h-Almost Conformal η-Ricci–Bourguignon Solitons and Spacetime Symmetry in Barotropic Fluids Within f($\mathcal{R}\mathbf{,}\mathcal{T}$) Gravity

Abstract

We investigate the geometric and physical properties of the h-almost conformal $\eta$-Ricci–Bourguignon soliton and its gradient form by employing a barotropic equation of state within the framework of $f(\mathcal{R},\mathcal{T})$ gravity. We derive this barotropic equation of state under the assumption that the matter content of $f(\mathcal{R},\mathcal{T})$ gravity is modeled by a barotropic perfect fluid. We also examine the way in which these soliton structures both reveal and limit the underlying symmetries of the spacetime geometry. Furthermore, we obtain modified Poisson and Liouville equations associated with these solitons in such a gravitational setting. Additionally, we explore certain harmonic aspects of the h-almost conformal $\eta$-Ricci–Bourguignon soliton on a spacetime filled with a barotropic perfect fluid, considering a harmonic potential function $\mathrm{\Psi }$. Finally, we present physical interpretations of the conformal pressure $\tilde{p}$ in the context of the h-almost conformal $\eta$-Ricci–Bourguignon soliton within $f(\mathcal{R},\mathcal{T})$ gravity.

1. Introduction

In recent developments within alternative theories of gravitation, significant attention has been devoted to the construction of solvable gravitational field equations. A particularly fruitful strategy in this direction relies on prescribing suitable forms for the Lagrangian density associated with a perfect fluid. By imposing physically motivated constraints, or by considering special fluid configurations, one is often able to simplify the governing dynamical equations while maintaining the fundamental coupling between matter and geometry. Such methodological choices not only enhance the analytical tractability of the modified field equations but also provide a deeper understanding of the interplay between relativistic fluids and generalized gravitational backgrounds [1,2,3,4,5].

Much of the modern treatment can be traced back to the seminal work of Brown [6], who established that, within the framework of general relativity (GR), the on-shell perfect fluid Lagrangian ${\mathcal{L}}_m$ (neglecting elastic contributions) reduces to

$${\mathcal{L}}_m=-\omega ,$$

where $\omega$ denotes the fluid energy density. In contrast, within modified gravity scenarios, it has been shown that the on-shell Lagrangian may also be consistently taken in the alternative form

$${\mathcal{L}}_m=\upsilon ,$$

with $\upsilon$ representing the fluid pressure. This duality in the admissible fluid Lagrangian plays a central role in shaping the corresponding modified field equations and their physical consequences. Such a duality may be viewed as expressing an underlying symmetry in the fluid Lagrangian, offering alternative yet consistent geometric formulations of matter fields.

In [7], an alternative formulation of the perfect fluid Lagrangian was introduced, expressed in terms of the hydrodynamic variables ${\eta }^{\alpha }$, $\rho$, and ${\mathcal{T}}^r$, together with the gravitational field variables $g_{uv}$. Here, ${\eta }^{\alpha }$ corresponds to the fluid four-velocity, $\rho$ denotes the rest-mass density, and ${\mathcal{T}}^r$ represents the rest temperature of the fluid. Along similar lines, the hydrodynamic equations describing a perfect fluid in the framework of general relativity (GR) have been reformulated within an Eulerian description, where the four-velocity is parametrized by six velocity potentials [8]. Subsequently, in [9], a matter Lagrangian of the form

$${\mathcal{L}}_m=-\rho (1+ϵ),$$

with $ϵ$ denoting the elastic potential, was proposed in order to derive the fluid equations of motion directly from a variational principle. This formulation highlights the flexibility of adopting different Lagrangian densities for perfect fluids, each tailored to emphasize distinct physical aspects of the system.

Despite these refinements, general relativity (GR) itself remains the most successful and widely accepted framework for probing the large-scale structure and dynamical evolution of the universe. Nevertheless, observational evidence has shown that GR, when applied in its standard form without the inclusion of dark energy, is insufficient to account for the accelerated expansion observed both in the early inflationary epoch and in the present cosmic acceleration phase. Moreover, GR does not fully resolve the deeper question concerning the fundamental nature of gravity. These limitations have motivated intensive research into various modifications of GR, aimed at constructing models capable of incorporating inflationary scenarios while simultaneously mimicking the dynamical effects of dark energy (DE). Such generalized frameworks also open new pathways for exploring the coupling between matter and geometry in regimes beyond the classical predictions of GR.

To address the inherent limitations of general relativity (GR) in explaining various cosmological observations, one of the conventional strategies has been to modify the Einstein field equations, originally formulated by Einstein [10,11]. These equations, when supplemented with the introduction of an additional hypothetical component—commonly referred to as dark matter—remain in good agreement with current astrophysical and cosmological data [12]. This framework has inspired a wide spectrum of mathematicians and physicists to develop extended and more sophisticated theories of gravitation, often derived from suitable generalizations of the Einstein–Hilbert action. Such efforts have given rise to an array of modified gravity models [13,14,15]. In addition to their phenomenological importance, modified gravity theories are also of interest from a theoretical standpoint, as they may be viewed as effective low-energy limits of an as yet unknown quantum theory of gravity [16].

A prominent instance of such an extension is the modification of the Einstein–Hilbert Lagrangian density into an arbitrary function $f\left(\mathcal{R}\right)$, where $\mathcal{R}$ denotes the Ricci scalar. This leads to the celebrated $f\left(\mathcal{R}\right)$ gravity framework, which has been extensively studied as a possible alternative to dark energy models. Even more general constructions arise when the gravitational Lagrangian is taken to depend not only on the Ricci scalar $\mathcal{R}$ but also on the trace of the energy–momentum tensor $\mathcal{T}$. This generalization gives rise to the $f(\mathcal{R},\mathcal{T})$ gravity theory, introduced by Harko et al. [17], which provides an additional degree of freedom by explicitly encoding a direct coupling between matter and curvature.

From the geometric standpoint, the universe can be modeled as a four-dimensional, time-oriented Lorentzian manifold $\mathrm{\Omega }$, belonging to the class of pseudo-Riemannian manifolds endowed with a Lorentzian metric g. Such a structure naturally provides the appropriate geometric framework for the study of relativistic cosmology. In most standard cosmological models, the matter content of the universe is typically described by a perfect fluid, and the resulting spacetime is thus referred to as a perfect fluid spacetime (PFST) [18]. This modeling choice not only captures the essential features of large-scale matter distribution but also allows for a tractable formulation of the field equations in both GR and its modified extensions.

Consequently, the energy–momentum tensor (EMT) $\mathcal{T}$ associated with a perfect fluid spacetime (PFST) is given by [18,19]

$${\mathcal{T}}_{uv}=\upsilon g_{uv}+(\omega +\upsilon ){\eta }_u{\eta }_v,$$

(1)

where $\upsilon$ and $\omega$ denote, respectively, the isotropic pressure and the energy density of the fluid, while $\eta$ represents a non-vanishing 1-form associated with the fluid four-velocity. This form of ${\mathcal{T}}_{uv}$ ensures compatibility with the assumptions of isotropy and homogeneity commonly invoked in cosmological modeling.

The concept of Ricci flow was first introduced in the early 1980s by Hamilton, who was inspired by the pioneering work of Eells and Sampson on harmonic map heat flow [20,21]. The Ricci flow prescribes the time evolution of a Riemannian metric $g\left(t\right)$ on a smooth manifold $({\mathrm{\Omega }}^{\tilde{n}},g)$ according to the geometric partial differential equation

$${\displaystyle \frac{\partial g\left(t\right)}{\partial t}}=-2\mathcal{R}ic,$$

(2)

where $\mathcal{R}ic$ denotes the Ricci curvature tensor associated with g. This flow deforms the metric in a way that tends to homogenize the curvature, and it has proved to be an indispensable tool in geometric analysis and topology.

Extending this notion, the concept of a Ricci soliton arises naturally in the study of self-similar solutions to the Ricci flow. On a semi-Riemannian manifold $({\mathrm{\Omega }}^{\tilde{n}},g)$, a Ricci soliton is defined by the condition

$$\mathcal{R}ic+\frac{1}{2}{\mathfrak{L}}_{\mathcal{E}}g={\lambda }_{1}g,$$

(3)

where ${\mathfrak{L}}_{\mathcal{E}}g$ denotes the Lie derivative of the metric tensor g with respect to a smooth vector field $\mathcal{E}$ and ${\lambda }_1\in \mathbb{R}$ is a constant parameter. Equation (3) captures the balance between the intrinsic Ricci curvature and the deformation of the geometry generated by the vector field $\mathcal{E}$. When ${\lambda }_1$ is positive, negative, or zero, the Ricci soliton is said to be shrinking, expanding, or steady, respectively, thereby providing canonical models of the long-term behavior of the Ricci flow. Ricci solitons often encode continuous geometric symmetry (for example via Killing fields) and so provide natural links between flow behavior and spacetime isometries.

According to [22], the notion of a conformal Ricci soliton extends the classical concept of Ricci solitons by incorporating a non-dynamical scalar field. It is defined by the equation

$${\mathfrak{L}}_{\mathcal{E}}g+2\mathcal{R}ic=\left({\displaystyle \frac{1}{n}}(\tilde{p}+2)-2\mu \right)g,$$

(4)

where ${\mathfrak{L}}_{\mathcal{E}}g$ denotes the Lie derivative of the metric tensor g with respect to a smooth vector field $\mathcal{E}$ on the n-dimensional manifold ${\mathrm{\Omega }}^n$, $\tilde{p}$ is a non-dynamical scalar field (which may vary with time), and $\mu \in \mathbb{R}$ is a fixed real constant. This formulation generalizes Equation (3) by allowing the soliton structure to interact explicitly with $\tilde{p}$, thereby producing a broader class of self-similar solutions to geometric flows.

A particularly important subclass arises when the potential vector field $\mathcal{E}$ is the gradient of a smooth scalar function f; i.e., $\mathcal{E}=\nabla f$. In this case, Equation (4) defines what is called a gradient conformal Ricci soliton. As in the classical setting, these solitons are further divided into shrinking, steady, or expanding types, according to the sign of the associated constant parameter ${\lambda }_1$: positive, zero, or negative, respectively. Such a classification reflects the asymptotic behavior of the manifold under the geometric flow, with shrinking solitons modeling collapsing geometries, steady solitons corresponding to equilibrium states, and expanding solitons describing geometries that dilate over time.

