Static compact objects in Einstein-Cartan theory
Paulo Luz Sante Carloni
gr-qc



paulo.luz@ist.utl.pt

Center for Mathematical Analysis, Geometry and Dynamical Systems,
Instituto Superior Técnico – IST, Universidade de Lisboa –
UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
Centro de Astrofísica e Gravitação - CENTRA, Departamento
de Física, Instituto Superior Técnico - IST, Universidade
de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
Centro de Matemática, Universidade do Minho, Campus de Gualtar,
4710-057 Braga, Portugal

sante.carloni@gmail.com

Centro de Astrofísica e Gravitação - CENTRA, Departamento
de Física, Instituto Superior Técnico - IST, Universidade
de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

We generalize the Tolman-Oppenheimer-Volkoff equations for space-times
endowed with a Weyssenhoff like torsion field in the Einstein-Cartan
theory. The new set of structure equations clearly show how the presence
of torsion affects the geometry of the space-time. We obtain new exact
solutions for compact objects with non-null intrinsic spin surrounded by vacuum,
explore their properties and discuss how these solutions should be
smoothly matched to an exterior space-time. We study how the intrinsic spin of matter changes
the Buchdahl limit for the maximum compactness of stars. Moreover,
under rather generic conditions, we prove that in the context of a
Weyssenhoff like torsion, no static, spherically symmetric compact
objects supported only by the intrinsic spin can exist. We also provide some
algorithms to generate new solutions.

Static compact objects in Einstein-Cartan theory
    Sante Carloni
    December 30, 2023
================================================





§ INTRODUCTION


Compact objects, in particular neutron stars, represent one of richest
environments to probe fundamental physics due their extreme gravitational
fields, densities and the state of the matter that composes them,
especially, at the core. The recent detection of gravitational waves
due to the coalescing of two orbiting neutron stars <cit.>
opened a new window to study their tidal deformations, allowing
the study of the properties of the matter fields that compose this
kind of objects. Nonetheless, the usage of neutron stars as a physics
laboratory is only possible if we have a deep knowledge of their properties.
In particular, it is important to understand how the intrinsic spin[We should remark that here, and in the following, the word “spin” will be used exclusively to represent the quantum spin of the particles that source the gravitational field equations. In no case the word spin will be associated to any form of rotation of the compact objects we will analyze.] of the fermionic
matter particles affects the behavior of such bodies.

In an astrophysical context, the effects of spin were first considered
when Chandrasekhar <cit.> established the maximum
mass an ideal white dwarf could hold due to the electron degeneracy
pressure, before it underwent continuous gravitational collapse (see
also Ref. <cit.> for the rotating case). In the subsequent
years, similar limits relying on the Pauli exclusion principle were
proposed for other types of compact objects, namely neutron stars,
showing that the spin of matter particles markedly influences astrophysical
objects (cf. e.g. Ref. <cit.>). Nevertheless, the
way in which the presence of intrinsic spin affects the properties
of astrophysical bodies, remains largely unknown.

In a affine theory of gravity, the gravitational field is represented
by the geometry of the space-time which is, in turn, is determined by
the energy and momentum of the matter fields. Mathematically, all
classical matter properties are described by an energy-momentum tensor
that acts as a source in the field equations.
Since spin can be considered as an intrinsic angular momentum of the
matter particles, one would expect that this property could also
be encoded in an energy-momentum tensor. However, in
the theory of General Relativity (GR) it appears immediately clear
that there is no obvious way to introduce the spin in a way that is
consistent with the conservation laws for the total angular momentum.
A way around this problem is to endow the space-time with additional
geometrical structure, providing extra degrees of freedom to model
spin and its relation with the gravitational field. This is the fundamental
idea behind the so-called Einstein-Cartan-Sciama-Kible (ECSK) theory
of gravity. In this theory the connection is not imposed be symmetric
so that, the anti-symmetric part of the connection defines an extra
tensor field: torsion. In this way, it is possible to impose a local
Poincaré gauge symmetry on the tangent space of each point of
the manifold such that the matter intrinsic spin can be related with
the torsion tensor field. Indeed, theories of gravity with a non-symmetric
connection (generically called Einstein-Cartan theories) were proposed
even before the discovery of spin. Sciama and Kibble <cit.>
introduced the idea of connecting the torsion tensor with the matter
intrinsic spin, paving the way to a geometrized treatment of spin.

Early works on the ECSK theory focused on the effects of spin on the
evolution of gravitational collapse and the possibility of avoidance
of singularities <cit.>. Only by the
end of the decade, solutions for spherically symmetric space-times
were found <cit.>. The solutions in Ref. <cit.>
were obtained by directly solving the field equations for the ECSK theory.
Such approach, though, leads to great difficulties in searching for
exact solutions. In this article we will adopt a different method
and consider the formalism provided by the 1+1+2 space-time decomposition
<cit.>. Covariant space-time
decomposition approaches were initially devised as a powerful tool
to explore the properties of cosmological models and their perturbations
(see e.g. <cit.>) and only
recently they have been employed to deal with space-times of astrophysical
interest. In ref. <cit.>, this approach was used
to construct - in the context of GR - a covariant version of the Tolman-Oppenheimer-Volkoff
equations. The new equations allowed to pinpoint the mathematical
nature of the problem of determining interior solutions for compact
objects and, for instance, the treatment of stars with anisotropic
pressure. Moreover, in the covariant language it was possible to define
algorithms to generate a number of new exact solutions, and to easily
obtain general theorems (like the ones <cit.>
in GR) which link apparently unrelated solutions.

In this article, we aim to study static compact objects in the context
of the ECSK theory, in particular, study how the presence of spin
affects the possible solutions. Moreover, we will also examine how
the boundary conditions imposed by the smooth junction of two space-times,
with possible non-null torsion, constraint the solutions.

The article is organized as follows: in Section <ref>
we define the 1+1+2 formalism and consider the decomposition of some
tensorial quantities; in Section <ref> we
describe the setup that we propose to study and provide the structure
equations; in Section <ref> we derive
the Tolman-Oppenheimer-Volkoff (TOV) equations for static, locally
rotationally symmetric space-times of class I and II in the presence
of a non-null torsion field; in Section <ref>
we generalize the conditions for the smooth junction of two space-times
with general torsion tensor fields and apply the results to the particular
considered setup; in Section <ref> exact solutions
are derived and studied; in Section <ref>
we provide a set of algorithms to generate new exact solutions from
previously known ones and in Section <ref> we we
summarize the results and conclude.

In this article we shall assume the metric signature (-+++)
and work in the geometrized units system where G=c=1.



§ THE 1+1+2 DECOMPOSITION


Consider a Lorentzian manifold of dimension 4 and a congruence of
time-like curves with tangent vector u. Without loss of generality
we can foliate the manifold in 3-hypersurfaces, V, orthogonal at
each point to the curves of the congruence, such that all quantities
are defined by their behavior along the direction of u and in V.
This procedure is usually called “1+3 space-time decomposition”.
Such decomposition of the space-time manifold relies on the existence
of a projector to the hypersurface V which can be naturally defined
as 

    h_αβ=g_αβ+u_αu_β ,

where g_αβ represents the space-time metric and u_αu^α=-1. the projector h_αβ has the following properties 

    h_αβ    =h_βα ,           h_αβh^βγ    =h_α^γ ,
    
    h_αβu^α    =0 ,           h_α^α    =3 .


The 1+1+2 decomposition <cit.>
builds from the 1+3 decomposition by defining a congruence of spatial
curves with tangent vector field e such that any quantity defined
in the sub-manifold V is defined by its behavior along e and
in the 2-surfaces W. We shall refer to W as “the sheet”.
As before, we can then define a projector onto W by 

    N_αβ=h_αβ-e_αe_β ,

where e_αe^α=1, and such that


    N_αβ    =N_βα ,           N_αβN^βγ    =N_α^γ ,
    
    N_αβu^α    =N_αβe^α=0 ,           N_α^α    =2 .


It is useful to introduce the following tensors 

    ε_αβγ    =ε_αβγσu^σ ,
    ε_αβ    =ε_αβγe^γ ,

derived from the covariant Levi-Civita tensor ε_αβγσ,
with the following properties 

    ε_αβγ    =ε_[αβγ] ,          ε_αβ    =ε_[αβ] ,
    ε_αβγu^γ    =0 ,          ε_αβu^α    =ε_αβe^α=0 ,
    ε_αβγε^μνγ    =h_α^μh_β^ν-h_β^μh_α^ν ,          ε_α^γε_βγ    =N_αβ ,
    ε_μναε^μνβ    =2h_α^β ,          ε_αβγ    =e_αε_βγ-e_βε_αγ+
                       +e_γε_αβ .


Using the results in appendix <ref>,
the covariant derivatives of the tangent vectors u and e can
be written as


    δ_αu_β=N_α^σN_β^γ∇_σu_γ    =N_αβ(1/3θ-1/2Σ)+Σ_αβ+ε_αβΩ ,
    
    D_αu_β=h_α^σh_β^γ∇_σu_γ    =δ_αu_β+(1/3θ+Σ)e_αe_β
        +2Σ_(αe_β)-ε_αλΩ^λe_β+e_αε_βλΩ^λ ,
    ∇_αu_β    =D_αu_β-u_α(𝒜e_β+𝒜_β) ,

and 

    δ_αe_β=    1/2N_αβϕ+ζ_αβ+ε_αβξ ,
    
    D_αe_β=    δ_αe_β+e_αa_β ,
    ∇_αe_β=     D_αe_β-u_αα_β-𝒜u_αu_β+(1/3θ+Σ)e_αu_β+
        +(Σ_α-ε_ασΩ^σ)u_β .


We shall also need to find the various contributions along u, e
and on W of the energy-momentum tensor 𝒯_αβ.
At this point we shall not assume 𝒯_αβ to
have any symmetry. Hence, 

    𝒯_αβ    =μ u_αu_β+p h_αβ+q_1αu_β+u_αq_2β+π_αβ+m_αβ
        =μ u_αu_β+Q_1αu_β+u_αQ_2β+Q_1 e_αu_β+
        +Q_2 u_αe_β+p_r e_αe_β+Π_1αe_β+e_αΠ_2β+
        +p_⊥ N_αβ+Π_αβ+ε_αβ𝕄 ,

with 

    q_1α    =-h_α^σu^γ𝒯_σγ ,          μ    =u^σu^γ𝒯_σγ ,
    
    q_2α    =-u^σh_α^γ𝒯_σγ ,           p     =1/3h^αβ𝒯_αβ ,
    
    Q_1α    =-N_α^σu^γ𝒯_σγ ,           p_r    =p+Π=e^σe^γ𝒯_σγ ,
    
    Q_2α    =-u^σN_α^γ𝒯_σγ ,           p_⊥    =p-1/2Π=1/2N^σγ𝒯_σγ ,
    Π_1α    =N_α^σe^γ𝒯_σγ ,           Q_1    =-e^σu^γ𝒯_σγ ,
    Π_2α    =e^σN_α^γ𝒯_σγ ,           Q_2    =-u^σe^γ𝒯_σγ ,
    π_αβ    =h_⟨α.^σh_.β⟩^γ𝒯_σγ ,          Π    =1/3𝒯_αβ(2e^αe^β-N^αβ) ,
    
    m_αβ    =h_[α.^σh_.β]^γ𝒯_σγ ,          𝕄    =1/2ε^μν𝒯_μν ,
    Π_αβ    =𝒯_{αβ} ,

where the angular and curly parentheses notation
is defined in Eq. (<ref>). Moreover,
the following relations are useful

    q_1,2α    =Q_1,2α+Q_1,2 e_α ,
    π_αβ    =Π_αβ+Π(e_αe_β-1/2N_αβ)+
        +Π_1(α.e_.β)+Π_2(α.e_.β) .


