MGD Dirac stars

The method of geometric deformation (MGD) is here employed to study compact stellar configurations, which are solutions of the effective Einstein-Dirac coupled field equations on fluid branes. Non-linear, self-interacting, fermionic fields are then employed to derive MGD Dirac stars, whose properties are analyzed and discussed. The MGD Dirac star maximal mass is shown to increase as an specific function of the spinor self-interaction coupling constant, in a realistic model involving the most strict phenomenological current bounds for the brane tension.


I. INTRODUCTION
The method of geometrical deformation (MGD) is a protocol emerging in the context of the AdS/CFT membrane paradigm, having general relativity (GR) as a limit that corresponds to rigid branes [1][2][3][4]. The MGD well describes compact stellar configurations in braneworld scenarios [5][6][7].
The MGD has the brane tension, σ, as a running parameter that encrypts Kaluza-Klein modes and gravity in the bulk as well. A rigid brane corresponds to the GR, σ → ∞, limit. As the brane emulates the universe we live in, cosmological expansion can be implemented from the brane deformation, due to a finite brane tension, into the warped additional dimension of the bulk. In fact, realistic brane-world models take into account a variable brane tension, that is proportional to the brane temperature, according to the Eötvös law [8][9][10].
Besides, the MGD was employed in the context of the generalized uncertainty principle to study the MGD black hole thermal spectrum [43].
The so called Dirac stars encompass compact stellar distributions generated by fermionic fields [44,45]. Exact solutions for self-gravitating Dirac fields were studied in Ref. [46]. Refs. [47][48][49] also scrutinize fermionic fields in gravity. Besides, Ref. [50] discussed Dirac stars generated by non-Abelian gauge fields. The main aim of this work is to study MGD compact stellar configurations, that are solutions of the effective Einstein-Dirac coupled field equations. The MGD-decoupling method is employed to correct the effective energy-momentum tensor on the brane by fermionic background effects. Non-linear, self-interacting, massive spinor fields are used to derive physical properties of MGD Dirac stellar configurations, with respect to both the finite brane tension and the spinor self-interaction coupling constant. The MGD Dirac star stability is also discussed. This paper is organized as follows: Sect. II is devoted to introduce a brief review, regarding the MGD applied to stellar distributions on a fluid brane, ruled by a variable tension that encodes the cosmological evolution. In Sect. III the Einstein-Dirac coupled system of ODEs is solved for self-interacting fermionic fields, having the MGD as a natural input. Then, the solutions of the coupled system of ODEs represent MGD Dirac stars, whose observational features are scrutinized and illustrated in several regimes. In Sect. IV the conclusions and perspectives are presented.

II. THE MGD SETUP AND FLUID BRANES
The MGD procedure is constructed to derive high energy corrections to GR [2,4,24]. Fluid branes have a variable tension that emulates cosmological evolution [8,10]. The extended MGD has been recently employed to derive, in the context of the quantum portrait of black holes, the strictest brane tension bound σ 2.81 × 10 6 MeV 4 [19].
where G µν = R µν − 1 2 Rg µν is the Einstein tensor and Λ stands for the cosmological running parameter on the brane. The energy-momentum tensor in (1) can be split into the following components, The term T µν denotes the energy-momentum encoding all kind of matter on the brane and E µν is the electric component of the bulk Weyl curvature tensor, taking account the average over the two edges of the Z 2 symmetric brane, whereas S µν is a tensor whose intricate expression involves quadratic terms of energy-momentum of the brane. Besides, the term L µν comes from an eventual asymmetric embedding of the brane into the bulk, whereas P µν regards the pull back of fields in the bulk that are beyond the standard model, encompassing moduli fields, dilatons and quantum radiation, for instance [10].
Stellar configurations that can be modeled by solutions of Eq. (1) are static and spherically symmetric, described by a metric of type The MGD metric was initially derived in Refs. [4,11,12], reading for a stellar configuration of mass M , where [11] f (r) = 1 and is the effective stellar configuration radius, for ρ(r) being the compact stellar configuration energy density [2]. The brane tension σ in Eq. (4) governs the influence of bulk effects onto the brane, carrying information about the bulk Weyl fluid bathing the brane [17]. After neglecting the infinitesimal values of higher order powers of the small observational brane tension, the MGD metric can be rewritten as [2,11] In the general-relativistic limit σ → ∞, the Schwarzschild metric is recovered from the MGD metric.