In 1979, Bourguignon [23] introduced a further generalization of the Ricci flow, now known as the Ricci–Bourguignon flow (RB flow). On a Riemannian manifold $({\mathrm{\Omega }}^n,g)$, this flow is governed by the evolution equation

$${\displaystyle \frac{\partial g\left(t\right)}{\partial t}}=-2\left(\mathcal{R}ic-\mathcal{R}\mathrm{\Gamma }g\right),$$

(5)

where $\mathcal{R}\in \mathbb{R}$ is a fixed parameter and $\mathrm{\Gamma }$ denotes the scalar curvature of the metric g. The RB flow interpolates between the classical Ricci flow ($\mathcal{R}=0$) and scalar curvature-normalized flows ($\mathcal{R}\ne 0$), thereby offering a flexible framework to analyze the interplay between Ricci curvature and the global scalar geometry of the manifold. These generalized flows not only govern the geometric evolution but also highlight underlying symmetry principles that persist or break under deformation.

The evolution equation defined in Equation (5) admits diverse geometric interpretations, depending on the specific choice of the parameter $\mathrm{\Gamma }$. As discussed in [24], the following cases arise:

(i)

$\mathrm{\Gamma }=\frac{1}{2}$, which yields the Einstein tensor $\mathcal{R}ic-\frac{\mathcal{R}}{2}g$, leading to the so-called Einstein soliton;

(ii)

$\mathrm{\Gamma }=\frac{1}{n}$, corresponding to the traceless Ricci tensor $\mathcal{R}ic-\frac{\mathcal{R}}{n}g$;

(iii)

$\mathrm{\Gamma }=\frac{1}{2(n-1)}$, which produces the Schouten tensor $\mathcal{R}ic-\frac{r}{2(n-1)}g$, giving rise to the Schouten soliton;

(iv)

$\mathrm{\Gamma }=0$, which reduces to the classical Ricci tensor $\mathcal{R}ic$, thereby recovering the standard Ricci soliton.

These distinct choices of $\mathrm{\Gamma }$ highlight how the Ricci–Bourguignon flow serves as a unifying framework that encompasses several well-known curvature structures as particular cases.

In the same work [24], the author introduced the concept of Ricci–Bourguignon solitons (RB solitons), which naturally extend the classical Ricci soliton by incorporating the Bourguignon curvature modification. Formally, a Riemannian manifold $({\mathrm{\Omega }}^n,g)$ is called a Ricci–Bourguignon soliton if there exists a smooth vector field $\mathcal{E}$ on ${\mathrm{\Omega }}^n$ such that

$$\mathcal{R}ic+\frac{1}{2}{\mathfrak{L}}_{\mathcal{E}}g=\left({\gamma }_1+\mathcal{R}\mathrm{\Gamma }\right)g,$$

(6)

where ${\gamma }_1\in \mathbb{R}$ is a constant and $\mathcal{R}\mathrm{\Gamma }$ encodes the Bourguignon adjustment. Equation (6) characterizes a self-similar geometric deformation of the metric under the combined influence of Ricci curvature, the scalar curvature contribution, and the flow generated by the vector field $\mathcal{E}$. This generalized soliton framework thus provides a broader setting in which Einstein, traceless Ricci, Schouten, and classical Ricci solitons all appear as limiting cases.

In the particular situation where the vector field $\mathcal{E}$ is the gradient of a smooth scalar function f, i.e., $\mathcal{E}=\nabla f$, the soliton is referred to as a gradient Ricci–Bourguignon soliton (GRBS). In this case, Equation (6) specializes to

$$\mathcal{R}ic+\nabla \nabla f=\left({\gamma }_1+\mathcal{R}\mathrm{\Gamma }\right)g,$$

(7)

where $\nabla \nabla f$ denotes the Hessian of f.

Building upon this framework, we now introduce a broader generalization. Let $({\mathrm{\Omega }}^n,g)$ be an n-dimensional complete Riemannian or pseudo-Riemannian manifold. We say that $({\mathrm{\Omega }}^n,g,h,\mathcal{E},{\gamma }_1)$ constitutes an h-almost conformal Ricci–Bourguignon soliton (henceforth h-ACRB soliton) if there exists a smooth vector field $\mathcal{E}$ on ${\mathrm{\Omega }}^n$ such that

$${\displaystyle \frac{h}{2}}{\mathfrak{L}}_{\mathcal{E}}g+\mathcal{R}ic=\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{2}{n}\right)+\mathcal{R}\mathrm{\Gamma }\right)g,$$

(8)

where $h,{\gamma }_1\in C^{\infty }\left({\mathrm{\Omega }}^n\right)$ are smooth functions and $\tilde{p}$ denotes a scalar field, which may depend on time but is regarded as non-dynamical in this setting. Depending on the sign of ${\gamma }_1$, the soliton is classified as shrinking if ${\gamma }_1>0$, steady if ${\gamma }_1=0$, and expanding if ${\gamma }_1<0$. This generalized formulation incorporates a richer interaction between Ricci curvature, conformal deformations, and the external scalar field $\tilde{p}$. If the distinguished vector field $\mathcal{E}$ further assumes the gradient form $\mathcal{E}=\nabla f$ for some smooth function f on ${\mathrm{\Omega }}^n$, then Equation (8) reduces to

$$h\nabla \nabla f+\mathcal{R}ic=\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{2}{n}\right)+\mathcal{R}\mathrm{\Gamma }\right)g,$$

(9)

and in this case the resulting structure is called an h-almost gradient conformal Ricci–Bourguignon soliton. Recently, Azami investigated h-almost conformal $\omega$-Ricci–Bourguignon solitons in relativistic spacetimes, and Azami, De, and Jafari studied generalized Z-solitons in perfect fluid spacetimes within $f(R,T)$ gravity. These works complement our approach and place the present results within the broader context of soliton studies in modified gravity theories [25,26].

Extending this framework further, we introduce the notion of an h-almost conformal $\eta$-Ricci–Bourguignon soliton. An n-dimensional complete Riemannian or pseudo-Riemannian manifold ${\mathrm{\Omega }}^n$ is said to admit an h-almost conformal $\eta$-Ricci–Bourguignon soliton, denoted by the quintuple $({\mathrm{\Omega }}^n,g,h,{\gamma }_1,{\gamma }_2)$, if there exists a smooth vector field $\mathcal{E}$ such that

$${\displaystyle \frac{h}{2}}{\mathfrak{L}}_{\mathcal{E}}g+\mathcal{R}ic=\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{2}{n}\right)+\mathcal{R}\mathrm{\Gamma }\right)g+{\gamma }_2\eta \otimes \eta ,$$

(10)

where h and ${\gamma }_1$ are smooth scalar functions on ${\mathrm{\Omega }}^n$, ${\gamma }_2\in \mathbb{R}$ is a constant, $\eta$ is a 1-form on ${\mathrm{\Omega }}^n$, and $\tilde{p}$ is a non-dynamical scalar field. The additional $\eta \otimes \eta$ anisotropic term can break global isotropy while selecting residual symmetry directions in the manifold.

The sign of ${\gamma }_1$ once again determines the dynamical nature of the soliton: the structure is called shrinking if ${\gamma }_1>0$, steady if ${\gamma }_1=0$, and expanding if ${\gamma }_1<0$.

If the vector field $\mathcal{E}$ is taken to be the gradient of a smooth potential function f, i.e., $\mathcal{E}=\nabla f$, then the defining relation (10) reduces to the gradient form

$$h\nabla \nabla f+\mathcal{R}ic=\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{2}{n}\right)+\mathcal{R}\mathrm{\Gamma }\right)g+{\gamma }_2\eta \otimes \eta ,$$

(11)

and in this case, the manifold $({\mathrm{\Omega }}^n,g)$ is called a gradient h-almost conformal $\eta$-Ricci–Bourguignon soliton.

These generalized soliton structures unify and extend several well-known geometric flows. Specifically, the presence of the conformal factor h enriches the scaling behavior of the soliton, while the scalar fields ${\gamma }_1$ and $\tilde{p}$ allow for coupling with external matter or non-dynamical fields. Moreover, the additional $\eta \otimes \eta$ term introduces anisotropic contributions, resembling $\eta$-Ricci solitons, and the Bourguignon adjustment $\mathcal{R}\mathrm{\Gamma }$ provides a curvature modification parameter that interpolates between the Einstein, traceless Ricci, Schouten, and Ricci tensors.

Altogether, this framework provides a natural extension of classical Ricci solitons and opens new avenues for exploring self-similar solutions in curvature flows on both Riemannian and pseudo-Riemannian manifolds. In particular, analyzing the role of symmetry in the context of h-almost conformal $\eta$-RB solitons sheds light on how geometric constraints interact with physical matter fields in $f(R,T)$ gravity.

In the subsequent sections, we investigate the geometric properties and physical interpretations of h-almost conformal Ricci–Bourguignon solitons and their gradient counterparts. In particular, we derive curvature identities, analyze their behavior under specific assumptions on the potential function and the conformal factor h, and explore applications within the framework of modified gravity theories, especially in the setting of $f(\mathcal{R},\mathcal{T})$ gravity models.

Several authors have examined the geometric aspects of perfect fluid spacetimes in connection with various classes of solitons (see, for instance, refs. [22,24,27,28,29,30]). Motivated by these developments, the present work is devoted to the study of perfect fluid spacetimes satisfying a barotropic equation of state, within the framework of $f(\mathcal{R},\mathcal{T})$ gravity, under the additional assumption that the underlying Lorentzian metrics admit the structure of an h-almost conformal $\eta$-Ricci–Bourguignon soliton or its gradient analogue.

For the sake of clarity and uniformity, we shall adopt the following abbreviations throughout this paper. The symbol EFEs refers to Einstein’s Field Equations, which constitute the cornerstone of the General Theory of Relativity (GR). The term PFST designates perfect fluid spacetimes, while BPFST stands for perfect fluid spacetimes governed by a barotropic equation of state, abbreviated as BEoS. The notation SET denotes the stress–energy tensor, which represents the distribution of matter and energy. We also use DE to indicate dark energy, accounting for the accelerated expansion of the universe, and HPF to refer to a harmonic potential function arising in the study of certain geometric flows and solitons.

Finally, we denote by $({\mathrm{\Omega }}^4,g)$ a four-dimensional perfect fluid spacetime governed by a barotropic equation of state in the framework of $f(\mathcal{R},\mathcal{T})$ gravity. These notational conventions are introduced to streamline the presentation and render the subsequent exposition more concise and accessible.

