*3.2. The Cauchy Problem in the Presence of a Scalar Field*

We take the coupling with a Klein–Gordon scalar field into account. Again, we give sufficient conditions for the well-posedness of the related Cauchy problem. To this end, let us denote by *<sup>ψ</sup>* a Klein–Gordon scalar field with self-interacting potential *<sup>U</sup>*(*ψ*) = <sup>1</sup> 2 *m* 2*ψ* 2 . The corresponding stress–energy tensor is given by:

$$\mathcal{T}\_{\mathrm{ij}} = \frac{\partial \boldsymbol{\psi}}{\partial \mathbf{x}^{i}} \frac{\partial \boldsymbol{\psi}}{\partial \mathbf{x}^{j}} - \frac{1}{2} \boldsymbol{g}^{ij} \left( \frac{\partial \boldsymbol{\psi}}{\partial \mathbf{x}^{p}} \frac{\partial \boldsymbol{\psi}}{\partial \mathbf{x}^{q}} \boldsymbol{g}^{pq} + m^{2} \boldsymbol{\psi}^{2} \right). \tag{45}$$

The associated Klein–Gordon equation is expressed as:

$$"\vec{\nabla}\_{\dot{\jmath}} \frac{\partial \psi}{\partial x^{i}} g^{ij} = m^{2} \psi \,. \tag{46}$$

where <sup>∇</sup>˜ denotes the Levi–Civita covariant derivative induced by the metric *<sup>g</sup>ij*. The trace of tensor (45) is:

$$\mathcal{T} := \mathcal{T}\_{\mathrm{ij}} \mathcal{g}^{\mathrm{ij}} = -\frac{\partial \psi}{\partial \mathbf{x}^p} \frac{\partial \psi}{\partial \mathbf{x}^q} \mathcal{g}^{pq} - 2m^2 \psi^2 \,. \tag{47}$$

The trace (47) depends explicitly on the metric tensor *gij*. Because of this, the conformal transformation cannot be applied directly to the field equations (13), with the scalar field *ϕ* defined by (9). Indeed, if we proceed in this way, both the metric *gij* and *g*¯*ij* would appear in the conformally transformed equations (22). This difficulty can be overcome making use of the already mentioned

dynamical equivalence with *ω*<sup>0</sup> = − 3 2 Brans–Dicke gravity. The idea is then to discuss the Cauchy problem for a *ω*<sup>0</sup> = − 3 2 Brans–Dicke theory coupled with the given Klein–Gordon field *ψ*. The field equations of such a theory are the Einstein-like Equation (13), the Equation (19), and the Klein–Gordon Equation (46), where the scalar field *ϕ* is a dynamical variable related to the trace T through Equation (19). After implementing the conformal transformation (21), the Einstein-like Equation (13) assumes the simpler form (22). At the same time, recalling the relation:

$$
\boldsymbol{\Gamma}\_{\boldsymbol{ij}}{}^{\boldsymbol{h}} = \boldsymbol{\Gamma}\_{\boldsymbol{ij}}{}^{\boldsymbol{h}} + \frac{1}{2\varrho} \frac{\partial \varrho}{\partial \boldsymbol{x}^{\boldsymbol{j}}} \boldsymbol{\delta}\_{\boldsymbol{i}}^{\boldsymbol{h}} - \frac{1}{2\varrho} \frac{\partial \varrho}{\partial \boldsymbol{x}^{\boldsymbol{p}}} \boldsymbol{g}^{\boldsymbol{p}\boldsymbol{h}} \boldsymbol{g}\_{\boldsymbol{ij}} + \frac{1}{2\varrho} \frac{\partial \varrho}{\partial \boldsymbol{x}^{\boldsymbol{j}}} \boldsymbol{\delta}\_{\boldsymbol{j}}^{\boldsymbol{h}}.\tag{48}
$$

linking the Levi–Civita connection Γ˜ *<sup>h</sup> ij* associated with the metric *gij* to the Levi–Civita connection Γ¯ *<sup>h</sup> ij* induced by the conformal metric *g*¯*ij*, we can write the Klein–Gordon equation in terms of the conformal metric *g*¯*ij* as:

$$-\frac{\partial \psi}{\partial \mathbf{x}^{i}} \mathbf{g}^{ij} \frac{\partial \boldsymbol{\varrho}}{\partial \mathbf{x}^{j}} + \boldsymbol{\varrho} \nabla\_{j} \frac{\partial \boldsymbol{\psi}}{\partial \mathbf{x}^{i}} \mathbf{g}^{ij} = m^{2} \boldsymbol{\psi},\tag{49}$$

where <sup>∇</sup>¯ *j* indicates the covariant derivative associated with the conformal metric *g*¯*ij*. Analogously, we can express the trace T as function of *g*¯*ij*, that is:

$$\mathcal{T} = -\frac{\partial \psi}{\partial \mathbf{x}^p} \frac{\partial \psi}{\partial \mathbf{x}^q} \varphi \overline{\mathbf{g}}^{pq} - 2m^2 \psi^2 \,. \tag{50}$$

