Einstein-Cartan gravitational collapse of a homogeneous Weyssenhoff fluid

We consider the gravitational collapse of a spherically symmetric homogeneous matter distribution consisting of a Weyssenhoff fluid in the presence of a negative cosmological constant. Our aim is to investigate the effects of torsion and spin averaged terms on the final outcome of the collapse. For a specific interior spacetime setup, namely the homogeneous and isotropic FLRW metric, we obtain two classes of solutions to the field equations where depending on the relation between spin source parameters, $(i)$ the collapse procedure culminates in a spacetime singularity or $(ii)$ it is replaced by a non-singular bounce. We show that, under certain conditions, for a specific subset of the former solutions, the formation of trapped surfaces is prevented and thus the resulted singularity could be naked. The curvature singularity that forms could be gravitationally strong in the sense of Tipler. Our numerical analysis for the latter solutions shows that the collapsing dynamical process experiences four phases, so that two of which occur at the pre-bounce and the other two at post-bounce regimes. We further observe that there can be found a minimum radius for the apparent horizon curve, such that the main outcome of which is that there exists an upper bound for the size of the collapsing body, below which no horizon forms throughout the whole scenario.

the same manner as that in GR [29]. Furthermore, it is important to emphasize a feature that subsequently follows from our case study: the dynamics of the horizon formation (controlled by the Misner-Sharp energy) will be directly influenced by the dependence of the scale factor dynamics on the averaged spin square density and axial current fluid parameters.
The organization of this paper is then as follows. In section (II) we give a brief review on the background field equations of the EC theory in the presence of a Weyssenhoff fluid and a negative cosmological constant, which governs the dynamics of our collapsing process case study. Section (III) provides a family of solutions to the field equations, with some representing a collapse that leads to a spacetime singularity within a finite amount of time. In section (IV) we (i) study the dynamics of apparent horizon and induced spin effects on the formation or otherwise of trapped surfaces and (ii) show that a naked singularity might be formed as a possible outcome for a subset of collapsing scenarios. In section (V) we examine the curvature strength of the naked singularity and exhibit that the singularity is gravitationally strong in the sense of Tipler [30]. Our conclusions are presented in section (VI). In appendix (A) we present a suitable solution for an exterior region and subsequently we discuss therein the matching between interior and exterior regions.

II. EQUATIONS OF MOTION
The action for the EC theory can be written in the form [16] whereR and L m are the Ricci scalar (constructed from the affine connectionΓ α µν ) 5 and matter Lagrangian, respectively, and can be expressed, in general, as functions of independent background quantities, namely the metric field g µν and the torsion T α µν ; Λ is the cosmological constant. We take the metric signature as (+, −, −, −) and κ 2 ≡ 16πG. The presence of torsion in the spacetime macroscopic structure can, theoretically, be attributed to anticipated microscopic fermionic matter fields with spin-angular momentum degrees of freedom. In general, the torsion tensor T α µν is geometrically defined as the antisymmetric part of the affine connection In this paper, we take the matter part of the action to be described by a Weyssenhoff fluid [27], which macroscopically is a continuous medium but also conveys features that can be suitably associated to (averaged) spinor degrees of freedom of microscopic matter fields. Moreover, it has been shown that, with the assumption of the Frenkel condition (also known as Weyssenhoff condition), the setup may likewise be equivalently described by means of an effective fluid in a plain GR setting, where the effective energy momentum tensor contains additional (spin induced) terms [31]. More concretely, the Lagrangian for the matter content can then be subsequently decomposed as in that L SF contains the induced effects of a spinning fluid, which can be written in terms of a perfect fluid contribution and a characteristic spin part [32], while L AC will correspond to the contribution of a minimal coupling spinor axial current with torsion [33]. A motivation for this latter term is as follows. It is reasonable to consider, in a collapse process, with matter with spin-angular momentum features, contracting to highly dense regimes, that those spin effects may no longer be neglected and a non-zero axial current is expected. Therefore, we could write 6 where, explicitly, J µ = ψ γ 5 γ µ ψ and S µ = ǫ αβρµ T αβρ would supposedly be the corresponding (averaged) spinor axial current 7 and axial torsion vector, respectively. 5 The Ricci curvature scalarR is rewritten in terms of the usual Ricci curvature scalar R (constructed from the Christoffel connection) and the contortion K α µν , asR = R + K α ρα K ρλ λ − K αρλ K ρλα . 6 γ µ is defined by means of [γ µ , γ ν ] = 2g µν , γ 5 = iγ 0 γ 1 γ 2 γ 3 is a chiral Dirac matrix, ψ is a fermion field,ψ = ψ † γ 0 is the conjugate fermion field and ǫ αβρµ is the totally antisymmetric Levi-Civita tensor. 7 In a high energy limit, due to quantum effects, the averaged spin axial vector would be non-zero [33].
