# Does space-time torsion determine the minimum mass of gravitating particles?

## Abstract

We derive upper and lower limits for the mass–radius ratio of spin-fluid spheres in Einstein–Cartan theory, with matter satisfying a linear barotropic equation of state, and in the presence of a cosmological constant. Adopting a spherically symmetric interior geometry, we obtain the generalized continuity and Tolman–Oppenheimer–Volkoff equations for a Weyssenhoff spin fluid in hydrostatic equilibrium, expressed in terms of the effective mass, density and pressure, all of which contain additional contributions from the spin. The generalized Buchdahl inequality, which remains valid at any point in the interior, is obtained, and general theoretical limits for the maximum and minimum mass–radius ratios are derived. As an application of our results we obtain gravitational red shift bounds for compact spin-fluid objects, which may (in principle) be used for observational tests of Einstein–Cartan theory in an astrophysical context. We also briefly consider applications of the torsion-induced minimum mass to the spin-generalized strong gravity model for baryons/mesons, and show that the existence of quantum spin imposes a lower bound for spinning particles, which almost exactly reproduces the electron mass.

## 1 Introduction

In a series of papers published around one hundred years ago, Cartan proposed an extension of Einstein’s theory of general relativity in which the spin properties of matter act as an additional source for the gravitational field, influencing the geometry of space-time [1, 2, 3, 4]. In standard general relativity, space-time is described by a four-dimensional Riemannian manifold \(V_4\), and its source of curvature is assumed to be the energy-momentum tensor of the matter content. In [1, 2, 3, 4], Cartan generalized Riemannian geometry by introducing connections with torsion, as well as an extended rule of parallel transport, referred to today as the Cartan displacement. From a mathematical point of view, torsion and the Cartan displacement are deeply related to the group of affine transformations, representing a generalization of the linear group of translations.

*n*is the particle number density and

*m*is the mass of an individual particle with spin \(\sim \hbar \).

It is important to note that, in Einstein–Cartan theory, all forms of rotation, including the angular momentum of an extended macroscopic body, a mass distribution of particles with randomly distributed spins, or an elementary particle with quantum mechanical spin, generate a modification of the standard Riemannian geometry of general relativity via torsion effects. However, in the following, we will adopt the standard interpretation of Einstein–Cartan gravity, according to which the antisymmetric spin density of the theory is associated with the quantum mechanical spin of microscopic particles.

Thus, we use the term “spinning fluid” to refer to an extended body whose infinitesimal fluid elements possess nonzero orbital angular momentum density, derived from *SO*(3) invariance, and the term “spin fluid” to refer to the course-grained (continuum) approximation of a large collection of particles, each possessing quantum mechanical spin. Hence, a spin fluid may also be a spinning fluid, if it possess “extrinsic” angular momentum in additional to “intrinsic” *SU*(2) spin. However, in the following, we will restrict our analysis to bodies with zero net orbital angular momentum, but a nontrivial intrinsic spin density.

At the macro-level, this approach yields a realistic a model of stable, static, compact astrophysical objects, composed of elementary quantum particles, while, on the micro-level, we take the continuum spin-fluid model at face value and apply it to the study of elementary particles themselves. In the latter, elementary “point” particles are modeled as inherently extended bodies, and the resulting physical description qualitatively resembles that obtained in Dirac’s extensible model of the electron [14].

In [15], it was argued that the Big Bang singularity is only avoided due to the high degree of symmetry in the cosmological model used in [12, 13]. However, later studies demonstrated conclusively that, even in anisotropic cosmological models, the solutions of the Einstein–Cartan field equations do not lead to a singularity if the effect of torsion is greater than that of the shear [16]. In [17], it was shown that early-epoch inflation may occur such that the dominant contributions to the effective energy-momentum tensor are given by the matter spin densities. A cosmic no-hair conjecture was also proven in Einstein–Cartan theory by taking into account the effects of spin in the matter fluid [18]. If the ordinary matter forming the cosmological fluid satisfies the dominant and strong energy conditions, and the anisotropy energy \(\sigma ^2\) is larger than the spin energy \(S^2\), then all initially expanding Bianchi cosmologies – except Bianchi type IX – evolve toward the de Sitter space-time on a Hubble expansion time scale \(\sim \sqrt{3/\Lambda }/c\). Static solutions of Einstein–Cartan theory with cylindrical and spherical symmetry were studied in [19, 20, 21, 22, 23, 24, 25, 26].

