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

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

Christian G. Böhmer Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom    Piyabut Burikham High Energy Physics Theory Group, Department of Physics, Faculty of Science, Chulalongkorn University, Phyathai Rd., Bangkok 10330, Thailand    Tiberiu Harko Department of Physics, Babes-Bolyai University, Kogalniceanu Street, Cluj-Napoca 400084, Romania Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom    Matthew J. Lake The Institute for Fundamental Study, “The Tah Poe Academia Institute”,
Naresuan University, Phitsanulok 65000, Thailand
Thailand Center of Excellence in Physics, Ministry of Education, Bangkok 10400, Thailand
July 16, 2019
###### 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.

###### pacs:
04.20.Cv; 04.50.Gh; 04.50.-h; 04.60.Bc

## I 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 Cartan (). In standard general relativity, space-time is described by a four-dimensional Riemannian manifold , and its source of curvature is assumed to be the energy-momentum tensor of the matter content. In Cartan (), 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.

In Einstein-Cartan theory, matter sources space-time curvature as in general relativity. In addition, spin is postulated as the source of torsion in the Riemann-Cartan space-time manifold  spin-tor (). It is interesting to note that the concept of spin was introduced into theories of gravity, even before it was introduced into quantum mechanics, by Uhlenbeck and Goudsmit in 1925 Uhl (). Perfect fluids with spin were first studied by Weyssenhoff and Raabe Weyssenhoff:1947 () and are commonly referred to as Weyssenhoff fluids. (See Boe1 () for a detailed discussion of their physical geometric properties.) Later, an important development in the application of Einstein-Cartan gravity was the proposal by Kopczynski Kop1 () and Trautman Traut () that the spin contributions of a Weyssenhoff fluid may avert the initial singularity at the Big Bang, by stopping the collapse in closed cosmological models at a minimum radius cm. In Einstein-Cartan theory, this corresponds to a matter density g/cm, so that, in the case of a chaotic spin distribution,

 Rmin=(3Gℏ2n8mc4)1/3,ρmax=4m2c23π2Gℏ2, (1)

where is the particle number density and is the mass of an individual particle with spin .

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 a extended body whose infitesimal fluid elements possess nonzero orbital angular momentum density, derived from 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” 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 Dirac_ext ().

In Stew (), it was argued that the Big Bang singularity is only avoided due to the high degree of symmetry in the cosmological model used in Kop1 (); Traut (). 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 Kop2 (). In Gasp (), 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 H1 (). If the ordinary matter forming the cosmological fluid satisfies the dominant and strong energy conditions, and the anisotropy energy is larger than the spin energy , then all initially expanding Bianchi cosmologies - except Bianchi type IX - evolve toward the de Sitter space-time on a Hubble expansion time scale . Static solutions of Einstein-Cartan theory with cylindrical and spherical symmetry were studied in static ().

Realistic cosmological models in Einstein-Cartan theory were considered in T2 (), where it was shown that, by assuming the Frenkel condition Frenkel (), 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 Palle:1998 () using a covariant approach, and this approach was revised and extended in Brechet:2007a (); Brechet08 (). 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 T3 (), 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 , 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 T4 () 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 scaling that emerges naturally from a torsion dominated epoch) where 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 T5 (), while the gravitational collapse of a homogeneous Weyssenhoff fluid sphere, in the presence of a negative cosmological constant, was considered in T6 (). For recent investigations of the cosmology and astrophysics of Einstein-Cartan theory see T7 (); T8 (); T9 (); T10 (); T11 (); T12 (); T13 (); T14 (); T15 (); T16 (). In T17 () it was shown that by enlarging the Einsteinâ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 Fermiâs effective four-fermion interaction and its emergent W and Z bosons.