For the purpose of clarity, nonetheless the previous paragraph, let us in the following be more precise concerning the matter content in our case study and discuss therefore the equations of motion. Variation of the action with respect to the torsion [31] results in where T µ = T ρµ ρ and τ µνα being defined as Moreover, τ µνα SF = − 1 2 S µν u α , where u α is the fluid 4-velocity, therefore conveying the corresponding spin fluid contribution. S µν is subsequently designated as the antisymmetric spin density tensor, constituting the effective source of torsion and K µνα is the contortion, which has the following form in terms of torsion In the EC theory, in contrast to the metric, the torsion is not really a dynamical field: the left hand side of equation (5) contains no derivatives of the torsion tensor and indeed appears as a purely algebraic equation. Torsion can therefore be eliminated by replacing it with the spin density S µ ν and hence, implying a modification to Einstein's field equations, by means of additional characteristic (spin induced) matter terms. Using the Frenkel condition 8 , S µν u ν = 0, the torsion constraint equation (5) may be rewritten in the form We can at this point more advisedly show how the employed Weyssenhoff fluid can fit the Frenkel condition. More concretely, this condition results in an algebraic relationship between the spin density tensor and torsion, as which can also be retrieved directly from the formalism proposed in [34]. Therefore, by virtue of the Frenkel condition, this means that the only remaining degrees of freedom of the torsion are the traceless components of the torsion tensor. Furthermore, the axial torsion vector can be written with the assistance of equations (7) and (8), as It is now straightforward to obtain the dynamical equations of motion, varying the action (1), with respect to the dynamical field, g µν , which can be written as Substituting then for the contortion from equation (7) into the above equation and using equation (8), we subsequently get where J 2 = J µ J µ . The energy momentum tensor contains two contributions, from the axial current, T AC µν , and the spin fluid, T SF µν , which can be expressed, after employing a suitable spin averaging 9 , as where we considered for spin averaged quantities [31] S µν S µν = 2σ 2 , From a macroscopic point of view, the spin fluid can therefore be considered as a contribution from a usual perfect fluid, with associated energy density ρ SF and pressure p SF , plus, as induced from a microscopic perspective and suitably averaged, the first two terms in equation (14), which relate to characteristic spin contributions, coming into play. Inserting equations (13) and (14) into (12), together with the above spin averaging, we can finally obtain the dynamical field equations for our case study, as [31] G

III. SOLUTIONS TO THE FIELD EQUATIONS
Owing to the lack of satisfactory tools to deal with the global properties of Einstein's field equations and respective solutions, together with their severe non-linearity, no proofs of the weak or strong versions of CCC have yet been formulated. In this context, several authors have pointed that the Lemaître-Tolman-Bondi (LTB) spacetime [36] (the simplest generalization of the Friedmann-Robertson-Walker (FRW) model), is one of the candidates 10 for a counterexample to the CCC. Collapse settings studied so far imply the formation of a curvature singularity being either naked for certain initial LTB configurations and covered for other initial conditions. In [38] it was shown that the central singularity of the marginally bound LTB model can be shell-focusing and naked. Work along the same line has proved that this is true for the bound case [39]. The study of LTB collapse has been widely carried out by many authors [40] and the main results, found from these analyses in connection with the concept of singularities and CCC, are summarized as follows: (i) The collapse process always leads to a central curvature singularity; (ii) Depending on the nature of the initial data, the resulting singularity can be either locally or globally naked; (iii) A generic singularity forms in the sense that there exists an infinite number of outgoing non-spacelike geodesics terminating at the singularity in the past; (iv) The singularity is gravitationally strong in the sense of Tipler [30]. In this section, within the framework introduced in the previous section, we will find a class of collapse solutions which lead to the formation of a spacetime singularity and study the formation or avoidance of trapped surfaces till the occurrence of the singularity in the next section.There is a paucity of physically reasonable exact solutions due to intrinsic difficulties to overcome, once spacetimes with less symmetry (inhomogeneities and anisotropies being present) are employed, so we