Realistic cosmological models in Einstein–Cartan theory were considered in [27], where it was shown that, by assuming the Frenkel condition [28, 29], the theory may be equivalently reformulated as an effective fluid model in standard general relativity, where the effective energy-momentum tensor contains additional spin-dependent terms. The dynamics of Weyssenhoff fluids were studied by Palle [30] using a \(1+3\) covariant approach, and this approach was revised and extended in [31, 32]. An isotropic and homogeneous cosmological model in which dark energy is described by Weyssenhoff fluid, giving rise to the late-time accelerated expansion of the Universe, was proposed in [33], and observational constraints from Supernovae Type Ia were also discussed. These results show that, although the cosmological constant is still needed to explain current observations, the spin-fluid model contains some realistic features, and demonstrates that the presence of spin density in the cosmic fluid can influence the dynamics of the early Universe. Interestingly, for redshifts \(z > 1\), it may be possible to observationally distinguish the spin-fluid model and the standard “concordance” model of cold dark matter with a cosmological constant, assuming a spatially flat geometry.

In [34] it was argued that, while spin-fluid dark energy models are statistically admissible from the point of view of the SNIa analysis, stricter limits obtained from Cosmic Microwave Background and Big Bang Nucleosynthesis constraints indicate that models with density parameters scaling as \(a^{-6}(t)\) (a scaling that emerges naturally from a torsion dominated epoch) where *a*(*t*) is the time-dependent scale factor of the Universe, are essentially ruled out by observations. The effects of torsion in the framework of Einstein–Cartan theory in early-Universe cosmology were investigated in [35], while the gravitational collapse of a homogeneous Weyssenhoff fluid sphere, in the presence of a negative cosmological constant, was considered in [36]. For recent investigations of the cosmology and astrophysics of Einstein–Cartan theory see [37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In [47] it was shown that by enlarging the Einstein? Cartan Lagrangian with suitable kinetic terms quadratic in the gravitational gauge field strengths (torsion and curvature) one can obtain some new, massive propagating gravitational degrees of freedom. It was also pointed out that this model has a close analogy to Fermi’s effective four-fermion interaction and its emergent W and Z bosons.

Using an alternative approach, sharp bounds on the maximum mass–radius ratio for both neutral and charged, isotropic and anisotropic compact objects, in the presence of a cosmological constant, were rigorously derived in [54, 55, 56, 57, 58]. For fluids with isotropic pressure distributions and zero net charge (\(Q=0\)), in the absence of dark energy (\(\Lambda = 0\)), the maximum mass–radius ratio bound in all studies reduces to the classic result by Buchdahl, \(2GM/(c^2R) \le 8/9\) [53]. The Buchdahl compactness limit for a pure Lovelock static fluid star was obtained in [59], where it was shown that the limit follows from the uniform density Schwarzschild’s interior solution. For four-dimensional Einstein gravity, or for pure Lovelock gravity in \(d=3N+1\) dimensions, Buchdahl’s limit is equivalent to the criterion that the gravitational field energy exterior to the star is less than half its gravitational mass. The Buchdahl bounds for a relativistic star in the presence of the Kalb–Ramond field in four as well as in higher dimensions were derived in [60].

Since a small but positive cosmological constant is still required in Einstein–Cartan theory, in order to explain late-time accelerated expansion [33], these results must be generalized to include the effects of spin (in the matter fluid) and torsion (in the space-time) in order to obtain realistic mass limits, either for fundamental particles or compact astrophysical objects. Though upper mass–radius ratio bounds are most relevant to the latter, lower mass–radius ratio limits may be applied, theoretically, to the former. In this case, one must ask the question: what is the gravitational radius of a fundamental particle?

For charged particles, Eq. (3) gives rise to a classical minimum radius which, for \(\Lambda R^2 \ll 1\), reduces approximately to the result obtained by Bekenstein, \(R \gtrsim (3/4)Q^2/(Mc^2)\) [61]. Essentially, this reproduces (up to a numerical factor of order unity) the classical radius of a charged particle, obtained by equating its rest mass with its electrostatic self-energy in special relativity. Hence, it may also be taken as a measure of the minimum classical gravitational radius of a charged particle in general relativity. Interestingly, this is also the length scale at which renormalization effects become important for charged particles in QED [62, 63, 64, 65], suggesting a link between the gravitational and quantum mechanical theories.

*M*. This, in turn, is equivalent to the uncertainty in the distance from

*M*to its horizon, \(r_{ H} \sim l_\mathrm{dS}\). Minimizing \(\Delta x_\mathrm{total}\) with respect to either

*M*or

*r*and equating \(R_\mathrm{min} \simeq (\Delta x_\mathrm{total})_\mathrm{min}\) yields \(Q^2/(Mc^2) \lesssim (l_\mathrm{Pl}^2l_\mathrm{dS})^{1/3}\).

