# Collapse and dispersal of a homogeneous spin fluid in Einstein–Cartan theory

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## Abstract

In the present work, we revisit the process of gravitational collapse of a spherically symmetric homogeneous dust fluid which is described by the Oppenheimer–Snyder (OS) model (Oppenheimer and Snyder in Phys Rev D 56:455, 1939). We show that such a scenario would not end in a spacetime singularity when the spin degrees of freedom of fermionic particles within the collapsing cloud are taken into account. To this purpose, we take the matter content of the stellar object as a homogeneous Weyssenhoff fluid. Employing the homogeneous and isotropic FLRW metric for the interior spacetime setup, it is shown that the spin of matter, in the context of a negative pressure, acts against the pull of gravity and decelerates the dynamical evolution of the collapse in its later stages. Our results show a picture of gravitational collapse in which the collapse process halts at a finite radius, whose value depends on the initial configuration. We thus show that the spacetime singularity that occurs in the OS model is replaced by a non-singular bounce beyond which the collapsing cloud re-expands to infinity. Depending on the model parameters, one can find a minimum value for the boundary of the collapsing cloud or correspondingly a threshold value for the mass content below which the horizon formation can be avoided. Our results are supported by a thorough numerical analysis.

## Keywords

Apparent Horizon Gravitational Collapse Torsion Tensor Weak Energy Condition Affine Connection## 1 Introduction

One of the most important questions in a gravitational theory, such as general relativity (GR), and relativistic astrophysics is the gravitational collapse of a massive star under its own gravity at the end of its life cycle. A process in which a sufficiently massive star undergoes a continual gravitational collapse on exhausting its nuclear fuel, without achieving an equilibrium state [2]. According to the singularity theorems in GR [3, 4, 5], the spacetimes describing the solutions of the Einstein field equations in a typical scenario of the collapse would inevitably admit singularities.^{1} These theorems are based on three main assumptions under which the existence of a spacetime singularity is foretold in the form of geodesic incompleteness in the spacetime. The first premise is in the form of a suitable causality condition that ensures a physically reasonable global structure of the spacetime. The second premise is an energy condition that requires the positivity of the energy density at the classical regime as seen by a local observer. The third one requires that gravity be so strong that trapped surface^{2} formation must occur during the dynamical evolution of a continual gravitational collapse.

The first detailed treatment of the gravitational collapse of a massive star, within the framework of GR, was published by Oppenheimer and Snyder [1]. They concluded that gravitational collapse of a spherically symmetric homogeneous dust cloud would end in a black hole. Such a black hole is described by the presence of a horizon which covers the spacetime singularity. This scenario provides the basic motivation for the physics of black holes and the cosmic censorship conjecture (CCC) [8, 9, 10, 11]. This conjecture states that the spacetime singularities that develop in a gravitational scenario of the collapse are necessarily covered by the event horizons, thus ensuring that the collapse end-product is a black hole only. As no proof, nor a stringent mathematical formulation of the CCC has been available so far, a great deal of effort has been made in the past decades to perform a detailed study of several collapse settings in GR, in order to extend our understanding of this phenomenon.

While black hole physics has given rise to interesting theoretical as well as astrophysical progress, it is necessary, however, to investigate more realistic collapse settings in order to put black hole physics on a firm status. This is because the OS model is rather idealized and pressures as well as inhomogeneities within the matter distributions would play an important role in the collapse dynamics of any realistic stellar object. It is therefore of significant importance to broaden the study of gravitational collapse to more realistic models in order to deal with this question: what ways are there for possible departures in final outcomes, as opposed to the homogeneous dust cloud collapse? Within this context, several gravitational collapse settings have been investigated over the past years which represent the occurrence of naked singularities.^{3} Work along this line has been reported in the literature within a variety of models; among them we quote the role of inhomogeneities within the matter distribution on the final fate of gravitational collapse [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], collapse of a perfect fluid with heat conduction [23, 24, 25, 26], the effects of shear on the collapse end-product [27, 28, 29, 30], and the collapse process in the context of different gravitational theories [31, 32] (see also [33, 34] for recent reviews).