2. Barotropic Perfect Fluid Spacetime in $\mathit{f}\mathbf{(}\mathcal{R}\mathbf{,}\mathcal{T}\mathbf{)}$ Gravity Theory

In this section, we investigate perfect fluid spacetimes governed by a barotropic equation of state within the framework of $f(\mathcal{R},\mathcal{T})$ gravity. By allowing the gravitational Lagrangian to depend explicitly on both the Ricci scalar $\mathcal{R}$ and the trace $\mathcal{T}$ of the stress–energy tensor, this extended theory provides a richer geometric and physical structure than that encountered in classical general relativity. Our focus is placed on spacetimes filled with a barotropic perfect fluid, with the aim of analyzing how the barotropic equation of state influences the curvature, the field dynamics, and the evolution of the gravitational interaction. The results obtained here form the foundation for the subsequent study of self-similar solutions, such as solitons, and for understanding the delicate interplay between matter distribution and modified gravity effects in both cosmological and astrophysical contexts.

More precisely, we consider a four-dimensional differentiable manifold $({\mathrm{\Omega }}^4,g)$, assumed to satisfy the gravitational field equations arising from the $f(\mathcal{R},\mathcal{T})$ framework. In this setting, the gravitational Lagrangian is modeled as a smooth functional depending on the Ricci scalar $\mathcal{R}$ and on the trace $\mathcal{T}$ of the stress–energy tensor, thereby encoding in an explicit manner the backreaction of matter on the underlying geometry of spacetime.

Guided by the theoretical construction introduced in [17], we adopt a specific form for this functional, given by

$$f(\mathcal{R},\mathcal{T})=\mathcal{R}+2f\left(\mathcal{T}\right),$$

(12)

where $f\left(\mathcal{T}\right)$ remains an arbitrary function characterizing the dependence on $\mathcal{T}$. The presence of the extra term $2f\left(\mathcal{T}\right)$ in the gravitational action effectively modifies the standard coupling between matter fields and the curvature of spacetime. This modification opens up new avenues for exploring the dynamical behavior of cosmological and astrophysical systems within this extended gravity framework.

The modified Einstein–Hilbert action in this framework is given by

$$S_{eh}={\displaystyle \frac{1}{16\pi }}\int f(\mathcal{R},\mathcal{T})\sqrt{-g}{d}^{4}x+\int {\mathcal{L}}_m\sqrt{-g}{d}^{4}x,$$

(13)

where $f(\mathcal{R},\mathcal{T})$ is an arbitrary function of the Ricci scalar $\mathcal{R}$ and the trace $\mathcal{T}$, and ${\mathcal{L}}_m$ denotes the matter Lagrangian density.

The stress–energy tensor (SET) of the matter field is defined by

$${\mathcal{T}}_{uv}=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_m\right)}{\delta g^{uv}}}.$$

(14)

Since the matter part of the Lagrangian density ${\mathcal{L}}_m$ does not depend on derivatives of the metric tensor, Equation (14) can be equivalently written as

$${\mathcal{T}}_{uv}=-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_m\right)}{\delta g^{uv}}}={\mathcal{L}}_m{g}_{uv}-2{\displaystyle \frac{\partial {\mathcal{L}}_m}{\partial g^{uv}}}.$$

(15)

Now, we consider the case of a perfect fluid obeying a barotropic equation of state (BEoS), where the pressure $\upsilon$ is a function of the energy density $\omega$; i.e., $\upsilon =\upsilon \left(\omega \right)$. In this scenario, the Lagrangian density ${\mathcal{L}}_m$ depends only on the energy density $\omega$. Assuming that the current of matter is conserved, i.e.,

$${\nabla }_{\sigma }\left(\omega {\eta }^{\sigma }\right)=0,$$

it follows from [17,31] that

$$\delta \omega ={\displaystyle \frac{1}{2}}\omega (g_{uv}-{\eta }_u{\eta }_v)\delta g^{uv},$$

(16)

where ${\eta }^{\alpha }$ is the four-velocity of the fluid, defined in local coordinates $x^{\sigma }$ as

$${\eta }^{\alpha }={\displaystyle \frac{d{x}^{\alpha }}{ds}},$$

with $ds$ satisfying $d{s}^2=-c^{2}d{t}^2$, where t denotes the proper time of the fluid particles and $\omega c^2$ represents the rest-mass energy density.

In view of Equations (12) and (13), it follows from [17] that

$${\mathcal{T}}^{uv}=\omega {\displaystyle \frac{d{\mathcal{L}}_m}{d\omega }}{\eta }^u{\eta }^v+({\mathcal{L}}_m-\omega \frac{d{\mathcal{L}}_m}{d\omega })g^{uv}.$$

(17)

For a barotropic perfect fluid (BPF), we identify this expression with the usual form of the stress–energy tensor:

$${\mathcal{T}}^{uv}=-[ϵ\left(\omega \right)+\upsilon ]{\eta }^u{\eta }^v+\upsilon g^{uv},$$

(18)

where $ϵ\left(\omega \right)$ denotes the total energy density and $\upsilon$ the pressure, both considered as functions of the rest-mass energy density $\omega$.

Comparing coefficients leads to the following relations:

$$\begin{array}{cc}\hfill {\mathcal{L}}_m& =-ϵ\left(\omega \right),\hfill \\ \hfill {\displaystyle \frac{d{\mathcal{L}}_m}{d\omega }}& =-{\displaystyle \frac{ϵ\left(\omega \right)+\upsilon }{\omega }}.\hfill \end{array}$$

(19)

From Equations (18) and (19), we deduce that

$$ϵ\left(\omega \right)=c^2\omega +\omega \Delta ,$$

(20)

where $c^2$ is an integration constant representing the rest-mass energy per unit mass, and $\Delta$ is the elastic compression potential energy per unit mass. According to [31], the latter is given by

$$\Delta =\int {\displaystyle \frac{\upsilon }{{\omega }^2}}d\omega =\int {\displaystyle \frac{d\upsilon }{\omega }}-{\displaystyle \frac{\upsilon }{\omega }}.$$

(21)

Thus, the Lagrangian density of a barotropic perfect fluid (BPF) takes the form

$${\mathcal{L}}_m=-c^2\omega -\omega \Delta ,$$

(22)

which implies that the corresponding stress–energy tensor (SET) becomes

$${\mathcal{T}}^{uv}=-\left[\omega [c^2+\Delta ]+\upsilon \right]{\eta }^u{\eta }^v+\upsilon g^{uv},$$

(23)

subject to the constraints

$${\eta }^u{\nabla }_v{\eta }_u=0,{\eta }_u{\eta }^u=-1.$$

(24)

From the variation in the action (10) with respect to the metric tensor $g_{uv}$ and applying the principle of least action, the field equations of $f(\mathcal{R},\mathcal{T})$ gravity are obtained as

$$\begin{array}{cc}\hfill & f_{\mathcal{R}}(\mathcal{R},\mathcal{T})\mathcal{R}i{c}_{uv}-\frac{1}{2}f(\mathcal{R},\mathcal{T})g_{uv}+(g_{uv}{\nabla }_c{\nabla }^c-{\nabla }_u{\nabla }_v)f_{\mathcal{R}}(\mathcal{R},\mathcal{T})\hfill \\ \hfill & =8\pi {\mathcal{T}}_{uv}-f_{\mathcal{T}}(\mathcal{R},\mathcal{T}){\mathcal{T}}_{uv}-f_{\mathcal{T}}(\mathcal{R},\mathcal{T}){\mathbb{H}}_{uv},\hfill \end{array}$$

(25)

where $f_{\mathcal{R}}={\displaystyle \frac{\partial f}{\partial \mathcal{R}}}$ and $f_{\mathcal{T}}={\displaystyle \frac{\partial f}{\partial \mathcal{T}}}$ denote the partial derivatives of $f(\mathcal{R},\mathcal{T})$ with respect to $\mathcal{R}$ and $\mathcal{T}$, respectively. Here, ${\mathbb{H}}_{uv}$ is defined by

$${\mathbb{H}}_{uv}=-2{\mathcal{T}}_{uv}+g_{uv}{\mathcal{L}}_m-2g^{ij}{\displaystyle \frac{{\partial }^2{\mathcal{L}}_m}{\partial g^{uv}\partial g^{ij}}}.$$

(26)

In view of Equations (22), (25) and (26), the Ricci tensor can then be expressed as

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_{uv}& =\left[\frac{1}{2}\mathcal{R}+2\upsilon [f^{\prime }\left(\mathcal{T}\right)+4\pi ]+2f^{\prime }\left(\mathcal{T}\right)[c^2\omega +\Delta ]+f\left(\mathcal{T}\right)\right]g_{uv}\hfill \\ \hfill & -2(f^{\prime }\left(\mathcal{T}\right)+4\pi )[\omega (c^2+\Delta )+\upsilon ]{\eta }_u{\eta }_v,\hfill \end{array}$$

(27)

where $f^{\prime }\left(\mathcal{T}\right)$ denotes the derivative of f with respect to $\mathcal{T}$.

By contracting Equation (27), we obtain the following relation:

$$20\pi \upsilon +2f\left(\mathcal{T}\right)+4\pi \omega (c^2+\Delta )+f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]+\frac{1}{2}\mathcal{R}=0,$$

(28)

where $f^{\prime }\left(\mathcal{T}\right)$ denotes the derivative of f with respect to $\mathcal{T}$.

Therefore, for a barotropic perfect fluid spacetime (BPFST) in $f(\mathcal{R},\mathcal{T})$ gravity, the Ricci tensor takes the form

$$\mathcal{R}i{c}_{uv}=\chi g_{uv}+\psi {\eta }_u{\eta }_v,$$

(29)

where the scalar functions $\chi$ and $\psi$ are given by

$$\begin{array}{cc}\hfill \chi & =\frac{1}{2}\mathcal{R}+2\upsilon [f^{\prime }\left(\mathcal{T}\right)+4\pi ]+2f^{\prime }\left(\mathcal{T}\right)[c^2\omega -\Delta ]+f\left(\mathcal{T}\right),\hfill \\ \hfill \psi & =-2(f^{\prime }\left(\mathcal{T}\right)+4\pi )[\omega (c^2+\Delta )+\upsilon ].\hfill \end{array}$$

(30)

As a consequence, we state the following result.

