Junction conditions in perfect fluid $f(\mathcal{G},~{}T)$ gravitational theory

The action that describes the $f(\mathcal{G},~{}T)$ gravity can be obtained via a generalization of the Einstein-Hilbert action minimally coupled with an ordinary matter Lagrangian $\mathcal{L}_{\mathrm{m}}$ as

$$S=\frac{1}{2\kappa^{2}}\int_{\Omega}\sqrt{-g}\bigg{[}R+f\big{(}\mathcal{G},T\big{)}\bigg{]}\mathrm{d}^{4}x+S_{\mathrm{m}},$$

(1)

where $\Omega$ is a 4-dimensional space-time manifold on which a set of coordinates $x^{i}$ is defined, $S_{\mathrm{m}}=\int_{\Omega}\mathcal{L}_{\mathrm{m}}\sqrt{-g}d^{4}x$ is the matter action written in terms of a matter Lagrangian density $\mathcal{L}_{\mathrm{m}}$, $g$ is the determinant of the metric $g_{ij}$ written in terms of $x^{i}$ and with a signature $\left(+,-,-,-\right)$, $\kappa^{2}=8\pi G/c^{4}$ where $G$ is the gravitational constant and $c$ is the speed of light, and $f\left(\mathcal{G},T\right)$ is an arbitrary function of the Gauss-Bonnet invariant $\mathcal{G}=R^{2}-4R_{ij}R^{ij}+R_{ijpq}R^{ijpq}$, in this mathematical function $R=R^{i}_{i}$ is the Ricci scalar which is the trace of $R_{ij}$, as well as $R_{ij}$ and $R_{ijpq}$ denote Ricci, and the Riemann tensors, respectively. And the trace $T$ of the stress-energy tensor $T_{ij}$, the latter is defined as a modification of the matter Lagrangian w.r.t. the $g_{ij}$ as

$$T_{ij}=-\frac{2}{\sqrt{-g}}\left(\frac{\delta\left(\sqrt{-g}\mathcal{L}_{\mathrm{m}}\right)}{\delta g^{ij}}\right)=g_{ij}\mathcal{L}_{\mathrm{m}}-\frac{2\partial\mathcal{L}_{\mathrm{m}}}{\partial g^{ij}}.$$

(2)

Under the assumption that $T_{ij}$ depends solely on components of the metric $g_{ij}$ but not on its derivatives. The MFEs of $f(\mathcal{G},~{}T)$ gravity may be derived by varying Eq.(1) w.r.t. $g_{ij}$, from which one obtains

		$\displaystyle R_{ij}-\frac{1}{2}g_{ij}R=\kappa^{2}T_{ij}+\frac{1}{2}g_{ij}f+\mathcal{G}_{ij}f_{\mathcal{G}}-\left(T_{ij}+\mathbf{\Theta}_{ij}\right)f_{T}$
		$\displaystyle-4\left(R_{ipjq}\nabla^{p}\nabla^{q}-\frac{R}{2}\nabla_{i}\nabla_{j}-g_{ij}R^{pq}\nabla_{p}\nabla_{q}+R_{i}^{p}\nabla_{j}\nabla_{p}-R_{ij}\nabla^{2}\right.\left.+\frac{R}{2}g_{ij}\nabla^{2}+R_{j}^{p}\nabla_{i}\nabla_{p}\right)f_{\mathcal{G}},$		(3)

here we introduced the notation $f_{\mathcal{G}}\equiv\partial f/\partial\mathcal{G}$ and $f_{T}=\partial f/\partial T$ for the partial derivatives of the function $f\left(\mathcal{G},T\right)$, $\nabla^{2}\equiv g^{ij}\nabla_{i}\nabla_{j}$ for the d’Alembert operator, where $\nabla_{i}$ are the 4-dimensional covariant derivatives expressed in the form of $g_{ij}$, the tensor $\mathcal{G}_{ij}$ is defined as

$\displaystyle\mathcal{G}_{ij}\equiv-RR_{ij}+2R_{i}^{p}R_{pj}+2R_{ipjq}R^{pq}-R_{i}^{pq\delta}R_{jpq\delta},$

(4)

and we have defined the auxiliary tensor $\mathbf{\Theta}_{ij}$ in the form of variation of $T_{ij}$ w.r.t. $g_{ij}$ as

$$\mathbf{\Theta}_{ij}=g^{pq}\frac{\delta T_{pq}}{\delta g^{ij}}.$$

(5)

We will consider that the matter sector can be well characterized by a relativistic perfect fluid, i.e. $T_{ij}$ may be expressed as

$\displaystyle T_{ij}=(\rho+P)u_{i}u_{j}-Pg_{ij},$

(6)

where $\rho$ and $P$ are the energy density and the isotropic surface pressure, respectively, and $u^{i}$ is the fluid 4-velocity vector. The normalization property is satisfied by the vector $u^{i}$ as $g_{ij}u^{i}u^{j}=1$, from which one obtains a matter Lagrangian density of the form $\mathcal{L}_{\mathrm{m}}=-P$. Consequently, Eq.(5) yields

