Einstein–Cartan gravitational collapse of a homogeneous Weyssenhoff fluid
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Abstract
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 nonsingular 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 prebounce and the other two at postbounce 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.
1 Introduction
The final state of the gravitational collapse of a massive star is one of the challenges in classical general relativity (GR) [1]. A significant contribution has been to show that, under reasonable initial conditions, the spacetime describing the collapse process would inevitably admit singularities [2]. These singularities, can either be hidden behind an event horizon^{1} or visible to distant observers. In the former, a black hole forms as the end product of a continual collapse process, as hypothesized by the cosmic censorship conjecture^{2} (CCC) [7, 8, 9, 10] (see also [11, 12, 13, 14] for reviews on the conjecture). The latter are classified as naked singularities, whose existence in GR has been established under a variety of specific circumstances and for different models, with matter content of various types, e.g. scalar fields [15, 16, 17, 18], perfect fluids [19, 20, 21, 22, 23, 24], imperfect fluids [25, 26, 27, 28] and null strange quark fluids [29, 30]. The analysis has also been taken to wider gravitational settings, such as \(f(R)\) theories [31], Lovelock gravity [32] (see also [33, 34, 35, 36, 37, 38] for some recent reviews) and hypothesized quantum gravity theories [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68]. This is an interesting line of research because, the possible discovery of naked singularities may provide us with an opportunity to extract information from physics beyond transPlanckian regimes [69]; see e.g. ([70, 71, 72, 73, 74, 75, 76] for the possibility of observationally detecting naked singularities).
It is therefore well motivated to consider other realistic gravitational theories whose geometrical attributes (not present in GR) may affect the final asymptotic stages of the collapse. More concretely, could ingredients mimicking spin effects (associated with fermions) potentially influence the final fate of a collapse scenario? In fact, if spin effects are explicitly present then GR will no longer be the theory to describe the collapse dynamics. In GR, the energymomentum couples to the metric. However, when the spin of particles is introduced into the framework, it is expected to couple to a geometrical quantity related to the rotational degrees of freedom in the spacetime. This point of view suggests a spacetime which is nonRiemannian, namely generalizations of GR induced from the explicit presence of matter with such spin degrees of freedom [77, 78, 79]. One such framework, which will allow nontrivial dynamical consequences to be extracted is the Einstein–Cartan (EC) theory [79, 80] where the metric and torsion determine the geometrical structure of spacetime.^{3} The torsion can be interpreted as caused by microscopic effects, e.g., by fermionic fields which are not taken explicitly into account [78].
Within the context of EC theories, it has been shown that considering the induced repulsive effects extracted from (averaged) spin interactions, the BigBang singularity can be replaced by a nonsingular bounce, before which the universe was contracting and halts at a minimum but finite radius [88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100]. However, a curvature singularity as the final fate of a gravitational collapse process can still occur even if explicit spin–torsion and spin–spin repulsive interactions [101] are taken into account. The argument that has been put forward is that since photons neither produce nor interact with the spacetime torsion, the causal structure of an EC manifold, determined by light signals, is the same as in GR. Hence the singularity theorems in GR can be generalized to the EC theory by taking into account a combined energymomentum tensor which would include, by means of some suitable averaging procedure, spin contributions [102].
The results conveyed within this paper are twofold. We consider a spherically symmetric configuration in the presence of a negative cosmological constant [103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114] whose matter content is assumed to be a homogeneous Weyssenhoff fluid [115, 116, 117, 118, 119] that collapses under its own gravity. On the one hand, the first class of our solutions is shown to evolve toward a spacetime singularity where the role of the negative cosmological constant is to set up the gravitational attraction through a positive pressure term. Then, as the collapse proceeds, a repulsive pressure computed from averaged spin–spin and spin–torsion interactions, balances the inward pressure, preventing trapped surfaces from forming in the later stages. Thus, the resulting singularity could be at least locally naked. Moreover, it is pertinent to point out that our analysis shows that, depending on the spin and energy density parameters, trapped surfaces can either be formed or avoided. On the other hand, second class of solutions suggest that the spin contributions to the field equations may generate a bounce that averts the formation of a spacetime singularity. Let us furthermore note that, in contrast to some alternative theories of gravity e.g., the Gauss–Bonnet theory in which the Misner–Sharp energy is modified [120, 121], our approach will involve only the manipulation of the matter content. Hence the Misner–Sharp energy which is the key factor that determines the dynamics of the apparent horizon is defined in the same manner as that in GR [122, 123, 124].
