Holographic Dark Energy as a Source for Wormholes in Modified Gravity

TL;DR

This work investigates traversable wormholes within the linearized $f(\mathcal{R},\mathbb{T})$ gravity framework using three holographic dark-energy densities derived from Rényi, Moradpour, and Bekenstein–Hawking entropies. By adopting the linear model $f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$ and a constant redshift function, the authors obtain explicit shape functions and analytic expressions for the energy density and pressures associated with each HDE profile, then analyze throat conditions, asymptotic behavior, and energy conditions. They show that radial NEC is violated near the throat while tangential NEC is satisfied, and verify hydrostatic equilibrium via a generalized TOV equation where matter–geometry coupling provides an additional force that cancels with hydrostatic and anisotropic forces. Gravitational lensing in the strong-field regime is explored, yielding deflection coefficients and observable lensing parameters, and revealing that the exotic-matter content can be reduced relative to GR across all models. Overall, the results indicate that $f(\mathcal{R},\mathbb{T})$ gravity combined with holographic dark energy can support stable traversable wormholes with physically motivated energy distributions, including localized BH-like interiors, offering a promising avenue for wormhole physics without excessive exotic matter.

Abstract

Traversable wormhole solutions are explored in $f(\mathcal{R},\mathbb{T})$ gravity, a curvature--matter extension in which $\mathcal{R}$ is the Ricci scalar and $\mathbb{T}$ denotes the trace of the energy--momentum tensor. To generate explicit wormhole models, we prescribe holographic dark-energy densities based on entropy formalism proposed by Rényi, Moradpour, and Bekenstein--Hawking, namely \[ ρ_{\textit{R}} = \fracα{4α_1 r^4 c^2 κ}\ln\!\left(1+πα_1 r^2\right), \qquad ρ_{\textit{M}} = \fracα{4πr^2 c^2 κ\left(πα_1 r^2 + 1\right)}, \qquad ρ_{\textit{BH}} = \fracα{4 c^2 κr^2}, \] with $α$ and $β$ carrying dimensions of $L^{-2}$. The corresponding shape functions obtained from the field equations satisfy the standard throat and flare-out requirements for traversability. We then study how varying $α$ and $β$ affects (i) the balance of forces associated with equilibrium and (ii) the status of the energy conditions. In particular, the null energy condition is found to be violated, indicating that exotic matter (or an effective exotic sector) is required to support the wormhole geometry. The spatial structure of the solutions is further visualized through embedding surfaces.

Figures (5)

• Figure 1: The density given by Eq.\ref{['Re']}:The figure depicts how the shape function and associated energy conditions vary with the radial coordinate $r$, using representative parameter values listed in the plot.
• Figure 2: Two-dimensional embedding diagram $z(r)$ for the wormhole geometry. The solid red curve corresponds to the Rényi holographic model ($\alpha=0.3$, $\alpha_1=3$, $\beta_1=0.2$), the blue dashed curve to the Moradpour model with the same parameters, and the gray dashdotted curve to the Bekenstein--Hawking case. All curves are plotted for $r_{\textrm{t}h}=1$.
• Figure 3: Density from Eq. \ref{['Mad']}: variation of the shape function and energy conditions with $r$ for chosen parameters.
• Figure 4: The density given by Eq.\ref{['BH']}: Variation of the shape function and energy conditions for the non-asymptotically flat BekensteinHawking model, illustrating a local wormhole interior.
• Figure 5: Variation of the generalized Tolman--Oppenheimer--Volkoff (TOV) equation as a function of the radial coordinate $r$ for the first holographic model in $f(R,T)=R+\beta T$ gravity with $r_{\textrm{t}h}=1$, $\alpha=0.3$, $\alpha_1=3$, and $\beta_1=0.2$. The hydrostatic force $F_H$ (red dashed), anisotropic force $F_a$ (blue dash-dotted), and matter--geometry coupling force $F_c$ (gray dashed) are shown together with their sum $\Delta(r)=F_H+F_c+F_a$ (green dashed). Panels (a)--(c) correspond to different equations of state: (a) $p_r=\omega\rho$, (b) $p_t=n p_r$, and (c) $p_r=A/\rho^{\alpha_{\textrm{c}h}}$. Although the individual force components are nonzero and dominant near the throat, their exact cancellation yields $\Delta(r)\approx 0$ throughout the spacetime, confirming that the wormhole configurations satisfy the generalized TOV equilibrium condition.