*M*is of the same of order of magnitude as the electron mass, \( m_{e} = 9.109 \times 10^{-28} \, \mathrm{g}\). Alternatively, rearranging the expression above gives

Relation (6) was first obtained by Nottale using a renormalization group approach [75], following work by Zel’dovich, who suggested that the dark energy density should be associated with the gravitational binding energy of electron–positron pairs spontaneously created in the vacuum [76]. It was obtained independently in [77], with the use of Dirac’s Large Number Hypothesis [78, 79, 80, 81, 82] in the presence of a cosmological constant \(\Lambda >0\), and in [83] using information theory considerations. A summary of the existing derivations of Eq. (6) is given in [84]. We also note that Eq. (6) was used in [85, 86] as the basis of a cosmological model in which \(\Lambda \propto \alpha _e^{-6}\). From an observational perspective, it was shown in [87] that the value of the fine structure constant and the rate of the acceleration of the Universe are better described by coinciding dipoles than by isotropic and homogeneous cosmological models.

Upper and lower bounds on the mass–radius ratio for stable compact objects in extended gravity theories, in which modifications of the gravitational dynamics are described by a modified (effective) energy-momentum tensor, were obtained in [88], and their implications for holographic duality between bulk and boundary space-time degrees of freedom were investigated. The physical implications of the mass scale \(M_T = (m_\mathrm{Pl}^2m_\mathrm{dS})^{1/3} \simeq m_\mathrm{e}/\alpha _\mathrm{e}\) were considered in [92], where, using the Generalized Uncertainty Principle (GUP) [93, 94, 95, 96, 97, 98, 99], it was shown that a black hole with age comparable to the age of the Universe may form a relic state with mass \(M_T^{^{\prime }}= m_\mathrm{Pl}^2/M_\mathrm{T}\), rather than the Planck mass. The properties of the static AdS star were studied in [100], where it was shown that, holographically, the universal mass limit corresponds to the upper limit of the deconfinement temperature in the dual gauge picture.

The brief summary above illustrates the potential importance of both spin and torsion in the gravitational dynamics of the Universe and, also the fundamental importance of mass bounds for both macroscopic and microscopic objects. Such bounds have been derived for charged/uncharged, isotropic/anisotropic and classical/quantum objects, in the presence of dark energy and without. However, to date, most such bounds have been formulated within the context of general relativity or its analogues [74], or within a class of extended gravity theories which do not include torsion [88]. Thus, it is the purpose of the present paper to consider the problem of upper and lower mass–radius ratio bounds for compact objects in Einstein–Cartan theory, in the presence of a cosmological constant. This represents a generalization of previous work to the important case of torsion gravity.

Thus, we obtain a spin-dependent generalization of the Buchdahl limit for the maximum mass–radius ratio of stable compact objects, which incorporates the effects of both torsion and dark energy, and we rigorously prove that a lower bound exists for spin-fluid objects, even in the absence of a cosmological constant. In the latter case, the lower limit is determined solely by the spin of the particles. In addition, we derive upper bounds on the physical and geometric parameters that characterize the spin fluids using Ricci invariants. As a physical application of our results, we obtain absolute limits on the redshift for spin-fluid objects, which suggest that the observation of redshifts greater than two may indicate of the existence of space-time torsion. Hence, redshift observations can, at least in principle, detect the presence of torsion using compact objects. The implications of mass limits in a spin-generalized strong gravity theory, in which strong interactions and the properties of hadrons are investigated in a mathematical and physical framework analogous to Einstein–Cartan theory, are also briefly discussed. Bounds on the minimum mass of strongly interacting particles are obtained, and the role of spin in the mass relation is discussed.

This paper is organized as follows. The basic physical principles and mathematical formalism of Einstein–Cartan theory are briefly reviewed in Sect. 2. In Sect. 3, the gravitational field equations of Einstein–Cartan theory, in the presence of a cosmological constant, and for a static, spherically symmetric geometry are determined. The generalized Tolman–Oppenheimer–Volkoff equation is also obtained. The spin-generalized Buchdahl inequality, and maximum/minimum mass–radius ratio bounds for compact spin-fluid compact objects are derived in Sect. 4, and complementary bounds on the physical and geometric parameters obtained from the Ricci invariants are presented. Mass–radius ratio bounds in Einstein–Cartan theory with generic dark energy are derived in Sect. 5. The astrophysical implications of our results are presented and discussed in Sect. 6, where the upper limit for the gravitational redshift of compact objects is obtained. The implications of the lower mass–radius ratio bound for elementary particles are also discussed in the framework of an Einstein–Cartan type spin-generalized strong gravity theory. We briefly discuss and conclude our results in Sect. 7.