In this paper we will assume that the space-time is endowed with
a linear, metric compatible connection C_αβ^γ.
Such connection is characterized by the metric connection - the Christoffel
symbols - and the torsion tensor field 

    S_αβ ^γ=C_[αβ]^γ .

Using Eq. (<ref>) we can write the torsion
tensor field as 

    S_αβγ=ε_αβ ^μS̅_μγ+W_[α|γu_|β]+S_αβu_γ+u_[αX_β]u_γ ,

with 

    S̅_αβ    =1/2ε_αμνh_β^σS^μν_σ ,
    
    W_αβ    =2u^μh_α^νh_β^σS_μνσ ,
    
    S_αβ    =-h_α^μh_β^νu^σS_μνσ ,
    
    X_α    =2u^μh_α^νu^σS_μνσ .

Notice that the tensors defined in Eq. (<ref>)
are orthogonal to the tangent vector u.

Now, from the definition of the Riemann tensor, R_αβγ^δ:

    R_αβγ^δw_δ=∇_α∇_βw_γ-∇_β∇_αw_γ+2S_αβ^δ∇_δw_γ ,

where w_γ is an arbitrary 1-form; in the case of a Lorentzian
manifold with non-null torsion, the Riemann curvature tensor does
not possess the same symmetries as the torsion free case. This is
discussed very thoroughly for instance in Ref. <cit.> to
which we redirect the reader for further details. In theories with
torsion the Riemann tensor has the following properties: 

    R_αβγδ    =-R_βαγδ ,
    
    R_αβγδ    =-R_αβδγ ,
    
    R_[αβγ]^δ    =-2∇_[α.S_.βγ]^δ+4S_[αβ|^ρS_|γ]ρ^δ ,

and the modified second Bianchi identity 

    ∇_[α.R_.βγ]δ^ρ=Q_αβγδ^ρ ,

where 

    Q_αβγδρ=2S_[αβ|^σR_|γ]σδρ .


The previous properties of the Riemann tensor are completely general,
in particular they are valid for space-times of any dimension. Let
us now consider the particular case of a space-time of dimension 4
with torsion. In this case, R_αβγδ can be
written as the following sum 

    R_αβγδ=C_αβγδ+R_α[γ.g_.δ]β-R_β[γ.g_.δ]α-1/3R g_α[γ.g_.δ]β ,

where C_αβγδ represents the Weyl tensor, R_αβ=R_αμβ^μ
the Ricci tensor and R the Ricci scalar. In precence of torsion,
the Weyl tensor is still defined as the trace-free part of the curvature
tensor, but it does not retain all the other usual symmetries: 

    C_αβγδ    =-C_βαγδ ,
    
    C_αβγδ    =-C_αβδγ ,
    
    C_[αβγ]δ    =R_[αβγ]δ+R_[αβ.g_.γ]δ .

It will be useful for our purposes to give the relation between the
derivative of the Weyl tensor and the Riemann tensor. The generalization
of the formula given in Ref. <cit.> (see also Ref. <cit.>)
is 

    ∇_αC^γδβα    =1/4ε^μνλβQ_μνλσρε^σργδ+
        +3/2g^β[δ.Q^.γ]μν_μν+∇^[δ.R^.γ]β-1/6g^β[γ.∇^.δ]R ,

where Q_αβγδρ is given by Eq. (<ref>).

In what follows, we will need to find the 1+1+2 decomposition of the
Weyl tensor. We start by decomposing C_αβγδ
into its components along u and V as 

    C_αβγδ    =-ε_αβμε_γδνE^νμ-2u_αE_β[γ.u_.δ]+2u_βE_α[γ.u_.δ]-
        -2ε_αβμH^μ_[γ.u_.δ]-2ε_μγδH̅^μ_[α.u_.β] ,

where 

    E_αβ    =C_αμβνu^μu^ν ,
    
    H_αβ    =1/2ε_α^μνC_μνβδu^δ ,
    H̅_αβ    =1/2ε_α^μνC_βδμνu^δ ,

are the “electric” part and “magnetic” parts of the Weyl tensor,
respectively. Eq. (<ref>) generalizes
the results in Ref. <cit.> to the case of non-null torsion.
Note, however, that, differently from the torsionless case, there
are two different tensor quantities associated to the magnetic part
of the Weyl tensor: H_αβ and H̅_αβ.

From the results in Eq. (<ref>), we see
that in the presence of torsion the tensors E_αβ, H_αβ
and H̅_αβ have the properties: 

    H_αβ    =h_α^μh_β^νH_μν ,           H_αβ    =H_(αβ) ,
    H̅_αβ    =h_α^μh_β^νH̅_μν ,          H̅_αβ    =H̅_(αβ) ,
    
    E_αβ    =h_α^μh_β^νE_μν ,           E^α_α    =0 .

Therefore, E_αβ, may not be a symmetric tensor and H_αβ
and H̅_αβ do not have to be trace-free. On the
other hand, due to the properties of the Levi-Civita tensor, even
in the presence of torsion, the magnetic parts of the Weyl tensor
are symmetric under the exchange of indexes. These properties allow
us to decompose the tensors E_αβ, H_αβ
and H̅_αβ as 

    E_αβ    =ℰ(e_αe_β-1/2N_αβ)+ℰ_αe_β+e_αℰ̅_β+ℰ_αβ+ε_αβ𝔼 ,
    
    H_αβ    =1/2N_αβH+e_αe_βℋ+ℋ_αe_β+e_αℋ_β+ℋ_αβ ,
    H̅_αβ    =1/2N_αβH̅+e_αe_βℋ̅+ℋ̅_αe_β+e_αℋ̅_β+ℋ̅_αβ ,

with 

    ℰ    =E_μνe^μe^ν=-N^μνE_μν ,          ℰ_α    =N_α^μe^νE_μν ,
    𝔼    =1/2ε^μνE_μν ,          ℰ̅_α    =e^μN_α^νE_μν ,
    H    =N^μνH_μν ,          ℰ_αβ    =E_{αβ} ,
    ℋ    =e^μe^νH_μν ,          ℋ_α    =N_α^μe^νH_μν ,
    H̅    =N^μνH̅_μν ,          ℋ̅_α    =N_α^μe^νH̅_μν ,
    ℋ̅    =e^μe^νH̅_μν ,          ℋ_αβ    =H_{αβ} ,
                ℋ̅_αβ    =H̅_{αβ} .

where the curly parentheses notation is defined in Eq. (<ref>).



§ DECOMPOSITION OF THE FIELD EQUATIONS


We are now in position to apply that framework to study solutions
of the Einstein-Cartan theory, characterized by the following field
equations


    R_αβ-1/2g_αβR     =8π𝒯_αβ ,
    
    S^αβγ+2g^γ[αS^β]_μ^μ    =-8πΔ^αβγ ,

where 𝒯_αβ represents the canonical energy-momentum
tensor and Δ^αβμ the intrinsic hypermomentum,
found by varying, independently, the Einstein-Hilbert action with
respect to the metric and to the connection. We assume a null cosmological
constant.

From the second Bianchi identity (<ref>)
and the field equations (<ref>) and (<ref>)
we find the conservation laws for the canonical energy-momentum tensor:

    ∇_β𝒯_α^β=2S_αμν𝒯^νμ+1/8π(S_αμ^μR-S^μνσR_ασμν) .

Using Eq. (<ref>) we can introduce the intrinsic
hypermomentum tensor in the above conservation laws and recover the
result in Ref. <cit.> (see also Ref. <cit.>
for similar results derived in a more general context).

We will assume that the source for the above field equations is an
uncharged Weyssenhoff fluid <cit.>. The Weyssenhoff
fluid provides a semi-classical description of a perfect fluid composed
of fermions, such that the fluid is characterized by its energy density,
pressure and spin density. Its canonical energy-momentum tensor is
given by 

    𝒯_αβ=μ u_αu_β+p h_αβ-(𝒜e^μ+𝒜^μ)S_μαu_β ,

where μ and p represent the energy density and pressure of
the fluid, respectively.

Following Refs. <cit.>, the hypermomentum tensor
for the Weyssenhoff spin fluid can be written, in our conventions,
as 

    Δ^αβγ=-1/8πΔ^αβu^γ ,

where u represents the proper 4-velocity of an element of volume
of the fluid and the anti-symmetric spin density tensor, Δ^αβ,
verifies Δ^αβu_β=0. From Eq. (<ref>)
we find, 

    S^αβγ=Δ^αβu^γ ,

hence, comparing Eq. (<ref>) with Eq. (<ref>)
we see that the Weyssenhoff fluid model implies that the tensors S̅^αβ,
W^αβ and X^α, Eq. (<ref>),
are null and S^αβ=Δ^αβ. In this way,
the decomposition of the torsion tensor will coincide with the decomposition
of S^αβ. Taking into account that S_αβ≡ S_[αβ]
we find 

    S_αβ=ε_αβτ+2s_[α.e_.β] ,

with 

    τ    =1/2ε^μνS_μν ,
    
    s_α    =N_α^σe^γS_σγ .





 §.§ The symmetries of the problem


We are interested in solutions of the Einstein-Cartan theory that
are static and locally rotationally symmetric (LRS). Following Ref. <cit.>,
a space-time is said to be local rotational symmetric in a neighborhood
of a point q, B(q), if there exists a non-discrete
sub-group G of the Lorentz group in the tangent space of each q'∈ B(q)
which leaves u, the curvature tensor and their derivatives (up
to third order) invariant. Assuming G to be one-dimensional, we
can set at each point the vector field e to have the same direction
as an axis of symmetry. Then, LRS implies that all covariantly defined
space-like vectors must have the same direction of e - otherwise
they would not be invariant under G. Thus, the vector quantities
{ a_β, α_β, Σ_β, Ω_β ,A_β}
are null in such space-times. Also the shear tensors of the congruences
of curves associated with u and e projected onto the sheet:
Σ_αβ and ζ_αβ, must be null since,
there can not be any preferred direction at the sheet[It should be remarked here that the presence of a generic torsion
tensor field affects the definition of the kinematical quantities
<cit.>. See the Appendix <ref>
for further details. As such, in the presence of a general torsion,
LRS implies that the geometric shear vector fields Σ_g αβ
and ζ_g αβ must be null and not the quantities
Σ_αβ=σ_{αβ} and
ζ_αβ≡δ_{αe_β}. However,
as discussed in Appendix <ref>,
for a Weyssenhoff fluid those are equal hence, from here on out we
shall refer to Σ_αβ and ζ_αβ
as the shear tensors, onto the sheet, of the congruences associated
with u and e, being implicit that we assume the Weyssenhoff
model.].

From the definition of LRS space-times, the Riemann curvature tensor
must also be invariant under G therefore, the vector components
of the Weyl tensor {ℰ_1α, ℰ_2α, (ℋ_,2)_α}
must also be null. Since the Riemann tensor also depends on the torsion
tensor, the latter must also be invariant under the action of G.
Therefore, from Eqs. (<ref>) and (<ref>),
the tensor field s^α, Eq. (<ref>),
must be null. In light of this results and taking into account Eq. (<ref>)
we also conclude that for an LRS space-time the intrinsic hypermomentum
tensor is simply given by 

    Δ_αβ=ε_αβδ ,

where δ=1/2ε^αβΔ_αβ,
with the constraint τ=δ.