$\displaystyle\mathbf{\Theta}_{ij}=-2T_{ij}-Pg_{ij}.$

(7)

Introducing Eq.(7) in Eq.(3), one obtains the MFEs for the isotropic perfect-fluid configuration as

		$\displaystyle R_{ij}-\frac{R}{2}g_{ij}=\big{(}f_{T}+\kappa^{2}\big{)}T_{ij}+\mathcal{G}_{ij}f_{\mathcal{G}}+g_{ij}Pf_{T}+\frac{f}{2}g_{ij}$
		$\displaystyle-4\left(R_{ipjq}\nabla^{p}\nabla^{q}-\frac{R}{2}\nabla_{i}\nabla_{j}-g_{ij}R^{pq}\nabla_{p}\nabla_{q}+R_{i}^{p}\nabla_{j}\nabla_{p}-R_{ij}\nabla^{2}\right.\left.+\frac{R}{2}g_{ij}\nabla^{2}+R_{j}^{p}\nabla_{i}\nabla_{p}\right)f_{\mathcal{G}}.$		(8)

The differential terms on $f_{\mathcal{G}}$ may be expanded in the form of derivatives of $\mathcal{G}$ as well as $T$ with the use of the chain rule, as follows

$\displaystyle\partial_{i}f_{\mathcal{G}}=f_{\mathcal{G}\mathcal{G}}\partial_{i}\mathcal{G}+f_{\mathcal{G}T}\partial_{i}T,$

(9)

$\displaystyle\nabla_{i}\nabla_{j}f_{\mathcal{G}}=f_{\mathcal{G}\mathcal{G}}\nabla_{i}\nabla_{j}\mathcal{G}+f_{\mathcal{G}T}\nabla_{i}\nabla_{j}T+f_{\mathcal{G}TT}\partial_{i}T\partial_{j}T+f_{\mathcal{G}\mathcal{G}\mathcal{G}}\partial_{i}\mathcal{G}\partial_{j}\mathcal{G}+2f_{\mathcal{G}\mathcal{G}T}\partial_{(i}\mathcal{G}\partial_{j)}T.$

(10)

The fully extended field equations can now be conceived by plugging the expansions of Eqs.(9) and (10) into Eq.(8). However, we chose not to write the result explicitly because of its extensive nature.

In the framework of modified gravities featuring extra scalar degrees of freedom in comparison to GR, it is frequently effective to recast the geometrical representation of Eq.(1) into a dynamically equivalent scalar-tensor representation in which the additional scalar degrees of freedom are exchanged by scalar fields. In the scenario of $f\left(\mathcal{G},T\right)$ gravity, the theory has two additional scalar degrees of freedom, representing the function $f$ which shows two arbitrary dependencies in the quantities $\mathcal{G}$ and $T$, and thus the scalar-tensor demonstration of this gravity can be conceived via the introduction of two scalar fields, say $\Phi$ and $\Psi$, as well as an interaction potential $\mathbb{V}\left(\Phi,\Psi\right)$. Let us start this transformation by introducing two auxiliary fields $\alpha$ and $\beta$ into Eq.(1) which yield

$\displaystyle S=\frac{1}{2\kappa^{2}}\int_{\Omega}\sqrt{-g}\left\{R+f(\alpha,\beta)+\frac{\partial f}{\partial\alpha}\big{(}\mathcal{G}-\alpha\big{)}+\frac{\partial f}{\partial\beta}\big{(}T-\beta\big{)}\right\}d^{4}x+S_{\mathrm{m}}.$

(11)

The action in expression Eq.(11) is explicitly reliant on three independent physical quantities: $g_{ij}$ as well as $\alpha$ and $\beta$. Consequently, the equations of motion for the auxiliary fields may be found by varying Eq.(11) w.r.t. these fields, respectively, which yields

$$\frac{\partial^{2}f}{\partial\alpha^{2}}\bigg{(}\mathcal{G}-\alpha\bigg{)}+\frac{\partial^{2}f}{\partial\alpha\partial\beta}\bigg{(}T-\beta\bigg{)}=0,$$

(12)

$$\frac{\partial^{2}f}{\partial\alpha\partial\beta}\bigg{(}\mathcal{G}-\alpha\bigg{)}+\frac{\partial^{2}f}{\partial\beta^{2}}\bigg{(}T-\beta\bigg{)}=0,$$

(13)

where the function $f\left(\alpha,\beta\right)$ is assumed to satisfy the Schwartz theorem, i.e., its crossed partial derivatives are commutative. Equations (12) and (13) can be recast into a matrix form $\mathcal{A}\textbf{x}=0$ as

$$\mathcal{A}\textbf{x}=\begin{pmatrix}\frac{\partial^{2}f}{\partial\alpha^{2}}&\frac{\partial^{2}f}{\partial\alpha\partial\beta}\\ \frac{\partial^{2}f}{\partial\alpha\partial\beta}&\frac{\partial^{2}f}{\partial\beta^{2}}\end{pmatrix}\begin{pmatrix}\mathcal{G}-\alpha\\ T-\beta\end{pmatrix}=0.$$