## 2 Einstein–Cartan theory and the Weyssenhoff fluid

In the present section we briefly review Einstein–Cartan theory and the inclusion of particle spin as a source of gravity. We also derive the gravitational field equations in a spherically symmetric geometry, obtain the generalized Tolman–Oppenheimer–Volkoff equation describing the hydrostatic equilibrium of a massive object, and discuss some specific models of the spin.

### 2.1 Einstein–Cartan theory

Einstein–Cartan theory is a geometric extension of Einstein’s theory of general relativity, which includes the spin density of massive objects as a source of torsion in the space-time manifold. The influence of the spin on the geometric properties and structure of space-time is thus a central feature of the theory, with fermionic fields such as those of protons, neutrons and leptons providing natural sources of torsion [5, 6, 7, 8, 17, 27, 33, 34, 35, 36]. In standard general relativity, the source of curvature in the Riemannian space-time manifold \(V_4\) is the matter energy-momentum tensor. In Einstein–Cartan theory, the Riemannian space-time manifold is generalized to a Riemann–Cartan space-time manifold \(U_4\), with nonzero torsion, and the spin of the matter fluid is assumed to act as its source [5, 6, 7, 8]. Thus, in the Einstein–Cartan theory, the spin-density tensor locally modifies the geometry of space-time, inducing a new geometric property, torsion.

### 2.2 The Weyssenhoff spin fluid

## 3 Static spherically symmetric fluid spheres in the Einstein–Cartan theory

In the present section we write down the interior field equations for a static spherically symmetric geometry in Einstein–Cartan gravity, and we derive the Tolman–Oppenheimer–Volkoff equation, describing the hydrostatic equilibrium properties of spin-fluid spheres. Some simple models of the torsion field are also introduced.

### 3.1 Field equations of spin-fluid spheres

*r*only. The components of the matter energy-momentum tensor are

*r*as

### 3.2 Models for the torsion

In the present section we will briefly review some of the physical and geometrical models proposed to describe torsion in the framework of Einstein–Cartan theory.

#### 3.2.1 The constant torsion model

The simplest assumption one can make about the averaged microscopic spin density is that it has a constant value inside the fluid, so that \(S^2=S_0^2=\mathrm {constant}\). This choice simplifies the field equations considerably. However, one is faced with a serious drawback. Due to the algebraic field equations for torsion, the vacuum region of space-time must be torsion-free. Therefore, a physically viable star should satisfy the condition of vanishing torsion at the surface, in additional to the vanishing pressure which, in general relativity, defines the vacuum boundary. This is the most conservative model one can build.

If one assumes that the “vacuum” region contains some remnant torsion, for instance torsion on cosmological scales, then one could relax this condition and consider solutions where the torsion does not vanish at the boundary but instead takes, for example, the value of the cosmological background torsion.

#### 3.2.2 The general-relativistic conservation equation

As in the previous case, this poses serious problems to the theory. For linear and polytropic equations of state, the vanishing pressure surface coincides with the vanishing density surface. This means there exists some radius *R* where \(\rho (R)=p(R)=0\). Then (40) implies \(S(R)=0\) which appears consistent. However, the problematic point is that \(e^{-\nu (R)} = 0\). Therefore, the metric function \(e^{\nu }\) becomes divergent and the boundary of the star. Consequently, solutions of this type are also not desirable.

#### 3.2.3 The Fermion model

*w*of the equation of state.

The functional form of this torsion contribution is similar to Eq. (40), but is without any link to the metric functions. Consequently, this model is the most viable physical model discussed so far.

### 3.3 Constant density stars in Einstein–Cartan theory

*r*, are then described by

*R*is the radius of the star and \(p_{c}\) is the central pressure. By introducing a set of dimensionless variables \(\left( \eta , M_{e\mathrm ff}, P\right) \), defined according to

*P*are presented, for different values of \(B_2\), in Fig. 1.

As one can see from the figures, even in this simple case, the torsion has some small but observable effects on the global properties of compact astrophysical objects. The presence of torsion reduces the radius of the star from its general-relativistic dimensionless radius \(\eta _S\approx 1.06\) to a somewhat smaller value, \(\eta _S\approx 1.02\). This value is not very sensitive to the assumed values of the parameter \(B_2\). However, when looking at the behaviour of the solution near the vanishing pressure surface, some difference are clear, as one can see from Fig. 2.