here restrict ourselves to a special but manageable class of LTB models and parametrize the interior line element (which is recognized as of FLRW geometry) as [41] where R(t, r) = ra(t) is the physical radius of the collapsing star, with a(t) being the scale factor, R ′ (t, r) = ∂R(t, r)/∂r and dΩ 2 being the standard line element on the unit 2-sphere. With the above choice of the area radius, the resulting interior spacetime is homogeneous and isotropic for which the field equations read where H =Ṙ/R =ȧ/a is the rate of collapse. Since we are interested in a continual collapse process,Ṙ(t, r) must be negative. Notice that we may consider the cosmological constant term as vacuum energy density [42]. The continuity equation for the matter fluid is thereforė As we mentioned in sect. II, the geometry and matter content effectively induce a macroscopic perfect fluid contribution, with barotropic equation of state p SF = wρ SF , together with intrinsic spin contributions, that are present in the form of averaged quadratic terms of spin quantities (that may admit a possible microscopic representation). Such a microscopic matter scenario could be that of a concrete spin fluid, directly related to a particle content formed by, e.g., unpolarized fermions 11 . It is plausible to assume that the fermions participating in the collapse process behave as ultra-relativistic particles; thus, the number density (n f ) of a fermionic gas, employing the Fermi-Dirac distribution, can be approximated by n f ∝ a −3 (see [46] for more details). So, the square of spin density and axial current (which are proportional to n 2 f [31,32,35]) depend on the scale factor, as 12 However, we proceed with a general setup where J 2 = J 2 0 a n and σ 2 = σ 2 0 a n (n ∈ N − ), and in sect. IV we present the results corresponding to n = −6. Therefore, from (22) it is easy to obtain 13 the energy density ρ SF (a) = C 0 a −3(1+w) + α n + 3(1 + w) a n , w = −1 whence it follows that solving the equations (20) and (21) leads to where ρ iSF and a i being the initial values of the energy density and the scale factor, respectively, and 11 In a realistic description of the collapse of a star, whose matter content is fermion dominated, it is conceivable that the effective spinning fluid be polarized. Thus, a spin alignment due to the presence of strong magnetic fields (cf. [43]) may potentially affect the collapse dynamics and therefore, quite possibly, its final outcome. Moreover, from a macroscopic viewpoint, each particle in the cluster undergoing gravitational collapse may also have orbital angular momentum, so that the net effect of all the particles is to introduce a nonzero tangential pressure in the energy momentum tensor. Such rotational effects on the collapse procedure (e.g., gravitational collapse of a system of counter-rotating particles -the" Einstein cluster" [44]) have been studied in [45]. It is shown therein that, depending on the strength of the angular momentum, trapped surface formation can be avoided, thus allowing the singularity to be visible. 12 In general, σ 2 0 and J 2 0 are the source parameters for spin square density and axial current, respectively, and should not be confused with their initial values defined as σ 2 i = σ 2 0 a n i and J 2 i = J 2 0 a n i . 13 The choice w = −1 is discussed separately at the end of this section, because it corresponds to a non-singular case.
A collapsing solution can be found by imposing the following restriction on the barotropic index as a function of J 0 and σ 0 where the limiting value of w → 1 corresponds to J 2 0 → 0. We can solve for the rate of collapse as where We find the interior solution as where t i and H i are an initial time at which the collapse starts and the initial rate of the collapse and we have set arg arctan This solution exhibits a collapse process which terminates in a spacetime singularity at a finite amount of time, provided that 14 Λ < 0. The condition H 2 i > 0 puts the following restriction on the initial rate of collapse The singular epoch t s , which is defined as a(t s ) = 0, is given by at which the Kretschmann invariant R µναβ R µναβ and the effective energy density diverge. Regarding the case w = −1, equation (22) leads to the following expression for the energy density ρ SF = ρ iSF + α n (a n − a n i ).
From equations (20) and (21) we can solve for the collapse rate as (a n − a n i ) From the above equation we see that the collapse begins with an initially negative rate (the minus sign) and proceeds till the scale factor reaches a critical value, a c , at which H(a c ) = 0. Beyond that the collapse procedure may turn to an expanding phase which corresponds to the situation 15 in which H jumps from a − to a + branch [48]. This occurs at the time and the Kretchmann invariant and the energy density behave regularly, thus no singularity forms 16 .