Theorem 1.

The Ricci tensor of the spacetime $({\mathrm{\Omega }}^4,g)$ in $f(\mathcal{R},\mathcal{T})$ gravity has the explicit form

$$\begin{array}{cc}\hfill \mathcal{R}i{c}_{uv}& =\left[\frac{1}{2}\mathcal{R}+2\upsilon [f^{\prime }\left(\mathcal{T}\right)+4\pi ]+2f^{\prime }\left(\mathcal{T}\right)[c^2\omega -\Delta ]+f\left(\mathcal{T}\right)\right]g_{uv}\hfill \\ \hfill & -\left[2(f^{\prime }\left(\mathcal{T}\right)+4\pi )[\omega (c^2+\Delta )+\upsilon ]\right]{\eta }_u{\eta }_v.\hfill \end{array}$$

(31)

Remark 1.

The particular choice $f(R,T)=R+2f\left(T\right)$ employed above is taken for simplicity only. The structural property used in the proof of all results is that the function $f(R,T)$ is linear in R and that $f_T$ depends solely on T. Hence the same conclusions hold for the broader class

$$f(R,T)=\alpha R+\beta F\left(T\right),$$

where $\alpha ,\beta$ are constants and F is a differentiable function of T. Indeed, for this class $f_R=\alpha$ is constant, which eliminates extra geometric terms arising from derivatives of $f_R$. Consequently, the geometric structure of the Ricci tensor obtained is preserved; only the numerical coefficients appearing in the field equations are redefined by the constants α and β. This shows that the result is universal for the entire family above and is not restricted to the special case $\alpha =1,\beta =2$.

Remark 2.

An important and well-motivated special case of our general framework arises when adopting a linear functional form for $f\left(\mathcal{T}\right)$, namely $f\left(\mathcal{T}\right)=\alpha \mathcal{T}$, where α is a constant. In this case, the gravitational Lagrangian simplifies to

$$f(\mathcal{R},\mathcal{T})=\mathcal{R}+2\alpha \mathcal{T}.$$

This model is of particular physical interest for several reasons. First, it provides the simplest and most direct realization of a non-minimal coupling between geometry (through the Ricci scalar $\mathcal{R}$) and matter (through the trace of the stress–energy tensor $\mathcal{T}$). Second, it represents a natural extension of general relativity, which is recovered in the limit $\alpha \to 0$. The linear coupling introduces a deviation from GR that depends explicitly on the matter content, leading to modified dynamical behavior that may account for phenomena such as cosmic acceleration without invoking a cosmological constant. Owing to its analytical simplicity and physical transparency, this model has been widely investigated in the literature, allowing our subsequent results to be directly compared with a broad range of existing studies.

Corollary 1.

The scalar curvature $\mathcal{R}$ of the spacetime $({\mathrm{\Omega }}^4,g)$ is given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathcal{R}}{2}}& =-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\hfill \\ \hfill & -f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)].\hfill \end{array}$$

(32)

Now, in view of Equation (28), we have

$$\begin{array}{ccc}\hfill \omega (c^2+\Delta )+5\upsilon & =& {\displaystyle \frac{4f^{\prime }\left(\mathcal{T}\right)[c^2-\Delta ]+2f\left(\mathcal{T}\right)+\frac{1}{2}\mathcal{R}}{-(4\pi +f^{\prime }\left(\mathcal{T}\right))}}.\hfill \end{array}$$

(33)

Since the scalar curvature $\mathcal{R}$ is non-vanishing for perfect fluid spacetimes (PFSTs) with a barotropic equation of state (BEoS) in $f(\mathcal{R},\mathcal{T})$ gravity, it follows from Equation (33) that

$$\omega (c^2+\Delta )+\upsilon \ne 0.$$

Consequently, we arrive at the following statement, which formalizes this observation.

Theorem 2.

Consider a matter distribution described by a perfect fluid with a barotropic equation of state (PFST) in the framework of $f(\mathcal{R},\mathcal{T})$ gravity, where the scalar curvature $\mathcal{R}$ is assumed to be non-vanishing. Under these conditions, the resulting fluid cannot consistently be regarded as representing a dark matter component.

Corollary 2.

As a direct implication, when the matter sector in $f(\mathcal{R},\mathcal{T})$ gravity consists of a barotropic perfect fluid (BPF), the corresponding barotropic equation of state (BEoS) must necessarily coincide with the expression given in Equation (33).

Corollary 3.

Furthermore, within the context of $f\left(\mathcal{R}\right)$ gravity for a fluid configuration characterized by nonzero scalar curvature $\mathcal{R}$, the barotropic equation of state (BEoS) adopts the form

$$\omega \left(c^2+\Delta \right)+5\upsilon ={\displaystyle \frac{\mathcal{R}}{-8\pi }}.$$

Moreover, if the source fluid is of radiation type, the BEoS is

$$\omega (c^2+\Delta )=3\upsilon .$$

Then, from Equation (33), we derive

$$\begin{array}{cc}\hfill \upsilon & ={\displaystyle \frac{8f^{\prime }\left(\mathcal{T}\right)(c^2-\Delta )+4f\left(\mathcal{T}\right)+\mathcal{R}}{-16(4\pi +f^{\prime }\left(\mathcal{T}\right))}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \omega (c^2+\Delta )& ={\displaystyle \frac{24f^{\prime }\left(\mathcal{T}\right)(c^2-\Delta )+12f\left(\mathcal{T}\right)+3\mathcal{R}}{-16(4\pi +f^{\prime }\left(\mathcal{T}\right))}}.\hfill \end{array}$$

(35)

Corollary 4.

If the source matter of the spacetime $({\mathrm{\Omega }}^4,g)$ is of radiation type, then the pressure υ and the total energy density $\omega (c^2+\Delta )$ of the fluid are given by the expressions in Equations (34) and (35).

Again, for the case of a phantom barrier, where

$$\omega (c^2+\Delta )=-\upsilon ={\displaystyle \frac{8f^{\prime }\left(\mathcal{T}\right)(c^2-\Delta )+4f\left(\mathcal{T}\right)+\mathcal{R}}{8(4\pi +f^{\prime }\left(\mathcal{T}\right))}},$$

(36)

we state the following result.

Corollary 5.

If the source matter of the spacetime $({\mathrm{\Omega }}^4,g)$ is of phantom barrier type, then the pressure υ and the total energy density $\omega (c^2+\Delta )$ of the perfect fluid spacetime with a barotropic equation of state is given by Equation (36).

3. $\mathit{h}$-Almost Conformal $\mathit{\eta }$-Ricci–Bourguignon Solitons on BPFS in $\mathit{f}(\mathcal{R},\mathcal{T})$ Gravity

For a fixed vector field $\mathcal{E}=\zeta$, and by employing the explicit expression of the Lie derivative in Equation (10), we have

$$\begin{array}{ccc}\hfill \mathcal{R}ic(U,V)& =& \left\{{\gamma }_1+\mathcal{R}\mathrm{\Gamma }-{\displaystyle \frac{1}{2}}\left(\tilde{p}+\frac{1}{2}\right)\right\}g(U,V)\hfill \\ & & -{\displaystyle \frac{h}{2}}\left(g({\nabla }_U\zeta ,V)+g(U,{\nabla }_V\zeta )\right)+{\gamma }_2\eta \left(U\right)\eta \left(V\right).\hfill \end{array}$$

(37)

Contracting Equation (37) yields

$$\mathcal{R}={\displaystyle \frac{1}{1-4\mathrm{\Gamma }}}\left(4{\gamma }_1-\left(2\tilde{p}+1\right)-{\gamma }_2-h\mathrm{div}\zeta \right).$$

(38)

Moreover, combining Equations (24), (29) and (38), we deduce

$$4\chi -\psi ={\displaystyle \frac{1}{1-4\mathrm{\Gamma }}}\left(4{\gamma }_1-\left(2\tilde{p}+1\right)-{\gamma }_2-h\mathrm{div}\zeta \right).$$

(39)

Now, by evaluating Equations (29) and (37) at $U=V=\zeta$, we obtain two expressions:

$$\mathcal{R}ic(\zeta ,\zeta )=-\chi +\psi ,$$

and

$$\mathcal{R}ic(\zeta ,\zeta )=-{\gamma }_1+\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+{\gamma }_2.$$

Since $\mathcal{R}=4\chi -\psi$, equating these expressions and solving for ${\gamma }_1$ leads to

$${\gamma }_1={\displaystyle \frac{1}{3+4\mathrm{\Gamma }}}\left({\displaystyle \frac{1}{4}}(3+4\mathrm{\Gamma })(2\tilde{p}+1)+h\mathrm{div}\zeta +4\mathrm{\Gamma }{\gamma }_2+3(1-4\mathrm{\Gamma })\chi \right).$$

(40)

So, we confirm the following result.

Theorem 3.

If a perfect fluid spacetime with a barotropic equation of state in $f(\mathcal{R},\mathcal{T})$ gravity admits an $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1,\mathrm{\Gamma })$, then the soliton is classified as follows:

• Expanding, if

  $$\tilde{p}>{\displaystyle \frac{-2}{3+4\mathrm{\Gamma }}}\left(3(1-4\mathrm{\Gamma })\chi +\mathit{h}\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}},$$
• Steady, if

  $$\tilde{p}={\displaystyle \frac{-2}{3+4\mathrm{\Gamma }}}\left(3(1-4\mathrm{\Gamma })\chi +\mathit{h}\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}},$$
• Shrinking, if

  $$\tilde{p}<{\displaystyle \frac{-2}{3+4\mathrm{\Gamma }}}\left(3(1-4\mathrm{\Gamma })\chi +\mathit{h}\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}}.$$

Corollary 6.

If a perfect fluid spacetime with a barotropic equation of state in $f(\mathcal{R},\mathcal{T})$ gravity admits a traceless $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1)$, then the soliton is classified as follows:

• Expanding, if

  $$\tilde{p}>{\displaystyle \frac{-1}{2}}\left(h\mathrm{div}\zeta +1\right),$$
• Steady, if

  $$\tilde{p}={\displaystyle \frac{-1}{2}}\left(h\mathrm{div}\zeta +1\right),$$
• Shrinking, if

  $$\tilde{p}<{\displaystyle \frac{-1}{2}}\left(h\mathrm{div}\zeta +1\right).$$

Corollary 7.