Hence, the radius of the star with the torsion effects taken into account is of the order of \(R\approx 10.57\times \left( \rho _0/10^{15}\;\mathrm{g/cm^3}\right) ^{-1/2}\) km, while for the standard general-relativistic star we have \(R\approx 10.98\times \left( \rho _0/10^{15}\;\mathrm{g/cm^3}\right) ^{-1/2}\) km. This represents a discrepancy of less than 5%. The same effect can be seen in the numerical values of the masses of the stars. While for the general-relativistic case \(M_{\mathrm{eff}}\) is of the order of \(M_{\mathrm{eff}}\approx 0.38\), for stars in Einstein–Cartan theory \(M_{\mathrm{eff}}\) has a slightly smaller value of order \(M_{\mathrm{eff}}\approx 0.36\), which gives the corresponding masses values of order \(M\approx 2.65\times \left( \rho _0/10^{15}\;\mathrm{g/cm^3}\right) ^{-1/2}\;M_{\odot }\) and \(M\approx 2.52\times \left( \rho _0/10^{15}\;\mathrm{g/cm^3}\right) ^{-1/2}\;M_{\odot }\), respectively. This corresponds to roughly a 5% change in the mass due to torrion. A good knowledge of the equation of state of dense neutron matter, associated with high precision astronomical observations, may therefore lead to the possibility of discriminating Einstein–Cartan theory from general relativity in the study of compact astrophysical objects.

## 4 Buchdahl limits in Einstein–Cartan theory

In this section, we investigate the effects of the spin density of the matter fluid on the upper and lower mass limits, obtained via the generalized Buchdahl inequality in Einstein–Cartan theory. For a rapidly rotating object, the spherical symmetry is lost, and all physical/geometrical quantities show an explicit dependence on the angular coordinates. However, this may not be (necessarily) true in the case of particles carrying intrinsic quantum mechanical spin. Therefore, in the following, we will tentatively assume that the only effect of the spin and, hence, of the torsion of the space-time, is to modify the thermodynamic parameters of the matter fluid, so that they take the effective forms given by Eqs. (22) and (23), without influencing the spherical symmetry of the system. The upper and lower mass bounds can then be obtained in an analogous way to general relativity.

### 4.1 The Buchdahl inequality in Einstein–Cartan theory

The system of the stellar structure equations given by Eqs. (37), (38) and (49) must be considered together with an equation of state for the spin fluid, \(p=p(\rho )\), and subject to the boundary conditions \(p(R)=0 \), \(p(0)=p_{c}\), \(\rho (0)=\rho _{c}\), \(S^2(0)=0\) and \(S^2(R)=\sigma _0^2\), where \(\rho _{c}\) and \(p_{c} \) are the density and pressure at the centre of the sphere, respectively.

*r*. It therefore follows that the mean effective density of the matter distribution, \(\langle \rho _\mathrm{eff} \rangle =3m_\mathrm{eff}(r)/4\pi r^{3}\), located inside radius

*r*, does not increase either. Hence it follows that, as in standard general relativity, the condition

*r*inside the compact object. We note that this result does not depend on the sign of the cosmological constant term \(\Lambda \).

### 4.2 The maximum mass–radius ratio bound for spin-fluid spheres

### 4.3 The minimum mass–radius ratio bound for spin-fluid spheres – “particles”

*u*, defined as

*u*the condition

*f*(

*u*) has two real roots, and condition (70) can be reformulated as

### 4.4 Bounds on the physical parameters from the Ricci invariants

*r*, satisfying the conditions \(e^{\nu (0)}=\mathrm {constant}\ne 0\) and \(e^{\lambda (0)}=1\), then the Ricci invariants must also be non-singular functions throughout the spin-fluid distribution. Consequently, for a regular space-time, the invariants are non-vanishing at the origin \(r=0\). The invariant \(r_0={^s}\Sigma ={^s}\Sigma _{\mu }^{\mu }\) is given by

*r*, so that \(r_0(0)\ge r_0(R)\), and that both the matter energy density and the pressure vanish at the vacuum boundary of the sphere, we obtain the restriction

## 5 Mass–radius ratio bounds in Einstein–Cartan theory with generic dark energy

In this section, we consider the upper and lower mass–radius ratio bounds for spherical object in the presence of dark energy with generic equation of state, \(P_{0}=w_{0}\rho _{0}c^{2}\) where \(P_{0}~(\rho _{0})\) denotes the dark energy pressure (energy density) respectively. Note that \(\Lambda = \kappa ^{2}\rho _{0}\).

### 5.1 Generic mass–radius ratio bounds in Einstein–Cartan theory

*R*. The mass–radius ratio is then bounded by

### 5.2 Holographic implications of the maximum and minimum mass–radius ratio bounds

In the bulk space-time, torsion contributes negative energy density and pressure for \(S^{2}>0\) and vice versa. This is a unique characteristic of torsion which is different from both ordinary matter and dark energy. For nonzero \(\Lambda \), the bulk space-time has an asymptotic boundary. For \(\Lambda >0\) this is a cosmological horizon, the de Sitter horizon \(\sim \sqrt{3/\Lambda }\), whereas for \(\Lambda <0\) an asymptotically AdS boundary exists instead. The holographic implication of the maximum mass–radius bound is that the maximum information content of the bulk space is equal to the number of quantum gravity “bits” (i.e., Planck-sized patches \(\sim l_\mathrm{Pl}^2\)) on the boundary.

