*4.1. The Coupling to a Spin Fluid*

Let us consider a Weyssenhoff spin fluid with stress–energy and the spin tensors, respectively, given by [15,53,54]:

$$\mathcal{T}^{i\bar{j}} = \mathcal{U}^{i}\mathcal{P}^{\bar{j}} + p\left(\mathcal{U}^{i}\mathcal{U}^{j} - \mathcal{g}^{i\bar{j}}\right),\tag{103a}$$

and:

$$\mathcal{S}\_{\rm ij}{}^h = \mathcal{S}\_{\rm ij}! \mathcal{U}^h. \tag{103b}$$

where *U<sup>i</sup>* is the 4-velocity, *P <sup>j</sup>* denotes the 4-density of energy–momentum, <sup>S</sup>*ij* <sup>=</sup> −S*ji* is the spin density, and *p* is the pressure of the fluid. By means of the conservation laws for the spin (12b), which are equivalent to the antisymmetric part of Einstein-like equations (10a), we can express the stress–energy tensor (103a) as [54]:

$$\mathcal{T}\_{\vec{i}\vec{j}} = (\rho + p) \mathcal{U}\_{\vec{i}} \mathcal{U}\_{\vec{j}} - p \mathcal{g}\_{\vec{i}\vec{j}} - \mathcal{U}\_{\vec{i}} \hat{\mathcal{T}}\_{\hbar} \mathcal{S}^{\hbar}\_{\ \vec{j}} - \mathcal{U}\_{\vec{i}} \hat{\mathcal{V}}\_{\hbar} \left( \mathcal{S}\_{\vec{k}\vec{j}} \mathcal{U}^{\hbar} \right) \mathcal{U}^{\hbar} \tag{104}$$

where *<sup>ρ</sup>* :<sup>=</sup> *<sup>U</sup>iP<sup>i</sup>* and <sup>∇</sup>˜ *h* is the covariant derivative with respect to the Levi–Civita connection induced by the metric *<sup>g</sup>ij*. In view of the usual convective condition <sup>S</sup>*ijU<sup>j</sup>* <sup>=</sup> 0 [53,55] and the representation (11), it is easily seen that the vanishing at Σ of the quantities (89), (92) yields the explicit equations:

$$-\varepsilon \left[ \mathbb{K}\_{jq}{}^{p} \right] n\_p \mathbb{E}\_A^q n^j + \left[ \mathbb{K}\_{pq}{}^{p} \right] \mathbb{E}\_A^q = -\varepsilon \left[ \frac{1}{\rho} \mathbb{S}\_{qj}! \mathbb{U}\_p \right] n^p n^j \mathbb{E}\_A^q - \left[ \frac{1}{\rho} \frac{\partial \rho}{\partial x^q} \right] \mathbb{E}\_A^q = 0,\tag{105a}$$

$$\begin{split} \boldsymbol{\tilde{H}}\_{AB} + \boldsymbol{\epsilon} \left( - \left[ \boldsymbol{K}\_{\boldsymbol{j}\boldsymbol{q}}^{p} \right] \boldsymbol{n}\_{p} \boldsymbol{E}\_{A}^{q} \boldsymbol{E}\_{B}^{j} + \left[ \boldsymbol{K}\_{\boldsymbol{q}}^{qp} \right] \boldsymbol{n}\_{p} \boldsymbol{h}\_{AB} \right) = \\ \boldsymbol{\tilde{H}}\_{AB} + \boldsymbol{\epsilon} \left( \left[ \frac{1}{2\boldsymbol{q}} \left( \boldsymbol{\mathcal{S}}\_{\boldsymbol{j}\boldsymbol{q}} \boldsymbol{\mathcal{U}}^{p} + \boldsymbol{\mathcal{S}}\_{\boldsymbol{q}}^{p} \boldsymbol{\mathcal{U}}\_{\boldsymbol{j}} + \boldsymbol{\mathcal{S}}\_{\boldsymbol{j}}^{p} \boldsymbol{\mathcal{U}}\_{\boldsymbol{k}} \right) \right] \boldsymbol{n}\_{p} \boldsymbol{E}\_{A}^{q} \boldsymbol{E}\_{B}^{j} + \left[ \frac{1}{\boldsymbol{q}} \frac{\partial \boldsymbol{\rho}}{\partial \mathbf{x}^{p}} \right] \boldsymbol{n}^{p} \boldsymbol{h}\_{AB} \right) = \boldsymbol{0}. \end{split} \tag{105b}$$