In min1 () it was shown that, within the framework of classical general relativity, the presence of a positive cosmological constant implies the existence of a minimum density in nature, such that

 2GMc2≥Λ6R3,ρ=3M4πR3≥ρΛ≡Λc216πG. (2)

These results follow rigorously from the generalized Buchdahl inequality for the Einstein-Hilbert action with an additional (positive) cosmological constant term  min1 (). The generalized Buchdahl inequality for charge-neutral, spherically symmetric, gravitating objects in the presence of was first derived in G1 () and was shown to give rise to both maximum and minimum mass-radius ratios for stable compact objects. These results were further generalized to include the effects of charge g2 () and of an anisotropic interior pressure distribution g3 (). The effects of both charge and dark energy were considered in m2 (), yielding the lower mass-radius ratio bound

 M≥34Q2Rc2+ΛR3c212G. (3)

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 sharp (). For fluids with isotropic pressure distributions and zero net charge (), in the absence of dark energy (), the maximum mass-radius ratio bound in all studies reduces to the classic result by Buchdahl,  Buch ().

Since a small but positive cosmological constant is still required in Einstein-Cartan theory, in order to explain late-time accelerated expansion T3 (), 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 , reduces approximately to the result obtained by Bekenstein,  Bekenstein:1971ej (). 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 QED (), suggesting a link between the gravitational and quantum mechanical theories.

Thus, in Burikham:2015sro (), was identified with the total minimum positional uncertainty , obtained by combining the canonical quantum uncertainty with gravitational/dark energy effects due to the existence of a finite horizon in space-times with . In this model, a new form of minimum length uncertainty relation (MLUR), dubbed the “dark energy uncertainty principle’” or DE-UP for short, which explicitly includes the de Sitter length as well as the Planck length , was proposed:

 Δxtotal≥(Δxcanon.)min+Δxgrav≃√λCr+l2PlldSλCr. (4)

Here, denotes the minimum possible canonical quantum uncertainty of a wave packet, corresponding to a particle with Compton wavelength , that has been freely evolving for a time  Calmet:2004mp (); Calmet:2005mh (). The term represents an additional contribution to the total uncertainty, due to the superposition of gravitational field states, which correspond to the superposition of position states associated with . This, in turn, is equivalent to the uncertainty in the distance from to its horizon, . Minimizing with respect to either or and equating yields .

According to the model presented in Burikham:2015sro (), this gives the maximum possible charge-squared to mass ratio for a stable, charged, self-gravitating and quantum mechanical object in general relativity with a positive cosmological constant. Assuming saturation of this bound for a particle that exists in nature and setting then gives

 M≳αe(m2PlmdS)1/3=7.332×10−28g, (5)

where is the Planck mass, is the “de Sitter mass” and is the fine structure constant. The limiting value of is of the same of order of magnitude as the electron mass, . Alternatively, rearranging the expression above gives

 Λ≃ℏ2G2m6ec6e12=l4Plr6e≃10−56 cm−2, (6)

where cm is the classical electron radius. This representation of the cosmological constant in terms of the fundamental constants of nature is consistent with current observational constraints on the value of  Planck (). Interestingly, an analogous formula derived in the context of strong gravity theory strong () correctly predicts the order of magnitude value of the mass of the up quark, as the lightest known charged, quantum mechanical and strongly interacting particle Burikham:2017bkn ().

Relation (6) was first obtained by Nottale using a renormalization group approach Nottale (), 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 Zel'dovich:1968zz (). It was obtained independently in min2 (), with the use of Dirac’s Large Number Hypothesis LNH (); Ray:2007cc () in the presence of a cosmological constant , and in  Beck:2008rd () using information theory considerations. A summary of the existing derivations of Eq. (6) is given in Lake:2017ync (). We also note that the expression (6) was used in Wei () as the basis of a cosmological model in which . From an observational perspective, it was shown in Mariano () 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.

For charge-neutral particles, the only “available” radius is the Compton radius. Substituting into (2) then gives

 M≳mΛ≡√mPlmdS≃10−35g, (7)

as the minimum mass of a stable, charge-neutral, gravitating and quantum mechanical object in general relativity with  B1 (); Lake2017 (). This is consistent with current experimental bounds on the mass of the electron neutrino obtained from Planck satellite data Planck (). In addition, may be interpreted as the mass of an effective dark energy particle, associated with the Compton wavelength , which is of the order of mm. According to this model, the dark energy density is approximately constant over large distances, but becomes granular on sub-millimetre scales, and it is notable that that tentative hints of periodic variation in the gravitational field strength on this length scale have recently been observed submm ().