In the asymptotically AdS case, the bulk gravity theory has a dual gauge-theory description on the AdS boundary (see, for example, [102] and the references therein for a review of holographic duality and the AdS/CFT correspondence). It was found by Hawking and Page [103] that AdS space-time at finite temperature has a number of thermal phases distinguished by the existence and size of the black hole in the background. After the proposal of the AdS/CFT correspondence, Witten [104] argued that these AdS phases correspond to the thermal phases of the gauge theory (CFT) living on the AdS boundary and that the Hawking–Page phase transition between the thermal AdS and large-mass AdS black hole space-times corresponds to the deconfinement phase transition of the dual gauge theory. The dual gauge theory on the boundary will undergo a deconfinement phase transition when the temperature exceeds the Hawking–Page temperature of the AdS bulk.

Naturally, if thermal radiation in the AdS bulk sufficiently accumulates, gravitational collapse will occur and a black hole will be formed. Therefore, gravitational collapse in the AdS bulk holographically corresponds to the deconfinement phase transition of the dual gauge matter on the AdS boundary. Consequently, maximum mass bounds for static objects in the bulk inevitably correspond to the minimum possible deconfinement temperature on the boundary. It was found in [100] that there exists a universal upper mass limit for a fermionic star in AdS space, which corresponds to the universal maximum Hawking–Page transition temperature.

A remarkable consequence of the minimum mass–radius bound ratio induced by torsion is the statement that a fermionic particle with Planck radius must have a mass larger than the Planck mass \(m_\mathrm{Pl}=\sqrt{\hbar c/G}\). This can easily be shown as follows. Using the minimum mass–radius ratio bound in Eqs. (76) or (94) and the fermionic spin density in (41), and substituting \(R=R_\mathrm{Pl}=\hbar /m_\mathrm{Pl}c\), we simply obtain \(M_\mathrm{eff}>9m_\mathrm{Pl}/8\). Thus, torsion provides an alternative interpretation of the Planck mass as the minimum mass of the fermionic particle with Planck radius.

## 6 Astrophysical and particle physics applications

In the present section we briefly consider some astrophysical and particle physics applications of the spin-fluid mass–radius ratio bounds obtained in Einstein–Cartan theory. In particular we will point out the effect that the torsion of the spin fluid can have on the gravitational redshift of electromagnetic radiation emitted from the surface of compact stars. In addition, we will investigate the minimum mass–radius ratio bound in the framework of the strong gravity description of strong interactions, which offers an alternative (geometric) description of Yang–Mills type theories. In the latter case, we note that the strong gravity description is valid only approximately and as an effective theory for the gauge-singlet sector of QCD. Hence, it may be used as an effective theory to study confinement but not to describe scattering processes involving *SU*(3) color charge (see [74] for further explanation).

### 6.1 Gravitational redshift

*z*, defined in the Schwarzschild–de Sitter geometry as [49]

While the red shift bound (100) relates to astrophysical objects, an alternative application of the formalism developed in the present paper relates to the physics of fundamental particles – in particular, to alternative mathematical models of the strong interaction. One such model is based on the assumption that tensor fields may play an important role in the physical description of strong interactions. This approach is called “strong gravity” theory, and was initially proposed and developed in [70, 71, 72, 73]. (For alternative approaches to the geometrization of strong interactions see [105, 106, 107, 108].)

### 6.2 Spin-generalized strong gravity

Mathematically, strong gravity is formulated as a two-tensor theory of both the strong and the “ordinary” gravitational interactions, in which equations formally analogous to the Einstein field equations govern the behavior of the strong tensor field. The difference between the ordinary gravitational interaction and the strong gravity interaction results from the different numerical values of the coupling parameters, i.e. \(\kappa _f \simeq 1\) GeV\(^{-1}\) for the strong interaction versus \(k_g \simeq 10^{-19}\) GeV\(^{-1}\) for the Newtonian gravitational coupling [70]. (Here, we follow the notation used in [105, 106, 107, 108].)