IV. SPIN EFFECTS ON THE DYNAMICS OF APPARENT HORIZON
We are now in a position to examine whether the obtained singularity is hidden behind an horizon or is visible by distant observers. The singularity is covered within an event horizon if trapped surfaces emerge early enough before the singularity formation and may be visible if the apparent horizon, which is the outermost boundary of trapped surfaces, is failed or delayed to form during the collapse process. The key factor that determines the dynamics of the apparent horizon is the Misner-Sharp energy [50] which describes the mass enclosed within the shell labeled by r at the time t, and is defined as [29] It is worth mentioning that in our case study the effects of torsion show themselves as an addition to the energy momentum tensor, more precisely by means of taking into account the contributions of a fluid (associated with matter with spin) to the geometry. It thus affects the dynamics of the apparent horizon. The spacetime is said to be trapped, untrapped and marginally trapped if, respectively, The field equations (20) and (21) can then be rewritten as [51] 2M ′ (t, r) Let us remind that our choice of setting and matter content can be interpreted in that a supposedly microscopic nature of the spin density and axial current would allow their effects to be macroscopically exhibited, in such a way that the effective equation of state parameter, w, would be determined by the source parameters, J 2 0 and σ 2 0 . With regard to the fact that the matter Lagrangian contains the spin density and axial current terms, we expect that the Hamiltonian as well depends on σ 2 0 and J 2 0 . Therefore, based on the Fermi-Dirac distribution, the energy density and pressure (and as a result the effective barotropic index) of a Fermi gas could be expressed in terms of σ 2 0 and J 2 0 . Thus, we seek for the conditions on σ 2 0 and J 2 0 under which the trapped surfaces may be avoided and the resulting singularity could be at least locally naked. Integrating the first expression in equation (41), we have for our solution (we have henceforth set n = −6) From the above relation, it is seen that for σ 2 0 < 24J 2 0 , if the ratio (2M/R) < 1 at the initial time or equivalently which is required by the regularity condition [52], then it would stay less than one till the singular epoch, leading to the trapped surface avoidance. However, if σ 2 0 > 24J 2 0 , as the collapse proceeds, the ratio 2M/R turns to be greater than one and thus the trapped surface formation takes place earlier than the formation of the singularity, to subsequently cover it 17 . From the second part of equation (41), the effective pressure can be obtained as 16 It should be noted that singularities of the type of big-rip, sudden or even type III, do not happen herein, since ρ and p are finite at t = tc. A type IV singularity does not occur either, since the higher derivatives of H do not diverge at t = tc. We further note that the weak energy condition is violated in this special case (w = −1) while for the other sensible types mentioned in this footnote, the weak energy condition is satisfied (see [49] and references therein). 17 Since σ 2 i = σ 2 0 a −6 i and J 2 i = J 2 0 a −6 i , the conditions on formation or otherwise of trapped surfaces can be rewritten as σ 2 i > 24J 2 i or σ 2 i < 24J 2 i , respectively. The rest of the conditions for the physical reasonableness, i.e., (43), (45), (48), (49) and (51), can also be rewritten in terms of these parameters.
The initial data of the collapsing configuration can thus be chosen so that the effective pressure be positive at initial epoch, which can be achieved if 18 As the collapse proceeds, the first term in the right hand side of (44) dominates the second one and the effective pressure becomes negative for σ 2 0 < 24J 2 0 . Therefore, we can deduce that, at later stages of the collapse, the failure of formation of trapped surfaces is accompanied by a negative pressure [53]. From the second part of equation (41), it is seen that the occurrence of negative pressure impliesṀ < 0, which can also be verified from the time derivative of equation (42) for σ 2 0 < 24J 2 0 ; this can be interpreted as the star continuously radiating away its matter content as the collapse process advances. On the other hand, if σ 2 0 > 48J 2 0 , which satisfies the condition for trapped surface formation (σ 2 0 > 24J 2 0 ), the pressure is initially positive and remains positive up to the final stages of the collapse. For the limiting value σ 2 0 = 48J 2 0 , which corresponds to a macroscopic dust fluid, the collapse ends in a black hole with the exterior solution, after a suitable spacetime matching, as the Schwarzchild-anti-de Sitter spacetime (cf. Appendix (A) for more details ).
Although the pressure (44) is allowed to get negative values, the collapse process will only be physically reasonable if the energy conditions (weak, dominant and null) are preserved throughout the collapse. The weak energy condition implies that the energy density as measured by any local observer is non-negative. Thus, along any non-spacelike vector, the following conditions have to be satisfied We thus have If the first inequality holds initially, it will remain valid at later times, therefore it is enough to satisfy which is exactly the same as (33). Since Λ < 0, it clearly needs to have γ > 0. The condition γ > 0 is also sufficient for the validity of the second inequality. Therefore, the initial set up for the energy density of the spin fluid has to satisfy ρ iSF ≥ αa n i n + 3(1 + w) , n = −6.
This subsequently guarantees the validity of the weak energy condition. We note that the second inequality in (46) implies the validity of the null energy condition. Another energy condition, which is frequently used, is the dominant energy condition (ρ eff ≥ |p eff |), stating that, for any timelike observer, the local energy flow of matter is non-spacelike. This ensures that the sound velocity in a collapsing medium cannot exceed the speed of light. For our solution (30), the dominant energy condition holds if It is obvious that if the above condition is satisfied initially, it would hold during the final stages of the collapse. It is then sufficient that Figure 1 shows numerically the allowed region for σ 2 0 and J 2 0 , for a chosen set of values of ρ 0SF , a i , which satisfy condition (43), i.e., the regularity condition, positivity of the effective pressure at initial time, i.e., equation (45) together with the weak (48) and (49) and dominant energy conditions (51).
It is of interest to consider these two following special cases, in order to show what happens when the effect of one of the sources i.e., spin density or axial current, may be negligible. • If J 2 0 ≪ σ 2 0 , i.e. the effects which can be associated with high energy regimes are negligible, then the interaction term containing the axial spinor current (4) disappears; this case is effectively equivalent to stiff matter with w = 1 in the presence of a negative cosmological constant, from which we obtain and hence, as the singularity is reached, the apparent horizon forms to cover it. It should be noticed that the effect of spin is implicitly present in γ (see (31) and second term of (24)).