Now, an LRS space-time is said to be of class I (LRSI) if the congruence
of the curves associated with vector field e - defined to have
the same direction as the axis of symmetry - is hypersurface orthogonal.[
Following Ref. <cit.>, a space-time is said to be
LRS II when it has locally rotational symmetry and the vector fields
u and e are hypersurface orthogonal. Just so happens, in space-times
with null-torsion, an hypersurface orthogonal congruence has null
vorticity. As such, in literature, LRS II space-times are characterized
and usually referred as space-times with locally rotational symmetry
and vorticity free u and e vector fields. As was shown in Ref. <cit.>
this is not the case for space-times with non-null torsion where an
hypersurface orthogonal congruence does not have null vorticity. In
this article, we will follow the naming convention of Ref. <cit.>.
This has at least one advantage: when comparing results with the null
torsion case, we simply have to compare with the same named class;
for instance, static spherically symmetric space-times, with or without
torsion, always fall in the category of static LRS II space-times
.]
If the congruence of curves associated with the vector field u
is also hypersurface orthogonal, the space-time is said to be LRS
of class II (LRSII). From the results in Ref. <cit.>,
for a torsion tensor given by Eq. (<ref>)
we have that e will be hypersurface orthogonal if and only if 

    ξ=0 ,

and that u will be hypersurface orthogonal if and only if 

    Ω    =τ ,
    
    s_α    =0 ,

where we opted to highlight that s_α will also be null from
the imposition that the congruence of u is hypersurface orthogonal.

Before proceeding we should point out the fact that s_α=0
has an interesting effect on the nature of the Weyssenhoff fluid.
Comparing Eq. (<ref>) with
Eq. (<ref>) we conclude
that for a Weyssenhoff fluid the only non-null covariantly defined
quantities in Eq. (<ref>)
are μ, p and q_1α=-(𝒜e^μ+𝒜^μ)S_μα.
Now, since in an LRS space-time both 𝒜_α and s_α
are null, it implies that q_1α=-(𝒜e^μ+𝒜^μ)S_μα=0,
that is, the contributions of spin in the Weyssenhoff fluid model
for an LRS space-time, will not appear in the canonical energy-momentum
tensor. From this result, one might (wrongly) conclude that 
torsion has no role in the dynamics of the setup. In reality, torsion
will still markedly influence the behavior of the matter fields. Indeed,
for instance, when comparing to space-times with null torsion, where
LRS II space-times are necessarily irrotational (cf. e.g. Ref. <cit.>),
the presence of a non-null torsion of the form of Eq. (<ref>)
will induce a non-null vorticity of the congruence of curves associated
with u, Eq. (<ref>).
Thus, although in the considered setup spin does not appear in the
canonical energy-momentum tensor, it will still markedly change the
geometry of the space-time.

An additional assumption we will consider is that the space-time is
static. Now, a space-time is said to be stationary if it admits the
existence of a time-like Killing vector field Ψ. If the congruence
of time-like curves associated with Ψ are also hypersurface
orthogonal the space-time is said to be static. Given that the choice
of the vector field u is arbitrary, we can write, at each point,
Ψ=C u, where C=C(x^α) is a generic non-null
smooth function of the coordinates. The Killing equation ℒ_Ψg_αβ=0
in presence of torsion can be written as 

    ∇_(α.Ψ_.β)+2S_σ(αβ)Ψ^σ=0 ,

for any metric compatible connection. Assuming Eq. (<ref>),
contracting Eq. (<ref>) with h_μ^αh_ν^β
and h^αβ we have 

    {θ, Σ, Σ_α, Σ_αβ} =0 ,

and u^a∂_aC(x^α)=0. All is left now
is to impose the condition that Ψ is hypersurface orthogonal.
However, if u is hypersurface orthogonal, so is any Ψ=C u.
Hence, for the space-time to be static, Eqs. (<ref>)
must hold.

Lastly, computing the quantities: N_μ^αN_ν^γv^βR_αβγδv^δ,
ε_μ^αβR_αβγδv^δ
and ε^μγv^βR_αβγδe^δ,
we also find that in the considered setup 

    {𝔼, ℰ_αβ, ℋ_αβ,ℋ̅_αβ} =0 .

Therefore, gathering the previous results we find that stationary
LRS I or LRS II space-times permeated by an uncharged Weyssenhoff
fluid are characterized by the following set of quantities {μ, p, ϕ, Ω, 𝒜, τ, ℰ, H, H̅, ℋ, ℋ̅}.



 §.§ Structure equations


We are now in position to find the structure equations for stationary,
locally rotationally symmetric space-time filled by a Weyssenhoff
fluid in the case where the congruence of space-like curves associated
with e are hypersurface orthogonal, that is, in the case when ξ=0.
The non-trivial, independent propagation equations are 

    p̂+𝒜(μ+p)     =-1/4πτℋ̅ ,
    𝒜̂+𝒜(𝒜+ϕ)+2Ω^2    =4π(μ+3p) ,
    ϕ̂+1/2ϕ^2+ℰ    =-16π/3μ ,
    ℰ̂+3/2ℰϕ+ΩH+2(τ-Ω)ℋ̅    =8π/3μ̂ ,
    ℋ̂-1/2ϕ(H-2ℋ)+ℰ(3Ω-2τ)     =-8πΩ(μ+p)+
        +8π/3τ(μ+3p) ,
    
    2Ω̂+Ωϕ    =H ,

and the constraint equations

    ℰ+𝒜ϕ+2Ω^2    =8π/3(μ+3p) ,
    
    2𝒜(Ω-τ)-Ωϕ+ℋ    =0 ,
    Ω(ϕ-2𝒜)+H̅    =0 ,
    H+H̅+2ℋ̅    =0 .


Let us discuss the cases when the congruence associated with u
is either hypersurface orthogonal or not, separately. Consider the
cases when Ω≠τ. In such cases we find from Eqs. (<ref>)
- (<ref>) the following relation between
Ω and τ 

    (ϕ-𝒜)(τ-Ω)+τ̂-Ω̂=0 ,

leading us to conclude that the difference between τ and Ω
can be uniquely described by the behavior of the variables ϕ
and 𝒜. Notice that if at an initial instant Ω
and τ are different then, unless the term ϕ-A diverges,
there will be no point in which they are equal. Conversely, if Ω=τ
at a point these two quantities will be equal at any point.

The relations Ω=τ or Eq. (<ref>),
for stationary LRS II or LRS I space-times, respectively, have the
advantage of not depending directly of the magnetic components of
the Weyl tensor and they can replace one of Eqs. (<ref>)
- (<ref>). As we shall see, it is useful
to remove Eq. (<ref>).

Finally, to close the system we will need an equation of state that
relates the pressure of the fluid with its energy-density: p=p(μ);
and an equation that relates the energy-density of the fluid with
the intrinsic hypermomentum: δ=δ(p(μ),μ).



§ GENERALIZED TOV EQUATION FOR STATIONARY LRS I AND LRS II SPACE-TIMES


With the full set of structure equations we are finally in position
to make the derivation of the generalized Tolman-Oppenheimer-Volkoff
(TOV) equations. Let us start by introducing the scalar function 

    K=8π/3μ-ℰ+1/4ϕ^2-3Ω^2+2Ωτ ,

with the following property 

    K̂=-ϕ K ,

found from the structure equations. Eq. (<ref>)
generalizes the expressions in Refs. <cit.> [Notice that in Ref. <cit.> there is a small typographic
error.]. Moreover, since the Gauss equation is unchanged by the presence
of torsion, it is possible to prove that, in the cases when the vector
fields u and e are hypersurface orthogonal, the quantity K
represents the Gaussian curvature of the 2-sheet orthogonal to both
u and e.

Now, following the treatment in Refs. <cit.>, without
loss of generality, let us re-parameterize the integral curves of
e using a, in general non-affine, parameter ρ, such that
for an arbitrary scalar function F 

    F̂=ϕ F_,ρ .

In particular we have 

    K_,ρ=-K .


Introducing the following set of variables 

    𝕏    =ϕ_,ρ/ϕ ,           𝔹_1    =H/ϕ^2 ,           ℳ    =8πμ/ϕ^2 ,
    𝕐    =𝒜/ϕ ,          𝔹_2    =H̅/ϕ^2 ,          𝒫    =8πp/ϕ^2 ,
    𝔼    =ℰ/ϕ^2 ,          𝔻_1    =ℋ/ϕ^2 ,          Δ    =δ/ϕ ,
    𝕋    =τ/ϕ ,          𝔻_2    =ℋ̅/ϕ^2 ,
    𝕎    =Ω/ϕ ,          𝒦    =K/ϕ^2 ,

we can re-write Eqs. (<ref>) - (<ref>)
as 

    2𝕐_,ρ+2𝕐(𝕏+𝕐+1)     =ℳ+3𝒫-4𝕎^2 ,
    𝒦_,ρ+𝒦(2𝕏+1)     =0 ,
    𝒫_,ρ+𝒫(2𝕏+𝕐)+𝕐ℳ    =2𝕋𝕎(𝕏+𝕐)+2𝕋𝕎_,ρ ,
    
    2𝕎_,ρ+𝕎(2𝕏+1)     =𝔹_1 ,

with the constraints

    ℳ+3𝒫-3𝕐-3𝔼-6𝕎^2    =0 ,
    
    2ℳ+2𝕏+2𝒫-2𝕐-4𝕎^2+1     =0 ,
    
    4𝕐+4𝕎(2𝕋-𝕎)-4𝒫-4𝒦+1     =0 ,
    𝔻_1+𝕎(2𝕐-1)-2𝕐𝕋    =0 ,
    𝔹_2+𝕎(1-2𝕐)     =0 ,
    𝔹_1+𝔹_2+2𝔻_2    =0 ,
    𝕋    =Δ ,

and, depending on whether we are considering stationary LRS I or LRS
II space-times, we have the extra equation 

    𝕎_,ρ-𝕋_,ρ=(1-𝕐+𝕏)(𝕋-𝕎)    , if LRS I,
    𝕎=𝕋   , if LRS II.

The system is closed provided and equation of state such that 𝒫=𝒫(ℳ)
and a relation such that Δ=Δ(𝒫(ℳ),ℳ).

Now, using Eqs. (<ref>) and (<ref>)
to eliminate 𝕏 and 𝕐 in Eqs. (<ref>)
and (<ref>) we find

    𝒫_,ρ    =-𝒫^2+𝒫[7/4-3𝒦+𝕎(8Δ-7𝕎)]+
        +2Δ[(2𝒦-1)𝕎-4𝕎^2(Δ-𝕎)+𝕎_,ρ]+
        +ℳ(1/4-𝒦+𝒫-𝕎^2) ,
    𝒦_,ρ    =-2𝒦(3𝕎^2-2Δ𝕎+𝒦-ℳ-1/4) ,
    𝒫    =𝒫(ℳ) ,
    Δ    =Δ(𝒫(ℳ),ℳ) ,

and


    𝕎_,ρ-Δ_,ρ    =1/2(Δ-𝕎)×
       ×(1-2ℳ-2𝒫+4𝕎^2)
       , if LRS I
    𝕎=Δ   , if LRS II

which represents the covariant TOV equations. The system is completed
by the extra relations:

    𝒦-1/4+𝒫+𝕎(𝕎-2Δ)     =𝕐 ,
    𝒦-3/4-ℳ+𝕎(3𝕎-2Δ)     =𝕏 ,
    ℳ+𝕎(6Δ-9𝕎)-3(𝒦-1/4)     =3𝔼 ,
    
    2𝕎_,ρ+𝕎(2𝒦-2ℳ-4Δ𝕎+6𝕎^2-1/2)     =𝔹_1 ,
    𝕎(2𝒦+2𝒫-4Δ𝕎+2𝕎^2-3/2)     =𝔹_2 ,
    𝔹_1+𝔹_2+2𝔻_2    =0 ,
    
    𝕎(6Δ𝕎-2𝕎^2-4Δ^2+1)+
    
    +2(Δ-𝕎)(𝒦+𝒫-1/4)     =𝔻_1





 §.§ The static case


The full set of Eqs. (<ref>) - (<ref>)
completely describe the geometry of a stationary LRS I or LRS II space-time
filled by an Weyssenhoff fluid. Let us now consider the particular
cases when the space-time is static, that is the case when 𝕎=𝕋=Δ.