Tentatively, we apply a similar logic to Einstein–Cartan theory, replacing \(\kappa \equiv k_g \simeq 10^{-19}\) GeV\(^{-1}\) with \(\kappa _f \simeq 1\) GeV\(^{-1}\) in the field equations, and the corresponding Buchdahl-type inequalities, in order to construct a “spin-generalized strong gravity” theory. In principle, this should be capable of describing certain aspects of realistic strong physics – namely, gauge-singlet interactions, including effective models of confinement – to particles with spin. Strictly, such a generalization of strong gravity theory is in fact necessary to describe baryons (not just mesons) but, despite some early investigations [109, 110, 111, 112, 113], and, somewhat surprisingly, it has not thus far been fully explored in the literature.

*R*the value \(R=r_e=2.81\) fm we can reproduce almost exactly the mass of the electron as \(M_\mathrm{eff}\left( r_e\right) =9.27\times 10^{-28}\) g \(\simeq =m_e\). (The exact value of the electron mass is \(m_e=9.11\times 10^{-28}\) g.) The same result can be achieved by assuming a particle radius of about \(R=1\) fm, and by slightly modifying the value of \(G_f\) to \(G_f=0.30\times 10^{30}\) cm\(^3\)/ g s\(^2\).

- 1.
It is reasonable to assume that the overall spin density of the Universe has a negligible effect on the position of the asymptotic de Sitter horizon \(r_\mathrm{H}(t_0) \simeq l_\mathrm{dS} = \sqrt{3/\Lambda }\), so that the DE-UP (4) remains valid independently of Eq. (75).

- 2.
It is reasonable also to neglect the role of cosmological constant in Eqs. (3) and (75) so that the Bekenstein bound \(R \gtrsim Q^2/(Mc^2)\) and the strong gravity spin bound (76) remain valid independently of each other, and of the DE-UP (4).

- 3.
Since Eq. (104) comes from evaluating the spin bound (76) for the spin of an elementary fermion (103), points 1 and 2 imply that the DE-UP, the Bekenstein bound, and the strong gravity spin-bound (104) all hold independently of each other, at least approximately.

- 4.

*M*. On the other hand, in general relativity, a “particle” of mass

*M*cannot be localized to within a region greater than the associated Schwarzschild radius \(r_S = 2GM/c^2\). If the particle size is less than this distance, no signal from \(r < r_S\) can reach the outside world and gravitational self-trapping occurs. Setting \(\lambda _{C}(M) = r_S(M)\) yields \(M \simeq m_\mathrm{Pl}\), so that the Planck mass marks the boundary between the black hole and elementary particle regimes [116, 117, 118, 119, 120, 121]. In string gravity theory, the equivalent condition yields the “strong gravity Planck mass”

*G*is the Newtonian gravitational constant, which is exactly the value of \(G_f\) postulated in the strong gravity theory [70, 71, 72, 73]. Hence, using this representation for \(G_f\), we obtain for the minimum mass bound in strong gravity the expression

## 7 Discussions and final remarks

In the present paper, we have considered upper and lower mass–radius ratio bounds for compact spin-fluid spheres in Einstein–Cartan theory, in the presence of a dark energy density generated by a cosmological constant, and of a dark fluid satisfying a linear equation of state with coefficient \(0 < w_0 \le 1\). For simplicity, we assumed throughout that the ordinary thermodynamic density and pressure of the matter fluid also satisfies a linear barotropic equation of state. In our analysis, we derived explicit bounds for the mass–radius ratio \(2GM/c^2R\) as a function of the spin density of the compact object \(S^2\), and of the cosmological constant \(\Lambda \) or the generalized dark energy parameters. As our physical model for the spin, we adopted the Weyssenhoff fluid, which represents an unpolarized material (“fluid”) consisting of microscopic particles with randomly orientated intrinsic (quantum) spins.

The effects of the spin degree of freedom are extremely important for the cosmological evolution of the very early Universe and, in the spin-fluid dominated epoch, the energy density of the spin-fluid scales as \((1+z)^6\), where *z* is the cosmological redshift [33, 34]. However, in order to obtain a description consistent with observational data, the contribution of the spin fluid cannot dominate over the standard radiation term before the onset of BBN, i.e., before \(z\approx 10^8\). Nonetheless, by imposing BBN and CMB constraints, a limit of \(\Omega _{s,0}=-0.012\) for the density parameter of a spin fluid is still possible [33] at the \(1\sigma \) level. Though worthy of further study, in the present work we have considered only the effects of the spin fluid on compact astrophysical objects, in which the torsion gives just a small contribution to the matter energy density and pressure, and have not attempted to analyze the interesting case of the spin-fluid dominated cosmological epoch. Hence, the possibilities of the survival, inside high density stars, of the remnants of the initial torsion determined by spin fluids, and of avoiding the Big Bang singularity through torsion contributions to the gravitational field, remain consistent with present day cosmological observations.