The relation corresponding to (19) now links the scalar field *ϕ* to the Klein–Gordon field *ψ*, its partial derivatives *∂ψ ∂x i* , and the conformal metric *g*¯*ij*. Moreover, as it has been already pointed out, the quantity:

$$\mathcal{T}\_{\rm ij} := \frac{1}{\mathcal{q}} \Sigma\_{\rm ij} - \frac{1}{\mathcal{q}^3} V(\boldsymbol{\varphi}) \mathfrak{g}\_{\rm ij\prime} \tag{51}$$

represents an effective stress–energy tensor. On the other hand, the Klein–Gordon equation (46) implies the conservation laws <sup>∇</sup>˜ *<sup>j</sup>*T*ij* <sup>=</sup> 0, thus also identifying <sup>∇</sup>¯ *<sup>j</sup>*T¯ *ij* = 0 (see Proposition 2). This is a key point, allowing us to making use of harmonic coordinates and then to apply similar arguments as in [23,24,26].

More specifically, after rewriting the Einstein-like equations (22) in the equivalent form:

$$
\bar{R}\_{\rm ij} = \bar{T}\_{\rm ij} - \frac{1}{2}\bar{T}\,\mathfrak{F}\_{\rm ij} \tag{52}
$$

we adopt harmonic coordinates obeying the condition:

$$
\nabla\_p \nabla^p \mathbf{x}^i = -\mathbf{g}^{pq} \Gamma^i\_{pq} = \mathbf{0},
\tag{53}
$$

in such a way that equations (52) can be expresed as (see, for example, [23,26]):

$$
\delta^{pq} \frac{\partial^2 \mathbb{g}\_{ij}}{\partial \mathbf{x}^p \partial \mathbf{x}^q} = f\_{ij}(\mathbb{g}, \partial \mathbb{g}, \boldsymbol{\psi}, \partial \boldsymbol{\psi}),
\tag{54}
$$

where *fij* are suitable functions depending only on the metric *g*¯, the scalar field *ψ*, and their first order derivatives.

In addition to this, we suppose that Equation (19) is solvable with respect to the variable *ϕ*, and then to derive from Equation (19) itself a function of the form:

$$
\varphi = \varphi(\lg, \psi, \frac{\partial \psi}{\partial x^p} \frac{\partial \psi}{\partial x^q} \mathfrak{g}^{pq}),
\tag{55}
$$

expressing the scalar field *ϕ* as a suitable function of the metric *g*¯, the Klein–Gordon field *ψ*, and its first order derivatives. We notice that, in view of Equation (50), the dependence of *ϕ* on the derivatives of *ψ* is necessarily of the form indicated in Equation (55). Once again, the solvability with respect the scalar field *ϕ* to about Equation (19) depends on the explicit form of the potential *V*(*ϕ*) which is defined in terms of the function *f*(*R*) via the relation (14). Therefore, the possibility of solving Equation (19) with respect to *ϕ* can be taken as a rule to select viable *f*(*R*)-models. Moreover, from Equation (55), we obtain the identity:

$$\frac{\partial\varPhi}{\partial\mathbf{x}^{i}} = \frac{\partial\varPhi}{\partial\left(\frac{\partial\varPsi}{\partial\mathbf{x}^{\mathcal{S}}}\frac{\partial\varPsi}{\partial\mathbf{x}^{\mathcal{S}}}\mathfrak{S}^{\mathcal{S}t}\right)}2\frac{\partial\varPsi}{\partial\mathbf{x}^{\mathcal{O}}}\mathfrak{S}^{\mathcal{P}q}\frac{\partial^{2}\varPsi}{\partial\mathbf{x}^{i}\partial\mathbf{x}^{p}} + f\_{i}(\mathfrak{g},\partial\mathfrak{g},\psi,\partial\psi). \tag{56}$$

Inserting Equation (56) in Equation (49) and taking Equation (53) into account, we get the final form of the Klein–Gordon equation expressed as:

$$
\left(\mathfrak{g}^{ip} - \frac{2}{\mathfrak{q}} \frac{\partial \mathfrak{q}}{\partial \left(\frac{\partial \mathfrak{q}}{\partial \mathfrak{x}^{\mathfrak{s}}} \frac{\partial \mathfrak{y}}{\partial \mathfrak{x}^{\mathfrak{t}}} \mathfrak{g}^{\mathfrak{t}\mathfrak{t}}\right)} \frac{\partial \mathfrak{y}}{\partial \mathfrak{x}^{\mathfrak{t}}} \mathfrak{g}^{\mathfrak{t}\mathfrak{t}} \frac{\partial \mathfrak{y}}{\partial \mathfrak{x}^{\mathfrak{q}}} \mathfrak{g}^{pq}\right) \frac{\partial^{2} \mathfrak{y}}{\partial \mathfrak{x}^{\mathfrak{t}} \partial \mathfrak{x}^{p}} = f(\mathfrak{g}, \partial \mathfrak{g}, \psi, \partial \psi). \tag{57}
$$

In Equations (56) and (57), *f<sup>i</sup>* and *f* denote suitable functions of *g*¯*ij*, *ψ*, and their first order derivatives only.