Introducing the following quantities 

    ℳ    =ℳ-Δ^2 ,
    𝒫    =𝒫-Δ^2 ,
    ℰ    =𝔼+2/3Δ^2 ,

Eqs. (<ref>) are given by 

    𝒫_,ρ    =-𝒫^2+𝒫[ℳ+1-3(𝒦-1/4)]-
       -ℳ(𝒦-1/4) ,
    𝒦_,ρ    =-2𝒦(𝒦-1/4-ℳ) .
    𝕐    =𝒦-1/4+𝒫 ,
    𝕏    =𝒦-3/4-ℳ ,
    
    3ℰ    =ℳ-3(𝒦-1/4) ,

which match exactly the expressions found in the theory of General
Relativity ( cf. Ref. <cit.>) for an effective energy density
and pressure and the corrected electric part of the Weyl tensor: ℳ,𝒫
and ℰ. Note that the extra constraints for the magnetic
components of the Weyl tensor 

    2Δ_,ρ+2Δ(𝒦-ℳ-1/4)     =𝔹_1 ,
    𝔹_2+Δ[1-2𝒫-2(𝒦-1/4)]     =0 ,
    𝔹_1+𝔹_2+2𝔻_2    =0 ,
    𝔻_1    =Δ ,

imply that the geometry of the space-time is fundamentally different
from the corresponding one in General Relativity. Nonetheless, the
fact that Eqs. (<ref>) - (<ref>)
have the same form for the corrected quantities in Eq. (<ref>)
lead us to the notable result: 

At the level of the metric, all static, locally rotationally symmetric
of class II solutions of the theory of General Relativity for a perfect
fluid with energy momentum (𝒯^GR)_αβ=μ u_αu_β+ph_αβ,
are also solutions of Einstein-Cartan theory sourced by a Weyssenhoff
fluid with energy-momentum tensor (𝒯^EC)_αβ=(μ+δ^2/8π) u_αu_β+(p+δ^2/8π)h_αβ. 


It is important to stress that, because of the nature of the corrections
in Eq. (<ref>), solutions which
are unacceptable in General Relativity due, for example, to negative
energy densities or pressure, might still correspond to physically
acceptable ones in the Einstein-Cartan case.

In the rest of the article we will consider the case of static spherically
symmetric space-times, hence, we will study solutions of Eqs. (<ref>)
- (<ref>).



§ JUNCTION CONDITIONS


In the analysis of compact objects in a geometric theory of gravity
it is often necessary to model the space-time as two distinct manifolds
glued together at a common boundary. Such operation is usually performed
using Israel junction conditions <cit.>. The Israel procedure
was initially developed for space-times in the absence of torsion.
More recently, a number of works have been published in which the
junction conditions are generalized to space-times with torsion in
different contexts <cit.>. Here we will summarize the
results and extend them in light of the structure equations we have
just obtained.

Consider two Lorentzian manifolds with boundary: 𝒱^-
and 𝒱^+, matched at an hypersurface 𝒩,
forming a new manifold 𝒱. Let n represent the unit
normal to 𝒩, pointing from 𝒱^- to 𝒱^+,
and e_a be the tangent vectors to 𝒩. Here 𝒩
can be either time-like or space-like. Now, following Ref. <cit.>,
for the total space-time to be a valid solution of the the field equations
and to guarantee that at 𝒩 there is no surface layer,
the following conditions must be met: 


  * the induced metric at 𝒩, as seen from each space-time
𝒱^- and 𝒱^+, h_ab^±:=g_αβ^±e_a^αe_b^β,
must be the same, 

    [h_ab]_±=0 ;


  * the extrinsic curvature tensor of 𝒩 as seen from 𝒱^-
and 𝒱^+, Q_ab^±:=e_a^αe_b^β∇_α^±n_β
, is such that 

    [Q_ab]_±=0 ;


  * the torsion tensor verifies 

    h^αβ[S_μβ^μ]_±+ϵ h^αβ[n^μn^νS_βμν]_±=0 .


For simplicity we labeled a field Υ defined on the the sub-manifold
𝒱^+ or 𝒱^- as Υ_+≡Υ(𝒱^+)
or Υ_-≡Υ(𝒱^-), respectively
and use the notation [Υ]_± to represent the
difference of a field as measured from each sub-manifold at the matching
surface, i.e., [Υ]_±≡.Υ(𝒱^+)|_𝒩-.Υ(𝒱^-)|_𝒩.

Clearly, conditions (<ref>) - (<ref>)
reduce to the Israel conditions <cit.> in the limit of null
torsion. In that case, the junction conditions not only guarantee
that at the matching surface there is no thin-shell but also are necessary
and sufficient to guarantee that the singular part of the Riemann
tensor is null. However, this is not the case for torsional space-times.
In the presence of torsion, assuming only compatibility with the metric,
the Riemann tensor of the total space-time is given by[Here we mix the distribution associated to a tensor and the tensor
itself which is, strictly speaking, an abuse of language. Our conclusions,
however, are not influence by this issue. See e.g. Ref. <cit.>
and references therein for more details.] 

    R_αβγ^ρ    =θ(λ)R_αβγ^+^ρ+θ(-λ)R_αβγ^-^ρ+
       +δ(λ)(A_αβγ^ρ+B_αβγ^ρ) ,

where θ(λ) represents the Heaviside distribution,
δ(λ) the Dirac distribution, λ is
the parameter of the integral curves of n, adjusted such that the
matching surface is located at λ=0, R_αβγ^±^ρ
represent the Riemann tensors of the 𝒱^- and 𝒱^+
sub-manifolds and 

    A_αβγ^ρ    =ϵ(n_β[Γ_αγ^ρ]_±-n_α[Γ_βγ^ρ]_±) ,
    
    B_αβγ^ρ    =ϵ(n_β[K_αγ^ρ]_±-n_α[K_βγ^ρ]_±) ,

are the singular parts of the - total - Riemann tensor, with Γ_αγ^ρ
being the Christoffel symbols, K_αβ^γ≡ S_αβ^γ+S^γ_αβ-S_β^γ_α
the contorsion tensor and ϵ=n_μn^μ. We see that
in general the conditions (<ref>) - (<ref>)
do not guarantee that both the tensors in Eq. (<ref>)
and (<ref>) are null. A smooth junction
of two space-times has to imply that the discontinuities of all curvature
tensors across the matching surface have to be at most finite otherwise,
the space-time will be singular at 𝒩. In the torsion
free case, no condition on the tensor (<ref>)
is required as it is identically zero and therefore does not appear
in the Einstein equations. In the Eistein-Cartan case, however, even
imposing Eqs. (<ref>) - (<ref>),
the remaining singular part of the Riemann tensor will appear in the
structure equations, leading to a singularity in 𝒩. Differently
from the standard violation of Israel's condition, such singularity
can not be attributed to the presence of a thin shell since, Eqs. (<ref>)
- (<ref>) prevent the existence of a surface layer
at 𝒩. For this reason, in the following we will require
a completely smooth matching of the Riemann tensor on the boundary. It is a known result (see e.g. <cit.> for a clear
derivation) that the tensor A_αβγ^ρ, Eq. (<ref>),
is null if and only if [K_ab]_±=0, where K_ab:=e_a^αe_b^β∇̃_αn_β
represents the extrinsic curvature computed from the metric connection.
On the other hand, a necessary and sufficient condition for B_αβγ^ρ
to be null is given by 

    [K_αβ^ρ]_±=ϵ n_α[n^μK_μβ ^ρ]_±.


Thus we the arrive to the following proposition: 

Let 𝒱^- and 𝒱^+ be two Lorentzian manifolds
with boundary, endowed with a metric compatible, affine connection.
𝒱^- and 𝒱^+ can be smoothly matched
at a common, non-null, hypersurface 𝒩 when the following
three conditions are verified:


  * the induced metric at 𝒩 is such that 

    [h_ab]_±=0 ;


  * the jump of extrinsic curvature of 𝒩 is null, that is

    [Q_ab]_±=0 ;


  * the jump of the contorsion tensor at 𝒩 verifies 

    [K_αβ^ρ]_±=ϵ n_α[n^μK_μβ ^ρ]_± ,

where the vector field n represents the unit normal to 𝒩. 



Notice that writing Q_ab=K_ab-e_a^αe_b^βK_αβ^γn_γ
and using Eq. (<ref>), Eq. (<ref>)
is the same as imposing [K_ab]_±=0.



 §.§ 1+1+2 junction of static LRSII space-times with torsion


Let us now express conditions (<ref>) -
(<ref>) covariantly in the specific case
of two static LRS II space-times endowed with a torsion tensor field
of the form S_αβ ^γ=ε_αβu^γτ,
where τ is a generic function of the space-time coordinates.

In what follows we will be interested in the case when the interior
and exterior space-times are to be matched at a time-like hypersurface,
orthogonal to the vector field e. Then, condition (<ref>)
reads 

    [N_αβ-u_αu_β]_±=0 ,

where N_αβ verifies Eqs. (<ref>).
Using Eq. (<ref>), in the considered setup, Eq. (<ref>)
is simply 

    [1/2ϕ N_αβ-𝒜u_αu_β]_±=0 ,

which, contracting with the induced metric at 𝒩 and using
Eq. (<ref>), gives 

    [ϕ-𝒜]_±=0 .

From Eqs. (<ref>) - (<ref>)
we find that at the matching surface the following constraints have
to be met 

    [ϕ]_±    =0 ,
    [𝒜]_±    =0 ,

implying, for ϕ≠0, 

    [𝕐]_±=0 .

Given that e is continuous across 𝒩, we can integrate
Eq. (<ref>), finding K=k_0e^-ρ. Using
Eq. (<ref>) and (<ref>)
we have 

    [𝒦]_±=0 .

Using the previous results in Eq. (<ref>) we arrive
at 

    [8π p-δ^2]_±=0 .


Finally, for the specific type of torsion that we consider in this
article, condition (<ref>) imposes 

    [δ]_±=0 ,

then, from Eq. (<ref>),

    [p]_±=0 .


We have then found that for a smooth matching between two static LRS
II space-times endowed with a torsion tensor field of the form S_αβ ^γ=ε_αβu^γτ,
both the pressure of the fluid and the spin density, as seen from
each space-time, must match at 𝒩.



§ EXACT SOLUTIONS FOR STATIC LRS II SPACE-TIMES


Given the set of structure equations (<ref>)
- (<ref>) that describe the behavior of a static,
LRSII space-time filled by a Weyssenhoff fluid, let us now find and
study some exact solutions.

As was stated before, the system of structure equations is not closed
until an equation of state and an expression for the spin density
are provided. Let us then consider some particular relations for the
pressure, energy and spin densities of the fluid in order to gain
some insight into the behavior of compact objects in a fully relativistic
theory with non-null spin.

For the remaining of the article we will consider only the particular
case of spherically symmetric space-times. Moreover, in what follows
we will refer to static, spherically symmetric compact objects as
“stars”. Although this is an abuse of language, it is also a trend in the
the literature since such systems are expected to be a good model
for slowly varying astrophysical bodies.



 §.§ Effective constant energy-density and the Buchdahl limit


We start by considering the case of a system where the effective energy
density is assumed to be constant, that is 

    8πμ-δ^2=μ̃_0 ,

where μ̃_0∈ℝ. Notice that, contrary to the
case of null torsion, the above assumption does not have to imply
that the energy density, μ, is constant.

Using Eqs. (<ref>) and (<ref>)
we have 

    K(ρ)=e^-ρ/r_0^2 ,

where r_0 is an integration constant. Eq. (<ref>)
then yields 

    ℳ(ρ)=μ̃_0r_0^2e^ρ𝒦(ρ) .

