*4.2. The Coupling to a Dirac Field*

Let *ψ* be a Dirac field with Lagrangian function given by:

$$\mathcal{L}\_{\mathfrak{m}} = \left[ \frac{i}{2} \left( \bar{\psi} \gamma^{i} D\_{i} \psi - D\_{i} \bar{\psi} \gamma^{i} \psi \right) - m \bar{\psi} \psi \right]. \tag{116}$$

where *Diψ* = *∂ψ ∂x <sup>i</sup>* + *ω µν i σµνψ* and *Diψ*¯ = *∂ψ*¯ *∂x <sup>i</sup>* <sup>−</sup> *ψω*¯ *µν i σµν* are the covariant derivatives of the Dirac fields, *σµν* = <sup>1</sup> 8 - *γµ*, *γ<sup>ν</sup>* , *γ <sup>i</sup>* = *γ µ e i <sup>µ</sup>* with *γ <sup>µ</sup>* denoting Dirac matrices and where *m* is the mass of the Dirac field. In what follows, the notation for which:

$$
\gamma^{\mu}\gamma^{\nu}\gamma^{\lambda} = \gamma^{\mu}\eta^{\nu\lambda} - \gamma^{\nu}\eta^{\mu\lambda} + \gamma^{\lambda}\eta^{\mu\nu} + i\epsilon^{\mu\nu\lambda\tau}\gamma\_5\gamma\_{\tau\nu} \tag{117}
$$

is used. From (116), we derive the Dirac equations:

$$i\gamma^h D\_h \psi + \frac{i}{2} T\_h \gamma^h \psi - m\psi = 0,\tag{118}$$

where, due to to the fact that torsion is no longer totally antisymmetric, the torsion vector *T<sup>h</sup>* := *T j hj* is present. The stress–energy and the spin density tensors are given by [15,42]:

$$\mathcal{T}\_{\rm ij} = \frac{i}{4} \left( \overline{\psi} \gamma\_i D\_j \psi - D\_j \overline{\psi} \gamma\_i \psi \right) \,, \tag{119}$$

and:

$$\mathcal{S}\_{\mathrm{ij}}{}^{h} = -\frac{1}{4} \eta^{\mu \sigma} \varepsilon\_{\sigma \nu \lambda \tau} \left( \bar{\psi} \gamma\_5 \gamma^\tau \psi \right) e\_{\mu}^{h} e\_i^{\nu} e\_j^{\lambda} \,. \tag{120}$$

In what follows, we can systematically assume that *ψψ*¯ <sup>6</sup><sup>=</sup> 0. Indeed, if *ψψ*¯ <sup>=</sup> 0, the trace of the stress–energy tensor would be constantly zero and the theory would amount to an ECSK-like theory for which the solution of the junction conditions problem is already known [35]. Therefore, without loss of generality, we can limit ourselves to dealing with spinor fields of type-1 and type-2 according to the Lounesto classification [57–59].

Making use of representation (11), it is seen that in this case the vanishing at Σ of the quantities (89), (92) yields the conditions:

− *ǫ* h *K p jq* i *npE q A n <sup>j</sup>* + h *K p pq* i *E q <sup>A</sup>* = −*ǫ* h *K*ˆ *p jq* i *npE q A n <sup>j</sup>* + h *K*ˆ *p pq* i *E q <sup>A</sup>* = − 1 *ϕ ∂ϕ ∂x q E q <sup>A</sup>* = 0, (121a) *H*˜ *AB* + *ǫ* − h *K p jq* i *npE q A E j <sup>B</sup>* + h *K qp q* i *<sup>n</sup>phAB* = *H*˜ *AB* + *ǫ* − h *K*ˆ *p jq* i *npE q A E j <sup>B</sup>* + h *K*ˆ *qp q* i *nphAB* − h *S*ˆ *p jq* i *npE q A E j B* = *H*˜ *AB* + *ǫ* 1 *ϕ* S *p jq npE q A E j <sup>B</sup>* + 1 *ϕ ∂ϕ ∂x p n p <sup>h</sup>AB* = 0. (121b)