Now, Equations (54) and (57) form a second order quasi-diagonal system of partial differential equations for the unknowns *g*¯*ij* and *ψ*. The matrix of the principal parts of such a system is diagonal, and its elements are the differential operators:

$$\delta \bar{g}^{pq} \frac{\partial^2}{\partial x^p \partial x^q} \tag{58a}$$

and:

$$
\left(\mathfrak{F}^{ip} - \frac{2}{\mathfrak{p}} \frac{\partial \mathfrak{q}}{\partial \left(\frac{\partial \boldsymbol{\Psi}}{\partial \mathbf{x}^{\boldsymbol{\Psi}}} \frac{\partial \boldsymbol{\Psi}}{\partial \mathbf{x}^{\boldsymbol{\Psi}}} \mathfrak{g}^{\boldsymbol{\text{st}}}\right)} \frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{\text{x}}^{\boldsymbol{\Psi}}} \mathfrak{g}^{\boldsymbol{\text{ii}}} \frac{\partial \boldsymbol{\Psi}}{\partial \mathbf{x}^{\boldsymbol{\Psi}}} \mathfrak{g}^{pq}\right) \frac{\partial^{2}}{\partial \mathbf{x}^{\boldsymbol{\Psi}} \partial \mathbf{x}^{\boldsymbol{\Psi}}}.\tag{58b}
$$

The operator (58a) is the wave operator associated with the metric *g*¯*ij*, while the operator (58b) is very similar to the sound wave operator involved in the analysis of the Cauchy problem for GR coupled with an irrotational perfect fluid [23,46]. It follows that the Cauchy problem associated with the system of Equations (54) and (57) can be discussed borrowing arguments and results from [23,46]. More in detail, we recall that if the quadratic form associated with (58b) is of Lorentzian signature and, if the characteristic cone of the operator (58b) is exterior to the metric cone, the system (54) and (57) is causal and Leray hyperbolic [49,50]. Under these conditions, the associated Cauchy problem is well-posed in suitable Sobolev spaces. Still borrowing from [23,46], if the signature of *g*¯*ij* is (+ − −−), the above mentioned conditions are satisfied whenever the vector *∂ψ ∂x j g*¯ *ij* is timelike and the inequality:

$$-\frac{2}{\varrho} \frac{\partial \varrho}{\partial \left(\frac{\partial \Psi}{\partial \mathbf{x}^{\delta}} \frac{\partial \Psi}{\partial \mathbf{x}^{\delta}} \mathbf{g}^{st}\right)} \geq 0,\tag{59}$$

holds. Of course, when the signature of the metric *g*¯*ij* is (− + ++), the sign of inequality (59) has to be inverted. As it has been already remarked, the function (55) depends on the potential (14), which is determined by the explicit form of the function *f*(*R*). Therefore, we can adopt requirement (59) as a criterion to single out viable *f*(*R*)-models with torsion.

As an illustrative example, we consider again the model *f*(*R*) = *R* + *αR* 2 . From the relation *F* −1 (*X*) = *f* ′ (*X*)*X* − 2 *f*(*X*) = −*X*, the identity (*f* ′ ) −1 (*ϕ*) = *<sup>ϕ</sup>* <sup>−</sup> <sup>1</sup> 2*α* , and the expression (14), we easily obtain the effective potential:

$$V(\varphi) = \frac{1}{8\alpha}(\varphi - 1)^2 \varphi. \tag{60}$$

Equation (60), together with Equations (19) and (50), yields:

$$\varphi = \frac{\left(\frac{1}{2\alpha} + 2m^2 \psi^2\right)}{\left(\frac{1}{2\alpha} - \frac{\partial \psi}{\partial x^s} \frac{\partial \psi}{\partial x^t} \overline{\xi}^{st}\right)'}\,\,\,\tag{61}$$

which describes the scalar field *ϕ* as a function of the metric *g*¯*ij*, the Klein–Gordon field *ψ*, and its first order derivatives. By deriving (61), we have:

$$\frac{\partial \boldsymbol{\varrho}}{\partial \left(\frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{\chi^{e}}} \frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{\chi^{t}}} \boldsymbol{\mathcal{S}}^{\rm st}\right)} = \frac{\boldsymbol{\varrho}}{\left(\frac{1}{2a} - \frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{\chi^{e}}} \frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{\chi^{t}}} \boldsymbol{\mathcal{S}}^{\rm st}\right)}.\tag{62}$$

If the metric *g*¯*ij* has signature (+ − −−), we see that the requirement (59) is fulfilled if *α* < 0 and if *g*¯ *pq ∂ψ ∂x q* is a time-like vector field. On the contrary, when the signature of *g*¯*ij* is (− + ++), *α* has to be positive.

### **4. The Junction Conditions**

In this section, we address the junction conditions issue within the framework of *f*(*R*)-gravity with torsion. As mentioned in the Introduction, the junction condition problem is crucial for any theory of gravitation; for instance, in order to join together the interior with the exterior region of a relativistic star, we need to know how matching different solutions of the field equations of the theory at a given hypersurface. After deriving general junction conditions, in order to highlight the main differences with respect to ECSK theory, we give two illustrative examples. For reasons of greater clarity and better readability, the proposed examples are presented in two separate subsections.