If a perfect fluid spacetime with a barotropic equation of state in $f(\mathcal{R},\mathcal{T})$ gravity admits the Schouten $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1)$, then the soliton is classified as follows:

• Expanding, if

  $$\tilde{p}>{\displaystyle \frac{-6}{11}}\left(\chi +h\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}},$$
• Steady, if

  $$\tilde{p}={\displaystyle \frac{-6}{11}}\left(\chi +h\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}},$$
• Shrinking, if

  $$\tilde{p}<{\displaystyle \frac{-6}{11}}\left(\chi +h\mathrm{div}\zeta \right)-{\displaystyle \frac{1}{2}}.$$

Now, let $\zeta$ be a Killing vector field and $f\left(\mathcal{T}\right)=0$, so that $f(\mathcal{R},\mathcal{T})$ gravity reduces to $f\left(\mathcal{R}\right)$ gravity. Then, in view of Equations (33) and (38), we obtain the following.

Corollary 8.

If a perfect fluid spacetime with a barotropic equation of state in $f\left(\mathcal{R}\right)$ gravity admits an $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1,\mathrm{\Gamma })$, where ζ is a Killing vector field, then its barotropic equation of state (BEoS) is given by

$$\upsilon =-{\displaystyle \frac{\omega \left(c^2+\Delta \right)}{5}}+{\displaystyle \frac{4{\gamma }_1-(2\tilde{p}+1)}{40\pi (4\mathrm{\Gamma }-1)}}.$$

Corollary 9.

If a perfect fluid spacetime with a barotropic equation of state in $f\left(\mathcal{R}\right)$ gravity admits an $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1,\mathrm{\Gamma })$, where ζ is a Killing vector field, then the barotropic equation of state (BEoS) for a radiation-type source matter is given by

$$\upsilon ={\displaystyle \frac{4{\gamma }_1-(2\tilde{p}+1)}{64\pi (4\mathrm{\Gamma }-1)}}.$$

Corollary 10.

If a perfect fluid spacetime with a barotropic equation of state in $f\left(\mathcal{R}\right)$ gravity admits an $\mathit{h}$-almost conformal Ricci–Bourguignon soliton $(g,\zeta ,h,{\gamma }_1,\mathrm{\Gamma })$, where ζ is a Killing vector field, then the BEoS for a phantom barrier-type source matter is given by

$$\upsilon ={\displaystyle \frac{4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2}{32\pi (4\mathrm{\Gamma }-1)}}.$$

4. $\mathit{h}$-Almost Conformal RB Soliton on BPFST in $\mathit{f}(\mathcal{R},\mathcal{T})$ Gravity with $\mathit{\zeta }=\mathit{grad}\mathrm{\Phi }$

In this section, we let the velocity vector field $\zeta$ of the h-almost conformal $\eta$-Ricci–Bourguignon soliton be of gradient type; that is, $\zeta =\nabla \mathrm{\Phi }$, where $\mathrm{\Phi }$ is a smooth function on $({\mathrm{\Omega }}^4,g)$. Then, in view of Equations (11), (32) and (39), we conclude the following.

Theorem 4.

Let $({\mathrm{\Omega }}^4,g)$ be a barotropic perfect fluid spacetime (BPFST) in $f(\mathcal{R},\mathcal{T})$ gravity admitting a gradient h-almost conformal η-Ricci–Bourguignon soliton $(g,\nabla \mathrm{\Phi },{\gamma }_1,\mathrm{\Gamma },{\gamma }_2)$, where h is a nonzero constant. Then the modified Poisson equation in the context of $f(\mathcal{R},\mathcal{T})$ gravity satisfied by ζ is

$$\begin{array}{cc}\hfill h{\nabla }^2\left(\zeta \right)& =4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2\hfill \\ \hfill & -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right).\hfill \end{array}$$

Corollary 11.

Let $({\mathrm{\Omega }}^4,g)$ be a barotropic perfect fluid spacetime (BPFST) in $f\left(\mathcal{R}\right)$ gravity admitting a gradient h-almost conformal η-Ricci–Bourguignon soliton $(g,\nabla \mathrm{\Phi },{\gamma }_1,\mathrm{\Gamma },{\gamma }_2)$, where h is a nonzero constant. Then the modified Poisson equation in the context of $f\left(\mathcal{R}\right)$ gravity satisfied by ζ is

$$\begin{array}{cc}\hfill h{\nabla }^2\left(\zeta \right)& =4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right).\hfill \end{array}$$

In particular, if $\mathrm{\Phi }\in C^{\infty }\left(\mathrm{\Omega }\right)$, then for the vector field $\zeta$ we have

$$\mathrm{div}\left(\mathrm{\Phi }\zeta \right)=\zeta \left(\mathrm{d}\mathrm{\Phi }\right)+\mathrm{\Phi }\mathrm{div}\left(\zeta \right).$$

The function $\mathrm{\Phi }\in C^{\infty }\left(\mathrm{\Omega }\right)$ is called a last multiplier of the vector field $\zeta$ with respect to g if

$$\mathrm{div}\left(\mathrm{\Phi }\zeta \right)=0.$$

In this case, it gives rise to the so-called Liouville equation:

$$\zeta (\mathrm{d}ln\mathrm{\Phi })=-\mathrm{div}\left(\zeta \right),$$

which describes how $\mathrm{\Phi }$ compensates for the divergence of $\zeta$ (see [32]).

So, utilizing this fact and Equations (11) and (32), we state the following.

Theorem 5.

Let $({\mathrm{\Omega }}^4,g)$ be a barotropic perfect fluid spacetime (BPFST) in $f(\mathcal{R},\mathcal{T})$ gravity admitting a gradient h-almost conformal η-Ricci–Bourguignon soliton $(g,\nabla \mathrm{\Phi },{\gamma }_1,\mathrm{\Gamma },{\gamma }_2)$, where h is a nonzero constant. Then the modified Liouville equation in $f(\mathcal{R},\mathcal{T})$ gravity satisfied by Φ and ζ is

$$\begin{array}{cc}\hfill \zeta (\mathrm{d}ln\mathrm{\Phi })& ={\displaystyle \frac{1}{h}}[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2\hfill \\ \hfill & -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right)].\hfill \end{array}$$

Again, if $f\left(\mathcal{T}\right)=0$, then $f(\mathcal{R},\mathcal{T})$ gravity reduces to $f\left(\mathcal{R}\right)$ gravity. Thus we have the following.

Corollary 12.

Let $({\mathrm{\Omega }}^4,g)$ be a BPFST in $f\left(\mathcal{R}\right)$ gravity admitting a gradient h-almost conformal η-Ricci–Bourguignon soliton $(g,\nabla \mathrm{\Phi },{\gamma }_1,\mathrm{\Gamma },{\gamma }_2)$, where h is a nonzero constant. Then the modified Liouville equation in $f\left(\mathcal{R}\right)$ gravity satisfied by Φ and ζ is

$$\begin{array}{cc}\hfill \zeta (\mathrm{d}ln\mathrm{\Phi })& ={\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right)\right].\hfill \end{array}$$

5. Harmonic Aspect of Gradient $\mathit{h}$-Almost Conformal $\mathit{\eta }$-RB Soliton on BPFST in $\mathit{f}(\mathcal{R},\mathcal{T})$ Gravity

We consider a function $f:\mathrm{\Omega }\to \mathbb{R}$. Then f is said to be harmonic if ${\nabla }^{2}f=0$, where ${\nabla }^2$ is the Laplacian operator on $\mathrm{\Omega }$ [33]. Since $\zeta =\mathrm{grad}\left(f\right)$, then utilizing Theorem 4, we have the following.

Theorem 6.

Let $({\mathrm{\Omega }}^4,g)$ admit a gradient h-almost conformal η-Ricci–Bourguignon soliton. If f is harmonic, then the soliton in $f(\mathcal{R},\mathcal{T})$ gravity is

• Expanding if

  $$\begin{array}{cc}\hfill \upsilon & >{\displaystyle \frac{1}{(1-4\mathrm{\Gamma })(20\pi +5f^{\prime }\left(\mathcal{T}\right))}}((2\tilde{p}+1)+{\gamma }_2\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })(-2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5c^2\omega +\Delta (\omega +4)])).\hfill \end{array}$$
• Steady if

  $$\begin{array}{cc}\hfill \upsilon & ={\displaystyle \frac{1}{(1-4\mathrm{\Gamma })(20\pi +5f^{\prime }\left(\mathcal{T}\right))}}((2\tilde{p}+1)+{\gamma }_2\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })(-2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5c^2\omega +\Delta (\omega +4)])).\hfill \end{array}$$
• Shrinking if

  $$\begin{array}{cc}\hfill \upsilon & <{\displaystyle \frac{1}{(1-4\mathrm{\Gamma })(20\pi +5f^{\prime }\left(\mathcal{T}\right))}}((2\tilde{p}+1)+{\gamma }_2\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })(-2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5c^2\omega +\Delta (\omega +4)])).\hfill \end{array}$$

In the context of $f\left(\mathcal{R}\right)$-gravity theory, we also obtain the following.

Corollary 13.

If a BPFST in $f\left(\mathcal{R}\right)$ gravity admits a gradient h-almost conformal η-Ricci–Bourguignon soliton, and if f is harmonic, then the soliton in $f\left(\mathcal{R}\right)$ gravity is

• Expanding if

  $$\begin{array}{cc}\hfill \upsilon & >{\displaystyle \frac{1}{(1-4\mathrm{\Gamma })\left(20\pi \right)}}\left((2\tilde{p}+1)+{\gamma }_2+2(1-4\mathrm{\Gamma })(-4\pi \omega (c^2+\Delta )\right).\hfill \end{array}$$
• Steady if

  $$\begin{array}{cc}\hfill \upsilon & ={\displaystyle \frac{1}{(1-4\mathrm{\Gamma })\left(20\pi \right)}}\left((2\tilde{p}+1)+{\gamma }_2+2(1-4\mathrm{\Gamma })(-4\pi \omega (c^2+\Delta )\right).\hfill \end{array}$$
• Shrinking if

  $$\begin{array}{cc}\hfill \upsilon & <{\displaystyle \frac{1}{(1-4\mathrm{\Gamma })\left(20\pi \right)}}\left((2\tilde{p}+1)+{\gamma }_2+2(1-4\mathrm{\Gamma })(-4\pi \omega (c^2+\Delta )\right).\hfill \end{array}$$

6. Gradient $\mathit{h}$-Almost Conformal RB Soliton on BPFST in $\mathit{f}(\mathcal{R},\mathcal{T})$ Gravity

In this section, we focus on the soliton vector field $\zeta =\mathcal{D}\mathrm{\Psi }$, where $\mathrm{\Psi }$ is a smooth potential function and $\mathcal{D}$ denotes the gradient operator with respect to the metric g in the framework of $f(\mathcal{R},\mathcal{T})$ gravity. Substituting this choice of $\zeta$ into Equation (9), one obtains

$$h\mathrm{Hess}\left(\mathrm{\Psi }\right)+\mathcal{R}ic-\left\{{\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{1}{2}+\mathcal{R}\mathrm{\Gamma }\right)\right\}g=0,$$

(41)

where $\mathrm{Hess}\left(\mathrm{\Psi }\right)$ denotes the Hessian of $\mathrm{\Psi }$.