Splitting Equation (121b) in its symmetric and antisymmetric parts, we obtain the equations:

$$
\tilde{H}\_{AB} + \varepsilon \left[ \frac{1}{\varrho} \frac{\partial \varrho}{\partial \mathbf{x}^p} \right] n^p h\_{AB} = 0,\tag{122a}
$$

$$\left[\frac{1}{\rho}\mathcal{S}\_{j\dot{q}}{}^{p}\right]n\_{p}E\_{A}^{q}E\_{B}^{j} = 0.\tag{122b}$$

As an illustrative example, we suppose joining two axially symmetric spacetimes, solutions of the field equations resulting again from the model *f*(*R*) = *R* + *αR* 2 . More in detail, we assume that the metric tensors in both the regions <sup>M</sup><sup>−</sup> and <sup>M</sup><sup>+</sup> are of Lewis–Papapetrou kind, expressed in spherical coordinates as:

$$\mathcal{g}^{\pm}\_{ij}\,dx^{i}dx^{j} = -\mathcal{B}^{2}\_{\pm}(r^{2}\,d\theta^{2} + dr^{2}) - A^{2}\_{\pm}(-\mathcal{W}\_{\pm}\,dt + d\phi)^{2} + \mathcal{C}^{2}\_{\pm}\,dt^{2},\tag{123}$$

where all functions *A*±(*r*, *θ*), *B*±(*r*, *θ*), *C*±(*r*, *θ*), and *W*±(*r*, *θ*) depend on the *r* and *θ* variables only. We assume that <sup>M</sup><sup>+</sup> is empty, while <sup>M</sup><sup>−</sup> is filled with a Dirac field. We also suppose that in <sup>M</sup><sup>+</sup> the metric is the Kerr one. This is consistent with the fact that *R* + *αR* <sup>2</sup> gravity with torsion in vacuo is equivalent to GR and, therefore, admits the same solutions. In the Lewis–Papapetrou form (123), the coefficients of the Kerr metric are expressed as:

$$A\_+^2(r,\theta) = \left[a^2 + \frac{\left(-a^2 + m^2 + 2mr + r^2\right)^2}{4r^2}\right] \sin^2\theta + \frac{ma^2\left(-a^2 + m^2 + 2mr + r^2\right)\sin^4\theta}{r\left(\frac{\left(-a^2 + m^2 + 2mr + r^2\right)^2}{4r^2} + a^2\cos^2\theta\right)},\tag{124a}$$

$$B\_+^2(r, \theta) = \frac{a^2 \cos^2 \theta}{r^2} + \frac{1}{4} + \frac{m}{r} + \frac{3m^2 - a^2}{2r^2} + \frac{m^3 - a^2 m}{r^3} + \frac{a^4 - 2a^2 m^2 + m^4}{4r^4},\tag{124b}$$

$$C\_{+}^{2}(r,\theta) = \frac{m^{2}a^{2}\left(-a^{2} + m^{2} + 2mr + r^{2}\right)^{2}\sin^{4}\theta}{\left(\left(a^{2} + \frac{\left(-a^{2} + m^{2} + 2mr + r^{2}\right)^{2}}{4r^{2}}\right)\sin^{2}\theta + \frac{mr^{2}\left(-a^{2} + m^{2} + 2mr + r^{2}\right)\sin^{4}\theta}{r\left(\frac{\left(-a^{2} + m^{2} + 2mr + r^{2}\right)^{2}}{4r^{2}} + a^{2}\cos^{2}\theta\right)}\right)}} \times 1$$
 
$$1 \qquad \qquad \qquad \qquad \qquad \qquad \begin{aligned} 1 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \begin{aligned} m\left(-a^{2} + m^{2} + 2mr + r^{2}\right) \end{aligned} \end{aligned} \tag{124c}$$

