arXiv:1012.0154v1 [astro-ph.SR] 1 Dec 2010 The relativistic Feynman-Metropolis-Teller theory for white-dwarfs in general relativity Jorge A. Rueda, Michael Rotondo, Remo Ruffini, ∗ and She-Sheng Xue Dipartimento di Fisica and ICRA, Sapienza Universit` a di Roma, P.le Aldo Moro 5, I–00185 Rome, Italy and ICRANet, P.zza della Repubblica 10, I–65122 Pescara, Italy (Dated: December 2, 2010) The recently formulation of the relativistic Thomas-Fermi model within the Feynman-Metropolis- Teller theory for compressed atoms, is applied to the study of general relativistic white-dwarf equi- librium configurations. The equation of state, which takes into account the beta equilibrium and the Coulomb interaction between the nuclei and the surrounding electrons, is obtained as a function of the compression by considering each atom constrained in a Wigner-Seitz cell. The contribution of quantum statistics, weak and electromagnetic interaction is obtained by the determination of the chemical potential of the Wigner-Seitz cell. The further contribution of the general relativistic equilibrium of white-dwarf matter is expressed by the simple formula √ g00µws = constant, which links the chemical potential of the Wigner-Seitz cell µws with the general relativistic gravitational potential g00 at each point of the configuration. The configuration outside each Wigner-Seitz cell is strictly neutral and therefore no global electric field is necessary to warranty the equilibrium of the white-dwarf. These equations modify the ones used by Chandrasekhar by taking into due account the Coulomb interaction between the nuclei and the electrons as well as inverse beta decay. They also generalize the work of Salpeter by considering a unified self-consistent approach to the Coulomb interaction in each Wigner-Seitz cell. The consequences on the numerical value of the Chandrasekhar-Landau mass limit are presented. The modifications of the mass-radius relation for 4 He and 56 Fe white-dwarf equilibrium configurations are also presented. These effects should be taken into account in processes requiring a precision knowledge of the white-dwarf parameters. Keywords: Relativistic Thomas-Fermi model – equation of state of white-dwarf matter – white-dwarf equi- librium configurations in general relativity I. INTRODUCTION The necessity of introducing the Fermi-Dirac statis- tics in order to overcome some conceptual difficulties in explaining the existence of white-dwarfs leading to the concept of degenerate stars was first advanced by R. H. Fowler in a classic paper [1]. Following that work, E. C. Stoner [2] introduced the effect of special relativ- ity into the Fowler considerations and, using what later became known as the exclusion principle, generally at- tributed in literature to Wolfgang Pauli, he discovered the concept of critical mass of white-dwarfs [52] M Stoner crit = 15 16 √ 5π M 3 Pl µ 2 m 2 n ≈ 3.72 M 3 Pl µ 2 m 2 n , (1) where M Pl =  c/G ≈ 10 −5 g is the Planck mass, m n is the neutron mass, and µ = A/Z ≈ 2 is the average molecular weight of matter which shows explicitly the dependence of the critical mass on the chemical compo- sition of the star. Following the Stoner’s work, S. Chandrasekhar [3] at the time a 20 years old graduate student coming to Cambridge from India pointed out the relevance of de- scribing white-dwarfs by using an approach, initiated by E. A. Milne [4], of using the powerful mathematical ∗ Electronic address: ruffini@icra.it method of the solutions of the Lane-Emden polytropic equations [5]. The same idea of using the Lane-Emden equations taking into account the special relativistic ef- fects to the equilibrium of stellar matter for a degenerate system of fermions, came independently to L. D. Lan- dau [6]. Both the Chandrasekhar and Landau treatments were explicit in pointing out the existence of the critical mass M Ch−L crit =2.015 √ 3π 2 M 3 Pl µ 2 m 2 n ≈ 3.09 M 3 Pl µ 2 m 2 n , (2) where the first numerical factor on the right hand side of Eq. (2) comes from the boundary condition −(r 2 du/dr) r=R =2.015 (see last entry of Table 7 on Pag. 80 in [5]) of the n = 3 Lane-Emden polytropic equa- tion. Namely for M>M Ch−L crit , no equilibrium configu- ration should exist. This unexpected result created a wave of emotional reactions: Landau rejected the idea of the existence of such a critical mass as a “ridiculous tendency” [6]. Chan- drasekhar was confronted by a lively dispute with A. Ed- dington on the basic theoretical assumptions he adopted (see [7] for historical details). The dispute reached such a heated level that Chandrasekhar was confronted with the option either to change field of research or to leave Cambridge. As is well known he chose the second option transferring to Yerkes Observatory near Chicago where he published his results in his classic book [8]. Some of the basic assumptions adopted by Chan- drasekhar and Landau in their idealized approach were