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

TL;DR

The authors develop a covariant Herglotz variational formulation of $f(R,T)$ gravity, introducing a background one-form $\lambda_\mu$ that induces dissipative geometric effects through $\mathcal{L} = f(R,T) + \lambda_\mu s^\mu$. They derive generalized field equations with explicit dissipative contributions encoded in $K_{\mu\nu}$ and $H_{\mu\nu}$, and discuss conditions under which energy–momentum conservation can be restored. In the Newtonian limit they obtain a generalized Poisson equation with an effective coupling $k_{\text{eff}}^2 = (f_T+8\pi)/(2 f_R)$ and explicit $\lambda_\mu$-driven corrections to $\Phi$, using Mercury’s perihelion precession and light deflection to bound the Herglotz field; the deflection also exhibits a wavelength-dependent scaling that mirrors plasma-like behavior. Cosmologically, two simple models, $f(R,T)=R+\alpha T$ and $f(R,T)=R+\alpha T^{-1}$, are analyzed; with a linear effective EOS the linear model can reproduce cosmic chronometer data and late-time acceleration, offering an alternative to $\Lambda$CDM, while remaining compatible with observational constraints for suitable parameter choices.

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. Remarkably, 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 $f(R,T)=R+αT$ model, previously ruled out in the standard formulation due to its fixed deceleration parameter, becomes consistent with observations.

Figures (7)

• Figure 1: The Hubble function $H(z)$ as a function of redshift for the models $f(R,T)=R+\alpha T$ (left panel) and $f(R,T)=R+\alpha T^{-1}$ (right panel), shown for different initial values $\tensor{\Phi}{_{0}}$.
• Figure 2: Redshift evolution of the deceleration parameter $q(z)$ for the models $f(R,T)=R+\alpha T$ (left) and $f(R,T)=R+\alpha T^{-1}$ (right), displayed for several initial values of $\Phi_0$.
• Figure 3: Evolution of the dimensionless matter density $r(z)$ with redshift for the $f(R,T)=R+\alpha T$ (left) and $f(R,T)=R+\alpha T^{-1}$ (right) models, for various initial values of $\Phi_0$.
• Figure 4: Redshift dependence of the Herglotz contribution for the models $f(R,T)=R+\alpha T$ (left) and $f(R,T)=R+\alpha T^{-1}$ (right), plotted for different initial values of $\Phi_0$.
• Figure 5: Behavior of the $Om(z)$ diagnostic as a function of redshift for the $f(R,T)=R+\alpha T$ (left) and $f(R,T)=R+\alpha T^{-1}$ (right) models, shown for several initial values of $\Phi_0$.