III. DIRAC STARS ON FLUID BRANES
Compact gravitating configurations on fluid branes, modeled by the MGD with a background spinor field, ψ, of mass m, can be described by the Einstein field equations (1), coupled to a Dirac equation. To the action that generates the Einstein field equations, it must be added the spinor where self-interacting spinor fields are implemented by Here {γ µ } is the set of gamma matrices, that satisfies the Clifford algebra {γ µ , γ ν } = 2g µν I 4×4 , where ζ is a small parameter driving the MGD decoupling, and the fermionic energy-momentum tensor readsT (12) represents the brane Einstein field equations (1), now corrected by the MGD-decoupling for the spinor energy-momentum tensorT µν in Eq. (15).
In order to describe the background spinor field, the following two stationary ansätze for ψ are compatible with the metric (3) [44,45], given by where α(r), β(r) are real functions. When one replaces (16) into the field equations (12,13), it yields the coupled systems of ODEs involving also the MGD metric (3): where, denoting λ 0 = /mc, the following quantities are defined [45]: The mass of compact stellar configurations, M ∼ M 2 p m , is usually studied when the spinor field mass, m, is much smaller than the Chandrasekhar mass, In fact, as the most strict current bound on the variable brane tension is σ 2.81 × 10 6 MeV 4 [19], then in what follows we take the lower brane tension limit σ ∼ 3 × 10 6 MeV 4 , together with σ ∼ 10 9 MeV 4 and σ ∼ 10 12 MeV 4 , to study the physical differences among these cases. It is worth to emphasize that the general-relativistic limit corresponds to a rigid brane, making σ → ∞ and ζ → ∞. In Fig. 1, for fixed values of the spinor coupling constantλ, there is a peak of the mass, at some value ofẼ. The bigger the brane tension, the smaller the maximal mass is, for both analyzed values ofλ. Boson stars were studied in a similar context, where a maximal mass was identified to a transition point, splitting stable and unstable compact stellar configurations [51]. This aspect was emulated and explored for MGD and EMGD compact stellar configurations, from the point of view of the information entropy, by Refs. [18,19].
It is worth to study compact stellar configurations in the regime |λ|  Fig. 1 shows that the more |λ| increases, the bigger the maximal MGD stellar masses are. In addition, for fixed values ofẼ, the mass peaks are bigger, the lower the brane tension is.   Fig. 2 shows that the bigger the brane tension, the steeper the slope of each plot is. It indicates that more realistic models, involving observational values of the brane tension, yield a MGD Dirac star maximal mass that increases with |λ|, however in a lower rate. The general-relativistic limit σ → ∞, ζ → ∞ is acquired, being very close to the black line plot in Fig. 2, as indicated in Ref. [45]. It is worth to emphasize that, when σ → ∞, Fig. 2 can be described, for |λ| 1, by the interpolation expression This emulates the results in Ref. [45] in the MGD-decoupling context.
When coupling spinor fields to the MGD solutions, the resulting compact stellar configurations present distinct profiles, in the |λ| 1 and the |λ| ≈ 0 regimes. Fig. 3 illustrates the spinor fields profiles with respect to the (adimensional radius) of the MGD Dirac star, forλ = 0 andλ = −100. Similar to bosonic stellar configurations [45], Eqs. (17) - (19) can be made adimensional, at the |λ| 1 regime, by the mappings Ref. [45] showed that at a large |λ| regime, the fermionic field percolates a large range √ |λ| m . This implies that terms involvingα andβ can be disregarded. Hence, Eq. (17) can be led to an analogue of Eq. (25). Taking only leading terms in Eqs. (17,18) yields, whenβ α, Replacing it into Eq. (19) yields In the general-relativistic, σ → ∞ limit, the maximal mass was derived in Ref. [50], being 0.4132 |λ| by current experimental and observational data [17], together with the most strict bound for the brane tension σ 2.81 × 10 6 MeV 4 [19]. Therefore, due to the small physical values of these two parameters that rule the deformation process in the MGD, the numerical solutions plotted in Figs. 1 -4, for MGD Dirac stars, present qualitative profiles that are similar to the standard Dirac stars [45], as expected. The MGD-decoupling parameter ζ = 0.1 was adopted in all the numerical calculations. As discussed in Ref. [45], standard Dirac stars generated by linear spinors fields have tiny masses. Including non-linear spinor fields, in particular the self-interaction (11), circumvents this feature. It thus makes possible to approach astrophysical MGD Dirac stars. The MGD Dirac star maximal mass increases as a function of the spinor self-interaction coupling constant, |λ|, as illustrated in Fig. (2). Moreover, for the strictest phenomenological bound for the brane tension σ 2.81 × 10 6 MeV 4 , the MGD Dirac star maximal mass was shown to increase in a lower rate, compared to higher order values of the brane tension.
The spinor self-interaction coupling constant,λ, in Eq. (11), indicates stable compact selfgravitating configurations, whose maximal mass is given by Eq. (23). This value of the mass, alternatively written as 1.96(1 + ζ) × 10 5 |λ|M (MeV 2 /m) corresponds to the mass of a MGD Dirac star that has similar order of magnitude as the Chandrasekhar mass, for fermions with mass m ≈ 1 GeV. In the general-relativistic limit, when ζ → 0 and σ → 0, all the results in Ref. [45] are recovered.
For MGD bosonic stellar distributions formed by Bose-Einstein condensates of gravitons, Ref. [18] showed that a critical local point of the star information entropy, as a function of the stellar configuration mass, indicates a transition between stability and instability against linear perturbations. The extended MGD case was discussed in Ref. [19]. In the case here studied, there is a family of MGD Dirac stars, parametrized by central value of the spinor field β c component. The mass of a MGD Dirac star can be expressed in terms of β c , existing a local critical mass for each value of β c , given by the solution of Eq. (26). In addition to this analysis, the information entropy of MGD Dirac stars should take place to provide a final answer to their instability/stability conditions. The developments in Refs. [52,53] can shed new light on this important problem, that is beyond the scope of this paper.