Eq. (<ref>) allow us to
solve Eq. (<ref>), finding 

    𝒦(ρ)=3/12-4μ̃_0r_0^2 e^ρ+3𝒦_0e^-ρ/2 ,

where 𝒦_0 is yet another integration constant. Setting
𝒦_0=0 to avoid a conical singularity at ρ→-∞
<cit.>, the structure equations yield 

    p(ρ)-δ(ρ)^2/8π    =-μ̃_0(P_0+3√(3-μ̃_0r_0^2e^ρ))/24π(P_0+√(3-μ̃_0r_0^2e^ρ)) ,
    𝒜(ρ)     =-μ̃_0r_0e^ρ/2/√(3)(P_0+√(3-μ̃_0r_0^2e^ρ)) ,
    ϕ(ρ)     =2/r_0√(3)e^-ρ/2√(3-μ̃_0r_0^2e^ρ) ,
    ℰ    =-2/3δ^2 ,

where we have chosen the direction of e so that ϕ is positive,
and the value of the integration constant P_0 is to be determined
by the boundary conditions.

Let us now assume that relations (<ref>)
- (<ref>) describe the interior
of a compact object matched at a boundary 𝒩 to an exterior
space-time modeled by the Schwarzschild vacuum solution. From Eqs. (<ref>)
and (<ref>) we find that the
quantity in Eq. (<ref>)
must be zero at the boundary. Setting, without loss of generality,
the matching hypersurface to be at ρ=0, we find 

    P_0=-3√(3-μ̃_0r_0^2) .

The matching conditions, Eqs. (<ref>)
and (<ref>), imply that interior and
exterior observers agree on the value of circumferential radius of
𝒩, say r_0, and the Schwarzschild parameter, M,
is given by 

    M=μ̃_0r_0^3/6 .

Moreover, from condition (<ref>)
we find that the spin density must go to zero at the matching surface,
that is, δ(ρ=0)=0.

Given the previous results we are now in position to study some effects
arising from the presence of spin in compact objects. In the remaining
of this subsection, for clarity, we shall write the results in terms
of the circumferential radius r. Using the fact that, in the considered
setup, the quantity K, Eq. (<ref>), represents
the Gaussian curvature of the 2-sheet, we have that the parameter
ρ and r are related by 

    ρ=2ln(r/r_0) ,

where we have set the value of the arbitrary scaling factor to be
r_0.

Now, defining the central pressure p_c:=p(ρ→-∞),
from Eqs. (<ref>) and (<ref>),
we have


    p_c=-μ_c(1-√(1-2M/r_0))/1-3√(1-2M/r_0)+δ_c^2(1-2√(1-2M/r_0))/4π(1-3√(1-2M/r_0)) ,

where μ_c≡μ(ρ→-∞) and δ_c≡δ(ρ→-∞).
If we compared directly the above expression to a similar system in
GR, we would see that the second term on the right-hand side of Eq. (<ref>)
represents an explicit contribution due to the presence of spin. However,
there is a subtlety: Eq. (<ref>)
indicates that the presence of spin also modifies the matching radius
r_0 and the value of the Schwarzschild parameter M, making
it difficult to draw conclusions only on (<ref>).

A clearer idea of the differences between our case and GR can be obtained
by computing the maximum mass that can be held by a star with constant
radius. Considering Eq. (<ref>)
and if neither the densities μ_c and δ_c diverge,
the central pressure in Eq. (<ref>)
will go to infinity when r_0=9/4M or, using Eq. (<ref>),
when 

    M_max=4/9√(3π)(μ-δ^2/8π)^-1/2 .

This results makes it clear that, when compared to a system with the
same energy density μ in GR, the presence of spin increases the
maximum allowed mass.

In analogy with the calculation of the Buchdahl limit in GR we can
generalize this discussion to non constant μ̃. Consider
the quantity μ̃:=8πμ-δ^2 and assume it to be
non-negative and dμ̃/dr≤0, for r∈[0,r_0].
Following the same reasoning of Ref. <cit.> (see also
<cit.>) we can find an upper limit for the amount of mass a
star with constant radius can hold: 

    m(r_0)/r_0≤4/9 ,

with 

    m(r_0)=1/2∫_0^r_0μ̃(r) r^2dr .


At first sight, the expression in Eq. (<ref>)
matches the one found by Buchdahl <cit.> for GR. However,
there is a correction due to the presence of spin in the function
m(r_0), Eq. (<ref>),
leading us to conclude that for the same value of the circumferential
radius, r_0, a star can hold more matter in the presence of spin
than in the null-spin case. It is also worth mentioning that the quantity
m(r_0) agrees with the value of the Schwarzschild
parameter of the exterior space-time, therefore, the gravitational
mass of such objects is determined not only by the energy density,
μ, but also by the spin density, δ, which was expected
because of the specific way in which spin gravitates in our specific
Einstein-Cartan setup.

As a final comment, although a priori there is nothing that
forces δ^2 to be smaller than 8πμ, it is expected
that in stars - even neutron stars - δ^2≪μ (see Refs. <cit.>),
hence, μ̃≥0, as was assumed in the derivation of Eq. (<ref>).
On the other hand, the requirement that dμ̃/dr≤0 might
not be as physically reasonable as in the case of GR since, as we
will see bellow, the presence of spin allows for a richer possible
behavior for the matter variables.



 §.§ Spin held stars


In the previous subsection we have considered a classical model for
a relativistic star which is similar to the simplest model for this
type of objects in General Relativity. However the presence of spin
allows solutions which are not contemplated in the Einstein's theory.
The prototype of such objects is a star which is supported only by
the gravitation of the spin of the Weyssenhoff fluid. In the remaining
of the subsection, we will analyze this case and prove the following
result 

There are no static, spherically symmetric
solutions of the Einstein-Cartan theory sourced by a Weyssenhoff fluid with null isotropic pressure
that have all the following properties 


  * δ(r) is non-null for r∈[0,r_0[
and δ(r_0)=0, for some r_0>0; 

  * δ^2(r) is a monotonically decreasing function
for all r∈[0,r_0]; 

  * the spin and energy density functions: δ(r) and
μ(r), are at least of class C^1 and the function
𝒜(r) is differentiable for all r∈[0,r_0]; 

  * the function M(r):=1/2∫_0^r[8π μ(r)-δ^2(r)]x^2dx
is such that 2M(r)<r, for all r∈]0,r_0]. 



To prove Proposition <ref> we will consider
first the behavior of the quantities of interest in a neighborhood
of the center, r=0, and then on the boundary of the star. In doing
so, in order to make the reasoning more intuitive, we shall consider
here that the integral curves of the vector field e are parameterized
by the circumferential radius r.

Defining the quantities 

    μ̃(r)     =8π μ(r)-δ^2(r) ,
    p̃(r)     =8π p(r)-δ^2(r) ,

we find from the structure equations 

    r/2ϕ(r)p̃_,r    =-𝒜(μ̃+p̃) ,
    r/2ϕ(r)𝒜_,r    +𝒜^2+𝒜ϕ=1/2(μ̃+3p̃) ,
    p̃    =𝒜ϕ-K+1/4ϕ^2 ,

with 

    K(r)     =1/r^2 ,
    ϕ(r)     =2/r√(1-2M(r)/r) ,

and 

    M(r)=1/2∫_0^rμ̃(x)x^2dx ,

where, without loss of generality, we chose the direction of e
so that ϕ(r) is non-negative. Moreover, from Eqs. (<ref>)
and (<ref>) we find the useful relation

    𝒜ϕ=2M(r)/r^3+p̃ .


We will consider now the case of a static, spherically symmetric compact
object held entirely by spin, that is, the case when p(r)=0
and p̃(r)=-δ^2(r), smoothly
matched to an exterior space-time modeled by a vacuum solution of
the Einstein-Cartan field equations. Moreover, we will assume that
for r>0, 2M(r)<r, otherwise the scalar ϕ(r)
would take complex values.



  §.§.§ Behavior at the center


Assuming that the functions μ(r), δ^2(r)∈ C^1
we can write in a small enough neighborhood of r=0: 

    μ(r)     =μ(0)+μ_,r(0) r ,
    δ^2(r)     =δ^2(0)+(δ^2)_,r(0) r ,

where comma represents partial - or total - derivative with respect
to the variable in front. From Eqs. (<ref>),
we find that in a small enough neighborhood of r=0, the mass function
(<ref>) is described by 

    2M(r)=μ̃(0)/3r^3+1/4(μ̃_,r(0))r^4 .

In particular we find that M(r) goes to zero at least
as fast as r^3.

Now, Eqs. (<ref>) and (<ref>)
yield 

    2/r(1-2M(r)/r)dδ^2/dr=(2M(r)/r^3-δ^2)(μ̃-δ^2) .

In a region where r∈[0,ϵ[, with ϵ≪1,
the RHS of this equation takes values in ℝ, therefore

    (δ^2)_,r(0)=0 .

Repeating the same reasoning in Eq. (<ref>)
we find that


    𝒜(0)=0 .


Let us now assume that there exists an r_a>0 where for r∈]0,r_a[,
𝒜(r)>0. From Eq. (<ref>)
we will find that in this region μ̃+p̃≤0 which
implies that μ̃+3p̃<0. Then, from Eq. (<ref>)
we find that 𝒜_,r(r)<0, for all r∈]0,r_a[.
This, however, violates the initial hypothesis since, 𝒜(0)=0
and we assume that 𝒜(r)>0, for r∈]0,r_a[,
that is, 𝒜_,r(r) would have to be positive
for some r∈[0,r_a[ .

Another possibility is that for a region r∈[0,r_a],
𝒜(r)=0 and for r∈]r_a,r_c]
with r_c>r_a, 𝒜(r)>0. If this were
the case, since for r∈]r_a,r_c], 𝒜(r)>0
, there would exist a value r_b∈]r_a,r_c] such
that 𝒜_,r(r_b)>0. Using this in Eq. (<ref>),
at r=r_b we find 

    μ̃+3p̃>0⇒μ̃+p̃>0 ,

but from Eq. (<ref>) and imposing that
(δ^2)_,r≤0 we find that: .μ̃+p̃|_r=r_b≤0,
contradicting (<ref>).

Another possibility is that 𝒜(r)=0, for all
r∈[0,r_0]. From Eqs. (<ref>)
and (<ref>) this simply represents a vacuum solution
as such it does not represent a solution for a compact object.

Gathering this results we conclude that there exists an r_d>0
such that in the region [0,r_d[, 𝒜(r)≤0
and it must take negative values in some sub-region.



  §.§.§ Behavior at the boundary


Let us now define the boundary of the compact object as the hypersurface
at which the spin density goes to zero, that is, δ^2(r_0)=0.
In such hypersurface we have three possible behaviors for the function
𝒜: 


  * 𝒜(r_0)<0; 

  * 𝒜(r_0)>0; 

  * 𝒜(r_0)=0. 

Let us consider each case separately.



  
(1)The case 𝒜(r_0)<0


 

From Eq. (<ref>) we have that at
r=r_0 

    .𝒜ϕ|_r=r_0=2M(r_0)/r_0^3<0 .