• For the case in which σ 2 0 ≪ J 2 0 , the spin fluid would be diluted (i.e., the number density of interacting spinning particles ≈ 0). This corresponds to w → −1 (w = −1) where we have where ǫ = σ 2 0 8J 2 0 ≪ 1. It can thus be concluded that, as the singularity is approached, the ratio 2M/R stays finite and thus trapped surfaces are avoided during the collapse process. Similarly, as it noted in the above mentioned case, the effect of spin would be implicitly present in γ. However, this herein situation must be taken with care, in view of physical reasonableness conditions, e.g., satisfying energy conditions. One may ask whether the initial configurations of a collapse scenario that separately lead to trapped or untrapped regions could change when the spin effects become more manifest, i.e., how the value of n, the power of scale factor dependence of spin densities, may influence the contribution of initial data that results in trapped or untrapped regions. In figure 2, we have made a numerical integration to calculate the ratio of the area of trapped to untrapped regions as a function of ρ iSF and n. We see that for a constant value of n, the greater ρ iSF the greater this ratio. Also as the initial profile of the energy density grows while the value of n decreases, the contribution of the initial data that leads to a trapped region increases, as compared to those that lead to untrapped region. In figure 3 we have plotted separately the ratio of the area of trapped to untrapped regions as a function of n for fixed values of initial energy density of spin fluid (left plot) and of ρ iSF for fixed values of n (right plot). In the left plot, we see that, as ρ iSF increases, the curves for a same choice of n have higher values for the mentioned ratio, signaling that the domain where to find initial configurations that induce trapped region in the spacetime, increases, as compared to those that lead to untrapped regions. However, as n decreases to more negative values, all the curves tend to smaller ratios, showing that when the spin effects become more overriding, there is a broader initial data set from which the  (23)), which conveys an indication of the magnitude of the square of spin density and axial current presence. The initial value of the scale factor has been set to be ai = 0.4 and we have chosen the units so that κ = 1 and Λ = −1. collapse commences in such a way that trapped surfaces are avoided till the late stages of the scenario. As we see in the right plot, the ratio increases as the initial energy density of the spin fluid gets larger, implying that there is a wider initial data set from which, as the collapse begins, the trapped surface formation takes place; Whereas the more negative the value of n the wider the space of initial data for which trapped surfaces are failed to form till the singularity formation. This shows that the influence of spin effects is related to the value of n, which may establish the range of initial data in determining the final fate of the collapse. Let us now focus on the nature of the central singularity, which occurs at R(a(t s ), 0) = 0. This type of singularity is at least locally naked if there exist future-directed outgoing null rays with past endpoint at the singularity. This means that along such light signals the area radius increases, i.e., dR/dr > 0. From the line element we have for the null rays and thus, along the radial null geodesics, we get Using the standard procedure developed in [52], we introduce the auxiliary variables u and Y so that We then have the following expression in the limit close to the central singularity, as From equation (30) we can substitute forṘ(t, r) = ra(t)H to find Y 0 , as where 2 2k (2k−1) and in the last line we have kept the terms up to first order in the curly brackets. In order to have a finite value for Y 0 the terms including r must not diverge. This can be achieved if we set the power of r in the mth term 19 in the last expression above to be zero, which fixes η as where η m is the value of η for which the power of r in the mth term vanishes. Then, the power of r in all the lower order terms becomes positive, with η = η m , and thus the associated terms vanish in the limit r → 0. If equation (58) admits a solution (a real positive root Y = Y 0 ), then there exists a null trajectory reaching the singularity at r = 0 with a definite outgoing tangent Y 0 and thus the singularity is at least locally naked. Otherwise, the singularity cannot be naked and the collapse culminates in a black hole. Setting m = 2, i.e. up to second order in the last line of equation (58), we obtain in the region near the singularity The left plot of figure 4 shows the region where the condition for trapped surfaces avoidance (σ 2 0 < 24J 2 0 ), together with where the existence of the real positive values for Y 0 (60) is respected. However, we should be careful concerning the physical reasonableness of the gravitational collapse scenario and validity of the regularity condition as conveyed in section IV. Let us be more precise. The gray zone in the right plot of figure 4 identifies where (i) real values for Y 0 are to be found and (ii) also corresponds to those physically permitted values of σ 2 0 and J 2 0 according to equations (43), (45), (48), (49), (51). In other words, therein real positive roots for (58) exist and the conditions demanding the physical reasonableness of a collapse scenario and regularity condition are satisfied. The relevant feature is that, altogether, it implies that the resulting singularity can be at least visible to its neighboring observers, i.e., a locally naked singularity can be formed as the collapse end product.