(14)

The solution of the matrix system in Eq.(14) will be unique if and only if $\mathcal{A}$ has non-zero determinant, i.e., if the following condition is satisfied:

$$\frac{\partial^{2}f}{\partial\alpha^{2}}\frac{\partial^{2}f}{\partial\beta^{2}}\neq\left(\frac{\partial^{2}f}{\partial\alpha\partial\beta}\right)^{2}.$$

(15)

If above mentioned Eq.(15) is fulfilled, then the unique solution of established Eq.(14) is $\alpha=\mathcal{G}$ and $\beta=T$. When this solution is reintroduced into Eq.(11), then Eq.(1) is recovered, confirming the equivalence of the two representations of the theory. Now scalar fields $\Phi$ and $\Psi$, as well as the interaction potential $\mathbb{V}\left(\Phi,\Psi\right)$, may now be defined as follows

		$\displaystyle\Phi=f_{\mathcal{G}}\equiv\partial f/\partial\mathcal{G},\quad\Psi=f_{T}\equiv\partial f/\partial T,$		(16)
		$\displaystyle\mathbb{V}(\Phi,~{}\Psi)=\alpha\Phi+\beta\Psi-f(\alpha,\beta),$		(17)

which upon replacement into the action in Eq.(11) yields the modified Einstein Hilbert action of considered representation of the $f\left(\mathcal{G},T\right)$ gravity in the form

$\displaystyle S$

$\displaystyle=\frac{1}{2}\int_{\Omega}\frac{\sqrt{-g}}{\kappa^{2}}\left\{R+\Phi\mathcal{G}+\Psi T-\mathbb{V}\big{(}\Phi,\Psi\big{)}\right\}d^{4}x+S_{\mathrm{m}}.$

(18)

Since then, Eq.(18) is now explicitly reliant on three independent quantities: the metric $g_{ij}$, as well as $\Phi$ and $\Psi$. By varying Eq.(18) w.r.t. the metric $g_{ij}$, one may get the MFEs in the scalar-tensor demonstration, whereas the equations of motion for the scalar fields are conceived via the variation w.r.t. $\Phi$ and $\Psi$. These equations take the forms

		$\displaystyle R_{ij}-\frac{1}{2}(R+\Phi\mathcal{G}+\Psi T-\mathbb{V})g_{ij}+\Phi\mathcal{G}_{ij}+4\left(R_{ipjq}\nabla^{p}\nabla^{q}-\frac{R}{2}\nabla_{i}\nabla_{j}-g_{ij}R^{pq}\nabla_{p}\nabla_{q}+R_{i}^{p}\nabla_{j}\nabla_{p}\right.$
		$\displaystyle\left.-R_{ij}\nabla^{2}+\frac{R}{2}g_{ij}\nabla^{2}+R_{j}^{p}\nabla_{i}\nabla_{p}\right)\Phi=\kappa^{2}T_{ij}-\left(T_{ij}+\Theta_{ij}\right)\Psi,$		(19)

		$\displaystyle\mathbb{V}_{\Phi}=\mathcal{G},$		(20)
		$\displaystyle\mathbb{V}_{\Psi}=T,$		(21)

respectively, where we have introduced the notation $\mathbb{V}_{\Phi}\equiv\frac{\partial\mathbb{V}}{\partial\Phi}$ and $\mathbb{V}_{\Psi}\equiv\frac{\partial\mathbb{V}}{\partial\Psi}$ for the partial derivatives of interaction potential. It should be noted that Eq.(19) may be produced directly from Eq.(3) by introducing the definitions in expressions Eqs.(16) and (17) and the use of the following geometrical identity valid only in four dimensions

$$RR_{ij}-2R_{i}^{p}R_{pj}-2R_{ipjq}R^{pq}+R_{i}^{\ pqr}R_{jpqr}=\mathcal{G}_{ij}.$$

Following the same procedure as for the geometrical representation, i.e., assuming that the matter sector is characterized by an isotropic perfect fluid, which implies that the stress-energy tensor is Eq.(6), which implies that the matter Lagrangian is $\mathcal{L}_{\mathrm{m}}=-P$ as well as the auxiliary tensor $\mathbf{\Theta}_{ij}$ is given in the form of Eq.(7), Eq.(19) can be recast in a more convenient manner

		$\displaystyle R_{ij}-\frac{1}{2}(R+\Phi\mathcal{G}+\Psi T-\mathbb{V})g_{ij}+\Phi\mathcal{G}_{ij}+4\left(R_{ipjq}\nabla^{p}\nabla^{q}-\frac{R}{2}\nabla_{i}\nabla_{j}-g_{ij}R^{pq}\nabla_{p}\nabla_{q}+R_{i}^{p}\nabla_{j}\nabla_{p}\right.$
		$\displaystyle\left.-R_{ij}\nabla^{2}+\frac{R}{2}g_{ij}\nabla^{2}+R_{j}^{p}\nabla_{i}\nabla_{p}\right)\Phi=\left(\kappa^{2}+\Psi\right)T_{ij}+Pg_{ij}\Psi.$		(22)