$$\begin{split} r^{2} \left( \frac{\left( -a^{2} + m^{2} + 2mr + r^{2} \right)^{2}}{4r^{2}} + a^{2} \cos^{2} \theta \right)^{2} &\quad \frac{1}{r} \left( \frac{\left( -a^{2} + m^{2} + 2mr + r^{2} \right)^{2}}{4r^{2}} + a^{2} \cos^{2} \theta \right)^{2}} \\ W\_{+}(r, \theta) = \frac{ma \left( -a^{2} + m^{2} + 2mr + r^{2} \right) \sin^{2} \theta}{\left( a^{2} + \frac{\left( -a^{2} + m^{2} + 2mr + r^{2} \right)^{2}}{4r^{2}} \right) \sin^{2} \theta + \frac{\frac{ma^{2} \left( -a^{2} + m^{2} + 2mr + r^{2} \right) \sin^{4} \theta}{4r^{2}}}{r \left( \frac{\left( -\frac{a^{2} + m^{2} + 2mr + r^{2}}{4r^{2}} \right)^{2} + a^{2} \cos^{2} \theta} \right)} \end{split} \tag{124d}$$
 
$$r \left( \frac{(-a^{2} + m^{2} + 2mr + r^{2})}{4r^{2}} + a^{2} \cos^{2} \theta \right)^{\prime}$$

where *a* and *m* are the parameters entering the Kerr metric. We want to analyze the junction conditions at the hypersurface Σ : *r* = *r*<sup>0</sup> const. To this end, by using Equations (118) and (119), we preliminarily notice that in the regions <sup>M</sup><sup>−</sup> and <sup>M</sup><sup>+</sup> we have, respectively:

$$
\varphi^- = 1 + 2aR = 1 - 2aT = 1 - am\bar{\Psi}\psi,\tag{125a}
$$

and:

$$
\varphi^+ = 1.\tag{125b}
$$

In view of Equation (125), the constraint (121a) implies that the scalar *ψψ*¯ is forced to be constant on the hypersurface Σ. Moreover, it is easily seen that the requirement (122b) is equivalent to the conditions:

$$
\bar{\psi}\gamma\_5\gamma^0\psi\_{|\Sigma} = 0, \qquad \bar{\psi}\gamma\_5\gamma^2\psi\_{|\Sigma} = 0, \qquad \bar{\psi}\gamma\_5\gamma^3\psi\_{|\Sigma} = 0,\tag{126}
$$

which have to be satisfied at Σ by the spinor field *ψ*. The remaining condition (122a) can be discussed by rewriting it in the equivalent form:

$$\left[\left[\check{Q}\_{AB}\right] = -\frac{1}{2} \left[\frac{1}{\varrho} \frac{\partial \varrho}{\partial x^{h}}\right] n^{h} h\_{AB\prime} \tag{127}$$

where - *Q*˜ *AB* indicates the jump across Σ of the extrinsic curvatures induced by the metrics (123). Denoting by *A*˜ := *A* <sup>+</sup>(*r*0, *θ*) = *A* <sup>−</sup>(*r*0, *θ*), *B*˜ := *B* <sup>+</sup>(*r*0, *θ*) = *B* <sup>−</sup>(*r*0, *θ*), *C*˜ := *C* <sup>+</sup>(*r*0, *θ*) = *C* <sup>−</sup>(*r*0, *θ*), and *W*˜ := *W*+(*r*0, *θ*) = *W*−(*r*0, *θ*) for simplicity, we have that the non–zero components of - *Q*˜ *AB* are:

$$\left[\check{Q}\_{\theta\theta}\right] = -r\_0^2 \left[\partial\_r B\right],\tag{128a}$$

$$\left[\check{Q}\_{\Phi\Phi}\right] = -\frac{\check{A}}{\overline{\mathcal{B}}} \left[\partial\_{\prime} A\right] ,\tag{128b}$$

$$\mathbb{E}\left[\check{Q}\_{t\phi}\right] = \frac{\tilde{A}\left(2\tilde{\mathcal{W}}\left[\partial\_{r}A\right] + \tilde{A}\left[\partial\_{r}\mathcal{W}\right]\right)}{2\tilde{B}},\tag{128c}$$

$$\left[\check{Q}\_{lt}\right] = \frac{\tilde{\mathcal{C}}\left[\partial\_{r}\mathcal{C}\right] - \tilde{A}\tilde{\mathcal{W}}^{2}\left[\partial\_{r}A\right] - \tilde{A}^{2}\tilde{\mathcal{W}}\left[\partial\_{r}\mathcal{W}\right]}{\tilde{B}}.\tag{128d}$$