V. STRENGTH OF THE NAKED SINGULARITY
In order to make more concrete our discussion in the previous section, we need to investigate the curvature strength of the naked singularity, which is an important aspect of its physical nature and geometrical importance. The main underlying idea is to examine the rate of curvature growth along non-spacelike geodesics ending at the singularity, in the limit of approaching to it. The singularity is said to be gravitationally strong in the sense of Tipler [30] if every collapsing volume element is crushed to zero size at the singularity, otherwise it is known as weak. It is widely believed that when there is a strong curvature singularity forming, the spacetime cannot be extended through it and is geodesically incomplete. While, if the singularity is gravitationally weak, it may be possible to extend the spacetime through it classically. In order that the singularity be gravitationally strong, there must exist at least one non-spacelike geodesic with the tangent ξ µ , along which the following condition holds in the mentioned limit where R µν is the Ricci tensor and λ being an affine parameter which vanishes at the singularity. Let us now consider an outgoing radial null geodesic having the tangent ξ µ = dx µ /dλ = (ξ t , ξ r , 0, 0). We note that since ξ µ is an affinely parametrized null geodesic, we have From the null condition for ξ µ , with the help of spacetime metric (19), we can obtain while the geodesic equation results in the following differential equations ξ tξt + aȧ(ξ r ) 2 = 0, which gives the vector field tangent to null geodesics as Next we proceed to check the quantity given by (61) which, with the use of field equation (18), reads as where use has been made of equations (63) and (65) and the null energy condition (47). Subsequently, we find that Substituting for the rate of collapse from equation (30), we finally get Therefore, the strong curvature condition is fulfilled along the singular null geodesics and thus the naked singularity is gravitationally strong in the sense of [30].

VI. CONCLUDING REMARKS
The study of the end-state of a massive star, collapsing within its own gravity [52], benefits when, in some suitable manner, spinorial (e.g., fermions) degrees of freedom are taken into account. To our knowledge, the literature concerning this research line is somewhat scarce 20 , see e.g., [23]. Torsion is perhaps one of the important consequences of coupling gravity to fermions. In general, this leads to consider non-Riemannian spacetimes, where differences with respect to GR dynamics can be expected and hence explored. Supergravity with matter fields [15] provides an interesting such setting to explore: gravity will be directly coupled to fermions, explicitly present, inducing that the spacetime geometry will have torsion 21 . The well known and established CSK [16] theories can also be a starting point, where the classical Einstein-Maxwell-Dirac theory of spinors is formulated [55].
Nevertheless, the presence of fermionic fields may not bring an easily discussable set up to investigate the final outcome of a gravitational collapse. There are, however, other, perhaps more manageable, theories. They employ torsion but just to mimic effects of matter with spin degrees of freedom on gravitational systems.
We therefore took such workable approach in the herein paper. More precisely, we studied the gravitational collapse of a massive star, whose matter content was taken as a Weyssenhoff fluid [27] in the context of the EC theory [24]. A negative Λ included to provide an initially positive pressure, such that a collapsing process can initially be set up. Being more concrete, we have torsion in the geometry, thus producing a non-Riemannian spacetime, where no fermions appear explicitly (in the spacetime torsion terms). The torsion is not, however, a dynamical field, being therefore eliminated by means of algebraic expressions. Furthermore, the subsequently retrieved effective equations of motion are those of Einstein's, but with terms brought from the mentioned Weyssenhoff fluid, with those induced by (a suitably averaging of) spin terms, that can be suitably related to the torsion. In addition, we have restricted ourselves to a special but manageable class of LTB models, namely the marginal bound case (see [41] for more details), where the interior line element (which is recognized as of FLRW geometry) allows a particularly manageable framework to investigate. The corresponding effective energy momentum from a macroscopic perspective, has a perfect fluid contribution, plus those due induced from the spin interactions. In particular, the corresponding barotropic index is determined by source parameters, J 0 and σ 0 , that represent those (average) spin terms. A relevant feature is that this effective matter can, within specific conditions, convey a negative pressure effect. As a consequence, this may induce the avoidance of trapped surfaces formation. Investigating, the initial conditions dependence of the final outcome of collapse would allow to convey concrete predictions.
In a compact manner, our main results are as follows: • The reasonableness of our case study requires the initial setting of the effective energy density and the initial value of the scale factor to satisfy specific ranges. These are (i) the regularity condition on the absence of trapped surfaces at an initial epoch, (ii) the validity of the energy conditions and (iii) the positivity of the effective pressure at an initial time. When these conditions are satisfied, then formation or otherwise of trapped surfaces during the collapse process may occur if σ 2 0 > 24J 2 0 or σ 2 0 < 24J 2 0 , respectively. It is indeed the competition between the source parameters of the axial current, J 2 0 , and of the interacting spinning fluid, σ 2 0 , or equivalently between their initial values σ 2 i and J 2 i , together with the Λ < 0 set up, that decides the escape or entrapment of null rays and thus the final outcome of the collapse scenario.