Equation (105b) can be decomposed into its symmetric and antisymmetric parts, thus giving rise to the further conditions: 

$$\left[\frac{1}{2\varrho}\mathcal{S}\_{\dot{\jmath}q}\mathcal{U}^p\right]n\_pE^q\_AE^{\dot{\jmath}}\_B=0,\tag{106a}$$

$$\tilde{H}\_{AB} + \varepsilon \left( \left[ \frac{1}{2\varrho} \left( \mathcal{S}^p\_{\,\,q} \mathcal{U}\_{\,\,j} + \mathcal{S}^p\_{\,\,j} \mathcal{U}\_{\,\,q} \right) \right] n\_p E^q\_A E^j\_B + \left[ \frac{1}{\varrho} \frac{\partial \varphi}{\partial \mathbf{x}^p} \right] n^p \mathcal{U}\_{AB} \right) = 0. \tag{106b}$$

In order to illustrate a specific case, we imagine having to join together two static and spherically symmetric metrics:

$$ds\_{\pm}^2 = e^{\nu^\pm}dt^2 - e^{\lambda^\pm}dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right),\tag{107}$$

solutions of Equation (10) coupled to a spin fluid. It is convenient to rename the spherical coordinates as *x* 0 := *t*, *x* 1 := *r*, *x* 2 := *θ*, *x* 3 := *φ* in such a way that the 4-velocity of the fluid (supposed to be at rest in the chosen frame) is described by *U<sup>i</sup>* = *U*<sup>0</sup> *δ i* 0 , with *U*<sup>0</sup> = *e* − *ν* <sup>2</sup> , and the unit normal to the hypersurface Σ : *x* <sup>1</sup> = const. is given by *n <sup>i</sup>* = *n* 1 *δ i* <sup>1</sup> with *n* <sup>1</sup> = *e* − *λ* <sup>2</sup> . The functions *ν* and *λ*, as well as all the involved matter fields, depend only on the radial variable *r*.

According to the convective condition <sup>S</sup>*ijU<sup>j</sup>* <sup>=</sup> 0 and the stated spherical symmetry, we suppose that the spins of the particles composing the fluid are all aligned in the *r* direction; this means that only the components S<sup>23</sup> = −S<sup>32</sup> of the spin density are non-zero [55]. Under these conditions, the stress–energy tensor of the spin fluid assumes the usual form:

$$\mathcal{T}\_{\rm ij} = (\rho + p) \, \mathcal{U}\_{\rm i} \mathcal{U}\_{\rm j} - p \mathbf{g}\_{\rm ij}. \tag{108}$$

Using the above assumptions, it is easily seen that the constraints (105a), (106a) are automatically satisfied, while Equation (106b) reduces to:

$$
\tilde{H}\_{AB} + \varepsilon \left[ \frac{1}{\varrho} \frac{\partial \varrho}{\partial \mathbf{x}^p} \right] n^p h\_{AB} = 0. \tag{109}
$$

Recalling the identity *H*˜ *AB* = −*ǫ* -*Q*˜ *AB* − - *Q*˜ *hAB* [34], it is seen that Equation (109) relates the quantity <sup>h</sup> 1 *ϕ ∂ϕ ∂x p* i to the jump across Σ of the extrinsic curvature *Q*˜ *AB* associated with the Levi–Civita connection of the metric (107). We note that, in the case of ECSK theory, condition (109) becomes *H*˜ *AB* = 0, which is the same condition holding in General Relativity [34].