The organization of this paper is then as follows. In Sect. 2 we present a brief review on the background field equations of the EC theory in the presence of a Weyssenhoff fluid and a negative cosmological constant. Section 3 provides a family of solutions, some of which represent a collapse scenario that leads to a spacetime singularity within a finite amount of time. In Sect. 4 we study the dynamics of apparent horizon and induced spin effects on the formation of trapped surfaces and show that trapped surface avoidance can occur for a subset of collapse settings. We examine the curvature strength of the naked singularity in Sect. 4.2 and show that the singularity is gravitationally strong in the sense of Tipler [125, 126, 127]. A second class of solutions exhibiting a bounce is presented in Sect. 4.3 where we show how the presence of a spin fluid could affect the dynamics of the apparent horizon. In Sect. 5 we present a suitable solution for an exterior region and discuss therein the matching between interior and exterior regions. Finally, conclusions are drawn in Sect. 6.
2 Equations of motion
3 Solutions to the field equations
4 Spin effects on the collapse dynamics
4.1 Singular solutions
4.2 Strength of the naked singularity
4.3 Nonsingular solutions
In the right panel of Fig. 3, we plot the Hamiltonian constraint (26) throughout the dynamical evolution of the collapsing object as governed by (27). We see that this constraint is numerically satisfied with the accuracy of the order of \(10^{6}\) or less.
We also need to check if a dynamical horizon is formed during the whole contracting and expanding phases. Firstly, as we stated in Sect. 4.1, the regularity condition has to be respected at the time at which the collapse commences. Let us define the maximum radius \(r_{\Sigma _\mathrm{max}}\) in such a way that if \(r_{\Sigma }=r_{\Sigma _\mathrm{max}}\), then the regularity condition would be violated. Therefore, if the boundary is chosen so that \(r_{\mathrm{min}}<r_{\Sigma }<r_{\Sigma _\mathrm{max}}\), four horizons can form.
The existence of a minimum value for the size of the collapsing object can be translated as saying that the speed of collapse has to be limited. As the solid curve in the right panel in figure 6 shows, in the early stages of the collapse, the trajectory of the system in \((\dot{a},a)\) plane follows the dashed curve in which the spin effects are neglected. At later times, it deviates from this curve to reach the maximum value for the speed of collapse, i.e., at the first inflection point. After this time, the collapse progresses with a decreasing speed reaching the minimum value of the scale factor, after which the collapsing cloud turns into an expansion. The absolute value of the collapse velocity is bounded throughout the contracting and expanding phases. In this sense, there can be found a maximum value for the collapse velocity (as related to a minimum value for the surface boundary or a threshold mass) so that for \(\dot{a}>\dot{a}_{\mathrm{max}}\) the horizon equation is never satisfied (see the horizontal dotted line labeled as A). However, if \(\dot{a}=\dot{a}_{\mathrm{max}}\), two horizons could still appear, first one at the contracting and the second one at the expanding regimes. Both of these horizons form at the same value of the scale factor at inflection points (see the horizontal dotted line labeled as B). The third possibility is \(\dot{a}<\dot{a}_{\mathrm{max}}\), for which four horizons could appear at the four phases of dynamical evolution of the scenario (see the horizontal dotted line labeled as C). On the other hand, in contrast to these cases, the collapse velocity diverges when the spin effects are absent and the horizon equation is always satisfied, the dashed curve.