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 B1 (), and their implications for holographic duality between bulk and boundary space-time degrees of freedom were investigated. The physical implications of the mass scale were considered in B4 (), where, using the Generalized Uncertainty Principle (GUP) GUP1 (), it was shown that a black hole with age comparable to the age of the Universe may form a relic state with mass , rather than the Planck mass. The properties of the static AdS star were studied in Burikham:2014ova (), 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 both the potential importance of 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 Burikham:2017bkn (), or within a class of extended gravity theories which do not include torsion B1 (). 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 Sec. II. In Sec. III, 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 Section IV, and complimentary 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 Sec. V. The astrophysical implications of our results are presented and discussed in Section VI, 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 Section VII.

## Ii 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.

### ii.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 spin-tor (); Gasp (); T2 (); T3 (); T4 (); T5 (); T6 (). In standard general relativity, the source of curvature in the Riemannian space-time manifold is the matter energy-momentum tensor. In Einstein-Cartan theory, the Riemannian space-time manifold is generalized to a Riemann-Cartan space-time manifold , with nonzero torsion, and the spin of the matter fluid is assumed to act as its source spin-tor (). Thus, in the Einstein-Cartan theory, the spin-density tensor locally modifies the geometry of space-time, inducing a new geometric property, torsion.

In holonomic coordinates the torsion tensor is defined as the antisymmetric part of the affine connection spin-tor (); Gasp (); T2 (); T3 (); T4 (); T5 (); T6 (),

 Tλμν=~Γλ [μν]=12(~Γλ μν−~Γλ νμ), (8)

where a tilde denotes geometric objects in geometry In general relativity, the torsion tensor vanishes, due to the assumed symmetry of the connection in its two lower indices.

In Einstein-Cartan theory, the spin-connection 1-form can be split into two parts, a torsion free part (giving the usual spin-connection 1-form , which is related to the standard Christoffel symbol ) and a contortion 1-form , which is related to the the torsion of space-time, so that spin-tor (); Boe1 ()

 ~ωμν=ωμν+Kμν. (9)

The torsion vector and the contortion tensor are related via spin-tor (); Boe1 ()

 Tμ=Deμ=deμ+~ωμνeν=Kμν∧eν, (10)

where we have used the fact that is a torsion-free (Riemannian) connection. The above relation between torsion and contortion implies that their vector and axial vector components are related by

 T[μνλ]=K[μνλ],Tνμν=12Kννμ. (11)

The gravitational field equations of Einstein-Cartan theory are derived by varying the usual Einstein-Hilbert action,

 S=∫d4x√−g[−~R2k2+Lm], (12)

where is the gravitational coupling constant, is the Einstein-Cartan curvature scalar constructed by using the general asymmetric connection of the manifold, by taking the vielbein and the spin-connection as independent variables. Hence the field equations in Einstein-Cartan theory can be written as spin-tor (); Gasp ()

 Rμν−12Rδμν=κ2Σμν, (13)
 Tμνλ+δμνTααλ−δμλTανα=κ2sμνλ, (14)

where denotes the canonical energy-momentum tensor, and is the canonical spin - density tensor of the matter fluid. Note that, here, we have implicitly included the existence of a cosmological constant term, which, for convenience, is incorporated on the right-hand side of the field equations in the definition of . Where necessary, we will redefine the energy-momentum tensor such that , and in the following analysis we will write the -dependent terms explicitly. It is important to mention that the equation governing the torsion tensor is an algebraic equation, and therefore the torsion is cannot propagate beyond the matter distribution, as, for example, a torsion wave. Hence the torsion tensor does not vanish only inside material objects. On the other hand, Einstein’s field equations contain some additional terms that are quadratic in the torsion tensor spin-tor ().

