Nonminimal coupling of perfect fluids to curvature (0806.4434v2)

Abstract: In this work, we consider different forms of relativistic perfect fluid Lagrangian densities, that yield the same gravitational field equations in General Relativity. A particularly intriguing example is the case with couplings of the form $[1+f_2(R)]{\cal L}_m$, where $R$ is the scalar curvature, which induces an extra force that depends on the form of the Lagrangian density. It has been found that, considering the Lagrangian density ${\cal L}_m = p$, where $p$ is the pressure, the extra-force vanishes. We argue that this is not the unique choice for the matter Lagrangian density, and that more natural forms for ${\cal L}_m$ do not imply the vanishing of the extra-force. Particular attention is paid to the impact on the classical equivalence between different Lagrangian descriptions of a perfect fluid.

Nonminimal Coupling of Perfect Fluids to Curvature

The paper by Orfeu Bertolami and Jorge Paramos addresses the intriguing subject of nonminimal coupling of perfect fluids to curvature within the framework of modified theories of gravity, specifically focusing on $f(R)$ theories. The primary objective of this paper is to examine different Lagrangian densities of relativistic perfect fluids leading to equivalent gravitational field equations in General Relativity (GR) and to explore the peculiarities when nonminimal couplings are considered.

Overview of the Study

In the context of $f(R)$ gravity theories, it is well-documented that a nonvanishing covariant derivative of the energy-momentum tensor ($\nabla_\mu T^{\mu \nu} \neq 0$) can occur, resulting in deviations from geodesic motion and the emergence of an extra force. The paper explores the implications of such couplings on stellar equilibrium and compares them with scalar-tensor theories. A notable point of investigation is the non-uniqueness of matter Lagrangian densities for perfect fluids, questioning the traditional notion that the choice $L_m = p$ (where $p$ is the pressure) is the "natural" choice.

The authors assert that different choices for the matter Lagrangian density, such as $L_m = -\rho$ (energy density) or $L_m = -na$ (where $n$ is the particle number density and $a$ is the physical free energy), do not inherently lead to the vanishing of the extra force, stressing that these choices impact the classical equivalence in Lagrangian descriptions.

Theoretical Analysis and Implications

The paper revisits the equations of motion in scenarios involving curvature-matter coupling and demonstrates the non-uniqueness of Lagrangian densities. The authors rigorously examine the implications of different Lagrangian descriptions in both GR and modified gravity contexts by investigating perfect fluid models with non-minimal scalar curvature couplings.

A key contribution of this work is the elucidation of how different on-shell Lagrangian densities, despite being classically equivalent, may yield different gravitational field equations when non-minimal couplings are involved. This challenges traditional assumptions and sheds light on new theoretical aspects of gravitational physics.

Numerical Results and Claims

The authors provide a comprehensive derivation of the extra force and its dependence on different Lagrangian choices, notably highlighting that significant conclusions about a perfect fluid's motion and dynamics stem from the specific Lagrangian form utilized. The paper emphasizes that the nonminimal coupling of matter and curvature requires a careful reassessment of Lagrangian density equivalence.

Future Directions and Speculations

The insights provided by this work open avenues for future research in applying nonminimal curvature-matter coupling to various astrophysical and cosmological scenarios. Potential developments include employing velocity-potentials to further explore nonminimal coupling effects or extending this analysis to other modified gravity frameworks with intricate matter-curvature interactions.

In summary, this paper contributes a substantial theoretical understanding of the consequences and intricacies of nonminimal coupling in perfect fluid dynamics and modified gravity, offering a scholarly discourse on the assumptions underpinning classical relativistic fluids in alternative gravitational theories. The findings underscore the need for precise calculations when exploring these theoretical spaces, ensuring correct interpretations of the gravitational phenomena under paper.