Let us consider a hypersurface <sup>Σ</sup> which separates two different regions <sup>M</sup><sup>+</sup> and <sup>M</sup><sup>−</sup> of spacetime. To begin with, let us deal with the case in which the hypersurface Σ is either timelike or spacelike; the case of null hypersurface will be discussed later. Let us denote by *g* + *ij* , Γ + *h ij* and *g* − *ij* , Γ − *h ij* two solutions of the field equations (10), defined in <sup>M</sup><sup>+</sup> and <sup>M</sup>−, respectively. We want to discuss how to solder together at Σ the two given Einstein–Cartan geometries, in order to obtain a unique solution of the field equations on the whole spacetime.

To this end, we refer Σ to local coordinates *y <sup>A</sup>* (*A* = 1, . . . , 3), and we adopt a coordinate system *x i* , locally overlapping both <sup>M</sup><sup>+</sup> and <sup>M</sup><sup>−</sup> in an open set containing <sup>Σ</sup>. After that, considering the arc length *<sup>s</sup>* between any point *<sup>p</sup>* ∈ M and <sup>Σ</sup> along the geodesic normal to <sup>Σ</sup> (with respect to one of the two given metric tensors) and passing through *p* itself, we define a function *s* which, without loss of generality, can be set negative in <sup>M</sup>−, positive in <sup>M</sup>+, and equal to zero at <sup>Σ</sup>. Indicating by *<sup>n</sup> i* the unit normal (with respect to the chosen metric tensor) outgoing from Σ, one has the relations:

$$n^i d\mathbf{x}^i = n^i d\mathbf{s}, \qquad n\_{\mathbf{i}} = \epsilon \partial\_{\mathbf{i}} \mathbf{s} \qquad \text{and} \qquad n^i n\_{\mathbf{i}} = \epsilon,\tag{63}$$

where *<sup>ǫ</sup>* = 1 if <sup>Σ</sup> is spacelike, and *<sup>ǫ</sup>* = −1 if <sup>Σ</sup> is timelike. Moreover, given any geometric quantity *<sup>W</sup>* defined on both sides of the hypersurface Σ, we denote by:

$$\mathcal{E}\left[\mathcal{W}\right] := \mathcal{W}\left(\mathcal{M}^+\right)\_{|\Sigma} - \mathcal{W}\left(\mathcal{M}^-\right)\_{|\Sigma} \tag{64}$$

the jump of *W* across Σ. The issue of matching different geometries at a given hypersurface Σ is usually discussed in the framework of distribution-valued tensors [29,30,32,51,52]. In this regard, denoting by Θ(*s*) (with Θ(0) := 1) the Heaviside distribution, we introduce the following geometrical objects:

$$\mathfrak{g}\_{\dot{i}\dot{j}} = \Theta(\mathbf{s})\mathfrak{g}\_{\dot{i}\dot{j}}^{+} + \left(1 - \Theta(\mathbf{s})\right)\mathfrak{g}\_{\dot{i}\dot{j}}^{-} \tag{65a}$$

$$
\Gamma\_{\rm ij}{}^h = \Theta(s)\Gamma\_{\rm ij}{}^h + \left(1 - \Theta(s)\right)\Gamma\_{\rm ij}{}^h \,, \tag{65b}
$$

with the requirement that the quantities (65) define a solution of the field equations (10) in the distributional sense. To satisfy this request, the quantities (65) and all the the geometric quantities induced by them have to be well defined as distributions. In particular, this must apply to the Riemann and the Einstein tensors. Moreover, consistency between (65), (3) implies the identity:

$$
\Gamma\_{ij}{}^h = \Theta(\mathbf{s}) \left( \hat{\Gamma}\_{ij}{}^h - \mathcal{K}\_{ij}{}^h \right) + \left[ 1 - \Theta(\mathbf{s}) \right] \left( \hat{\Gamma}\_{ij}{}^h - \mathcal{K}\_{ij}{}^h \right), \tag{66}
$$

where Γ˜ *<sup>h</sup> ij* are the Christoffel coefficients associated with the metric (65a). By differentiating (65), we get the relations:

$$
\partial\_k \mathbf{g}\_{i\bar{j}} = \Theta(s) \partial\_k \mathbf{g}\_{i\bar{j}}^+ + (1 - \Theta(s)) \, \partial\_k \mathbf{g}\_{i\bar{j}}^- + \epsilon \delta(s) \left[ \mathbf{g}\_{i\bar{j}} \right] \, \mathbf{n}\_{k\nu} \tag{67a}
$$

$$
\partial\_k \Gamma\_{\dot{ij}}{}^h = \Theta(s) \partial\_k \Gamma\_{\dot{ij}}{}^h + (1 - \Theta(s)) \, \partial\_k \Gamma\_{\dot{ij}}{}^h + \epsilon \delta(s) \left[\Gamma\_{\dot{ij}}{}^h\right] n\_{k\prime} \tag{67b}
$$

where, referring the reader to [31,51,52] and references therein for the definition of the Dirac *δ*-function with support on the submanifold Σ : *s* = 0, we have used the identities *<sup>∂</sup><sup>s</sup> ∂x <sup>i</sup>* <sup>=</sup> *<sup>ǫ</sup>n<sup>i</sup>* and *<sup>d</sup>*Θ(*s*) *ds* = *δ*(*s*).