### ii.2 The Weyssenhoff spin-fluid

We adopt the Weyssenhoff fluid model Weyssenhoff:1947 () for the description of matter with nonzero spin. From a physical point of view the Weyssenhoff fluid represents a continuous macroscopic medium (fluid), which is characterized on microscopic scales by the spin of the matter – that is, by the individual spins of the particles which make up the “fluid”. The spin properties of the fluid, including the spin density, are described by an antisymmetric tensor  spin-tor (); T2 (); T3 (); Brechet08 () given by

 Sμν=−Sνμ, (15)

which is the source of the canonical spin - density tensor of the space-time, defined as

 sλμν=uλSμν, (16)

where we have introduced the four-velocity of the fluid element . The Weyssenhoff spin-fluid also satisfies another important condition, the Frenkel condition Frenkel (), which imposes the constraint that the intrinsic spin of the constituent particle of the fluid is space-like in the rest frame of the medium, so that

 Sμνuν=0. (17)

The Frenkel condition leads to an algebraic coupling between spin and torsion, which can be written as

 Tλμν=κ2uλSμν. (18)

This follows from the fact that the torsion tensor becomes trace-free and hence the second and third terms on the left-hand side of (14) vanish.

Mathematically, such a coupling also arises naturally when one performs the variation of the total action of the gravitational field–spinning fluid system T2 (). Thus, an important result in Einstein-Cartan theory is that the torsion contributions to the gravitational field equations are completely described by the spin density of the fluid. The spin-density scalar is an important and useful physical quantity, which is defined as  spin-tor (); T2 (); T3 ()

 S2≡12SμνSμν≥0. (19)

From a computational point of view the field equations of Einstein-Cartan theory simplify considerably for a perfect fluid source, reducing to the effective general-relativistic field equations with additional spin-dependent terms, and a spin field equation, respectively spin-tor (); T2 (); T3 (); T4 (); T5 (); T6 (). The gravitational field equations can be formulated in the Riemann geometry as

 Rμν−12gμνR+Λgμν=κ2SΣμν. (20)

Here is the cosmological constant and the effective energy-momentum tensor of the spin-fluid is

 SΣμν =(ρeffc2+peff)uμuν−peffgμν −2(gρλ−uρuλ)∇ρ[u(μSν)λ]. (21)

The effective mass density and the effective pressure are given by,

 ρeff =ρ−κ2S2=ρS, (22) peff =p−c2κ2S2=pS, (23)

where we introduce the torsional quantities and . In the presence of the cosmological constant the total energy density and total pressure becomes

 ρtot=ρeff+Λκ2=ρ−κ2S2+Λc2κ2, (24)
 ptot=peff−Λκ2=p−c2κ2S2−Λκ2. (25)

The spin field equation is given by,

 ∇λ(uλSμν)=2uρu[μ]∇|λ(uλS[ρ|ν]). (26)

If we assume the Frenkel condition (17), then the spin contribution to the energy-momentum tensor can be reformulated as Boe1 ()

 uα∇β(uβSαμ) =uαSαμ∇νuν+uαuβ∇βSαμ =uαuβ∇βSαμ=−(uβ∇βuα)Sαμ =−aαSαμ. (27)

In the third and fourth step of the derivation the Frenkel condition was necessary for simplifying the results, and we have introduced the acceleration of the fluid , defined by . In the following analysis, we restrict our study to the case for which the acceleration vanishes, . for the sake of simplicity, we also assume that the physical energy density and pressure of the matter satisfy the linear barotropic equation of state,

 p=wρc2, (28)

where is the equation of state parameter.