Equivalently, the above relation can be expressed in operator form as

$$h\nabla \mathcal{D}\mathrm{\Psi }+\mathcal{Q}-\left\{{\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{1}{2}+\mathcal{R}\mathrm{\Gamma }\right)\right\}I=0,$$

(42)

where I stands for the identity transformation and $\mathcal{Q}$ denotes the Ricci operator associated with g.

Thus, Equation (41) provides the tensorial form of the soliton structure, while Equation (42) encodes the same information in terms of the action of operators on vector fields. The equivalence between the two expressions highlights the interplay between the analytic formulation via the Hessian of $\mathrm{\Psi }$ and the geometric formulation involving the Ricci operator.

Contracting Equation (42) with respect to the metric g, and subsequently applying Corollary 1, we arrive at the following identity for the Laplacian of the potential function $\mathrm{\Psi }$:

$$\begin{array}{cc}\hfill \Delta \mathrm{\Psi }=& {\displaystyle \frac{1}{h}}[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2\hfill \\ \hfill & -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )-f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right)].\hfill \end{array}$$

(43)

Here, $\Delta \mathrm{\Psi }=\mathrm{div}\left(\mathcal{D}\mathrm{\Psi }\right)$ denotes the Laplacian of $\mathrm{\Psi }$ with respect to g.

Equation (43) reveals that the behavior of $\mathrm{\Psi }$ is governed not only by the intrinsic soliton parameters $(h,{\gamma }_1)$, but also by the fluid-mechanical variables $(\tilde{p},\upsilon ,\omega ,c^2,\Delta )$ and the functional dependence of the modified gravity term $f\left(\mathcal{T}\right)$ and its derivative $f^{\prime }\left(\mathcal{T}\right)$. This highlights the interplay between the geometric soliton structure and the underlying non-Einsteinian contributions.

Theorem 7.

Let $({\mathrm{\Omega }}^4,g)$ be a four-dimensional pseudo-Riemannian manifold admitting a gradient h-almost conformal Ricci–Bourguignon soliton in the framework of $f(\mathcal{R},\mathcal{T})$ gravity. Then the potential function Ψ of the soliton satisfies the Poisson-type Equation (43).

In particular, the soliton potential $\mathrm{\Psi }$ behaves as the solution of a Poisson equation where the source term depends linearly on the structural parameters of the gravitational modification $f\left(\mathcal{T}\right)$ and on the fluid variables. This result provides a bridge between the analytic properties of $\mathrm{\Psi }$ and the physical data encoded in $f(\mathcal{R},\mathcal{T})$ gravity.

Corollary 14.

Let a perfect fluid spacetime (PFST) with a barotropic equation of state in $f\left(\mathcal{R}\right)$ gravity admit a gradient h-almost conformal Ricci–Bourguignon soliton. Then the corresponding soliton potential function Ψ satisfies the Poisson-type equation

$$\begin{array}{cc}\hfill \Delta \mathrm{\Psi }=& {\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right)\right].\hfill \end{array}$$

Corollary 15.

Under the same assumptions, the potential function Ψ of the soliton satisfies the Laplace equation (i.e., $\Delta \mathrm{\Psi }=0$) if and only if the soliton parameter ${\gamma }_1$ obeys the balancing condition

$$\begin{array}{c}\hfill 4{\gamma }_1=(2\tilde{p}+1)+{\gamma }_2+2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right).\end{array}$$

In [33,34], it was established that the following barotropic equations of state (BEoS)

$$\upsilon =-\omega (c^2+\Delta ),\upsilon =\omega (c^2+\Delta ),3\upsilon =\omega (c^2+\Delta ),\upsilon =0,$$

correspond, respectively, to the dark matter era, the stiff matter era, the radiation era, and the dust matter era. Combining these physical scenarios with the general result of Theorem 7, one arrives at the following consequence.

Corollary 16.

If $({\mathrm{\Omega }}^4,g)$ admits a gradient h-almost conformal Ricci–Bourguignon soliton, then the potential function Ψ of the soliton satisfies a Poisson-type equation in $f\left(\mathcal{R}\right)$ gravity whose form depends on the cosmological era considered. Explicitly, one has the following.

f$\left(\mathcal{R}\right)$ gravity era	BEoS	Poisson’s equation
Dark matter era	$\upsilon =-\omega \left(c^2+\Delta \right)$	$\Delta \mathrm{\Psi }={\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(16\pi \omega (c^2+\Delta )\right)\right]$
Stiff matter era	$\upsilon =\omega \left(c^2+\Delta \right)$	$\Delta \mathrm{\Psi }={\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(24\pi \omega (c^2+\Delta )\right)\right]$
Radiation era	$3\upsilon =\omega \left(c^2+\Delta \right)$	$\Delta \mathrm{\Psi }={\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(32\pi \upsilon \right)\right]$
Dust matter era	$\upsilon =0$	$\Delta \mathrm{\Psi }={\displaystyle \frac{1}{h}}\left[4{\gamma }_1-(2\tilde{p}+1)-{\gamma }_2-2(1-4\mathrm{\Gamma })\left(-4\pi \omega (c^2+\Delta )\right)\right]$

Remark 3.

To ensure that the Poisson-type equation in this section admits a unique and physically meaningful solution for the potential Ψ, we assume, for instance, that the spacetime is asymptotically flat and impose the boundary condition $\mathrm{\Psi }\to 0$ as the spatial distance tends to infinity. Under these standard assumptions, the existence and uniqueness of the solution are guaranteed by the well-established analytical properties of the Laplace operator on Euclidean-type manifolds.

Again, starting from Equation (8), we obtain

$$h{\nabla }_U\mathcal{D}f+\mathcal{Q}U-\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+\mathcal{R}\mathrm{\Gamma }\right)U=0,$$

(44)

which directly implies that

$$\begin{array}{cc}\hfill h{\nabla }_V{\nabla }_U\mathcal{D}f& =-\left({\nabla }_V\mathcal{Q}\right)\left(U\right)-\mathcal{Q}\left({\nabla }_{V}U\right)\hfill \\ \hfill & +\left({\gamma }_1-\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+\mathcal{R}\mathrm{\Gamma }\right)\left({\nabla }_{V}U\right)+\mathrm{\Gamma }V\left(\mathcal{R}\right)U.\hfill \end{array}$$

(45)

Now, combining Equations (44) and (45), and recalling the curvature identity

$$\mathcal{R}(U,V)\mathcal{D}f={\nabla }_U{\nabla }_V\mathcal{D}f-{\nabla }_V{\nabla }_U\mathcal{D}f-{\nabla }_{[U,V]}\mathcal{D}f,$$

(46)

we deduce the equivalent expression

$$\mathcal{R}(U,V)\mathcal{D}f={\displaystyle \frac{1}{h}}\left(\left({\nabla }_V\mathcal{Q}\right)\left(U\right)-\left({\nabla }_U\mathcal{Q}\right)\left(V\right)\right)+{\displaystyle \frac{\mathrm{\Gamma }}{h}}\left(U\left(\mathcal{R}\right)V-V\left(\mathcal{R}\right)U\right).$$

(47)

Furthermore, by employing Equation (29) in index-free notation, one arrives at

$$\left({\nabla }_U\mathcal{Q}\right)\left(V\right)=U\left(\chi \right)V+U\left(\psi \right)\eta \left(V\right)\zeta +\psi \left({\nabla }_U\eta \right)\left(V\right)\zeta +\psi \eta \left(V\right){\nabla }_U\zeta .$$

(48)

This relation highlights the contribution of both the scalar functions $\chi$ and $\psi$, as well as the $(1,1)$-tensor $\eta$ and the vector field $\zeta$, in determining the behavior of the tensor field $\mathcal{Q}$ under covariant differentiation. It also emphasizes how the coupling terms involving $\eta$ and $\zeta$ enrich the curvature identity in the presence of an h-almost conformal Ricci–Bourguignon soliton structure.

In view of the preceding relation and by making use of Equation (48), we deduce from Equation (47) that

$$\begin{array}{cc}\hfill \mathcal{R}(U,V)\mathcal{D}f& ={\displaystyle \frac{1}{h}}\left(V\left(\chi \right)U-U\left(\chi \right)V\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{h}}\left(V\left(\psi \right)\eta \left(U\right)-U\left(\psi \right)\eta \left(V\right)\right)\zeta \hfill \\ \hfill & +{\displaystyle \frac{1}{h}}\psi \left(\left({\nabla }_V\eta \right)\left(U\right)-\left({\nabla }_U\eta \right)\left(V\right)\right)\zeta \hfill \\ \hfill & +{\displaystyle \frac{1}{h}}\psi \left(\eta \left(U\right){\nabla }_V\zeta -\eta \left(V\right){\nabla }_U\zeta \right)\hfill \\ \hfill & +{\displaystyle \frac{\mathrm{\Gamma }}{h}}\left(U\left(\mathcal{R}\right)V-V\left(\mathcal{R}\right)U\right).\hfill \end{array}$$

(49)

Next, contracting Equation (49) with respect to the vector field U, we obtain

$$\mathcal{R}ic(V,\mathcal{D}f)={\displaystyle \frac{V\left(\psi \right)}{2h}}-{\displaystyle \frac{3V\left(\chi \right)}{h}}-{\displaystyle \frac{1}{h}}\zeta \left(\psi \right)\eta \left(V\right)-{\displaystyle \frac{1}{h}}\psi \left(\left({\nabla }_{\zeta }\eta \right)\left(V\right)+\eta \left(V\right)\mathrm{div}\zeta \right).$$

(50)