Making use of Equation (67), as well as of the identities Θ<sup>2</sup> (*s*) = <sup>Θ</sup>(*s*) and <sup>Θ</sup>(*s*)(<sup>1</sup> − <sup>Θ</sup>(*s*)) = 0, it is easily seen that the Levi–Civita contribution to the connection Γ *h ij* contains a singular term having expression:

$$\frac{1}{2}\mathcal{g}\_{|\Sigma}^{+\hbar k}\left(\left[g\_{ik}\right]n\_{\rangle}+\left[g\_{jk}\right]n\_{i}-\left[g\_{i\bar{j}}\right]n\_{k}\right)\epsilon\delta(s).\tag{68}$$

Requirement (65b) implies then the vanishing of the term (68); thus,

$$\begin{bmatrix} \mathbf{g}\_{ij} \end{bmatrix} = \mathbf{0},\tag{69}$$

amounting to the fact that the two metrics have to coincide on the hypersurface Σ. In addition, from Equation (67b), we get the expression of the Riemann tensor of the the connection (65b):

$$\boldsymbol{\mathcal{R}}\_{q\dot{i}j}^{p} = \boldsymbol{\Theta}(\boldsymbol{s})\boldsymbol{\mathcal{R}}\_{q\dot{i}j}^{+p} + \left(1 - \boldsymbol{\Theta}(\boldsymbol{s})\right)\boldsymbol{\mathcal{R}}\_{q\dot{i}j}^{-p} + \boldsymbol{\delta}(\boldsymbol{s})\boldsymbol{A}\_{q\dot{i}j'}^{p} \tag{70}$$

where we have denoted by:

$$A^{p}\_{\
u j\dot{j}} := \mathfrak{e}\left( \left[ \Gamma\_{jq}^{\ \ p} \right] n\_{\dot{i}} - \left[ \Gamma\_{i\dot{q}}^{\ \ p} \right] n\_{\dot{j}} \right) \tag{71}$$

the tensor connected with the presence of the *δ*-function term in the Riemann tensor (70). Once again, decomposition (3) can be used, so that we can rewrite the tensor (71) as the sum:

$$A^{p}\_{\
quig} = \tilde{A}^{p}\_{\
quig} + \tilde{A}^{p}\_{\
quig\prime} \tag{72}$$

where:

$$\boldsymbol{\tilde{A}}\_{\;\;\boldsymbol{q}ij}^{p} = \boldsymbol{\varepsilon} \left( \left[ \boldsymbol{\tilde{\Gamma}}\_{\;\;\boldsymbol{q}}^{\;\;\;p} \right] \boldsymbol{n}\_{\!\;i} - \left[ \boldsymbol{\tilde{\Gamma}}\_{\;\;\boldsymbol{q}}^{\;\;\;p} \right] \boldsymbol{n}\_{\!\;\boldsymbol{j}} \right) \tag{73}$$

and:

$$\boldsymbol{\bar{A}}\_{\
u \dot{\boldsymbol{q}} \dot{\boldsymbol{j}}}^{p} = \boldsymbol{\varepsilon} \left( \left[ -\boldsymbol{\mathcal{K}}\_{\dot{\boldsymbol{j}} \dot{\boldsymbol{q}}}^{p} \right] \boldsymbol{n}\_{\dot{\boldsymbol{i}}} + \left[ \boldsymbol{\mathcal{K}}\_{\dot{\boldsymbol{i}} \dot{\boldsymbol{q}}}^{p} \right] \boldsymbol{n}\_{\dot{\boldsymbol{j}}} \right) \tag{74}$$

are quantities related to the Levi–Civita and contortion, respectively.

The continuity of the metric tensor across the hypersurface Σ implies that its derivatives may have discontinuities only along the normal direction. Then, there exists a tensor field on Σ:

$$k\_{\rm ij} := \varepsilon \left[ \partial\_h g\_{\rm ij} \right] n^h,\tag{75}$$

such that:

$$\left[\partial\_{\hbar}g\_{i\bar{j}}\right] = k\_{i\bar{j}}n\_{\hbar}.\tag{76}$$

From Equation (76), we get the expressions:

$$\left[\left[\hat{\Gamma}\_{ij}\right]^h\right] = \frac{1}{2}\left(k^h{}\_j n\_i + k^h{}\_i n\_j - k\_{ij} n^h\right),\tag{77}$$

which, inserted into Equation (73), yield the explicit representation:

$$\tilde{A}^{p}\_{\ q\dot{q}j} = \frac{\epsilon}{2} \left( k^{p}{}\_{\ j} n\_{q} n\_{\dot{l}} - k^{p}{}\_{\ i} n\_{q} n\_{\dot{j}} - k\_{q\dot{l}} n^{p} n\_{\dot{l}} + k\_{q\dot{l}} n^{p} n\_{\dot{l}} \right) \,. \tag{78}$$

