Herglotz-type $f(R,T)$ gravity

TL;DR

This paper introduces a novel formulation of $f(R,T)$ gravity using the Herglotz variational principle, incorporating dissipative effects, which modifies gravitational dynamics and offers new explanations for cosmological acceleration.

Contribution

The paper develops a Herglotz variational approach to $f(R,T)$ gravity, extending the theory to include dissipation and providing new insights into gravitational and cosmological phenomena.

Findings

Herglotz contributions modify the Newtonian potential and are constrained by Solar System tests.

The approach allows the linear $f(R,T)=R+eta T$ model to fit observational data, unlike in standard formulations.

Herglotz vector can act as an effective cosmological constant, explaining accelerated expansion.

Abstract

The non-conservation of the energy-momentum tensor in $f(R,T)$ gravity can be interpreted as an effective manifestation of dissipation. Motivated by this, we propose a new formulation of $f(R,T)$ gravity based on the Herglotz variational principle, which extends the usual {Hamilton} variational principle to dissipative systems by allowing the Lagrangian to depend explicitly on the action. The resulting gravitational field equations extend those of $f(R,T)$ gravity by including Herglotz contributions. In the Newtonian limit, these contributions modify the gravitational potential, allowing us to constrain the Herglotz vector through Mercury's perihelion precession and the relativistic light deflection. The Herglotz corrections lead to a scaling law consistent with observations from the Cassini spacecraft. Examining two representative cosmological models, the Herglotz vector effectively reduces to a single function that, under suitable conditions, can play the role of a cosmological constant, providing an alternative mechanism for the Universe's accelerated expansion. Within the Herglotz variational approach, the linear $f(R,T)=R+\alpha T$ model, previously ruled out in the standard formulation due to its fixed deceleration parameter, becomes consistent with observations {for a certain choice of the Herglotz vector.