Therefore, from Eq. (<ref>) there exists a region
]r_f,r_g[ where 

    μ̃(r)<0 ,

then μ̃+p̃<0, in that region. From Eq. (<ref>),
to guarantee that the spin density is a monotonically decreasing function
of r, we find that 𝒜(r)≥0, for r∈]r_f,r_g[.
So, either 𝒜(r)=0 ∧ 𝒜_,r(r)=0
for all r∈]r_f,r_g[, that is, the function 𝒜(r)
takes the value zero and stays zero for all r∈]r_f,r_g[;
or 𝒜(r)>0 for some r∈]r_f,r_g[.
The former case is not possible: from Eq. (<ref>),
μ̃(r)+3p̃(r)=0, hence, μ̃(r)≥0,
for all r∈]r_f,r_g[, which contradicts the inequality
(<ref>). As for the latter - the
case when 𝒜(r)>0, for some r∈]r_f,r_g[
- in the previous sub-section it was shown that for some sub-region
of [0,r_d[, 𝒜(r)≤0, therefore
the region ]r_f,r_g[ cannot be a sub-region of [0,r_d[.
With this said, since 𝒜(r) is a differentiable
function, there exists a region with, say, r=r_e<r_0, where
𝒜(r_e)>0 ∧ 𝒜_,r(r_e)>0.
Then, from Eq. (<ref>) 

    .μ̃+3p̃|_r=r_e>0⇒.μ̃+p̃|_r=r_e>0 .

However, substituting this results in Eq. (<ref>)
we find: (δ^2)_,r(r_e)>0, which
contradicts the assumption that the spin density is a monotonically
decreasing function.



  
(2) The case 𝒜(r_0)>0 

 

For the case when 𝒜(r_0)>0, we
can simply repeat the proof in the previous sub-subsection and conclude
in the same way that the assumptions are violated in a region. We
just remark that the point with radial coordinate r=r_e, in the
proof, can always be chosen such that r_e<r_0 since, for whatever
the value of 𝒜(r_0)>0, there is a point
where 0<𝒜(r<r_0)<𝒜(r_0).



  
(3)The case 𝒜(r_0)=0 

 

In this the case when 𝒜(r_0)=0
we have, from Eq. (<ref>) that 

    M(r_0)=0 .

From this we have three possibilities:


  * μ̃(r)=0, for r∈[0,r_a]; 

  * μ̃(r)<0, for r∈]0,r_a]; 

  * μ̃(r)>0, for r∈]0,r_a]; 

for some r_a>0.

Let us consider each case individually.

(a) In the case when μ̃(r)=0, for r∈[0,r_a]
we have from Eq. (<ref>) that 𝒜(r)<0.
However, using this result in (<ref>) we
see that it implies that the spin density is an increasing function
of r, violating the hypothesis.

(b) In the case when the corrected energy density is such that μ̃(r)<0,
for r∈]0,r_a], from (<ref>) we
have that the mass function is negative, in this region. From Eq. (<ref>)
we than conclude that 𝒜(r)<0, r∈]0,r_a].
However, going back to Eq. (<ref>) we find
that the spin density is an increasing function of r, violating
the hypothesis.

(c) Finally, consider the case when μ̃(r)>0,
for r∈]0,r_a]. From (<ref>), this
implies that the mass function is positive in this region. Since Eq. (<ref>)
must be verified, there must be a region where μ̃(r)<0.
We can then repeat the arguments of the case 𝒜(r_0)<0,
which lead to the conclusion that the hypothesis would be violated
in some region inside the star.

Gathering the previous results we have proven the result in Proposition
<ref>.

We end this Section by remarking that if instead of imposing p̃=-δ^2
we only imposed that p<δ^2, that is, the thermodynamical
pressure is always smaller than correction due to the spin density,
then all the previous results are valid if p̃_,r≥0.
Notice, however, that in this scenario this condition, simply measures
the gradient of the quantity 8π p-δ^2.



 §.§ Reconstructing exact solutions


As in the case of the theory of General Relativity, when torsion is
present it is possible to generate exact solutions via reconstruction
algorithms <cit.>. The idea is to assign a given metric
tensor and deduce the corresponding behavior of the energy density,
pressure and spin density.

Analyzing Eqs. (<ref>) - (<ref>)
and using Eqs. (<ref>) and (<ref>),
shows that differently from the case of anisotropic compact objects
in General Relativity <cit.>, the structure equations cannot
be solved for the spin density. This implies that the reconstruction
algorithm can only be used if an additional relation is provided,
either relating the spin density to the other matter variables or
an equation of state for matter.

In the following we will show some applications of this algorithms
which return some interesting solutions from a physical point of view.



  §.§.§ Connecting the spin density to the energy density: “Buchdhal stars”


A natural additional relation is to have the spin density to be proportional
to the energy density of the Weissenhoff fluid. In this case, however,
the junction conditions that we have seen in Sec. <ref>
pose the problem to have both the energy density and the pressure
to be zero at the boundary. A class of solutions which are devised
to have exactly this property was given by Buchdhal <cit.>.
We will now reconstruct this solutions in the case of equations (<ref>)
- (<ref>).

Consider a spherically symmetric space-time characterized by the line
element 

    ds^2=-A(w)dt^2+B(w)dw^2+C(w)(dθ^2+sin^2θ dφ^2) ,

where 

    η(w)     =(a-1)sin(Rw)/Rw ,           A(w)     =a(1+a-η)/1+a+η ,
    
    B(w)     =(1+a+η)/a(1+a-η)          C(w)     =w^2(1+a+η)^2/4a^2 ,

and w is connected to ρ by the relation 

    e^ρ=w^2/4a^2(1+a+η)^2 .

Notice that the circumferential radius, r, vanishes when w=0.

From Eqs. (<ref>), (<ref>), (<ref>)
and (<ref>), assuming 𝕎=Δ and Δ^2=γℳ,
we find


    ℳ    =2𝒦_,ρ+4𝒦^2-𝒦/4(1-γ)𝒦 ,
    𝒫    =Y-𝒦+1/4-2𝒦_,ρ+4𝒦^2-𝒦/4(1-γ)𝒦 ,
    
    0     =(2Y+1)𝒦_,ρ-4𝒦^2-𝒦[4Y_,ρ+4(Y-1)Y-1] .

The form of Y and 𝒦 that satisfies the constraint
(<ref>) can be found directly from their definition
in a general coordinate system (see Refs. <cit.>)

    Y     =1/2CA_,w/AC_,w=(1+a)wη_,w/2(η-a-1)(1+a+η+wη_,w) ,
    𝒦    =BC/(C_,w)^2=a(1+a+η)/(1+a-η)(1+a+η+wη_,w)^2 .

Using Eqs. (<ref>), (<ref>) and (<ref>),
the energy density and the pressure are then given by 

    μ    =aR^2η(3η-2-2a)/8π(γ-1)(1+a+η)^2 ,
    
    p     =aR^2η[2γ(2η-a-1)-η]/8π(γ-1)(1+a+η)^2 .


As said, this family of solutions have, by construction, the property
that the pressure, energy and spin densities all vanish at a particular
hypersurface. In Figures <ref> - <ref> we
present the behavior of these quantities for a few combinations of
the parameters, showing that the values of the parameters a and
γ have a direct impact in the profile of the densities, whereas,
the parameter R defines the value when the matter variables go
to zero. Moreover, from the plots it is clear that the presence of
spin markedly changes the type of behavior the matter may have. In
particular, for certain values of the parameters a and γ
the functions μ, p or δ might not be monotonically
decreasing functions of the coordinate w.










  §.§.§ Connecting the spin density to the pressure


Another option that reduces the number of conditions related to the
junction is to associate the spin density to the pressure. This choice,
which at first might appear unnatural, corresponds to the case in
which the spin depends on the equation of state. We can imagine that
particles with spin will create different structures
not unlikely to the ones that characterize the crystalline phases
of water ice (see e.g. Ref. <cit.>). Our ansatz refers to this
kind of effects.

The reconstruction equations in this case, setting Δ^2=γ𝒫,
read 

    ℳ=    4𝒦^2+2𝒦_,ρ-𝒦/4(1-γ)𝒦-γ[𝒦_,ρ+𝒦(4𝒦-2Y-1)]/2(1-γ)𝒦 ,
    𝒫=    1-4𝒦+4Y/4(1-γ) ,
    
    0=    (2Y+1)𝒦_,ρ-4𝒦^2-𝒦[4Y_,ρ+4(Y-1)Y-1] .


Let us now consider a metric in which the (0,0) coefficient, A,
is given by 

    A=A_0(a+br_0^2e^ρ)^2 ,

where a,b and r_0 are arbitrary constants. From the definition
of Y one obtains 

    Y=1/2A_,ρ/A=br_0^2e^ρ/a+br_0^2e^ρ ,

and from Eq. (<ref>) it follows that 

    𝒦=(a+3br_0^2e^ρ)^2/3/𝒦_0e^ρ+4(a+3br_0^2e^ρ)^2/3 ,

where 𝒦_0 is an integration constant. In terms of the
area radius r, this result corresponds to the line element 

    ds^2=-A(r)dt^2+B(r)dr^2+r^2 (dθ^2+sin^2θ dφ^2),

with 

    A(r)     =A_0(a+br^2)^2,
    
    B(r)     =(1+c r^2/(a+3br^2)^2/3)^-1 .

The energy density and the pressure are given by 

    μ    =b^2r^2[5c(1-4γ)r^2-12γ(a+3br^2)^2/3]/8π(γ-1)(a+br^2)(a+3br^2)^5/3
       -4ab[γ(a+3br^2)^2/3+2c(2γ-1)r^2]/8π(γ-1)(a+br^2)(a+3br^2)^5/3+
       +a^2c(3-4γ)/8π(γ-1)(a+br^2)(a+3br^2)^5/3 ,
    
    p     =4b(a+3br^2)^2/3+ac+5bcr^2/8π(1-γ)(a+br^2)(a+3br^2)^2/3 .


We give in Figure <ref> the behavior of this solution for
specific values of the parameters, showing the existence of an hypersurface
where both p and δ^2 vanish, so that we can smoothly
match such solution with a vacuum exterior space-time.



Another example,based on the same assumptions, can be given considering

    A=A_0(a+√(c-be^ρ))^2 ,

which corresponds to 

    Y=-be^ρ/2√(c-be^ρ)(a+√(c-be^ρ)) ,

Eq. (<ref>) then gives 

    𝒦=cψ(a√(c-be^ρ)-2be^ρ+c)/(c-be^ρ)[4ψ(a√(c-be^ρ)-2be^ρ+c)-b de^ρ] ,

with 

    ψ=(√(a^2+8c)+a+4√(c-be^ρ)/√(a^2+8c)-a-4√(c-be^ρ))^a/√(a^2+8c) .

Using the area radius r, we find the following solution for the
metric (<ref>)


    A     =A_0[a+y(r)]^2 ,
    
    B     =4c[ay(r)+2y(r)^2-c]/y(r)^2[4ay(r)+8y(r)^2+d ψ(r)(y(r)^2-c)-4c] ,

where 

    y(r)     =√(c-br^2/r_0^2) ,
    ψ(r)     =(√(a^2+8c)+a+4y(r)/√(a^2+8c)-a-4y(r))^a/√(a^2+8c) ,

with the following expressions for the energy density and pressure
of the fluid


    μ    =b d γ[6y^5+7ay^4+2y^3(a^2-3c)-4acy^2]ψ/16π c(γ-1)r_0^2(a+y)[c-y(a+2y)]^2+
       +b d γ c^2(2y+a)ψ/16π c(γ-1)r_0^2(a+y)[c-y(a+2y)]^2-
       -bd[6y^4+3ay^3-5cy^2+2c^2]ψ/32π c(γ-1)r_0^2[c-y(a+2y)]^2+
       +b[2γ(2a+3y)-3(a+y)]/8π cr_0^2(γ-1)(a+y) ,
    
    p     =b d y(ay-2c+3y^2)ψ/32π c(γ-1)r_0^2(a+y)(ay-c+2y^2)+
       +b(a+3y)/8π cr_0^2(γ-1)(a+y) ,

In Figure <ref> we show the behavior of this solution for
specific values of the parameters. Notice that also this solution
admits the existence of a common hypersurface where both p and
δ vanish.