Take the space-time to be given by a manifold $(\mathbf{M},g_{ij})$ which can be divided into two distinct regions, a region $\mathbf{M}^{+}$ described by a metric $g_{ij}^{+}$ written in terms of a coordinate system $x^{i}_{+}$, and a region $\mathbf{M}^{-}$ described by a metric $g_{ij}^{-}$ written in terms of a coordinate system $x^{i}_{-}$. These two regions are separated by a hyper-surface $\Sigma$, on both sides of which we specify a coordinate system $y^{\mu}$, the direction perpendicular to $\Sigma$ is excluded from Greek indices. The projection vectors from the four-dimensional manifold $\mathbf{M}$ to the three-dimensional hyper-surface $\Sigma$ may then be expressed as follows: $e^{i}_{\mu}=\partial x^{i}/\partial y^{\mu}$, and we define the normal unit vector orthogonal to the hyper-surface and pointing from the region $\mathbf{M}^{-}$ to the region $\mathbf{M}^{+}$ as $n^{i}$, which implies that the result $e^{i}_{\mu}n_{i}=0$ holds. Schematic construction of our considered scenario can be found in Figure 1. The displacement from $\Sigma$ along the geodesic congruence generated by the normal vector $n^{i}$ is parameterized by an affine parameter $l$, i.e., one can write

$$dx^{i}=n^{i}dl,\qquad n_{i}=\varepsilon\partial_{i}l,$$

(23)

where $\varepsilon\equiv n^{i}n_{i}$ can be either $1$ or $-1$ depending on the normal vector being spacelike or timelike, respectively. Furthermore, we chose the affine parameter $l$ in such a way to guarantee that $l>0$ in the region $\mathbf{M}^{+}$, $l<0$ in the region $\mathbf{M}^{-}$, and $l=0$ at the hyper-surface $\Sigma$.

A suitable mathematical approach to deal with these situations where the space-time can be divided into different regions is the distribution formalism. In this formalism, any regular quantity $Y$ defined in the entire space-time manifold $\mathbf{M}$ may be expressed in terms of the Heaviside distribution function $\mathbf{\Theta}(l)$ i.e.,

$\displaystyle Y=Y^{+}\mathbf{\Theta}(l)+Y^{-}\mathbf{\Theta}(-l),$

(24)

where $Y^{\pm}$ denote the forms of the quantity $Y$ in the regions $\mathbf{M}^{\pm}$, respectively. The Heaviside function is thus defined as $0$ in the region $l<0$, as $1$ in the region $l>0$, and as $\frac{1}{2}$ at the point $l=0$, which corresponds to $\Sigma$. We noted in this formulism the derivative of the Heaviside function and the following properties:

$$\mathbf{\Theta}^{2}(l)=\mathbf{\Theta}(l),~{}~{}\frac{d}{dl}\mathbf{\Theta}(l)=\delta\left(l\right),~{}~{}\mathbf{\Theta}(l)\mathbf{\Theta}(-l)=0,$$

(25)

where $\delta\left(l\right)$ is the Dirac-delta distribution function [57]. We also noted that the product $\mathbf{\Theta}(l)\delta\left(l\right)$ is not defined as a distribution. Furthermore, the jump of the quantity $Y$ across the hyper-surface $\Sigma$ as well as the value of the quantity $Y$ at the hyper-surface $\Sigma$ are defined as

$$[Y]=Y^{+}|_{\Sigma}-Y^{-}|_{\Sigma}.$$

(26)

$$Y^{\Sigma}=\frac{1}{2}\left(Y^{+}|_{\Sigma}+Y^{-}|_{\Sigma}\right).$$

(27)

If a quantity $Y$ is continuous, then its jump satisfies $\left[Y\right]=0$ and one obtains $Y^{+}|_{\Sigma}=Y^{-}|_{\Sigma}=Y^{\Sigma}$. Thus, by construction, the normal vector $n^{i}$ and the projection vectors $e^{i}_{\mu}$ satisfy the property $\left[n^{i}\right]=\left[e_{\mu}^{i}\right]=0$.

We now have introduced all the necessary tools for the analysis that follows. Let us now focus on the general scenario for which a thin-shell is allowed to emerge at the separation hyper-surface $\Sigma$ and use the distribution formalism described above to deduce the corresponding JCs of the theory. For the line element to be fully specified on the entire space-time manifold $\mathbf{M}$ and, in particular, on both sides of the hyper-surface $\Sigma$, in the distribution formalism, we express $g_{ij}$ as

$\displaystyle g_{ij}=g_{ij}^{+}\mathbf{\Theta}(l)+g_{ij}^{-}\mathbf{\Theta}(-l).$

(28)

The partial derivatives of $g_{ij}$ are as follows:

$$\partial_{w}g_{ij}=\partial_{w}g_{ij}^{+}\mathbf{\Theta}(l)+\partial_{w}g_{ij}^{-}\mathbf{\Theta}(-l)+\varepsilon\left[g_{ij}\right]n_{w}\delta(l).$$