By contraction of Equation (78), we have:

$$\check{A}\_{q\bar{j}} := \check{A}\_{\;\;q\bar{p}\,\!j}^{p} = \frac{\varepsilon}{2} \left( k^{p}{}\_{\;\;j} n\_{q} n\_{p} - k n\_{q} n\_{\bar{j}} - k\_{q\bar{j}} \varepsilon + k\_{q\bar{p}} n^{p} n\_{\bar{j}} \right) \tag{79}$$

and:

$$\vec{A} := \tilde{A}^q{}\_q = \epsilon \left( k\_{pq} n^p n^q - \epsilon k \right), \tag{80}$$

with *k* := *kijg ij*. Making use of Equations (56), (80), we introduce the tensor:

$$\tilde{A}\_{q\dot{j}} = \tilde{A}\_{q\dot{j}} - \frac{1}{2}\tilde{A}\mathbf{g}\_{q\dot{j}} = \frac{\varepsilon}{2}\left(k\_{\dot{j}}^p n\_{\dot{q}} n\_p - k n\_{\dot{q}} n\_{\dot{j}} - k\_{q\dot{j}}\varepsilon + k\_{q\dot{p}} n^p n\_{\dot{j}}\right) - \frac{\varepsilon}{2}\left(k\_{\text{st}} n^s n^t - \varepsilon k\right)\mathbf{g}\_{q\dot{j}}\tag{81}$$

which represents the *δ*-function part of the Einstein tensor, generated by Levi–Civita connection. Tensor (81) is symmetric and tangent to the hypersurface Σ. In fact, it is a straightforward matter to verify that *H*˜ *qjn <sup>j</sup>* = 0. If we denote by *E i A* := *<sup>∂</sup><sup>x</sup> i ∂y A* , the tensor *H*˜ *qj* can be expressed as *H*˜ *qj* = *H*˜ *ABE q A E j B* , with [34]:

$$
\tilde{H}\_{AB} := \tilde{H}\_{q\bar{j}} \mathbb{E}\_A^q \mathbb{E}\_B^j = -\frac{1}{2} k\_{q\bar{j}} \mathbb{E}\_A^q \mathbb{E}\_B^j + \frac{1}{2} k\_{pq} h^{pq} h\_{AB\prime} \tag{82}
$$

where *h pq* := *g pq* <sup>−</sup> *<sup>ǫ</sup><sup>n</sup> pn <sup>q</sup>* and *hAB* := *gijE i A E j B* are the projection operator and the induced metric on the hypersurface Σ, respectively.

Analogously, we can single out the contributions given by contortion to the *δ*-function part of the Einstein tensor. By contraction, from Equation (74), we in fact:

$$\boldsymbol{\bar{A}}\_{q\boldsymbol{j}} := \boldsymbol{\bar{A}}\_{q\boldsymbol{p}\boldsymbol{j}}^{p} = \boldsymbol{\varepsilon} \left( \left[ -\boldsymbol{\mathcal{K}}\_{\boldsymbol{j}q}^{\boldsymbol{p}} \right] \boldsymbol{n}\_{p} + \left[ \boldsymbol{\mathcal{K}}\_{pq}^{\boldsymbol{p}} \right] \boldsymbol{n}\_{\boldsymbol{j}} \right) \tag{83}$$

and:

$$\vec{A} := \vec{A}^q{}\_q = \mathfrak{L}\mathfrak{e}\left[\boldsymbol{K}\_{pq}^{\;p}\right]n^q. \tag{84}$$

By means of expressions (83), (84), we define the tensor:

$$\bar{H}\_{q\dot{j}} := \bar{A}\_{q\dot{j}} - \frac{1}{2}\bar{A}\mathbf{g}\_{q\dot{j}} = \varepsilon \left( \left[ -\mathbf{K}\_{jq}^{~p} \right] n\_p + \left[ \mathbf{K}\_{pq}^{~p} \right] n\_{\dot{j}} \right) - \varepsilon \left[ \mathbf{K}\_{\rm st}^{~s} \right] n^t \mathbf{g}\_{q\dot{j}}.\tag{85}$$

which, in general, is neither symmetric nor tangent to the hypersurface Σ. All the obtained results allow us to express the effective stress–energy tensor appearing on the right-hand side of Equation (10a) in the form:

$$\hat{T}\_{\vec{q}\vec{j}} = \Theta(\mathbf{s}) \left[ \frac{1}{\varrho} \mathcal{T}\_{\vec{q}\vec{j}} + \frac{1}{\varrho} \mathcal{U}(\mathcal{T}) \mathcal{g}\_{\vec{q}\vec{j}} \right]^{+} + (1 - \Theta(\mathbf{s})) \left[ \frac{1}{\varrho} \mathcal{T}\_{\vec{q}\vec{j}} + \frac{1}{\varrho} \mathcal{U}(\mathcal{T}) \mathcal{g}\_{\vec{q}\vec{j}} \right]^{-} - \delta(\mathbf{s}) H\_{\vec{q}\vec{j}} \,\tag{86}$$

where:

$$H\_{q\mathbf{j}} = \tilde{H}\_{q\mathbf{j}} + \tilde{H}\_{q\mathbf{j}}.\tag{87}$$

and where, for simplicity, we have denoted by *<sup>U</sup>*(<sup>T</sup> ) :<sup>=</sup> <sup>1</sup> 2 [ *f*(*R*(T )) − *f* ′ (*R*(T ))*R*(T ))].