## Iii 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.

### iii.1 Field equations of spin-fluid spheres

As a starting point in our analysis we assume that the interior line element for a spin-fluid is spherically symmetric, so that

 ds2=eνc2dt2−eλdr2−r2(dθ2+sin2θdφ2), (29)

where the metric tensor components and are functions of the radial coordinate only. The components of the matter energy-momentum tensor are

 SΣ00=ρS,SΣ11=SΣ22=SΣ33=−pS. (30)

The field equations describing the interior of a static spin-fluid sphere in Einstein-Cartan theory then take the form

 −e−λ(1r2−λ′r)+1r2−Λ =c2κ2(ρ−κ2S2), (31) −e−λ(ν′r+1r2)+1r2−Λ =−κ2(p−c2κ2S2), (32) p′=−12(ρc2+p)ν′ +κ2S2(ν′+2S′S), (33)

where Eq. (33) follows from the conservation of the effective energy-momentum tensor, . Eq. (31) can be immediately integrated to give

 e−λ=1−2Gmeff(r)c2r−Λ3r2, (34)

where we have defined the effective mass inside radius as

 meff(r) =4π∫r0ρS¯r2d¯r =4π∫r0[ρ(¯r)−κ2S2(¯r)]¯r2d¯r. (35)

By substituting Eqs. (33) and (34) into Eq. (32) we obtain the generalized Tolman-Oppenheimer-Volkoff equation, describing the equilibrium of spin-fluid spheres in Einstein-Cartan theory as

 (36)

We note that this equation can also be conveniently written using the quantities and . It then simplifies to

 dpSdr=−(ρSc2+pS)r2(1−2Gmeff/c2r−Λr2/3)× {(κ2/2)[pS−(2/3)(Λ/κ2)]r3+Gmeff/c2}, (37) dmeffdr=4πr2ρS. (38)

Formally this system of equations cannot be distinguished from the corresponding equations in the absence of torsion.

### iii.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.

#### iii.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 . 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.

#### iii.2.2 The general-relativistic conservation equation

A second form of the spin scalar can be obtained by imposing the condition that the thermodynamic parameters of the spin-fluid still satisfy the standard general relativistic conservation equation , see static (), which gives the radial spin variation equation

 ν′+2S′S=0. (39)

In turn, this fixes the spin dependence of the metric as

 S2=S20e−ν=S20e∫2dpρc2+p=S20ρ2w/(1+w), (40)

where is an arbitrary constant of integration, and we have used the linear barotropic equation of state (28).

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 where . Then (40) implies which appears consistent. However, the problematic point is that . Therefore, the metric function becomes divergent and the boundary of the star. Consequently, solutions of this type are also not desirable.

#### iii.2.3 The Fermion model

A similar dependence of the spin on the energy density can be obtained as follows static (); Gasp (). We assume that the compact object consists of an ideal fluid made of fermions and that there is no overall polarisation of the spins. It was shown in Gasp () that the contribution to the energy-momentum tensor then takes the form

 S2=18ℏ2⟨n2⟩=18ℏ2A−2/(1+w)wρ2/(1+w), (41)

where the matter is assumed to satisfy the linear barotropic equation of state (28). Here, is a dimensional constant depending on the parameter 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.

### iii.3 Constant density stars in Einstein-Cartan theory

Constant density stars, with are important toy models for estimating general relativistic/modified gravity effects on stellar properties. In the following, we briefly investigate the properties of constant density stars in Einstein-Cartan theory. For simplicity we assume first that the cosmological constant can be neglected, setting in the following analysis. In order to close the system of equations, we consider a pressure dependent “equation of state” for the torsion

 S2=βpℓ, (42)

where and are constants. The variation of the effective mass and thermodynamic pressure, as functions of the radial coordinate , are then described by

 dmeff(r)dr=4π[ρ0−κ2βpl]r2, (43)

together with the corresponding TOV equation

 dpdr=−(ρ0c2+p−2c2κ2βpℓ){(κ2/2)(p−c2κ2βpℓ)r3+Gmeff/c2}r2(1−2Gmeff/c2r)(1−c2κ2βℓpℓ−1). (44)

