f(R,L_m) gravity (1008.4193v2)

Abstract: We generalize the $f(R)$ type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar $R$ and of the matter Lagrangian $L_m$. We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert--Einstein Lagrange density are also derived.

• The paper introduces a generalized f(R, L_m) gravity model that derives modified field equations using the metric formalism, revealing non-geodesic motion of particles.
• It explores an exponential dependence on the Ricci scalar and matter Lagrangian, suggesting a dynamically evolving cosmological constant as an alternative to dark energy.
• The study highlights potential equivalence principle violations, offering new avenues for experimental tests that could distinguish these models from standard general relativity.

Overview of ($f(R, L_m)$) Gravity

The paper authored by Tiberiu Harko and Francisco S. N. Lobo investigates a generalized form of modified gravity, denoted as $f(R, L_m)$, where the gravitational Lagrangian is a function of both the Ricci scalar $R$ and the matter Lagrangian $L_m$. This research builds upon and extends the foundational principles of $f(R)$ gravity models, which modify the Einstein-Hilbert action to potentially elucidate cosmic phenomenology like the late-time acceleration of the universe and dark matter dynamics without invoking additional fields or particles.

Field Equations and Particle Motion

In this model, the gravitational field equations are derived using the metric formalism. A significant consequence of this theoretical framework is the resulting non-zero covariant divergence of the energy-momentum tensor, indicating non-geodesic motion for test particles. Notably, this motion involves an extra force orthogonal to the particle's four-velocity. This non-geodesic nature implies potential violations of the equivalence principle, presenting opportunities to test these models, especially given suggestions that equivalence principle violations might occur due to dark matter and dark energy interactions.

Furthermore, the Newtonian limit of these equations is explored, and a method to derive the energy-momentum tensor is proposed. This is crucial for understanding the classical limits of this extended theory and its concordance with existing empirical data within the solar system regime.

A Specific Model: Exponential Dependency

The paper explores a specific $f(R, L_m)$ model where the action features an exponential dependency on the Ricci scalar and the matter Lagrangian. The field equations for this model suggest a dynamically evolving cosmological constant, offering a potential mechanism to account for the universe's observed accelerated expansion. This approach juxtaposes the cosmological constant's static nature in standard general relativity.

Theoretical and Practical Implications

The $f(R, L_m)$ gravity model, by coupling geometry and matter more intricately than traditional frameworks, opens up several theoretical avenues. The potential violations of the equivalence principle and the distinct field equations for this theory could lead to observational signatures distinguishable from standard general relativity or other modified gravity theories. Practically, these models might yield insights into cosmic phenomena traditionally attributed to dark energy and dark matter, offering explanations without resorting to unknown particles.

Conclusion

By presenting a maximal extension of $f(R)$ gravity theories, this paper contributes to the ongoing discourse on modified gravity models, providing a richer framework for linking gravitational dynamics with matter effects. Future theoretical work may focus on further exploring the cosmological and astrophysical implications of $f(R, L_m)$ models, while empirical investigations could leverage violations of the equivalence principle as a new testing ground for gravity theories. As such, this work holds promise for broadening our understanding of gravitational phenomena at both cosmic and local scales.