Before finishing this section we remark that, as shown by Figs. <ref> - <ref>, in all considered cases it is possible to find values of the parameters for which all the thermodynamical quantities and spin density are positive, hence, all the classical energy conditions are valid.




§ GENERATING THEOREMS


As discussed in Ref. <cit.>, the form of the structure equations
(<ref>) - (<ref>) is especially
useful to find algorithms for generating new solutions from previous
known ones.

Consider a solution for the structure equations (<ref>)
- (<ref>) characterized by the functions 

    {𝒫_0,ℳ_0,Δ_0,K_0,ℰ_0,𝕏_0,𝕐_0,(𝔹_1)_0,(𝔹_2)_0,(𝔻_1)_0,(𝔻_2)_0} .

Given the quantities 

    ℳ    =ℳ_0+ℳ_1 ,
    𝒫    =𝒫_0+𝒫_1 ,
    Δ^2    =Δ_0^2+Δ_1^2 ,
    𝒦    =𝒦_0+𝒦_1 ,

where {𝒫_1,ℳ_1,Δ_1,𝒦_1}
are sufficiently smooth arbitrary functions, let us search conditions
on the deforming functions to so that the set {𝒫,ℳ,Δ,𝒦,ℰ,𝕏,𝕐,𝔹_1,𝔹_2,𝔻_1,𝔻_2}
is a solution of the structure equations.

Substituting Eq. (<ref>)
in Eq. (<ref>) we find

    ∂_ρ𝒦_1+2𝒦_1^2+2𝒦_0(Δ_1^2-ℳ_1)-
    
    -𝒦_1(2ℳ_0-4𝒦_0-2Δ_0^2+2Δ_1^2-2ℳ_1+1/2)     =0 .

This equation has the form of a Riccati differential equation to which,
in general, there are no known closed form solutions. We can, nonetheless,
consider particular cases so that the previous equation reduces to
a Bernoulli differential equation, where general closed form solutions
exist.



 §.§ Case 1


Let us first consider that 

    ℳ_1=Δ_1^2 .

In this case, Eq. (<ref>)
can be readily integrated for 𝒦_1, such that 

    𝒦_1=0  𝒦_1(ρ)=Exp[∫_ρ_0^ρΛ dx]/𝒦_⋆+2∫_ρ_0^ρExp[∫_y_0^yΛ dx] dy ,

where 𝒦_⋆ is an integration constant and 

    Λ=2ℳ_0-2_0^2-4+1/2 .

Using Eq. (<ref>) in Eq. (<ref>)
we find


    ∂_ρ𝒫_1+𝒫_1(3𝒦_0+3𝒦_1-ℳ_0+2𝒫_0-Δ_0^2-7/4)+
    
    +𝒫_1^2+1/4ℱ    =0 ,

with

    ℱ    =4ℳ_0(Δ_1^2+𝒦_1)-8(𝒫_0+𝒫_1)Δ_1^2+12𝒫_0𝒦_1+
        +4Δ_0^2Δ_1^2-4∂_ρΔ_1^2+4Δ_1^4+7Δ_1^2-
        -12Δ_1^2𝒦_0-16Δ_0^2𝒦_1-12Δ_1^2𝒦_1 .

For Eq. (<ref>) to reduce
to a Bernoulli like differential equation we will require ℱ(ρ)=0,
that is 

    ∂_ρΔ_1^2-Δ_1^4-𝒦_1(ℳ_0+3𝒫_0-4Δ_0^2)-
    
    -Δ_1^2(7/4+ℳ_0-3𝒦_0-3𝒦_1-2𝒫_0-2𝒫_1+Δ_0^2)     =0 ,

which, by setting 𝒦_1=0 or ℳ_0+3𝒫_0-4Δ_0^2=0,
can be formally solved, such that

    Δ_1^2(ρ)=0  Δ_1^2(ρ)=Exp[∫_ρ_0^ρΦ dx]/Δ_⋆-∫_ρ_0^ρExp[∫_y_0^yΦ dx] dy ,

where Δ_⋆ is an integration constant and 

    Φ=ℳ_0-3𝒦_0-3𝒦_1-2𝒫_0-2𝒫_1+Δ_0^2+7/4 .

Consequently, from Eq. (<ref>),
we find 

    𝒫_1(ρ)=0  𝒫_1(ρ)=Exp[∫_ρ_0^ρΓ dx]/𝒫_⋆+∫_ρ_0^ρExp[∫_y_0^yΓ dx] dy ,

with

    Γ=ℳ_0-2𝒫_0+Δ_0^2-3𝒦_0-3𝒦_1+7/4 ,

and 𝒫_⋆ is an integration constant.

Before we conclude this Section, we should stress that Eqs. (<ref>),
(<ref>) and
(<ref>) present
two possible solutions for the considered functions and all combinations
of those solutions verify the structure equations with ℳ_1=Δ_1^2,
leading, a priori, to distinct solutions.



 §.§ Case 2


Another possibility to solve Eq. (<ref>)
is the case when 

    (𝒦_0+𝒦_1)(2Δ_1^2-2ℳ_1)=G(ρ)𝒦_1+Q(ρ)𝒦_1^2 ,

where G(ρ) and Q(ρ) are sufficiently
smooth, arbitrary functions. Setting

    2Δ_1^2-2ℳ_1    =𝒦_1Q(ρ) ,
    
    G(ρ)     =𝒦_0Q(ρ) ,

and substituting Eqs. (<ref>)
and (<ref>) in Eq. (<ref>)
we find

    ∂_ρ𝒦_1+𝒦_1[2Δ_0^2-2ℳ_0+4𝒦_0+𝒦_0Q(ρ)-1/2]+
    
    +[2+Q(ρ)]𝒦_1^2    =0 ,

which, provided an expression for Q(ρ) can be solved
for 𝒦_1, or vice-versa.

Now, to solve the remaining equations for the functions Q, 𝒫_1
and Δ_1, we will consider that the original solution is
such that ℳ_0=𝒫_0=Δ_0=0, that is,
the original space-time is described by a vacuum solution of the field
equations. From Eq. (<ref>) we then find 

    ∂_ρ𝒫_1+𝒫_1^2+𝒫_1[3𝒦_0+3𝒦_1-7/4]+𝒥(ρ)=0 ,

where 

    𝒥(ρ)     =-∂_ρΔ_1^2+Δ_1^4+1/2Q(ρ)𝒦_1[𝒫_1-𝒦_0-𝒦_1+1/4]
        +Δ_1^2(-2𝒫_1-1/2Q(x)𝒦_1-3𝒦_0-3𝒦_1+7/4) .

As before, to reduce Eq. (<ref>)
to a Bernoulli differential equation we will impose 𝒥(ρ)=0.
Unfortunately, this equation itself is also not possible to solve
in general since it has the form of a Riccati differential equation.
Let us then further impose the last term in the first line of the previous equation
to be zero. Solving for 𝒦_1, we have 

    𝒦_1=𝒫_1-𝒦_0+1/4 ,

where we have ignored the solutions where Q=0  𝒦_1=0
since they lead to a particular case of subsection <ref>.

Considering the constraint that originally we have a vacuum solution,
substituting Eq. (<ref>)
in Eq. (<ref>)
we find, 

    Q=-8(2∂_ρ𝒫_1+4𝒫_1^2+𝒫_1)/(4𝒫_1+1)(4𝒫_1-4𝒦_0+1) .


Gathering the previous results we find the following expressions for
the remaining perturbations 

    𝒫_1    =e^ρ/𝒫_⋆+4e^ρ ,
    Δ_1^2=0  Δ_1^2    =Exp[-∫_ρ_0^ρΦ dx]/Δ_⋆-∫_ρ_0^ρExp[-∫_y_0^yΦ dx] dy ,
    ℳ_1    =Δ_1^2+2∂_ρ𝒫_1+4𝒫_1^2+𝒫_1/4𝒫_1+1 ,

where 𝒫_⋆ and Δ_⋆ are integrating
constants and 

    Φ=2𝒫_1+1/2Q(x)𝒦_1+3𝒦_0+3𝒦_1-7/4 .

Notice that we did not consider the case when 𝒫_1=0
since it would lead to the case when Q(ρ)=0, which,
as mentioned before, represents a particular case of subsection <ref>.
Let us also remark that, for solutions generated using the above equations,
the functional form of the pressure, 𝒫≡𝒫_1,
is independent of the original solution and completely determined
up to a constant. Moreover, notice that the pressure - in such solutions
- is only null when ρ→-∞.



 §.§ Case 3


Let us now consider the deformations in Eq. (<ref>)
with the extra constraint 

    𝕐    =𝕐_0 ,

that is, we will impose that the function 𝕐 is unchanged
between the original and the perturbed space-time. This is a generalization
of the deformations considered in Refs. <cit.>,
for non-null spin density. Substituting Eqs. (<ref>)
and (<ref>) in Eq. (<ref>)
we find that 

    𝒫_1=Δ_1^2-𝒦_1 .

Using Eqs. (<ref>),
(<ref>) and (<ref>)
in Eqs. (<ref>) and (<ref>) we
find the following relations for ℳ_1 and 𝒦_1


    ℳ_1    =𝒦_1(2𝕐_0+3)/2𝕐_0+1+Δ_1^2 ,
    𝒦_1    =Exp[-∫_ρ_0^ρΦ dx]/𝒦_⋆-∫_ρ_0^ρ4/2𝕐_0+1Exp[-∫_y_0^yΦ dx] dy-𝒦_0 ,

where 

    Φ    =𝒦_0(6+4𝕐_0)/2𝕐_0+1+2Δ_0^2-2ℳ_0-1/2 ,

and 𝒦_⋆ is an integration constant. Eqs. (<ref>)
- (<ref>) generalize the results
in Ref. <cit.> in the presence of a non-null spin density
[Notice that there is an error in the expression for ℳ_1
in Ref. <cit.>. The correct expression is found by setting
Δ_0^2=Δ_1^2=0 in Eq. (<ref>).].

Contrary to the previous cases, the Eqs. (<ref>)
- (<ref>) do not completely determine
the system since the function Δ_1^2 is unconstrained.
Notice that 𝒦≡𝒦_1 is determined uniquely
by the unperturbed solution and Δ_1^2 will only affect
ℳ_1 and 𝒫_1. Therefore, provided an
unperturbed solution, the metric of the perturbed space-time is completely
determined by Eqs. (<ref>)
and (<ref>). As already pointed
out, Δ_1^2 will not only affect the energy density and
the pressure of the fluid but also the Weyl tensor components. Thus,
although the metric of the space-time is independent of Δ_1^2,
the geometry is profoundly influenced by the presence of spin.



§ CONCLUSIONS


In this paper we have used the 1+1+2 formalism to derive the structure
equations for LRS I and LRS II, stationary space-times with a Weyssenhoff
like torsion field in the context the ECSK theory of gravity. The
structure of the covariant equations show in detail how the spin interacts
with the space-time via the torsion tensor. In particular, the presence
of a torsion tensor field separates the magnetic part of the Weyl
tensor in two distinct tensors, which behave differently. Even in
the case of static LRS II space-times, the magnetic parts of the Weyl
tensor do not vanish and some of its components depend on both the
value and spatial derivative of the spin density. This suggests, in
particular, that the effects of spin on the matter fluid, even in
the regimes expected to be found in neutron stars, may not be negligible,
as it was previously thought  (see e.g. <cit.>), even in the case in which the contribution to the spin is very small.