On the other hand, by using Equation (29), the same component of the Ricci tensor may be written as

$$\mathcal{R}ic(V,\mathcal{D}f)=\chi V\left(f\right)+\psi \eta \left(V\right)\zeta \left(f\right).$$

(51)

In particular, if we fix $V=\zeta$, the above two expressions specialize to

$$\mathcal{R}ic(\zeta ,\mathcal{D}f)={\displaystyle \frac{3\zeta \left(\psi \right)}{2h}}-{\displaystyle \frac{3\zeta \left(\chi \right)}{h}}+{\displaystyle \frac{\psi }{h}}\mathrm{div}\zeta ,$$

(52)

and

$$\mathcal{R}ic(\zeta ,\mathcal{D}f)=(\chi -\psi )\zeta \left(f\right).$$

(53)

By equating these two representations of $\mathcal{R}ic(\zeta ,\mathcal{D}f)$, we finally arrive at the relation

$$3\zeta \left(\psi \right)-6\zeta \left(\chi \right)+2\psi \mathrm{div}\zeta =2h(\chi -\psi )\zeta \left(f\right).$$

(54)

This identity provides a differential constraint linking the functions $\chi$ and $\psi$, the vector field $\zeta$, and the potential function f. In particular, it shows how the divergence of $\zeta$ and the derivatives of the scalar functions $\chi$ and $\psi$ along $\zeta$ must balance the interaction term involving f in the soliton structure. Now, let $\zeta$ be a Killing vector field; that is, ${\mathfrak{L}}_{\zeta }g=0$. If, in addition, the scalar functions $\chi$ and $\psi$ are invariant along $\zeta$, i.e., $\zeta \left(\chi \right)=\zeta \left(\psi \right)=0$, then it follows that $\mathrm{div}\zeta =0$ as well. Consequently, from the preceding identity we obtain the alternative

$$\chi =\psi \mathrm{or}\zeta \left(f\right)=0.$$

Thus, it is natural to distinguish between the following two cases.

Case I. Suppose that $\chi =\psi$ while $\zeta \left(f\right)\ne 0$. In this situation, Equation (30) reduces to

$$\upsilon ={\displaystyle \frac{-\mathcal{R}-4f^{\prime }\left(\mathcal{T}\right)\left(c^2\omega -\Delta \right)}{12f^{\prime }\left(\mathcal{T}\right)+48\pi }}-{\displaystyle \frac{\omega c^2+\omega \Delta }{3}},$$

(55)

which provides an explicit expression for $\upsilon$ in terms of the scalar curvature R, the scalar function $\omega$, and the potential T.

Case II. Suppose instead that $\chi \ne \psi$ while $\zeta \left(f\right)=0$. Differentiating the relation $g(\zeta ,\mathcal{D}f)=0$ covariantly along an arbitrary vector field U and making use of Equation (44) together with Equation (29), we obtain

$$g({\nabla }_U\zeta ,\mathcal{D}f)=-{\displaystyle \frac{1}{h}}\left(\chi +\psi -{\gamma }_1+\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+\mathcal{R}\mathrm{\Gamma }\right)\eta \left(U\right).$$

(56)

Since $\zeta$ is a Killing vector field, we recall that

$$g({\nabla }_{\zeta }U,V)+g(U,{\nabla }_{\zeta }V)=0\mathrm{for}\mathrm{all}U,V.$$

In particular, fixing $V=\zeta$, we obtain

$$g(U,{\nabla }_{\zeta }\zeta )=0,$$

because $g({\nabla }_U\zeta ,\zeta )=0$ holds identically. Hence, we deduce that

$${\nabla }_{\zeta }\zeta =0,$$

That is, the integral curves of the Killing vector field $\zeta$ are geodesics.

From the preceding relation it follows immediately that

$${\gamma }_1=\chi +\psi +\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+\mathcal{R}\mathrm{\Gamma }.$$

(57)

In view of Equation (30), this identity further implies that

$$\begin{array}{cc}\hfill {\gamma }_1& ={\displaystyle \frac{\mathcal{R}(1+2\mathrm{\Gamma })}{2}}+\frac{1}{2}\left(\tilde{p}+\frac{1}{2}\right)+2\upsilon \left(f^{\prime }\left(\mathcal{T}\right)+4\pi \right)\hfill \\ \hfill & +2f^{\prime }\left(\mathcal{T}\right)\left(c^2\omega -\Delta \left(\omega \right)\right)+f\left(\mathcal{T}\right)-2\left(f^{\prime }\left(\mathcal{T}\right)+4\pi \right)\left(\omega (c^2+\Delta \left(\omega \right))+\upsilon \right).\hfill \end{array}$$

(58)

Thus, the soliton parameter ${\gamma }_1$ is expressed explicitly in terms of the scalar curvature $\mathcal{R}$, the equation of state parameter $\tilde{p}$, the potential $\mathcal{T}$, the scalar function $\omega$, the auxiliary variable $\upsilon$, and the constants $c,\mathcal{R},\mathrm{\Gamma }$.

Theorem 8.

Let $({\mathrm{\Omega }}^4,g)$ be a pseudo-Finsler spacetime admitting a gradient h-almost conformal Ricci–Bourguignon soliton. If the velocity vector field ζ is Killing and the scalars χ and ψ are invariant along ζ, then the soliton parameter ${\gamma }_1$ is determined by Equation (58).

Corollary 17.

Suppose that a perfect fluid spacetime (PFST) with a barotropic equation of state in $f\left(R\right)$ gravity admits a gradient h-almost conformal Ricci–Bourguignon soliton. If the velocity vector field ζ is Killing and the scalars χ and ψ are invariant along ζ, then the character of the soliton is determined by the scalar curvature R as follows:

• Expanding, if

  $$\mathcal{R}>{\displaystyle \frac{16\pi \omega \left(c^2-\Delta \left(\omega \right)\right)-\left(\tilde{p}+\frac{1}{2}\right)}{1+2\mathrm{\Gamma }}},$$
• Steady, if

  $$\mathcal{R}={\displaystyle \frac{16\pi \omega \left(c^2-\Delta \left(\omega \right)\right)-\left(\tilde{p}+\frac{1}{2}\right)}{1+2\mathrm{\Gamma }}},$$
• Shrinking, if

  $$\mathcal{R}<{\displaystyle \frac{16\pi \omega \left(c^2-\Delta \left(\omega \right)\right)-\left(\tilde{p}+\frac{1}{2}\right)}{1+2\mathrm{\Gamma }}}.$$

Corollary 18.

Let the metric of BPFST in $f\left(\mathcal{R}\right)$ gravity admit a gradient h-almost conformal Ricci–Bourguignon soliton. If ζ is Killing and the scalars χ and ψ are invariant along ζ, then we have the following.

$BPFST$	EoS	Nature of soliton
Dark matter era	$\upsilon =-\omega (c^2+\Delta )$	$\mathcal{R}⪌{\displaystyle \frac{16\pi (\upsilon +\omega (c^2+\Delta )-(\tilde{p}+\frac{1}{2})}{(1+2\mathrm{\Gamma })}}$
Stiff matter era	$\upsilon =\omega (c^2+\Delta )$	$\mathcal{R}⪌{\displaystyle \frac{16\pi (\upsilon -(\tilde{p}+\frac{1}{2})}{(1+2\mathrm{\Gamma })}}$
Radiation era	$3\upsilon =\omega (c^2+\Delta )$	$\mathcal{R}⪌{\displaystyle \frac{48\pi (\upsilon -(\tilde{p}+\frac{1}{2})}{(1+2\mathrm{\Gamma })}}$
Dust matter era	$\upsilon =0$	$\mathcal{R}⪌{\displaystyle \frac{16\pi (\omega (c^2+\Delta )-(\tilde{p}+\frac{1}{2})}{(1+2\mathrm{\Gamma })}}$

7. Harmonic Aspect of a Gradient h-Almost Conformal Ricci–Bourguignon Soliton on BPFST in $\mathit{f}(\mathcal{R},\mathcal{T})$ Gravity

A smooth function $\mathrm{\Psi }:\mathrm{\Omega }\to \mathbb{R}$ is said to be harmonic if it satisfies the condition $\Delta \mathrm{\Psi }=0$, where $\Delta$ denotes the Laplacian operator on $\mathrm{\Omega }$ [33]. In the framework of $f(\mathcal{R},\mathcal{T})$ gravity, the presence of a harmonic potential function allows for a refined classification of gradient h-almost conformal Ricci–Bourguignon solitons. From Theorem 7, we deduce the following result.

Theorem 9.

Let $({\mathrm{\Omega }}^4,g)$ be a spacetime in $f(\mathcal{R},\mathcal{T})$ gravity that admits a gradient h-almost conformal Ricci–Bourguignon soliton with harmonic potential function Ψ. Then the soliton is classified according to the following conditions:

• Expanding if

  $$\begin{array}{cc}\hfill 2\left(\tilde{p}+\frac{1}{2}\right)>& -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })\left(f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right),\hfill \end{array}$$
• Steady if

  $$\begin{array}{cc}\hfill 2\left(\tilde{p}+\frac{1}{2}\right)=& -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })\left(f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right),\hfill \end{array}$$
• Shrinking if

  $$\begin{array}{cc}\hfill 2\left(\tilde{p}+\frac{1}{2}\right)<& -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & +2(1-4\mathrm{\Gamma })\left(f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\right).\hfill \end{array}$$

The classification hinges on the relative size of the conformal pressure $\tilde{p}$ compared with a threshold determined by the parameters $\mathcal{R},\mathrm{\Gamma },\upsilon ,\omega ,c^2,\Delta$ and the functions $f\left(\mathcal{T}\right),f^{\prime }\left(\mathcal{T}\right)$. In particular, the harmonicity of $\mathrm{\Psi }$ eliminates additional Laplacian terms and simplifies the soliton equations, thereby yielding a sharper criterion for distinguishing expanding, steady, and shrinking solitons.

For the special case $f\left(\mathcal{T}\right)=0$, one recovers the framework of $f\left(\mathcal{R}\right)$ gravity. In this setting, the influence of the trace term $\mathcal{T}$ disappears, and the classification criteria become simpler. From Corollary 14, we deduce the following consequence.

Corollary 19.