5 Exterior solution
The gravitational collapse setting studied so far deals with the interior of the collapsing object. We found two classes of solutions, where for the singular ones, depending on the spin source parameters and initial energy density the apparent horizon can be avoided. However, the absence of apparent horizon in the dynamical process of collapse does not necessarily imply that the singularity is naked [154]. In fact, the singularity is naked if there exist future pointing null geodesics terminating in the past at the singularity. These geodesics have to satisfy \({\mathrm{d}t}/{\mathrm{d}r}=a(t)\) in the interior spacetime so that the area radius must increase along these geodesics. As discussed in [15], this situation cannot happen since the singularity occurs at the same time for all collapsing shells. However, this process, to be completely discussed, may require a suitable matching to an exterior region whose boundary \(r=r_{\Sigma }\) is the surface of the collapsing matter that becomes singular at \(t=t_s\), into which null geodesics can escape. Employing the junction conditions [155, 156], our aim here is to complete the full spacetime geometry presented for the spherically symmetric gravitational collapse via matching the homogeneous interior spacetime to a suitable exterior spacetime.
Now, from the first part of (26), we see that for specific values of \(n\) and \(w\), taken from the brown region of Fig. 1, the collapse velocity tends to infinity. Thus, there is no minimum value for \(r_{\Sigma }\) (or correspondingly a minimum mass for the collapsing volume) so that the horizon can be avoided. In contrast, for \(n\) and \(w\) taken from the yellow region of Fig. 1, the speed of collapse stays bounded until the singularity time at which the scale factor vanishes. This means that to satisfy the horizon condition in the limit of approach to the singularity, the boundary of the volume must be taken at infinity which is physically irrelevant. Thus we can always take the surface boundary so that the apparent horizon is avoided. Therefore, if the collapse velocity is bounded we can take the boundary surface to be sufficiently small so that the formation of horizon is avoided during the entire phase of contraction. Furthermore, the null geodesic that has just escaped from the outermost layer of the mass distribution of the cloud (\(r_{\Sigma },t_s\)) can be extended to the exterior region exposing the singularity to external observers. For bouncing solutions, as the right panel of Fig. 6 shows, \(\dot{a}\) remains finite throughout the collapsing and expanding phases, thus by a suitable choice of the boundary surface, the apparent horizon is failed to cover the bounce.
6 Concluding remarks
The study of the endstate of matter gravitationally collapsing becomes quite interesting when averaged spin degrees of freedom and torsion are taken into account. To our knowledge, the literature concerning this line of research is somewhat scarce,^{17} see e.g., [101]. Torsion is perhaps one of the important consequences of coupling gravity to fermions. In general, this leads to nonRiemannian spacetimes where departures from the dynamics of GR would be expected and should be explored. The well known and established CSK [78] theories can also be a starting point.^{18} Nevertheless, the explicit presence of fermionic fields may not provide a simple enough setup to investigate the final outcome of a gravitational collapse. There are, however, other, perhaps more manageable scenarios. They employ torsion just to mimic the effects of matter with spin degrees of freedom on gravitational systems.
It was in that precise context that we have therefore considered the approach presented in this paper. More precisely, we studied the gravitational collapse of a cloud whose matter content was taken as a Weyssenhoff fluid [115, 116, 117, 118, 119] in the context of the EC theory [102], i.e., with torsion. A negative \(\Lambda \) was included to provide an initially positive pressure, so that a collapse process could initially be set up. The torsion is not, however, a dynamical field, allowing it to be eliminated in favor of algebraic expressions.
In addition, we have restricted ourselves to a special but manageable spacetime model where the interior region line element is a FLRW metric, allowing a particularly manageable framework to investigate. The corresponding effective energymomentum from a macroscopic perspective has a perfect fluid contribution plus those induced from averaged spin interactions. 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 the formation of trapped surfaces, from one hand, and the possibility of singularity removal from the other hand.

For singular solutions (\(\ell >0\)), the formation or otherwise of trapped surfaces not only depend on the equation of state parameter but also on the spin density divergence term (\(n\)). Therefore, from determining the initial setting subject to (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, trapped surfaces can either develop (for \(n<2, w>1/3\)) or be avoided (\(2<n<0, w<1/3\)) throughout the collapse.