The system of Eqs. (43) and (44) must be integrated subject to the boundary conditions , and , where is the radius of the star and is the central pressure. By introducing a set of dimensionless variables , defined according to

 r =c√4πGρ0η=10.362×(ρ01015g/cm3)−1/2×ηkm, meff =c3√4πG3ρ0Meff= 6.999×(ρ01015g/cm3)−1/2×MeffM⊙, (45)

and , and by denoting

 Bl(l)=c2κ2β(ρ0c2)l−1, (46)

the structure equations (43)-(44) can be rewritten in dimensionless form as

 dMeff(η)dη=[1−Bl(l)Pl(η)]η2, (47)

and

 [1−lBl(l)Pl−1(η)]dP(η)dη=−[1+P(η)−2Bl(l)Pl(η)]{P(η)[1−Bl(l)Pl−1(η)]η3+Meff(η)}η2(1−2Meff/η). (48)

Eqs. (47) and (48) must be integrated subject to the boundary conditions , , and , where defines the vacuum boundary of the star. Hence, in order to obtain the boundary condition for the pressure at the center of the star, we need to fix the equation of state at . In the following, we assume that the central matter satisfies the Zeldovich, or “stiff” equation of state, so that . This choice fixes the central value of the dimensionless pressure as .

In the following, for simplicity, we will consider only the case , for which . We note that, for the choice , the coefficient is given by . Hence, for the cases considered, the numerical values of are of the order of s. The variation of the dimensionless mass and of the dimensionless pressure are presented, for different values of , 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 to a somewhat smaller value, . This value is not very sensitive to the assumed values of the parameter . 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 km, while for the standard general relativistic star we have 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 is of the order of , for stars in Einstein-Cartan theory has a slightly smaller value of order , which gives the corresponding masses values of order and , 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.

## Iv 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.

### iv.1 The Buchdahl inequality in Einstein-Cartan theory

The gravitational properties of a compact, static, spin-fluid sphere can be described in Einstein-Cartan theory by the spherically symmetric gravitational structure equations,

 dmeffdr=4πρSr2, (49)
 dpSdr=−(ρSc2+pS)[Gmeffc2+κ22(pS−2Λ3κ2)r3]r2(1−2Gmeffc2r−13Λr2), (50)

and

 dνdr=2[Gmeffc2+κ22(pS−2Λ3κ2)r3]r2(1−2Gmeffc2r−13Λr2). (51)

Eqs. (49)–(51) must be considered together with an equation of state for the spin-fluid, , and subject to the boundary conditions , , , and , where and are the density and pressure at the centre of the sphere, respectively. With the use of Eqs. (49)-(51), it is straightforward to show that the metric function , which is positive everywhere within the interior, , for all , satisfies the following differential equation G1 ():

 1r√1−2Gmeff(r)c2r−13Λr2ddr⎡⎣√1−2Gmeff(r)c2r−13Λr21rdζ(r)dr⎤⎦=ζ(r)rddrGmeff(r)c2r3. (52)

This equation is formally analogous to its general-relativisitc counterpart, with effective spin-dependent quantities taking the place of standard thermodynamic variables.

As a next step in our analysis, we adopt the fundamental assumption that the effective density does not increase with increasing radial distance . It therefore follows that the mean effective density of the matter distribution, , located inside radius , does not increase either. Hence it follows that, as in standard general relativity, the condition

 ddrmeff(r)r3<0, (53)

must hold independently of the equation of state of the matter. This is a crucial assumption in the following analysis, and we note that there may be torsion models of stars which do not satisfy this assumption. Consequently, our results would not apply in such cases. By introducing a new independent variable, defined as G1 ()

 ξ=∫r0r′(1−2Gmeff(r′)c2r′−13Λr′2)−12dr′, (54)

we obtain, from Eq.(52), the fundamental result that in Einstein-Cartan theory all stellar-type spin-fluid distributions with negative density gradient satisfy the condition

 d2dξ2(eν(ξ)2)<0,∀r∈[0,R]. (55)

With the use of the mean value theorem, it follows that

 ddξ(eν(ξ)2)≤eν(ξ)2−eν(0)2ξ, (56)

or, by taking into account that , we obtain

 ddξ(eν(ξ)2)≤eν(ξ)2ξ. (57)

