Further matters in space-time geometry: $f(R,T,R_{μν}T^{μν})$ gravity (1304.5957v3)

Abstract: We consider a gravitational model in which matter is non-minimally coupled to geometry, with the effective Lagrangian of the gravitational field being given by an arbitrary function of the Ricci scalar, the trace of the matter energy-momentum tensor, and the contraction of the Ricci tensor with the matter energy-momentum tensor. The field equations of the model are obtained in the metric formalism, and the equation of motion of a massive test particle is derived. In this type of models the matter energy-momentum tensor is generally not conserved, and this non-conservation determines the appearance of an extra-force acting on the particles in motion in the gravitational field. The Newtonian limit of the model is also considered, and an explicit expression for the extra-acceleration which depends on the matter density is obtained in the small velocity limit for dust particles. We also analyze in detail the so-called Dolgov-Kawasaki instability, and obtain the stability conditions of the model with respect to local perturbations. A particular class of gravitational field equations can be obtained by imposing the conservation of the energy-momentum tensor. We derive the corresponding field equations for the conservative case by using a Lagrange multiplier method, from a gravitational action that explicitly contains an independent parameter multiplying the divergence of the energy-momentum tensor. The cosmological implications of the model are investigated for both the conservative and non-conservative cases, and several classes of analytical solutions are obtained.

Insightful Overview of the f(R,T,R_{\mu\nu}) Gravity Theory

The paper "Further Matters in Space-time Geometry: f(R,T,R_{\mu\nu}) Gravity" by Zahra Haghani et al. explores an extension of the f(R) gravitational theories by considering a framework where the Lagrangian depends not only on the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$, but also on the contraction of the Ricci tensor $R\_{\mu\nu}$ with $T\_{\mu\nu}$. This development aims to provide more insights into the possible deviations from General Relativity (GR) and address the challenges related to dark matter and dark energy without the need for exotic forms of matter.

Field Equations and Particle Motion

The authors derived the field equations in the metric formalism for this class of theories, achieving a particular emphasis on the non-conservation of the energy-momentum tensor. This non-conservation translates into an additional force that acts on test particles within the gravitational field. This force is contingent upon the Ricci tensor, indicating significant departures from geodesic motion, especially when there is a strong matter-geometry coupling. This distinct feature could potentially address cosmic phenomena such as the galaxy rotation curves, normally attributed to dark matter.

The Newtonian Limit and Stability Analysis

In examining the Newtonian limit, the paper formulates the modified Poisson equation and identifies the extra-acceleration phenomenon, which depends on the matter density. The analysis of the Dolgov-Kawasaki instability reveals the conditions for the theory's stability against local perturbations. These conditions ensure the model's viability in predicting cosmological behaviors consistent with observational evidence while avoiding fatal instabilities characteristic of certain f(R) models.

Energy-momentum Conservation and Lagrange Multipliers

The paper explores a scenario where the energy-momentum tensor is conserved through a Lagrange multiplier method. By introducing a vector field in the gravitational action, the authors manage to impose the conservation condition effectively. This step ensures consistency with standard physical expectations and allows exploring cosmological solutions both with and without energy-momentum conservation.

Cosmological Implications

The cosmological potential of the theory is investigated extensively, resulting in various exact analytical and approximate solutions. Notably, when considering specific functional forms such as $f = R + \alpha R\_{\mu\nu} T\_{\mu\nu}$, the theory leads to accelerated expansion akin to a de Sitter phase, possibly offering an alternative explanation for the late-time acceleration of the Universe—usually attributed to dark energy. The theory's versatility allows for distinct cosmological dynamics depending on the specific form chosen for $f$.

Numerical Solutions and Theoretical Prospects

The authors provide insights into future research directions, suggesting that the framework could be expanded to explore further cosmological and astrophysical phenomena. The inclusion of terms involving the contraction of $R\_{\mu\nu}$ with $T\_{\mu\nu}$ opens new perspectives on long-standing issues in gravitational physics, potentially impacting our understanding of fundamental interactions at cosmic scales.

This paper thus contributes significantly to the ongoing exploration of modified theories of gravity, expanding the theoretical toolkit available for addressing the unresolved challenges in cosmology. Future research in this direction could unveil more comprehensive understandings of the interplay between matter and geometry, and how it defines the structure and evolution of the Universe.