A special case in which the equation of state of spin fluid is \(p_{\mathrm{SF}}=\rho _{\mathrm{SF}}\) was considered separately and it was found that no singularity occurs. This very unorthodox case can be thought of as a stationary state.

The set of collapse solutions can be categorized through the sixdimensional space of the parameters (\(J_i^2,\sigma _i^2,n,w,\) \(a_i,\rho _{i_{\mathrm{SF}}}\)) so that the first two are related to initial values for spin source parameters (note that \(J_i^2=J_0^2a_i^n\) and \(\sigma _i^2=\sigma _0^2a_i^n\)). The next two parameters are the rate of divergence of spin density and barotropic index and the last two are the initial values of the scale factor and energy density. Each point from this space represents a collapse process that can be either led to a spacetime singularity or a nonsingular bounce. Determining the suitable ranges for this set of initial data is not straightforward and so, for the sake of clarity, we have to deal with the twodimensional subspaces by fixing four of the above parameters. However, we could infer that among the allowed sets of the initial data we can always pick up those for which trapped surfaces are prevented (in singular solutions) during the collapse scenario (see the regions in Fig. 1), where we have fixed the same initial values for energy density and scale factor.

Depending on the initial value of energy density and the source parameters related to spin–spin contact interaction and axial current, singular (\(\ell >0\) and \(C_0>0\)) and nonsingular (\(\ell <0\) and \(C_0>0\)) solutions can be found. In the former the singularity occurs sooner than the case in which the spin correction term is neglected (see the left panel in Fig. 2). For the nonsingular scenario, the collapse process halts at a finite value of the scale factor and then turns to expansion.
It is also interesting to note that, beside the Frenkel condition we employed here, we could consider the possibility of relaxing it, therefore allowing to take a more general matter content. 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.
Finally, we would like to present a few possible additional subsequent lines of exploration.
Although being a wider setup with respect to GR, it could be fruitful to generalize action (1). More concretely, replacing the cosmological constant by some scalar matter. This would allow for the establishment of limits for the dominance of any matter component (and associated intrinsic effects) toward a concrete gravitational collapse outcome where, for example, bosonic and fermionic matter would be competing. Perhaps more challenging would be to employ a Weyssenhoff fluid description that could have different features whether we use \(s=\frac{1}{2}\) fermion or a Rarita–Schwinger field with \(s=\frac{3}{2}\) spin angular momentum. The gravitational theory of such latter particles in the presence of torsion has been discussed in [185].
Footnotes
 1.
There is a recent discussion by Hawking [3] arguing that this role is played instead by the apparent horizon, which is formed during the collapse process and is responsible for concealing the singularity to the outside observers.
 2.
The CCC is categorized into two types, the weak cosmic censorship conjecture (WCCC) and the corresponding strong version (SCCC). WCCC states that there can be no singularity communicating with asymptotic observers, thus forbidding the occurrence of globally naked singularities, while SCCC asserts that timelike singularities never occur, prohibiting the formation of locally naked singularities [4, 5]. Whereas the CCC is concerned with stability of solutions to Einstein’s field equations, there is a second class of censorship conjectures [6] which asserts that all naked singularities are in some sense gravitationally weak.
 3.
Somewhat related to such settings, let us mention the teleparallel theories of gravity [81, 82], as well as the Lyra theory [83, 84, 85]. The latter concerning the propagation of torsion by means of expressing it as the gradient of a scalar field. This scalar field can be suitably regarded as a gauge function, describing a torsion potential or as a Brans–Dicke scalar field which couples nonminimally to the curvature [86, 87].
 4.
For the sake of generality, we keep \(\kappa \) and \(\Lambda \) throughout the equations but in plotting the diagrams we set the units so that \(c=\hbar =\kappa =1\) and \(\Lambda =1\).
 5.
This term can be represented by an effective four fermion interaction which, together with a part from a Dirac Lagrangian, can be realized as Nambu–JonaLasinio effective action in \(4\)D.
 6.
In more detail, \(\mathcal {L}_{\mathrm{AC}}\) can be associated to a chiral interaction that corresponds to the coupling of contorsion to the massless fermion fields due to a massless Dirac Lagrangian in a curved background.