Due to Equations (125) and (128), the non-trivial equations of (127) result to have explicit expression:

$$\frac{[\partial\_{\varGamma}B]}{\tilde{B}} = -\frac{\alpha m}{2(1 - am\tilde{\psi}\psi\_{|\varSigma})}\partial\_{\varGamma}(\tilde{\psi}\psi)\_{|\varSigma'}\tag{129a}$$

$$\frac{[\partial\_{\prime}A]}{\tilde{A}} = -\frac{\alpha m}{2(1 - \alpha m \bar{\psi}\psi\_{|\Sigma})} \partial\_{r} \left(\bar{\psi}\psi\right)\_{|\Sigma\prime} \tag{129b}$$

$$\frac{2\left[\partial\_{\boldsymbol{r}}A\right]}{\tilde{A}} + \frac{\left[\partial\_{\boldsymbol{r}}\mathcal{W}\right]}{\tilde{\mathcal{W}}} = -\frac{am}{(1 - am\tilde{\psi}\psi\_{|\boldsymbol{\Sigma}})}\partial\_{\boldsymbol{r}}\left(\tilde{\psi}\psi\right)\_{|\boldsymbol{\Sigma}\boldsymbol{r}}\tag{129c}$$

$$\frac{\tilde{\mathcal{C}}\left[\partial\_{\mathcal{I}}\mathcal{C}\right]-\tilde{A}\tilde{\mathcal{W}}^{2}\left[\partial\_{\mathcal{I}}A\right]-\tilde{A}^{2}\tilde{\mathcal{W}}\left[\partial\_{\mathcal{I}}\mathcal{W}\right]}{\tilde{\mathcal{C}}^{2}-\tilde{A}^{2}\tilde{\mathcal{W}}^{2}}=-\frac{am}{2(1-am\tilde{\Psi}\psi\_{|\Sigma})}\partial\_{\mathcal{I}}\left(\tilde{\Psi}\psi\right)\_{|\Sigma}.\tag{129d}$$

From Equation (129), it is seen that the jumps of the *r*-derivatives of quantities *A* ±, *B* ±, and *C* ± have to satisfy the relations:

$$\frac{[\partial\_r A]}{\tilde{A}} = \frac{[\partial\_r B]}{\tilde{B}} = \frac{[\partial\_r \mathbb{C}]}{\tilde{\mathbb{C}}} = -\frac{am}{2(1 - am\bar{\psi}\psi\_{|\Sigma})} \partial\_r \left(\bar{\psi}\psi\right)\_{|\Sigma} = \frac{1}{2\rho} \frac{\partial \rho}{\partial(\bar{\psi}\psi)} \partial\_r \left(\bar{\psi}\psi\right)\_{|\Sigma'} \tag{130}$$

while the function *W*(*r*, *θ*) has to be of class C 1 .

In conclusion, it is shown that, in the non–linear case *<sup>f</sup>*(*R*) <sup>6</sup><sup>=</sup> *<sup>R</sup>* <sup>+</sup> *<sup>λ</sup>*, the scalar field *ψψ*¯ is also involved in the characterization of the junction conditions. In particular, the derivatives of the metric components with respect to the coordinate *r* can have some jumps at the hypersurface Σ, connected with the *r*–derivative of the scalar quantity *ψψ*¯ . This is a difference from the linear case *f*(*R*) = *R* + *λ* (ECSK theory), where, instead, the metric has to be at least of class C 1 .