• We further examined the nature of the singularity by considering a null geodesic coming out of it. We concluded that the outgoing tangent to this geodesic is positive definite and thus the resulting singularity, depending on the parameters σ 2 0 and J 2 0 , can be at least locally naked. In this case the null rays will be initially emerging from the singularity and traveling for a while but they begin to re-converge and fall back into the singularity without actually reaching a faraway observer. Thus, only the strong version of CCC is violated but the weak form is preserved, i.e., the singularity would not be globally naked. To prove that, it is necessary to show that there exist infinite non-spacelike geodesics with positive definite tangent reaching distant observers and ending at the singularity [57].
• The spin effects become more effective for more negative values of the parameter n, see figure 2. Therefore, as it has been therein depicted, we expect that, among the whole set of possible initial data, our freedom to choose those for which trapped surfaces are averted will increase, as n takes more negative values. This should be contrasted to those that lead to a continual gravitational collapse in which trapped surface formation takes place.
• A special case, in which the equation of state of spin fluid was p SF = −ρ SF , was considered separately and it was found that no singularity occurs.
Let us emphasize that it was usually expected that torsion (as supposedly produced by the spin-angular momentum of fermionic matter) inducing a repulsive pressure, would prevent the formation of singularities [22], [58]. However, such repulsive spin effects of matter may merely be, as our results seem to suggest, that trapped surfaces formation are avoided throughout the dynamical process of the gravitational collapse, i.e., not only do they not avoid the singularity formation (either hidden behind an event horizon or visible), but also, in the case in which a naked singularity forms, it can be gravitationally strong. The validity of null energy condition guarantees the validity of the strong curvature condition.
Furthermore, as we near in closing this paper, there are a few features employed herein that may require some additional elaboration. Firstly, we could consider the possibility of relaxing the Frenkel condition, therefore taking a more general matter content (cf. section II this paper). However, if such a modification is employed, then the number of degrees of freedom of the torsion tensor would increase. Seemingly bringing a more complicated setting to deal with, we also expect that extra (torsion related) terms will appear in the field equations. These can potentially affect the dynamics of the collapse and the final outcome. Secondly, it can be pointed that the spacetime set up we assumed for the gravitational collapse, is a rather specific case. Nonetheless, the scenario herein investigated is rich enough (by means of the EC context and the presence of torsion as well as of the Weyssenhoff fluid) to provide quite interesting new physical results. However, for a realistic isolated body such as a dense star, we would expect that the density distribution would typically grow when the regions near the center are approached and reduces once we move away from the center. Thirdly, the final outcome of the collapse setting of such a dust cloud was studied in [37], where the effects of inhomogeneous density distribution were also taken into account, and a broad class of collapse solutions exhibiting strong curvature naked singularities has been found. Investigating the gravitational collapse of an inhomogeneous Weyssenhoff fluid, beyond the herein presented model, could be valuable but surely with difficulties: in addition to find (exact or even approximate) solutions for an inhomogeneous setting, taking averages of spin densities (and other related quantities) in the presence of the inhomogeneities is not straightforward. Beside the effects of inhomogeneity, it is worth mentioning that shear effects of the collapsing matter can also play a relevant role in determining the collapse end state. In [59], it was shown that shear effects are important in determining the final outcome of a gravitational collapse setting.
Finally, we would like to present a few possible subsequent lines of exploration, motivated by the research work that substantiated this paper. Besides the direction pointed in footnote 11., which may require some degree of spatial anisotropy to be considered (either geometrically and matter-like, i.e., not a perfect fluid), it could also be of interest to further explore that J 0 is related to a fermionic vacuum expected value, asψψ is proportional to the number density of fermions. The axial current has been pointed in the literature as responsible for Lorentz violation. Constraints have been put on some of the torsion components due to recent Lorentz violation investigations [60]. Moreover, although being a wider set-up with respect to GR, it could be fruitful to enlarge action (1). More concretely, replacing the cosmological constant by some scalar matter. This would allow to establish the limits of dominance of any matter component (and associated intrinsic effects), towards a concrete gravitational collapse outcome, where, e.g., bosonic and fermionic matter (mimicked or not) would be competing. Perhaps more challenging, it would be curious to employ a Weyssenhoff fluid description that could have different features whether we use s = 1 2 fermion or a Rarita-Schwinger field, with s = 3 2 spin angular momentum. The gravitational theory of such particles in the presence of torsion has been discussed in [61]. Hence, how would gravitationally collapsing matter, with dominant fermionic content, with either s = 3 2 field features or instead s = 1 2 , differ concerning the physical outcome, by means of a Weyssenhoff fluid-like description?