On the other hand, though GR has emerged as a highly successful theory of gravitation, it suffers from the occurrence of spacetime singularities under physically reasonable conditions. It is therefore plausible to seek for the alternative theories of gravitation whose geometrical attributes are not present in GR. This allows for the inclusion of more realistic matter fields within the structure of stellar objects, in order to cure the singularity problem. In this regard, since the realistic stars are made up of fermions, it would be difficult to reject the role of intrinsic angular momentum (spin) of fermions in collapse studies. As we shall see, the inclusion of spin of fermions and thus its possible effects on the collapse dynamics could be of significant importance, specially at the late stages of the collapse setting where these effects could go against the gravitational attraction to ultimately balance it. In such a scenario the collapse may no longer terminate in a spacetime singularity and instead is replaced by a bounce, a point at which the contraction of matter cloud stops and an expanding phase begins. However, if the spin effects are explicitly present, then GR will no longer be the relevant theory to describe the collapse dynamics. In GR, the energy–momentum tensor couples to the metric, while in the presence of fermions, it is expected that the intrinsic angular momentum is coupled to a geometrical quantity related to the rotational degrees of freedom in the spacetime, the so called spacetime torsion. This obviously is not possible in the ambit of GR so that one is forced to modify the theory in order to introduce torsion and relate it to the spin degrees of freedom of fermions. This point of view suggests a spacetime manifold which is non-Riemannian. One such framework, within which the inclusion of spin effects of fermions can be worked out and thus will allow non-trivial dynamical consequences to be extracted is the Einstein–Cartan (EC) theory [35, 36, 37, 38, 39]. Within this context, many cosmological models have been found in which the unphysical big bang singularity is replaced with a bounce at a finite amount the scalar factor [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. From another perspective, the research of the recent years has shown that in the final stages of a typical scenario of the collapse where a high energy regime governs, the effects of quantum gravity would regularize the singularity that happens in the classical model [55, 56]. In cosmological settings, it is shown that non-perturbative quantum geometric effects in loop quantum cosmology would replace the classical singularity by a quantum bounce in the high energy regime where the loop quantum modifications are dominant [57, 58, 59, 60, 61]. However, since the full quantum theory of gravity has not yet been discovered, investigating the repulsive spin effects of fermions, which is more physically reasonable and confirmed observationally, on the final state of collapse could be well motivated. The organization of this paper is as follows: In Sect. 2 we give a brief review on the field equations in EC theory and the phenomenological Weyssenhoff model. In Sect. 3, we study the collapse dynamics in the presence of spin effects and the possibility of singularity removal. Finally, our conclusions are drawn in Sect. 4.

## 2 Einstein–Cartan theory

^{4}However, the spin corrections are significant only at the late stages of gravitational collapse of a compact object, where super-dense regions of extreme gravity are involved. Therefore, we have a good motivation to investigate the collapse process of material fluid sources which are endowed with spin. Let us now apply Eq. (6) to estimate the influence of spin in the case of the Weyssenhoff fluid, which generalizes the perfect fluid of GR to the case of non-vanishing spin. This model of the fluid was first studied by Weyssenhoff and Raabe [67, 68, 69] and extended by Obukhov and Korotky in order to build cosmological models based on the EC theory [70]. In the model presented in this paper we employ an ideal Weyssenhoff fluid which is considered as a continuous medium whose elements are characterized by the intrinsic angular momentum (spin) of particles. In this model the spin density is described by the second-rank antisymmetric tensor \(S_{\mu \nu }=-S_{\nu \mu }\). The spin tensor for the Weyssenhoff fluid is then postulated to be