 7.
The axial current has been pointed out in the literature as responsible for the Lorentz violation. Constraints have been imposed on some of the torsion components due to recent Lorentz violation investigations [131].
 8.
\(\gamma ^{\mu }\) is defined by \(\left[ \gamma ^{\mu },\gamma ^{\nu }\right] =2g^{\mu \nu }\), \(\gamma ^5=\frac{i}{\sqrt{g}}\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}\) is a chiral Dirac matrix, \(\psi \) is a fermion field, \(\bar{\psi }=\psi ^{\dagger }\gamma ^{0}\) is the conjugate fermion field and \(\epsilon _{\alpha \beta \rho \mu }\) is the totally antisymmetric LeviCivita tensor.
 9.
This translates as saying that the intrinsic spin contribution (in the form of the antisymmetric spin density tensor) of a matter field is spacelike in the rest frame of the fluid.
 10.
The spin–spin and spin–torsion interactions are only significant over microscopic ranges, i.e., at sufficiently high matter densities. This means that the EC theory does not directly challenge general relativity at large scales. In order to take into account the macroscopic effects of spin contributions within the framework of EC theory, a suitable averaging of the spin is assumed [133]. It is worth mentioning that in the process of taking the average of a spherically symmetric isotropic system of randomly oriented spin particles, the average of the spin density tensor is assumed to vanish, \(\langle S^{\mu \nu }\rangle \)=0, but for the spin squared terms \(\langle S^{\mu \nu }S_{\mu \nu }\rangle \ne 0\).
 11.
In a collapse setting whose matter content is explicitly fermion dominated it is conceivable that the effective spinning fluid might be polarized. Thus, a spin alignment due to the presence of strong magnetic fields (cf. [135, 136, 137, 138]) 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 energymomentum tensor. Such rotational effects on the collapse process (e.g., gravitational collapse of a system of counterrotating particles  the“ Einstein cluster”[139, 140]) have been studied in [141, 142, 143]. It is shown there that trapped surface formation can be avoided, and so the singularity can be visible, if the angular momentum is strong enough.
 12.
In general, \(\sigma _0^2\) and \(J_0^2\) are the source parameters for the squared spin density and axial current, respectively, and should not be confused with their initial values defined as \(\sigma _i^2=\sigma _0^2 a_i^n\) and \(J_i^2=J_0^2 a_i^n\).
 13.
The choice \(w=1\) is discussed separately at the end of Sect. 4.3, because it corresponds to a nonsingular case.
 14.
Through the whole paper, we are excluding this case unless it is explicitly mentioned.
 15.
It should be noted that bigrip, sudden or even type III singularities do not happen here since \(\rho \) and \(p\) are finite at \(t=t_c\). A type IV singularity does not occur either since the higher derivatives of \(H\) do not diverge at \(t=t_c\) [153].
 16.
It should be noticed that if no bounce occurs and the collapse goes beyond \(a_c\), the effective energy density of the collapsing object would be negative and, as a result, the weak energy condition will be violated. The gravitational collapse of regions with negative energy density has been discussed in the literature, mainly in the context of topological black holes [151]. It has been claimed that topology changing processes, due to quantum fluctuations of spacetime, would be a possible mechanism for such behavior. However, this discussion is beyond the scope of this paper.
 17.
With respect to the initial cosmological singularity, there seems to have been made more efforts in analyzing it when fermionic terms impose modifications to the classical equations (explicitly by means of fermionic degrees of freedom being present or induced by means of some averaged quantities); see e.g., ([88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 128, 129, 130, 132, 133, 166, 167, 168, 169]).
 18.
A collapse setting was introduced in [170] where the nonminimal coupling of classical gravity to fermions results in the singularity avoidance.
Notes
Acknowledgments
The authors are grateful to D. Malafarina and R. Goswami for useful discussions and to F. W. Hehl for helpful correspondence. A. Ranjbar also would like to thank F. Canfora and J. Zanelli for their interesting comments.
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