Let the BPFST in $f\left(\mathcal{R}\right)$ gravity, satisfying the Einstein field equations without a cosmological constant, admit a gradient h-almost conformal Ricci–Bourguignon soliton with harmonic potential function Ψ. Then the soliton can be classified as follows:

• Expanding if

  $$\begin{array}{cc}\hfill 2\left(\tilde{p}+\frac{1}{2}\right)>& -2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right),\hfill \end{array}$$
• Steady if

  $$2\left(\tilde{p}+\frac{1}{2}\right)=-2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right),$$
• Shrinking if

  $$2\left(\tilde{p}+\frac{1}{2}\right)<-2(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right).$$

In the $f\left(\mathcal{R}\right)$ gravity case, the classification depends solely on the conformal pressure $\tilde{p}$ and the parameters $\mathcal{R},\mathrm{\Gamma },\upsilon ,\omega ,c^2$, and $\Delta$. The absence of $f\left(\mathcal{T}\right)$ and its derivative significantly reduces the complexity of the equilibrium condition. Consequently, the role of the harmonic potential function $\mathrm{\Psi }$ is even more transparent, as it ensures the soliton dynamics are governed by a balance between geometric contributions and fluid parameters without additional coupling from $\mathcal{T}$.

Recall that a smooth function $\mathrm{\Psi }$ on a semi-Riemannian manifold $\mathrm{\Omega }$ is said to be harmonic, subharmonic, or superharmonic if it satisfies

$$\Delta \mathrm{\Psi }=0,\Delta \mathrm{\Psi }\ge 0,\mathrm{or}\Delta \mathrm{\Psi }\le 0,$$

respectively. Combining this fact with Theorem 7, we arrive at the following characterization of the soliton parameter.

Theorem 10.

Let ${\mathrm{\Omega }}^4$ admit a gradient h-almost conformal Ricci–Bourguignon soliton with potential function Ψ. Then Ψ is harmonic, subharmonic, or superharmonic if and only if the soliton parameter ${\gamma }_1$ satisfies the following, respectively:

• Harmonic:

  $$\begin{array}{cc}\hfill {\gamma }_1& ={\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\hfill \end{array}$$
• Subharmonic:

  $$\begin{array}{cc}\hfill {\gamma }_1& \ge {\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\hfill \end{array}$$
• Superharmonic:

  $$\begin{array}{cc}\hfill {\gamma }_1& \le {\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -2f\left(\mathcal{T}\right)-4\pi \omega (c^2+\Delta )\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })f^{\prime }\left(\mathcal{T}\right)[5\upsilon +5c^2\omega +\Delta (\omega +4)]\hfill \end{array}$$

The theorem shows that the harmonicity type of the potential function $\mathrm{\Psi }$ is fully determined by the soliton parameter ${\gamma }_1$. In the harmonic case, ${\gamma }_1$ is exactly balanced by the contributions of the conformal pressure $\tilde{p}$, the fluid parameters $(\upsilon ,\omega ,c^2,\Delta )$, and the functions $f\left(\mathcal{T}\right),f^{\prime }\left(\mathcal{T}\right)$. For the subharmonic and superharmonic cases, ${\gamma }_1$ is bounded below or above by the same threshold, respectively. This provides a direct analytic criterion linking the geometric soliton structure with the analytic behavior of the potential function.

Corollary 20.

Let ${\mathrm{\Omega }}^4$ in $f\left(\mathcal{R}\right)$ gravity admit a gradient h-almost conformal Ricci–Bourguignon soliton with potential function Ψ. Then Ψ is classified as follows:

• Harmonic if

  $$\begin{array}{c}\hfill {\gamma }_1={\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right)\end{array}$$
• Subharmonic if

  $$\begin{array}{c}\hfill {\gamma }_1\ge {\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right)\end{array}$$
• Superharmonic if

  $$\begin{array}{c}\hfill {\gamma }_1\le {\displaystyle \frac{1}{4}}(2\tilde{p}+1)+{\displaystyle \frac{1}{2}}(1-4\mathrm{\Gamma })\left(-20\pi \upsilon -4\pi \omega (c^2+\Delta )\right)\end{array}$$

Compared with Theorem 10, the absence of the functions $f\left(\mathcal{T}\right)$ and $f^{\prime }\left(\mathcal{T}\right)$ in $f\left(\mathcal{R}\right)$ gravity leads to a substantially simpler threshold condition for ${\gamma }_1$. In this framework, the harmonicity type of the potential function $\mathrm{\Psi }$ depends only on the conformal pressure $\tilde{p}$, the geometric parameters $\mathcal{R},\mathrm{\Gamma }$, and the fluid quantities $\upsilon ,\omega ,c^2,\Delta$. This highlights the fact that the trace contribution $\mathcal{T}$ plays a decisive role in complicating the analytic structure of the soliton equations in the more general $f(\mathcal{R},\mathcal{T})$ setting.

8. Physical Significance of Conformal Pressure

According to [35], the time-dependent scalar field $\tilde{p}$ is referred to as the conformal pressure. In the context of classical fluid mechanics, the physical pressure is the essential quantity responsible for maintaining the incompressibility condition of a fluid. By contrast, the conformal pressure $\tilde{p}$, which satisfies $\tilde{p}\ge 0$, plays a more subtle role. In particular, $\tilde{p}$ assumes negative values outside an equilibrium point and vanishes precisely at equilibrium. This behavior indicates that $\tilde{p}$ can be interpreted as a generalized restoring term in the dynamical evolution of the system. In geometric terms, the underlying metric g corresponds to an equilibrium configuration when it reduces to an Einstein metric, thus producing a nonlinear restoring force that stabilizes the spacetime geometry.

In light of Equation (40), one can derive a more explicit expression for the conformal pressure in the framework of $f(\mathcal{R},\mathcal{T})$ gravity. This leads to the following theorem, which connects the geometry of the spacetime with the thermodynamical properties of the fluid.

Theorem 11.

Let $({\mathrm{\Omega }}^4,g,{\gamma }_1,{\gamma }_2)$ be a perfect fluid spacetime with a barotropic equation of state in $f(\mathcal{R},\mathcal{T})$ gravity. If the spacetime admits an h-almost conformal η-Ricci–Bourguignon soliton, then the conformal pressure $\tilde{p}$ is given explicitly by

$$\tilde{p}={\displaystyle \frac{2}{3+4\mathrm{\Gamma }}}\left((3+4\mathrm{\Gamma }){\gamma }_1-h\mathrm{div}\zeta -4\mathrm{\Gamma }{\gamma }_2-3(1-4\mathrm{\Gamma })\chi \right)-{\displaystyle \frac{1}{2}}.$$

The above relation shows that $\tilde{p}$ depends linearly on the geometric quantities ${\gamma }_1$, ${\gamma }_2$, and $\chi$ and the divergence of $\zeta$, while being modulated by the parameter $\mathrm{\Gamma }$. In particular, for $\mathrm{\Gamma }=0$ the contribution of ${\gamma }_2$ vanishes, whereas for $\mathrm{\Gamma }=\frac{-3}{4}$ the term involving ${\gamma }_1$ disappears. Such cancellations highlight the special physical regimes in which the conformal pressure simplifies considerably.

Corollary 21.

Suppose a perfect fluid spacetime with a barotropic equation of state in $f(\mathcal{R},\mathcal{T})$ gravity admits an h-almost conformal η-Ricci–Bourguignon soliton $({\mathrm{\Omega }}^4,g,{\gamma }_1,{\gamma }_2)$. Then the metric g represents an equilibrium configuration; i.e., it reduces to an Einstein metric if and only if the following relation holds:

$$4\left((3+4\mathrm{\Gamma }){\gamma }_1-h\mathrm{div}\zeta -4\mathrm{\Gamma }{\gamma }_2-3(1-4\mathrm{\Gamma })\chi \right)=3+4\mathrm{\Gamma }.$$

Corollary 22.

Under the same assumptions, if the metric g is an equilibrium point, then it plays the role of a nonlinear restoring force in the evolution of the conformal pressure $\tilde{p}$. In particular, departures from equilibrium are counterbalanced by the geometry of the spacetime, ensuring that the pressure tends to stabilize around the Einstein configuration.

It is worth noting that special values of the parameter $\mathrm{\Gamma }$ yield simplified equilibrium conditions. For instance, when $\mathrm{\Gamma }=0$ the contribution of the parameter ${\gamma }_2$ vanishes, while for $\mathrm{\Gamma }=\frac{-3}{4}$ the term involving ${\gamma }_1$ disappears. In both situations, the equilibrium condition reduces to a more tractable relation among the remaining parameters. These cases correspond to special dynamical regimes in which the geometry of the spacetime admits a more direct stabilization mechanism for the conformal pressure.

9. Conclusions

In this work, we have undertaken a systematic study of barotropic perfect fluid spacetimes within the framework of $f(\mathcal{R},\mathcal{T})$ gravity, focusing on the geometric and physical implications of h-almost conformal $\eta$-Ricci–Bourguignon solitons and their gradient counterparts. By deriving explicit conditions under which such spacetimes admit expanding, steady, or shrinking soliton structures, we have established analytic criteria linking the fluid parameters, conformal pressure, and scalar curvature with the soliton dynamics. The results demonstrate that the inclusion of the trace term T substantially enriches the mathematical structure of the soliton equations compared to the $f\left(\mathcal{R}\right)$ gravity case, thereby highlighting the crucial role of the matter–geometry coupling.

Furthermore, we have shown that the conformal pressure $\tilde{p}$ plays the role of a restoring force in the dynamical evolution of the system, stabilizing the underlying metric towards Einstein configurations. This interpretation provides a deeper physical insight into the interplay between the fluid equation of state and the self-similar evolution of the spacetime geometry. The harmonic, subharmonic, and superharmonic classifications of the potential function $\mathrm{\Psi }$ offer an analytic framework that links the geometric soliton structure with thermodynamical features of the fluid.

Overall, the findings of this paper contribute to the broader understanding of modified gravity theories by revealing how soliton dynamics in $f(\mathcal{R},\mathcal{T})$ gravity can model different cosmological scenarios, including radiation, stiff matter, dust, and phantom eras. These results may serve as a foundation for further investigations into the stability of self-similar solutions, the role of conformal pressures in cosmic evolution, and potential applications in modeling dark energy and dark matter alternatives within extended theories of gravity.

It is worth noting that the method adopted in this work can be naturally connected with the symmetry approach based on Noether’s theorem, which is commonly employed to determine suitable functional forms of $f(R,T)$ in modified gravity theories. Establishing such a link may provide additional constraints and a systematic way to select physically meaningful models. This potential connection will be explored in future investigations.