In the derivatives of the metric, there appears to be a component proportional to $\delta(l)$, which is problematic. Indeed, when one tries to establish the Christoffel symbols $\Gamma^{p}_{ij}$ related to the metric $g_{ij}$, these terms result in the occurrence of products of the type $\mathbf{\Theta}(l)\delta(l)$ in $\Gamma^{p}_{ij}$, which will result in terms proportional to $\delta\left(l\right)$ at $\Sigma$ since $\mathbf{\Theta}(0)=\frac{1}{2}$. These terms are problematic when one defines the Riemann tensor $R^{i}_{\ jpq}$, as it depends on products of Christoffel symbols and thus it depends on terms of the type $\delta^{2}(l)$, that is singular in considered distribution formalism. To avoid the existence of these highly problematic products, the continuity of the metric $g_{ij}$ must be imposed i.e.,

$\displaystyle\left[g_{ij}\right]=0.$

(29)

This expression applicable to the coordinate system $x^{i}$ only. Furthermore, we can simply convert this into a coordinate invariant statement: $0=[g_{ij}]e_{\mu}^{i}e_{\nu}^{j}=[g_{ij}e_{\mu}^{i}e_{\nu}^{j}];$ this last step comes as a result of the property $\left[n^{i}\right]=\left[e_{\mu}^{i}\right]=0$. The metric intrinsic to the hyper-surface $\Sigma$ is produced by confining the line element to displacements restricted to the hyper-surface. Consequently the parametric equations $x^{i}=x^{i}\left(y^{\mu}\right)$, we have the vectors $e^{i}_{\mu}=\frac{\partial x^{i}}{\partial y^{\mu}}$ are tangent to curves contained in $\Sigma$. Now, for displacements within separation hyper-surface $\Sigma$ we have

	$\displaystyle\mathrm{d}s^{2}$	$\displaystyle=g_{ij}\mathrm{d}x^{i}\mathrm{~{}d}x^{j}$
		$\displaystyle=g_{ij}\left(\frac{\partial x^{i}}{\partial y^{\mu}}\mathrm{~{}d}y^{\mu}\right)\left(\frac{\partial x^{i}}{\partial y^{\nu}}\mathrm{~{}d}y^{\nu}\right)$
		$\displaystyle=\mathbf{h}_{\mu\nu}\mathrm{~{}d}y^{\mu}\mathrm{~{}d}y^{\nu}$

where $\mathbf{h}_{\mu\nu}=g_{ij}e_{\mu}^{i}e_{\nu}^{j}$ is $1st$ fundamental form, of the hyper-surface $\Sigma$. We will refer to such objects as three-tensors. The argument above can be extended to the induced metric $\mathbf{h}_{\mu\nu}=g_{ij}e^{i}_{\mu}e^{j}_{\nu}$ at the hyper-surface $\Sigma$. The induced metric from the exterior can be written as $\mathbf{h}_{\mu\nu}^{+}=g_{ij}^{+}e_{\mu}^{i}e_{\nu}^{j}$, whereas the induced metric from the interior is $\mathbf{h}_{\mu\nu}^{-}=g_{ij}^{-}e_{\mu}^{i}e_{\nu}^{j}$. Since we have established that $\left[g_{ij}\right]=0$, this implies that $\mathbf{h}_{\mu\nu}^{+}-\mathbf{h}_{\mu\nu}^{-}=0$ and we obtain the $1st$ JC of the theory in the form

$\displaystyle\left[\mathbf{h}_{\mu\nu}\right]=0.$

(30)

The JC obtained in Eq.(30) corresponds also to the $1st$ JC of GR, and it is true for a wide range of metric theories of gravity, including the $f(\mathcal{G},T)$ we study in this work. Following Eq.(29), the partial derivatives of $g_{ij}$ take the regular form

$\displaystyle\partial_{w}g_{ij}=\partial_{w}g_{ij}^{+}\mathbf{\Theta}(l)+\partial_{w}g_{ij}^{-}\mathbf{\Theta}(-l),$

(31)

and the Christoffel symbols are well-defined in the distribution formalism. One can now proceed to construct the remaining geometrical quantities in the distribution formalism, namely the Riemann tensor $R_{\ pij}^{q}=\partial_{i}\Gamma_{pj}^{q}-\partial_{j}\Gamma_{pi}^{q}+\Gamma_{ri}^{q}\Gamma_{pi}^{r}-\Gamma_{rj}^{q}\Gamma_{pi}^{r}$, the Ricci tensor $R_{ij}=R^{p}_{\ ipj}$, and the Ricci scalar $R=R^{i}_{i}$. These quantities take the forms

$$R_{ijpq}=R^{+}_{ijpq}\mathbf{\Theta}(l)+R^{-}_{ijpq}\mathbf{\Theta}(-l)+\bar{R}_{ijpq}\delta(l),$$

(32)

$$R_{ij}=R^{+}_{ij}\mathbf{\Theta}(l)+R^{-}_{ij}\mathbf{\Theta}(-l)+\bar{R}_{ij}\delta(l),$$