In contrast to standard general relativity, we have not found universal limits – which are independent of both \(S^2\) and \(\Lambda \) – for the mass–radius ratio of compact spin-fluid objects. In particular, we found that the spin density plays a key role in determining the minimum mass–radius ratio of a stable, compact, neutral spin-fluid sphere, which we identify with a simgle elementary particle by setting \(S^2 \sim \hbar /R^3\). Crucially, we found that such a limit exists for \(S^2 > 0\), even in the absence of dark energy (\(\Lambda =0\)).

However, while the lower bound on the mass–radius ratio may be applicable to elementary particles, the upper bound is of relevance to astrophysical objects. Our analysis shows that the surface red shift of such objects is strongly modified due to the presence of spin, which affects both the energy density and the pressure distribution inside the fluid sphere, as well as by the presence of non-vanishing spin density at the vacuum boundary. In general, the mass–radius ratio limits depend on the value of the surface spin density, so that different physical models of the spin could lead to very different upper and lower bounds.

A general feature of the parameters which characterize the physical properties of spin-fluid compact objects in Einstein–Cartan theory is that their absolute limiting values depend on both \(\Lambda \) and \(S^2\). These include the minimum/maximum mass–radius ratios, and the surface red shift of the system. Tentatively, we have extended our results to the field of the elementary particles via the strong gravity approach initiated in [70, 71, 72, 73], which has been proposed as an effective geometric description of strong interactions. By rescaling the Newtonian gravitational constant *G* to its strong gravity analogue \(G_f \simeq 10^{30}\;G\), we obtained a mass–spin–radius relation for minimum-mass particles in spin-generalized strong gravity theory, \(M_{\mathrm{eff}}\propto G_fS^2R^3\). Evaluated numerically, this is of the same order of magnitude as the mass of the electron. On the other hand, this result also implies the existence of a minimum mass density, given by \(\rho _{\mathrm{eff}}=M_{\mathrm{eff}}/R^3\propto S^2\), which is fully determined by the spin density of the object. Hence, at least at the level of fermionic elementary particles, both mass and mass density appear as manifestations of the essentially quantum property of intrinsic rotation (spin), which has no classical analogue.

However, in applying the spin-fluid model to elementary particles, we note an intrinsic drawback of our analysis. In this case, we take the spin-fluid description at “face value” as a model of nuclear matter, rather than as a continuum approximation as in the astrophysical case. Although, in principle, this is theoretically viable, we note that for particles the spin does have an overall polarisation, i.e. spins are either “up” or “down”. Whilst, on purely dimensional grounds, we may expect the same or quantitatively similar results will hold, even when the polarisation is explicitly accounted for, it is worth pointing out that our existing model does not explicitly capture this (physical) feature of the elementary constituents of matter.

The possible role of the Einstein–Cartan theory in the physics of elementary particles has also been recently emphasized in [114], where it was proposed that, by using a non-linear extension of the Dirac equation known as the Hehl-Datta equation, obtained within the Einstein–Cartan–Sciama–Kibble generalization of general relativity, one can solve two of the major fundamental problems in theoretical physics [114]: why no elementary fermionic particles exist in the mass range between the electroweak scale and the Planck scale, and what is the nature of the energy counterbalancing the divergencies of the electrostatic and strong force energies of point-like charged fermions near the Planck scale? By using an S-matrix approach, as well as some semiclassical considerations, an equation giving the radius \(r_x\) of an elementary particle of mass \(m_x\) can be derived in the form \(m_xc^2=e^2/r_x-\left( G/r_x^3\right) \left( \hbar /2c\right) ^2\), which for \(m_x=m_e\) correctly reproduces the electron radius. On the other hand, in our approach based on the Einstein–Cartan formulation of strong gravity, describing strong interactions, particle masses are naturally generated at the electroweak energy scale, due to their explicit dependence on the quantum mechanical spin density, whose numerical value is fixed by the fundamental laws of quantum mechanics. In fact, the second term in the mass equation for \(m_x\) in [114] is (almost) the same as the minimum mass given by Eq. (105), written in Newtonian gravity, but with an important sign difference.

In conclusion, the methods developed in the present analysis provide theoretical tools that could aid the experimental detection of the presence of torsion in the natural world, on both astrophysical and elementary particle scales.

## Notes

### Acknowledgements

We would like to thank the anonymous referee for comments and suggestions that helped us to improve our manuscript. The work of BH is partly supported by COST Action CA15117 (Cosmology and Astrophysics Network for Theoretical Advances and Training Actions), which is part of COST (European Cooperation in Science and Technology). TH would like to thank the Yat Sen School of the Sun Yat Sen University in Guangzhou, P. R. China, for the kind hospitality offered during the preparation of this work. ML is supported by a Naresuan University Research Fund individual research grant.

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