From Equation (86), it follows that the request that the Einstein-like equations (10a) have a smooth transition across the hypersurface Σ is then equivalent to require that the tensor *Hqj* vanishes at Σ. Therefore, the remaining junction conditions can be obtained by imposing the vanishing of all projections of the tensor *Hqj* on Σ. About this, we have:

• the completely orthogonal projection of *Hqj* on Σ is automatically zero:

$$
\hbar H\_{q\dot{j}} n^q n^{\dot{j}} = \hbar\_{q\dot{j}} n^q n^{\dot{j}} = -\varepsilon \left[ K\_{\dot{j}}^{\
q p} \right] n\_p n\_q n^{\dot{j}} = 0 \tag{88}
$$

because *H*˜ *qj* is tangent to Σ and the contorsion is antisymmetric in the last two indexes;

• the tangent–orthogonal projection of *Hqj* is:

$$
\hbar \, H\_{q\bar{j}} \mathcal{E}\_A^q n^{\bar{j}} = \tilde{H}\_{q\bar{j}} \mathcal{E}\_A^q n^{\bar{j}} = -\varepsilon \left[ \mathcal{K}\_{\bar{j}q}^{~\bar{p}} \right] n\_p \mathcal{E}\_A^q n^{\bar{j}} + \left[ \mathcal{K}\_{p\bar{q}}^{~\bar{p}} \right] \mathcal{E}\_A^q. \tag{89}
$$

According to [35], the quantity in Equation (89) results in the jump of trace of the projection on Σ of the contorsion tensor. In fact, it is easily seen that the identity:

$$\left[\boldsymbol{\mathbb{K}}\_{jq}\prescript{p}{}{\mathbf{h}}\_{l}^{j}\mathbf{h}\_{p}^{i}\boldsymbol{E}\_{A}^{q}\right] = \left[\prescript{p}{}{\mathbf{K}}\_{jq}^{p}\left(\prescript{j}{}{\mathbf{}}\_{p}^{j} - \epsilon n\_{p}n^{j}\right)\prescript{q}{}{\mathbf{E}}\_{A}^{q}\right] = -\boldsymbol{\varepsilon}\left[\prescript{p}{}{\mathbf{K}}\_{jq}^{p}\right]n\_{p}n^{j}\mathbf{E}\_{A}^{q} + \left[\prescript{}{\mathbf{K}}\_{pq}\prescript{p}{}{\mathbf{}}\_{A}^{q}\right]\mathbf{E}\_{A}^{q}\tag{90}$$

holds.

• the orthogonal–tangent projection of *Hqj* is zero:

$$\hbar H\_{q\bar{j}} \mathbf{E}\_A^j n^q = \hbar\_{q\bar{j}} \mathbf{E}\_A^j n^q = \mathbf{e} \left( - \left[ \mathbf{K}\_{jq}^{~p} \right] n\_p n^q \mathbf{E}\_A^j + \left[ \mathbf{K}\_{p\bar{q}}^{~p} \right] n\_{\bar{j}} \mathbf{E}\_A^j n^q \right) = \mathbf{0},\tag{91}$$

in view of the antisymmetry properties of the contorsion tensor and the orthogonality between the vectors *n <sup>i</sup>* and *E i A* ;

• the totally tangent projection of *Hqj* is given by:

$$\boldsymbol{H}\_{\boldsymbol{q}\boldsymbol{j}}\boldsymbol{E}\_{A}^{q}\boldsymbol{E}\_{B}^{j} = \boldsymbol{\tilde{H}}\_{\boldsymbol{q}\boldsymbol{j}}\boldsymbol{E}\_{A}^{q}\boldsymbol{E}\_{B}^{j} + \boldsymbol{\tilde{H}}\_{\boldsymbol{q}\boldsymbol{j}}\boldsymbol{E}\_{A}^{q}\boldsymbol{E}\_{B}^{j} = \boldsymbol{\tilde{H}}\_{AB} + \boldsymbol{\varepsilon}\left(-\left[\boldsymbol{\boldsymbol{K}}\_{\boldsymbol{j}\boldsymbol{q}}^{\boldsymbol{P}}\right]\boldsymbol{n}\_{p}\boldsymbol{E}\_{A}^{q}\boldsymbol{E}\_{B}^{j} + \left[\boldsymbol{\boldsymbol{K}}\_{\boldsymbol{q}}^{\boldsymbol{Q}\boldsymbol{P}}\right]\boldsymbol{n}\_{p}\boldsymbol{h}\_{AB}\right). \tag{92}$$

Summarizing everything, it is seen that the vanishing of the tensor *Hqj* needs the quantities (89), (92) to be zero at Σ. In particular, as it happens in GR, it can be shown that the condition *HqjE q A E j <sup>B</sup>* = 0 is connected with the vanishing of the jump of the extrinsic curvature across Σ. To clarify this point, let us introduce the quantity:

$$Q\_{AB} := \left(\nabla\_i n\_{\dot{j}}\right) E\_A^{\dot{j}} E\_{B\prime}^{\dot{i}} \tag{93}$$

which generalizes the notion of extrinsic curvature for an arbitrary linear connection (3). From Equation (93) together with Equations (3) and (77), we get the relation:

$$\mathbb{E}\left[Q\_{AB}\right] = \left[\nabla\_i n\_j \mathbb{E}\_A^j \mathbb{E}\_B^i\right] = \left[\nabla\_i n\_j\right] \mathbb{E}\_A^j \mathbb{E}\_B^i = \frac{\mathfrak{c}}{2} k\_{i\bar{j}} \mathbb{E}\_A^i \mathbb{E}\_B^j + \left[\mathbb{K}\_{\bar{j}i}{}^h\right] n\_h \mathbb{E}\_A^i \mathbb{E}\_B^j. \tag{94}$$