(33)

$$R=R^{+}\mathbf{\Theta}(l)+R^{-}\mathbf{\Theta}(-l)+\bar{R}\delta(l),$$

(34)

where the quantities $\bar{R}_{ijpq}$, $\bar{R}_{ij}$ and $\bar{R}$ are given by

$$\bar{R}_{ijpq}=\left[\mathcal{K}_{\mu\nu}\right]\left(e^{\mu}_{i}e^{\nu}_{q}n_{j}n_{p}-e^{\mu}_{i}e^{\nu}_{p}n_{j}n_{q}-e^{\mu}_{j}e^{\nu}_{q}n_{i}n_{p}+e^{\mu}_{j}e^{\nu}_{p}n_{i}n_{q}\right),$$

(35)

$$\bar{R}_{ij}=-\left(\varepsilon\left[\mathcal{K}_{\mu\nu}\right]e^{\mu}_{i}e^{\nu}_{j}+n_{i}n_{j}\left[\mathcal{K}\right]\right),$$

(36)

$$\bar{R}=-2\varepsilon\left[\mathcal{K}\right],$$

(37)

here $\mathcal{K}_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\nabla_{i}n_{j}$ indicates the extrinsic curvature of $\Sigma$, also known as the second fundamental form, and $\mathcal{K}=\mathcal{K}_{\mu}^{\mu}$ is its trace. From the expressions above it is clear that the singular parts of the Riemann and the Ricci tensors will disappear if and only if the $\left[\mathcal{K}_{\mu\nu}\right]=0$, i.e., the jump of the extrinsic curvature disappears, and the singular portion of the Ricci scalar will vanish if and only if the jump of the trace of the extrinsic curvature disappears, i.e., $\left[\mathcal{K}\right]=0$.

To proceed with our analysis, we also need to find the explicit form of the Gauss-Bonnet invariant $\mathcal{G}=R^{2}-4R^{ij}R_{ij}+R^{ijpq}R_{ijpq}$ in the distribution formalism. Using the results obtained in Eqs.(32) to (34) into the expression for $\mathcal{G}$, one expects the Gauss-Bonnet invariant to take a form of the type:

$$\mathcal{G}=\mathcal{G}^{+}\mathbf{\Theta}(l)+\mathcal{G}^{-}\mathbf{\Theta}(-l)+\bar{\mathcal{G}}\delta\left(l\right)+\hat{\mathcal{G}}\delta^{2}\left(l\right),$$

(38)

because the terms corresponding to $\delta\left(l\right)$ are present in $R_{ijpq}$, $R_{ij}$ and $R$. However, this calculation shows that the term proportional to $\delta\left(l\right)$, i.e.,

$$\hat{\mathcal{G}}=\bar{R}^{2}-4\bar{R}_{ij}\bar{R}^{ij}+\bar{R}_{ijpq}\bar{R}^{ijpq},$$

vanishes identically. Using this fact one can thus write the Gauss-Bonnet invariant in the distribution formalism as

$$\mathcal{G}=\mathcal{G}^{+}\mathbf{\Theta}(l)+\mathcal{G}^{-}\mathbf{\Theta}(-l)+\bar{\mathcal{G}}\delta\left(l\right),$$

(39)

here $\bar{\mathcal{G}}$ can be expressed in the form of the geometrical quantities given previously as

$$\bar{\mathcal{G}}=R^{\Sigma}\bar{R}-4R_{ij}^{\Sigma}\bar{R}^{ij}+R^{\Sigma}_{ijpq}\bar{R}^{ijpq}.$$

(40)

Although the term $\bar{\mathcal{G}}$ is not problematic by itself, one must note that we are working in the $f\left(\mathcal{G},~{}T\right)$ gravitational theory. This shows arbitrary dependence of the function $f$ in $\mathcal{G}$, one can expect that this function admits a Taylor series expansion in $\mathcal{G}$, which will feature power-laws of $\mathcal{G}$, thus giving rise to singular terms of the form $\delta^{2}\left(l\right)$. To prevent the occurrence of these terms, one must impose $\bar{\mathcal{G}}=0$. According to Eqs.(35) to (37), one verifies that for $\bar{\mathcal{G}}$ to vanish it is necessary to impose that $\left[\mathcal{K}_{\mu\nu}\right]=0$. One thus obtains the second JC in the form

$$\left[\mathcal{K}_{\mu\nu}\right]=0.$$

(41)

This is a highly restrictive JC that is not present in GR or any of the other previously extensions of GR except for the particular case of smoothly match, i.e., matching without thin-shells, and it is linked to an arbitrary dependence of the action in $\mathcal{G}$. This condition implies that all the quantities $\bar{R}_{ijpq}$, $\bar{R}_{ij}$ and $\bar{R}$ vanish, consequently all the terms proportional to $\delta\left(l\right)$ in Eqs.(32) to (34) vanish, and these geometrical quantities become regular at $\Sigma$. We thus obtain the regular forms