The 1+1+2 equations were then used to derive the covariant Tolman-Oppenheimer-Volkoff
equations for ECSK gravity for LRSI and LRSII space-times. In the
case of LRSII space-times, the equations are structurally very similar
to the ones of GR. Indeed this similarity allows to recast them into
the same form of the GR TOV equations via a redefinition of the matter
variables and the electric part of the Weyl tensor. As a consequence
we found that at the level of the metric it is possible to map static,
locally rotationally symmetric solutions of class II from the Einstein-Cartan
theory to the ones of the theory of General Relativity. Moreover,
due to this mapping and the re-scale in the matter variables, some
GR solutions which are physically irrelevant become, in the context
of ECSK gravity, interesting.

When we examine in detail the physical properties of physically relevant
solutions, the differences between the the Einstein-Cartan theory
and GR become once more evident. This is particularly true looking
at junction conditions. We found that the requirement that all of
the components of the Riemann tensor have finite discontinuities across
the separation surface 𝒩 leads to additional constraints
with respect to the tornsionless case. This is especially evident
when looking at the structure equations for stationary LRSI and LRS
II space-times sourced by a Weyssenhoff fluid. In these equations
the magnetic parts of the Weyl tensor depend explicitly on the derivatives
of the torsion tensor and the classical Israel junction conditions
of GR do not guarantee these terms to be finite across 𝒩.
As consequence of the generalized junction conditions, in the considered
setup, observers at the interior and exterior space-times must measure
the same value for the spin density at 𝒩, turning the
task of finding physically relevant solutions even more daunting.

Using the full set of structure equations and boundary conditions
provided by the junction formalism, we were able to study various
properties of possible solutions. We started by analyzing how the
presence of spin changes the Buchdahl limit for the maximum compactness
of a star. We concluded that the spin-geometry coupling allows stars
with a given circumferential radius to hold more matter than the corresponding
GR ones. Next we considered the case of static, spherically symmetric
compact objects entirely held by the matter spin, smoothly matched
to a vacuum exterior. This scenario was expected to represent a good
model for cold neutron stars, where the thermodynamical pressure is
negligible when compared to the spin density. We found, surprisingly,
that such objects cannot be simultaneously static, spherically symmetric
and smoothly matched to a vacuum exterior. This is a strong result
and it is necessary to discuss in detail the hypothesis that led to
such conclusion. More specifically our conclusion may not be valid
if:


  * the spin density is not a monotonically decreasing function of the
radial coordinate inside the star; 

  * we consider a non-vacuum exterior space-time; 

  * we replace the uncharged Weyssenhoff fluid model; 

  * we allow the presence of a thin shell. 

The first possibility might lead to a total energy density and a pressure
density which is not monotonically decreasing. While this is not a
strong enough reason to discard this case, we expect these oscillation
to make the solution unstable under small perturbations. The second
case suggests that if ECSK theory had a non trivial vacuum (vortical)
solution, one could smoothly match the interior to it, bypassing the
requirement of the spin density to vanish at an hypersurface. At present
there is no evidence that such solution might/should exist. Indeed
the theory is expected to reduce to GR in vacuum. For what concerns
hypothesis (iii), the Weyssenhoff fluid can be advocated to be a good
model for the matter fluids that might constitute cold neutron stars.
However, in this work we made the simplifying assumption that the
fluid is electrically neutral. If instead a charged Weyssenhoff fluid
model is considered, we expect that other effects will appear - such
as anisotropic pressure - which may drastically change the behavior
of the fluid. As for the last possibility, although a smooth junction
with a vacuum exterior might represent a more reasonable scenario,
it might be argued that neutron stars may have a well defined surface,
therefore it is not completely unreasonable to consider the presence
of a thin shell of matter at the matching surface.

On top of the zero pressure solution considered above, we have also
considered solutions in which pressure is non zero. Using reconstruction
algorithms we have been able to obtain various classes of solutions
for the interior of static, spherically symmetric compact objects
that can be smoothly matched to a Schwarzschild exterior. One family
of those solutions, which we dubbed Buchdahl stars, represent a very
interesting scenario: they admit the existence of a common hypersurface
where the pressure, spin density and energy density all vanish. This
models, studied for the first time by Buchdahl for gaseous stars in
GR <cit.>, represents the scenario where the fluid that composes
a star will smoothly dissipate away from a denser core and transition
to vacuum. These solutions also provided a key example for the effects
that spin may have on the behavior of the fluid. Figures. <ref>
- <ref> clearly exemplify that even if the spin density
is much smaller than the other matter variables, it allows for a much
richer behavior for the fluid.

The natural question that emerges is about the stability of these solutions. 
Because of the non trivial role of the magnetic part of the Weyl tensor, no standard “zeroth order” (and Newtonian based) criterion is necessarily valid in our case. In cases in which the spin density is decreasing, one very heuristic criterium of stability of our solutions is to guarantee (as we have done) that both energy density and pressure of the fluid are decreasing functions of the radial coordinate when the spin is small at least in a non empty set of values of the parameters. However, a complete study of the stability of the solutions we have found requires a more careful study, which will be the topic of a series of future works. 

Finally, as in the case of GR, also in ECSK theory it is possible
to derive generating theorems. In this work we have presented several
algorithms to generate new exact solutions from previously known ones.
We should stress here that the results we obtained followed from the
simple idea of finding conditions so that the Riccati differential
equations would reduce to Bernoulli equations. Although this scheme
allowed us to find various generating algorithms, we make no claim
that we have exhausted all possibilities for finding new ones. On
this note, the integrability conditions for Riccati type equations
in Refs. <cit.> were also considered. However, these
did not lead useful results in the considered context.

PL is grateful to IDPASC and FCT-Portugal for financial support through
Grant No. PD/BD/114074/2015. SC is supported by the Fundação
para a Ciência e Tecnologia through project IF/00250/2013 and
acknowledge financial support provided under the European Union's
H2020 ERC Consolidator Grant “Matter and strong-field gravity: New
frontiers in Einstein's theory” grant agreement No. MaGRaTh646597. 
We also thank the anonymous referee for useful suggestions.







§ COVARIANTLY DEFINED QUANTITIES FOR THE DERIVATIVES OF THE TANGENT
VECTORS


Using the definitions of the projector operators onto the hypersurfaces
V and W, let us show how the covariant derivatives of the tangent
vector fields v and e can be uniquely decomposed in their components
along u, e and W.



 §.§ Decomposition on the sheet W


Let us first consider the projected covariant derivatives of the tensors
u and e on the sheet. These can be uniquely decomposed as


    δ_αu_β≡ N_α^σN_β^γ∇_σu_γ=1/2N_αβθ̃+Σ_αβ+ε_αβΩ ,

where 

    θ̃    =δ_αu^α ,
    Σ_αβ    =δ_{αu_β} ,
    Ω    =1/2ε^σγδ_σu_γ ,

and 

    δ_αe_β=1/2N_αβϕ+ζ_αβ+ε_αβξ ,

with 

    ϕ    =δ_αe^α ,
    ζ_αβ    =δ_{αe_β} ,
    ξ    =1/2ε^σγδ_σe_γ ,

where the curly brackets represent the projected symmetric part without
trace of a tensor in W, that is, for a tensor ψ_αβ,

    ψ_{αβ}=[N^μ_(αN_β) ^ν-N_αβ/2N^μν]ψ_μν .

Using the 2-form volume ε_αβ a completely
anti-symmetric tensor defined on the sheet, ψ_[αβ],
can be written as 

    ψ_[αβ]=ε_αβ(1/2ϵ^γσψ_γσ) .

This property was used in Eq. (<ref>).



 §.§ Decomposition on V


The decomposition of the projected covariant derivatives of u^α
onto V, is given by 

    D_αu_β=h_α^σh_β^γ∇_σu_γ=1/3h_αβθ+σ_αβ+ω_αβ ,

with 

    θ    =h^αβD_αu_β ,
    σ_αβ    =D_⟨α.u_.β⟩ ,
    ω_αβ    =h^σ_[αh_β]^γD_σu_γ ,

where we used the angular brackets to represent the projected symmetric
part without trace of a tensor on V, that is, for a tensor, ψ_αβ,

    ψ_⟨αβ⟩=[h^μ_(αh_β)^ν-h_αβ/3h^μν]ψ_μν .

The scalar and tensor quantities in Eqs. (<ref>)
- (<ref>) can themselves be further
decomposed in their contributions exclusively on W and along e,
such that 

    θ=θ̃+θ̅ ,

where θ̃ is defined in Eq. (<ref>)
and 

    θ̅=-u_β(e^αD_αe^β)=-u_βê^β ;


    σ_αβ=Σ_αβ+2Σ_(αe_β)+Σ(e_αe_β-1/2N_αβ) ,

with 

    Σ_αβ    =σ_{αβ} ,
    Σ_α    =N_α^γe^βσ_γβ ,
    Σ    =e^αe^βσ_αβ=-N^αβσ_αβ ,

and 

    ω_αβ=ε_αβΩ-ε_αλω^λe_β+e_αε_βλω^λ ,

where Ω is given in Eq. (<ref>)
and 

    ω^λ=1/2ε^μνλD_μu_ν ,

which can be itself decomposed as 

    ω^λ=Ω e^λ+Ω^λ  ,with Ω^λ=N_α^λω^α=1/2N_α^λε^μναD_μu_ν ,

therefore, equivalently, 

    ω_αβ=ε_αβγ(Ω e^γ+Ω^γ) .

The quantities θ, Σ, θ̃ and θ̅
are not independent, in fact: 

    θ̅    =1/3θ+Σ ,
    θ̃    =2/3θ-Σ ;

as such, when setting up the 1+1+2 formalism only two are chosen.
The convention followed here uses the variables θ and Σ.

For the projected covariant derivative of the vector field e on
V we have 

    D_αe_β=h_α^σh_β^γ∇_σe_γ=δ_αe_β+e_αa_β ,

where δ_αe_β is given by Eq. (<ref>)
and 

    a_α=e^μD_μe_α=ê_α .





 §.§ Decomposition on the full manifold


Finally, we can decompose the total covariant derivatives of u^α
and e^α, such that 

    ∇_αu_β=-u_α(𝒜e_β+𝒜_β)+D_αu_β ,

with 

    𝒜    =-u_γu^μ∇_μe^γ=-u_γė^γ ,
    𝒜_α    =N_αβu̇^β ,

and 

    ∇_αe_β    =D_αe_β-u_αα_β-𝒜u_αu_β+[1/3θ+Σ]e_αu_β+
       +[Σ_α-ε_ασΩ^σ]u_β ,

where 

    α_μ=h_μ^σė_σ .





 §.§ The actual physical kinematical variables


As discussed in Refs. <cit.>, the presence
of a generic torsion field will affect the definition of the kinematical
quantities that characterize a congruence of curves, such that, θ,
σ_αβ and ω_αβ, Eqs. (<ref>)
- (<ref>), in general, do not represent
the actual geometric - physical - expansion, shear and vorticity of
the time-like congruence to which u is tangent. These, however,
are related with the actual kinematical quantities by 

    θ_g    =θ+W_σ^σ ,
    σ_g αβ    =σ_αβ+W_⟨αβ⟩ ,
    ω_g αβ    =ω_αβ+W_[αβ] ,

where we have used the index g to represent the physical -
geometric - kinematical quantities and the definition of angular brackets
is given in Eq. (<ref>). In the
same way the presence of the torsion field will modify the kinematical
quantities ϕ, ζ_αβ and ξ, Eqs. (<ref>).
In particular we have the following relations 

    ϕ_g    =ϕ+2S_γμνe^γN^μν ,
    ζ_g αβ    =ζ_αβ+2S_γμνe^γN_⟨α|^μN_|β⟩^ν ,
    ξ_g    =ξ+S_γμνe^γε^μν .

In the particular setup that we propose to study - Weyssenhoff like
torsion - the extra terms in the RHS of Eqs. (<ref>)
and (<ref>) that depend explicitly
of the torsion tensor will be null, therefore, in our case, the indicated
quantities will correspond to the actual geometric kinematical quantities. 
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