Because of Equations (8) and (9), in general, the condition (109) involves the derivatives of matter fields. To see this point more in detail, we again take the model *f*(*R*) = *R* + *αR* 2 into account. Due to Equation (108), from the trace equation (7) and the definition (9), we have the relations:

$$-R = \mathcal{T} = \rho - \mathfrak{B}p,\tag{110}$$

and:

$$
\varphi = 1 + 2\mathfrak{a} \left( 3p - \rho \right). \tag{111}
$$

Moreover, it is easy to verify that:

$$\check{Q}\_{00} = \frac{1}{2} \frac{\partial \nu}{\partial r} e^{\nu - \frac{\lambda}{2}}\_{|r = r\_0} \tag{112}$$

is the only non–vanishing component of the extrinsic curvature *Q*˜ *AB* induced by the metric (107) on the hypersurface Σ : *r* = *r*<sup>0</sup> const.. In view of this, requirement (109) is seen to reduce to the following two conditions:

$$\left[\frac{2\alpha\left(3\frac{\partial p}{\partial r} - \frac{\partial \rho}{\partial r}\right)}{1 + 2\alpha\left(3\rho - \rho\right)}\right] = 0,\tag{113a}$$

and:

$$
\left[\frac{\partial \nu}{\partial r}\right] = 0.\tag{113b}
$$

As an even more specific example, we suppose to have to joining together the interior spacetime <sup>M</sup><sup>−</sup> of a star with spin properties, with the exterior region <sup>M</sup><sup>+</sup> assumed empty. In such a circumstance, we have T + *ij* = 0 and S + *h ij* <sup>=</sup> 0, and in <sup>M</sup><sup>+</sup> the field equations (10) are identical to the Einstein equations (without cosmological constant) in vacuo; their unique solution *g* + *ij* , Γ + *h ij* is then given by the Schwartzchild metric:

$$g\_{ij}^{+}dx^{i}dx^{j} = \left(1 - \frac{2M}{r}\right)dt^{2} - \left(1 - \frac{2M}{r}\right)^{-1}dr^{2} - r^{2}\left(d\theta^{2} + \sin^{2}\theta \, d\phi^{2}\right),\tag{114}$$

together with its Levi–Civita connection Γ + *h ij* <sup>=</sup> <sup>Γ</sup>˜ <sup>+</sup> *<sup>h</sup> ij* . Consequently, the junction conditions (69), (113) assume the explicit form:

$$e^{\nu^{-}(r\_0)} = \left(1 - \frac{2M}{r\_0}\right), \qquad e^{\lambda^{-}(r\_0)} = \left(1 - \frac{2M}{r\_0}\right)^{-1},\tag{115a}$$

$$\left(\frac{\partial v^{-}}{\partial r}\right)\_{|r=r\_{0}} = \frac{2M}{r\_{0}\left(r\_{0} - 2M\right)}, \qquad \left(3\frac{\partial p^{-}}{\partial r} - \frac{\partial \rho^{-}}{\partial r}\right)\_{|r=r\_{0}} = 0. \tag{115b}$$

On Equation (115), some comments are in order. Due to the second equation (115b), at Σ the spin fluid must behave like a sort of radiation, having a barotropic factor of the form *w* = *∂p* − *∂ρ*− |*r*=*r*<sup>0</sup> = 1/3 at the boundary *r* = *r*0. This fact is quite general: for all static and spherically symmetric solutions (107) of *<sup>f</sup>*(*R*)-gravity with torsion, the condition −3 *∂p* − *<sup>∂</sup><sup>r</sup>* + *∂ρ*− *∂r* |*r*=*r*<sup>0</sup> = *<sup>∂</sup>*<sup>T</sup> − *∂r* |*r*=*r*<sup>0</sup> = 0 is always sufficient (together with (113b)) to fulfill the requirement (109), and it becomes necessary also whenever *∂ϕ*− *∂*T |*r*=*r*<sup>0</sup> 6= 0 (like in the case *f*(*R*) = *R* + *αR* 2 , where *∂ϕ*<sup>−</sup> *∂*T |*r*=*r*<sup>0</sup> = −2*α*). On the other hand, whenever the condition *∂ϕ*<sup>−</sup> *∂*T |*r*=*r*<sup>0</sup> = 0 is imposed, it yields a relation between density and pressure at the separation hypersurface, which constraints the equation of state [56].