	$\displaystyle R_{ijpq}$	$\displaystyle=R^{+}_{ijpq}\mathbf{\Theta}(l)+R^{-}_{ijpq}\mathbf{\Theta}(-l),$		(42)
	$\displaystyle R_{ij}$	$\displaystyle=R^{+}_{ij}\mathbf{\Theta}(l)+R^{-}_{ij}\mathbf{\Theta}(-l),$		(43)
	$\displaystyle R$	$\displaystyle=R^{+}\mathbf{\Theta}(l)+R^{-}\mathbf{\Theta}(-l),$		(44)
	$\displaystyle\mathcal{G}$	$\displaystyle=\mathcal{G}^{+}\mathbf{\Theta}(l)+\mathcal{G}^{-}\mathbf{\Theta}(-l).$		(45)

Now consider the differential terms in the MFEs, see Eq.(8). Following the chain rule in Eq.(10), one verifies that these $2nd$-order differential terms in $f_{\mathcal{G}}$ can be expressed in the form of $1st$ and $2nd$-order differential terms of $\mathcal{G}$, and thus we must compute these derivatives in the considered formalism. Applying the $1st$-order derivative of Eq.(45) one gets the form:

$\displaystyle\partial_{i}\mathcal{G}=\partial_{i}\mathcal{G}^{+}\mathbf{\Theta}(l)+\partial_{i}\mathcal{G}^{-}\mathbf{\Theta}(-l)+\varepsilon[\mathcal{G}]n_{i}\delta(l).$

(46)

According to Eq.(10), the $2nd$-order differential terms in $f_{\mathcal{G}}$ contain products of the form $\partial_{i}\mathcal{G}\partial_{j}\mathcal{G}$ which, according to the result just derived for $\partial_{i}\mathcal{G}$, will feature singular products of the form $\delta^{2}(l)$. To prevent the occurrence of these terms, one verifies that the Gauss-Bonnet invariant must be continuous across $\Sigma$, i.e., we can write the $3rd$ JC of the theory as

$\displaystyle[\mathcal{G}]=0.$

(47)

This JC is also absent in GR and is linked to an arbitrary dependency of the action in the Gauss-Bonnet invariant $\mathcal{G}$, and thus it is expected to arise also in other similar theories like e.g. the simpler $f\left(\mathcal{G}\right)$ theory. Under this condition, the partial derivatives of $\mathcal{G}$ simplify to

$\displaystyle\partial_{i}\mathcal{G}=\partial_{i}\mathcal{G}^{+}\mathbf{\Theta}(l)+\partial_{i}\mathcal{G}^{-}\mathbf{\Theta}(-l).$

(48)

One can now take the $2nd$-order covariant derivatives of the Gauss-Bonnet invariant $\mathcal{G}$, i.e., which now take the form

$\displaystyle\nabla_{i}\nabla_{j}\mathcal{G}=\nabla_{i}\nabla_{j}\mathcal{G}_{+}\mathbf{\Theta}(+l)+\nabla_{i}\nabla_{j}\mathcal{G}_{-}\mathbf{\Theta}(-l)+\varepsilon n_{i}\left[\partial_{j}\mathcal{G}\right]\delta(l).$

(49)

Let us investigate the matter section of the MFEs. In the previous calculations, we have shown the $2nd$-order covariant derivatives of $\mathcal{G}$ in expression Eq.(49), feature terms proportional to $\delta\left(l\right)$ which linked to the occurrence of a perfect fluid thin-shell at $\Sigma$. In this representation $\mathcal{S}_{ij}$ denoting the stress-energy tensor of the thin-shell, one can write the stress-energy tensor $T_{ij}$ in the distribution formalism as

$\displaystyle T_{ij}=T_{ij}^{+}\mathbf{\Theta}(l)+T_{ij}^{-}\mathbf{\Theta}(-l)+\mathcal{S}_{ij}\delta(l).$

(50)

The quantity $\mathcal{S}_{ij}$ is a four-dimensional tensor representing the thin-shell. This quantity can be expressed as a three-dimensional $\mathcal{S}_{\mu\nu}$ tensor at $\Sigma$ via the projection

$\displaystyle\mathcal{S}_{ij}=\mathcal{S}_{\mu\nu}e_{i}^{\mu}e_{j}^{\nu}.$

(51)

The trace $T$ can thus be expressed in the distribution formalism as:

$$T=T^{+}\mathbf{\Theta}(l)+T^{-}\mathbf{\Theta}(-l)+\mathcal{S}\delta(l),$$

here $\mathcal{S}=\mathcal{S}^{i}_{i}$ is the trace of the stress-energy tensor of the thin-shell. Now, since the function $f\left(\mathcal{G},T\right)$ features an arbitrary dependence on the quantity $T$, one can expect in general that this function admits a Taylor series expansion in $T$, which will feature power-laws of this quantity. These power-laws will then feature the usual singular terms of the form $\delta^{2}(l)$ that must be removed to preserve the definiteness of the function. Thus, one concludes that the trace $\mathcal{S}$ must vanish, i.e., the $4th$ JC takes the form