Vaidya radius. We assume that Σ is endowed with an intrinsic line element given by Here y a = {τ, θ, φ}, (a = 0, 2, 3) are the coordinates of Σ with τ being the time coordinate defined on it, and we have chosen the angular coordinates θ and φ to be continuous. The governing equations of hypersurface Σ in the coordinates X µ ± are given by Using the above two equations, the interior and exterior induced metrics on Σ − and Σ + take the form, respectively and We assume that there is no surface stress-energy or surface tension at the boundary (see e.g. [74] for study of junction conditions for boundary surfaces and surface layers) and then, according to junction conditions, for a smooth matching in the absence of surface layer, the induced metrics, when approaching Σ from the exterior or interior spacetimes, must obey which gives where an overdot denotes d/dt. Next, we need to compute the components of the extrinsic curvature of the interior and exterior hypersurfaces. The unit spacelike normal vector fields to these hypersurfaces are given by Let us take X µ = X µ (y a ) as the parametric equations of Σ. The extrinsic curvature or second fundamental form of the hypersurface Σ is a three-tensor, defined as [75] K ab ≡ e ν a e µ b ∇ µ n ν , where e ν a = ∂X ν /∂y a are the basis vectors tangent to the hypersurafce Σ (tetrads defined on Σ that bring the three and four dimensional quantities in connection) and the covariant derivative is taken with respect to Christoffel symbols, γ νµ . The above expression can be re-written into the following form as Now, if we take X µ + (y a ) and X µ − (y a ) as parametric relations for the hypersurfaces Σ + and Σ − on the exterior and interior regions, we get, respectively Here we should be careful on the second term containing the asymmetric affine connection defined in (2). Since in EC theory, torsion cannot propagate outside the spin matter distribution [76], the connection for the exterior region is precisely the Christoffel symbol, whose non-vanishing components are v vv where ", R ≡ ∂/∂R", ", v ≡ ∂/∂v" and f = f (v, R). In order to calculate the nonzero components of (A12) we proceed by noting that Substituting the above expressions into equation (A12) and having in mind that ∂θ/∂τ = ∂φ/∂τ = 0, we finally get In the process of obtaining the extrinsic curvature of Σ, proceeding from the interior region, we should note that the connection through which the extrinsic curvature tensor is calculated is no longer symmetric, owing to the presence of spin matter. Therefore, we need to begin witĥ Substituting for the torsion tensor from equation (8) and rearranging the terms, we find Γ µ− να = µ να + κ 2 2 2 ǫ µ ρ ν α + ǫ µ ρ α ν − ǫ µ ρ να J ρ + 1 2 S να u µ − S µ α u ν − S µ ν u α .
A subsequently suitable spacetime averaging reveals that the second term in square brackets gets vanished since the connection is linear with respect to the spin density tensor. However, the first term may not generally become zero since the axial current is a timelike vector field. Substituting for the averaged affine connection into the minus sign of (A12) we have from which we readily find that, the third term in parentheses vanishes due to antisymmetrization property of Levi-Civita tensor and the partial derivatives. After a straightforward calculation we find However, due to the presence of the third term in (A18) there may remain other components of the extrinsic curvature tensor though the spacetime is spherically symmetric. Let us calculate them to show that these terms vanish too. The (t, θ) component reads since J ρ has only a time component, the Levi-Civita tensor vanishes for the two repeated indices and we get The remaining component can be calculated in the same way as since both the Levi-Civita tensors are equal but with the opposite signs. Using (A8), the continuity of the extrinsic curvatures across Σ implies the following relations fv +Ṙ = 1, v 2 (f f ,R + f ,v )v + 3f ,RṘ + 2 vR −Ṙv = 0.
Taking derivatives of (A23) and the first part of (A8) we have 2Ṙv =ḟv 2 , 2ṘR = −ḟv 2Ṙ + fv , hence we can construct the following relation asRv Substituting (A26) and the first part of (A25) into (A24), we finally find which clearly shows that f (v, R) must be a function of R only. Solving equations (A23) and first part of (A8) we get the four-velocity of the boundary, as seen from an exterior observer, as where a minus sign forṘ has been chosen since we are dealing with a collapse setting. From the second component of the above vector field and the interior solution (30), we find, for a smooth matching of the interior and exterior spacetimes, that the Vaidya mass must be equal to the interior mass function, as where M 0 = (1/12)κ 2 C 0 r 3(1+w) Σ . Therefore, the exterior spacetime reads It is seen that for σ 2 0 = 48J 2 0 , the exterior spacetime turns to Schwarzchild-anti de Sitter metric in retarded null coordinates [66,77,78], and for σ 2 0 > 24J 2 0 , a dynamical horizon evolves to cover the singularity. For σ 2 0 < 24J 2 0 , as we saw in section (IV), the apparent horizon may fail to form till the singularity formation. From equations (42) and (A29) and the first component of (A28), it can be easily checked that the radial null geodesic that has escaped from the singularity at t s , can be extended to the exterior spacetime as the Vaidya null geodesic. Hence a naked singularity may be produced in the gravitational collapse of the homogeneous Weyssenhoff fluid.