In terms of our initial variables we therefore obtain the inequality

 (G/c2)meff(r)+(κ2/2)[pS(r)−2Λ3κ2]r3r3√1−2Gmeffc2r−13Λr2≤⎡⎢⎣∫r0r′(1−2Gmeff(r′)c2r′−13Λr′2)−12dr′⎤⎥⎦−1. (58)

Since, as already pointed out, for stable compact objects the mean density does not increase outwards, it follows that

 meff(r′)r′≥meff(r)r(r′r)2,∀r′≤r. (59)

For convenience, we now introduce the dimensionless variable , defined as

 α(r)=1+c26GΛr3meff(r). (60)

Moreover, we assume that in Einstein-Cartan theory, in the presence of a cosmological constant, the condition G1 ()

 α(r′)meff(r′)r′≥α(r)meff(r)r(r′r)2, (61)

or, equivalently

 (1+c26GΛr′3meff(r′))meff(r′)r′≥ (1+c26GΛr3meff(r))meff(r)r(r′r)2,∀r∈[0,R],

holds inside any compact spin-fluid object. In fact, the validity of Eq. (IV.1) is independent of the sign of the cosmological constant and is generally valid for all spin-fluid matter distributions with decreasing density profiles. Hence, we can evaluate the right-hand side of Eq. (58) in the following way:

 ∫r0r′(1−2Gmeff(r′)c2r′−13Λr′2)12dr′ = ∫r0r′(1−2α(r′)Gmeff(r′)c2r′)12dr′≥∫r0r′[1−2α(r)Gmeff(r)c2r(r′r)2]12dr′ (63) = c2r32α(r)Gmeff(r)⎡⎢⎣1−(1−2α(r)Gmeff(r)c2r)12⎤⎥⎦.

Finally, with the use of Eq. (58), Eq.  (63) yields the Buchdahl inequality for compact gravitating spheres in Einstein-Cartan theory,

 Gmeff(r)c2+κ22(pS−2Λ3κ2)r3√1−2Gmeff(r)c2r−13Λr2≤2Gmeff(r)c2(1+c26GΛr3meff(r))1−√1−2Gmeff(r)c2r−13Λr2, (64)

which is valid for all inside the compact object. We note that this result does not depend on the sign of the cosmological constant term .

### iv.2 The maximum mass-radius ratio bound for spin-fluid spheres

Let us consider first the case and . By evaluating Eq. (64) at the vacuum boundary of the object , we obtain

 1√1−2GMc2R≤2⎡⎣1−(1−2GMc2R)12⎤⎦−1, (65)

where is the total mass of the star, leading to the well-known Buchdahl limit for the maximum mass of stable, zero-spin density compact objects Buch (),

 2GMc2R≤89. (66)

For , , Eq. (64) leads, instead, to the following upper limit for the mass-radius ratio of compact spin-fluid sphere:

 2GMeffc2R≤(1−13ΛR2)⎡⎢ ⎢ ⎢ ⎢ ⎢⎣1−19(1+3pS(R)⟨ρeff(R)⟩c2−2Λκ2⟨ρeff(R)⟩c2)2(1−13ΛR2)(1+pS(R)⟨ρeff(R)⟩c2)2⎤⎥ ⎥ ⎥ ⎥ ⎥⎦, (67)

where is the total mass of the object, and we have denoted .

### iv.3 The minimum mass-radius ratio bound for spin-fluid spheres – “particles”

The Buchdahl inequality (64) can be rewritten in the equivalent form

 √1−2GMeffc2R−13ΛR2≥GMeffc2R+κ22pS(R)R2−13ΛR23GMeffc2R+κ22pS(R)R2. (68)

By introducing a new variable , defined as

 u=GMeffc2R+16ΛR2, (69)

Eq. (68) can be rewritten as

 √1−2u≥u+a3u+a, (70)

where we have defined

 a=κ22