Comparing Equations (82), (92) and (94), it is then an easy matter to prove the identity:

$$H\_{AB} := H\_{\text{kj}} \mathbf{E}\_A^k \mathbf{E}\_B^j = -\epsilon \left( [\mathbf{Q}\_{AB}] - [\mathbf{Q}] \, h\_{AB} \right) \,. \tag{95}$$

where [*Q*] := [*QAB*] *h AB*. It follows that the requirements *HAB* = 0 and [*QAB*] = 0 at Σ are equivalent.

The request of vanishing of the quantities (89), (92) involves the Levi–Civita connection and the spin tensor (via the contorsion tensor) but also the trace of the energy–impulse tensor and its first derivatives. This is because of the torsional contributions given by the non-linearity of the function *f*(*R*) and it represents a significant difference from the ECSK theory. In order to better clarify this last aspect, in the next subsections, we illustrate two examples dealing with the spin fluid and the Dirac field.

Before doing this, for the sake of completeness, we briefly outline also the case of null hypersurface. Then, let Σ be a null hypersurface described by an equation Φ(*x i* ) = 0, where Φ is a smooth function. We suppose that <sup>M</sup><sup>+</sup> and <sup>M</sup><sup>−</sup> correspond to the domains where <sup>Φ</sup> is positive and negative, respectively. Again, we discuss the matching on Σ of two solutions of the field equations in the form:

$$\mathcal{g}\_{\dot{i}\dot{j}} = \Theta(\Phi)\mathcal{g}\_{\dot{i}\dot{j}}^{+} + (1 - \Theta(\Phi))\mathcal{g}\_{\dot{i}\dot{j}}^{-} \tag{96a}$$

$$
\Gamma\_{\vec{ij}}{}^{\hbar} = \Theta(\Phi)\Gamma\_{\vec{ij}}{}^{\hbar} + \left(1 - \Theta(\Phi)\right)\Gamma\_{\vec{ij}}{}^{\hbar}.\tag{96b}
$$

The null normal vector is defined as *n<sup>i</sup>* = *α* <sup>−</sup>1*∂i*Φ, where *α* is a suitable non–zero function on Σ. By means of analogous arguments to those given above, it is immediately seen that the metric tensor (96a) has to be continuous across the hypersurface Σ, namely [*gij*] = 0. Following a usual procedure, let us then introduce a transverse vector field *N<sup>i</sup>* satisfying the requirements *Nin<sup>i</sup>* = 1 and *NiN<sup>i</sup>* = 0. We have the relations [*n i* ] = [*N<sup>i</sup>* ] = 0. We also introduce the transverse metric:

$$h\_{\rm ij} = g\_{\rm ij} - n\_{\rm i} \mathbf{N}\_{\rm j} - n\_{\rm j} \mathbf{N}\_{\rm i}. \tag{97}$$

Due to the continuity of the metric tensor across Σ, its derivatives may have discontinuities only along the transverse direction. This implies the existence of a tensor field *γij* on Σ, such that:

$$\gamma\_{\rm ij} = [\partial\_{\rm s} g\_{\rm ij}] \mathcal{N}^s \qquad \Longleftrightarrow \qquad [\partial\_{\rm s} g\_{\rm ij}] = \gamma\_{\rm ij} \eta\_{\rm s}. \tag{98}$$

By Equation (98), we can express the jump of the Christoffel symbols as:

$$\mathbb{E}\left[\widetilde{\Gamma}\_{ij}{}^h\right] = \frac{1}{2}\left(\gamma^h{}\_j n\_i + \gamma^h{}\_i n\_j - \gamma\_{i\bar{j}} n^h\right). \tag{99}$$

Making use of Equation (99) and following the identical procedure illustrated above, it is easily seen that the *δ*-function part of the Einsein tensor is now given by the sum:

$$H^{i\bar{j}} = \tilde{H}^{i\bar{j}} + \tilde{H}^{i\bar{j}},\tag{100}$$

where:

$$\tilde{H}^{ij} := \frac{\alpha}{2} \left( \gamma\_h^i n^h n^j + \gamma\_h^j n^h n^i - \gamma\_h^h n^i n^j - \gamma\_{hk} n^h n^k g^{ij} \right) \tag{101}$$

represents the contribution due to Levi–Civita terms, and:

$$H^{ij} := \mathfrak{a}\left(-\left[\mathcal{K}^{jih}\right]n\_h + \left[\mathcal{K}\_h^{ih}\right]n^j + \left[\mathcal{K}\_h^{hk}\right]n\_k\mathcal{g}^{ij}\right) \tag{102}$$

represents the contribution given by contorsion terms. As in the case of spacelike or timelike hypersurfaces, smooth transition across the hull hypersurface Σ at the level of Einstein-like equations requires the vanishing of the tensor (100).
