexts of condensed matter physics. Historically, the perturbative study of an equilibrium con- figuration of a given perfect barotropic and irrotational Newtonian fluid using an ‘‘effective space-time metric’’ has been formerly addressed by Unruh [1] and further extended by Visser [2] in view of possible analog contexts in which the so-called ‘‘Hawking effect’’ is probed, leading to black hole evaporation. Only recently, there has been experimental evidence of this effect in analog metrics experimentally associated with optical media stimulated by laser pulses [3] and in the case of shallow water waves [4], drawing the general attention on this new point of view. In short, the analog geometry approach is characterized by the fact that the equations satisfied by the perturbed quan- tities in fluids or nonlinear optical media are unified in a linear second-order hyperbolic equation with nonconstant coefficients conveniently cast in the form of a massless scalar field motion on a curved geometry and leading in this way to an ‘‘effective gravity’’ (we refer to Refs. [5–7] for an overview on the subject). This theoretical framework has been extended to the study of irrotational perturbations of spherically symmetric self-gravitating polytropic con- figurations for different values of the polytropic index n, considering the backreaction of hydrodynamics and gravi- tation [8]. Further extensions [9] were then introduced in order to include the background vorticity via a general- ization of the Clebsch formalism [10], by perturbing the distorted analytical configuration of a uniformly rotating polytrope with index n ¼ 1 found by Williams [11]. Here, we focus now on a generalization of these previous studies on classical matter by taking into account a nonpolytropic equation of state (a barotrope, in short) for a totally degen- erate quantum Fermi electron gas in spherical symmetry, i.e., a white dwarf (WD) [12]. These systems exhibit a number of differences in comparison with polytropic con- figurations. First of all, the relativistic properties of the equation of state play a significant role, even in the case of small mass configurations. Moreover, the quantum nature of these totally degenerate configurations leads to a finite mass configuration, totally shrunk into a point of infinite density (Chandrasekhar’s limit mass). Another important characteristic of these quantum systems is the absence of an homology property of the solutions, present instead in the case of the Lane-Emden polytropic stars. Stated in a more physical way, each mass has its proper density dis- tribution which cannot be inferred from a density configu- ration for a system of different mass. There are many aspects which suggest then a study of this system by using the analog geometry approach. In order to do so, we shall first need to find a numerical and analytical approximate solution for the nonlinear equation describing the stellar density. This task is accomplished by introducing a com- pact parametric Pade ´ approximated solution, which allows PHYSICAL REVIEW D 83, 064039 (2011) 1550-7998= 2011=83(6)=064039(15) 064039-1 Ó 2011 American Physical Society