$\displaystyle\mathcal{S}=0.$

(52)

This JC is related to the arbitrary reliance of the action in $T$ and it is also featured in other well-know theories e.g. $f\left(T\right)$ gravity and both the metric and the Palatini formulation of $f\left(R,T\right)$ gravity. Following this condition, the trace $T$ simplifies to

$\displaystyle T=T^{+}\mathbf{\Theta}(l)+T^{-}\mathbf{\Theta}(-l).$

(53)

Turning now to the differential terms in the MFEs, and similarly to what happens to the differential terms in $\mathcal{G}$, the $2nd$-order covariant derivatives of $f_{\mathcal{G}}$ can be rewritten in terms of $1st$ and $2nd$ order derivatives of $T$ via the chain rule in Eq.(10). Applying the $1st$-derivatives of $T$ one obtains $\partial_{i}T=\partial_{i}T^{+}\mathbf{\Theta}(l)+\partial_{i}T^{-}\mathbf{\Theta}(-l)+\varepsilon[T]n_{i}\delta(l)$. In Eq.(10) one verifies that products of the form $\partial_{i}T\partial_{j}T$ are present in the $2nd$-order differential terms of $f_{\mathcal{G}}$, which give rise to singular products of the form $\delta^{2}(l)$. This products can be avoided by imposing the continuity of the trace $T$, i.e., the $5th$ JC takes the form

$\displaystyle[T]=0.$

(54)

This JC is also associated with the arbitrary dependence of the action in $T$ and it is also present e.g. in both formulations of $f\left(R,T\right)$ gravity (i.e., metric and Palatini). Under this condition, the $1st$ derivatives of $T$ simplify to

$\displaystyle\partial_{i}T=\partial_{i}T^{+}\mathbf{\Theta}(l)+\partial_{i}T^{-}\mathbf{\Theta}(-l).$

(55)

Consequently, applying the $2nd$-order covariant derivatives of $T$ one obtains the form

$\displaystyle\nabla_{i}\nabla_{j}T=\nabla_{i}\nabla_{j}T_{+}\mathbf{\Theta}(l)+\nabla_{i}\nabla_{j}T_{-}\mathbf{\Theta}(-l)+\varepsilon n_{i}\left[\partial_{j}T\right]\delta(l).$

(56)

Now we have constructed all the necessary quantities in distribution formalism to obtain the singular part of the field equations and derive an explicit expression for the stress-energy tensor of the thin-shell in terms of the geometrical quantities. To do so, we introduce Eqs.(28), (42), (43), (44), (45), (48), (49), (50), (53), (55), and (56), into the MFEs in Eq.(8), we project the result onto $\Sigma$ by utilizing the projection vectors $e^{i}_{\mu}$, and discard all non-singular terms. The result is as follows

	$\displaystyle\left(k^{2}+f_{T}\right)\mathcal{S}_{\mu\nu}$	$\displaystyle=$	$\displaystyle 4\varepsilon\left(e^{i}_{\mu}e^{j}_{\nu}R_{\ i\ j}^{p\ q}-R^{pq}\textbf{h}_{\mu\nu}\right)n_{p}\left(f_{\mathcal{G}\mathcal{G}}\left[\partial_{q}\mathcal{G}\right]+f_{\mathcal{G}T}\left[\partial_{q}T\right]\right)$		(57)
		$\displaystyle+$	$\displaystyle 4\varepsilon\big{(}e_{\mu}^{i}e_{\nu}^{j}R_{ij}-\frac{1}{2}R\mathbf{h}_{\mu\nu}\big{)}n^{p}\left(f_{\mathcal{G}\mathcal{G}}\left[\partial_{p}\mathcal{G}\right]+f_{\mathcal{G}T}\left[\partial_{p}T\right]\right).$

We can now use the trace of Eq.(57) and the fact that $\mathcal{S}=0$ to recast the $4th$ JC in a more convenient form as

$$f_{\mathcal{G}\mathcal{G}}\left[\partial_{p}\mathcal{G}\right]+f_{\mathcal{G}T}\left[\partial_{p}T\right]=0.$$

Consequently, the right-hand side of Eq.(57) disappears identically, which implies that $\mathcal{S}_{\mu\nu}=0$. This result indicates that under this formalism, thin-shells can not exist in the $f\left(\mathcal{G},T\right)$ gravitational theory. This is a very restrictive result that forces all the junctions between two different space-times at a given hyper-surface $\Sigma$ to be smooth in this theory. The whole set of JCs for the geometrical representation of the $f\left(\mathcal{G},T\right)$ theory may thus be written as

$$\centering\left.\begin{split}\left[\mathbf{h}_{\mu\nu}\right]=0,\\ [\mathcal{K}_{\mu\nu}]=0,\\ [\mathcal{G}]=0,\\ [T]=0,\\ f_{\mathcal{G}\mathcal{G}}\left[\partial_{w}\mathcal{G}\right]+f_{\mathcal{G}T}\left[\partial_{w}T\right]=0.\\ \end{split}~{}~{}\right\}\@add@centering$$

(58)