## 3 Spin effects on the collapse dynamics and singularity removal

The study of gravitational collapse of a compact object and its importance in relativistic astrophysics was initiated since the work of Datt [72] and OS [1] (see also [73] for a pedagogical discussion). This model, which simplifies the complexity of such an astrophysical scenario, describes the process of gravitational collapse of a homogeneous dust cloud with no rotation and internal stresses in the framework of GR. Assuming the interior geometry of the collapsing object to be that of the FLRW metric, they investigated the dynamics of the continuous gravitational collapse of such a matter distribution under its own weight and showed that for an observer comoving with the fluid, the radius of the star crushes to zero size and the energy density diverges in a finite proper time. For this idealized model, which showed that a black hole is developed as the collapse end-state, the only evolving portion of the spacetime is the interior of the collapsing object, while the exterior spacetime remains like that of the Schwarzschild solution with a dynamical boundary. However, in more realistic scenarios, the dynamical evolution of a collapse setting would be significantly different in the later stages of the collapse where the inhomogeneities are introduced within the densities and pressures. These effects could alter the dynamics of the horizons and, consequently, the scenario describing the fate of collapse [74].

Let us consider the dust fluid (\(w=0\)) for which the solution of (\(k=0\)) clearly represents an expanding solution. For the case (\(k<0\)) the collapse velocity is non-real which is physically implausible [76]. Thus the only remaining case is \(k>0\) for which we are to investigate the collapse dynamics for large and small values of the scale factor, i.e., the early and late stages of the collapse process, respectively.

### 3.1 Numerical analysis

## 4 Concluding remarks

We studied the process of gravitational collapse of a massive star whose matter content is a homogeneous Weyssenhoff fluid in the context of EC theory. Such a fluid is considered as a perfect fluid with spin correction terms that stem from the presence of intrinsic angular momentum of fermionic particles within a real star. The main objective of this paper was to show that, contrary to the OS model, if the spin contributions of the matter sources are included in the gravitational field equations, the scenario of the collapse does not necessarily end in a spacetime singularity. The spin effects can be negligible at the early stages of the collapse, while as the collapse proceeds, these effects would play a significant role in the final fate of the scenario of the collapse. This situation can be compared to the singularity removal for a FLRW spacetime in the very early universe, as we go *backwards in time* [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. We showed that in contrast to the homogeneous dust collapse which leads inevitably to the formation of a spacetime singularity, the occurrence of such a singularity is avoided and instead a bounce occurs at the end of the contracting phase. The whole evolution of the star experiences four phases two of which are in the contracting regime and the other two ones are in the post-bounce regime. While, in the homogeneous dust case without spin correction terms, the singularity is necessarily dressed by an event horizon, the formation of such a horizon can always be prevented by suitably choosing the surface boundary of the collapsing star. This signals that there exists a critical threshold value for the mass content, below which no horizon would form. The same picture can be found in [85] where the non-minimal coupling of gravity to fermions is allowed. Besides the model presented here, non-singular scenarios have been reported in the literature within various models such as \(f(R)\) theories of gravity in the Palatini [86] and metric [87] formalisms, non-singular cosmological settings in the presence of a spinning fluid in the context of EC theory [88], bouncing scenarios in brane models [89, 90, 91, 92, 93], and modified Gauss–Bonnet gravity [94] (see also [95] for recent review). While the spacetime singularities could generically occur as the end-product of a continual gravitational collapse, it is widely believed that in the very final stages of the collapse where the scales are comparable to the Planck length and extreme gravity regions are dominant, quantum corrections could generate a strong negative pressure in the interior of the cloud to finally resolve the classical singularity [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107]. Finally, as we come near the end of this paper we should point out that quantum effects due to particle creation could possibly avoid the cosmological [108, 109, 110, 111] as well as astrophysical singularities [112].

## Footnotes

- 1.
These are the spacetime events where the metric tensor is undefined or is not suitably differentiable, the curvature scalars and densities are infinite and the existing physical framework would break down [6].

- 2.
A trapped surface is a closed two-surface on which both in-going as well as out-going light signals normal to it are necessarily converging [7].

- 3.
In this case, the horizons are delayed or failed to form during collapse, as governed by the internal dynamics of the collapsing object. Then the scenario where the super-dense regions are visible to external observers occurs, and a visible naked singularity forms [2].

- 4.
We note that if the spin is switched off, the field Eq. (6) reduces to the ordinary Einstein’s field equation.

## Notes

### Acknowledgments

The authors would like to sincerely thank the anonymous referee for constructive and helpful comments to improve the original manuscript.

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