Holographic Dark Energy as a Source for Wormholes in Modified Gravity

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(πα1 r^2 + 1\right)}, \qquad ρ{\textit{BH}} = \fracα{4 c^2 κr^2}, ] with $α$ and $β$ carrying dimensions of $L<sup>{-2}$. 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.

• The paper identifies three explicit traversable wormhole solutions in f(R, T) gravity, each driven by distinct holographic dark energy (HDE) profiles based on Rényi, Moradpour (Tsallis-inspired), and Bekenstein–Hawking entropy formalisms.
• All three HDE profiles satisfy the throat and flare-out conditions for wormhole traversability, though the Bekenstein–Hawking profile results in a non-asymptotically flat solution, thus requiring an external vacuum match,
• The modified gravity sector reduces exotic matter requirement with the Bekenstein–Hawking profile minimizing it by roughly 32% compared to GR.

Overview

This paper constructs explicit traversable wormhole solutions in $f(\mathcal{R},\mathbb{T})$ gravity, where the gravitational Lagrangian depends on both the Ricci scalar $\mathcal{R}$ and the trace $\mathbb{T}$ of the energy–momentum tensor. The matter source is taken to be holographic dark energy (HDE), with three distinct density profiles derived from different entropy formalisms: Rényi entropy, the Moradpour (Tsallis-inspired non-additive) construction, and the standard Bekenstein–Hawking area law. The central result is that, within the linear model $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$ with a constant redshift function, all three profiles yield shape functions satisfying the throat and flare-out conditions for traversability, while the radial null energy condition (NEC) is violated near the throat and the tangential NEC is satisfied throughout (2601.22577).

Theoretical framework

The authors work in the metric formulation of $f(\mathcal{R},\mathbb{T})$ theory with the Levi-Civita connection, adopting the action

$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$

with $\kappa = 8\pi G/c^4$. Following Harko et al., the matter Lagrangian is set to $\mathcal{L}_m = p$, which fixes $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$. The paper explicitly acknowledges that this choice is not unique—an alternative such as $\mathcal{L}_m = -\rho$ would alter numerical coefficients and parameter bounds—though it argues that the qualitative NEC violation at the throat persists under any consistent choice of $\mathcal{R}$0.

A key structural feature of the theory is that $\mathcal{R}$1 generically; energy–momentum conservation holds only when $\mathcal{R}$2. This non-conservation generates an additional force term in the generalized Tolman–Oppenheimer–Volkoff (TOV) equation, originating from the non-minimal matter–geometry coupling. To retain analytic tractability, the authors restrict to the linear realization $\mathcal{R}$3, originally introduced by Xu et al., noting that nonlinear choices render the effective energy–momentum tensor too intricate for closed-form solutions.

Wormhole geometry and field equations

The spacetime is static and spherically symmetric,

$\mathcal{R}$4

with $\mathcal{R}$5 and $\mathcal{R}$6. Traversability requires $\mathcal{R}$7, $\mathcal{R}$8, asymptotic flatness ($\mathcal{R}$9), and a finite redshift function to preclude horizons; the constant redshift choice $\mathbb{T}$0 is adopted throughout. For the linear model, the field equations reduce to algebraic relations among $\mathbb{T}$1, $\mathbb{T}$2, $\mathbb{T}$3, and derivatives of the shape function, with regularity requiring $\mathbb{T}$4, where $\mathbb{T}$5.

Three holographic dark energy models

Rényi HDE: The density $\mathbb{T}$6 derives from Rényi entropy, appropriate for systems with nonlocal interactions and fractal characteristics. Matching to the field equations yields a closed-form shape function involving logarithms and arctangent functions, with the flare-out condition imposing an upper bound $\mathbb{T}$7 on the normalization. Positivity of the density holds for $\mathbb{T}$8, $\mathbb{T}$9, and the profile decays as $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$0, ensuring asymptotic flatness.

Moradpour HDE: Replacing Bekenstein–Hawking entropy by a Tsallis-type non-additive form gives $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$1, decaying as $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$2. The resulting shape function again satisfies all geometric constraints, with a corresponding upper bound $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$3 on $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$4.

Bekenstein–Hawking HDE: The conventional profile $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$5 produces a shape function whose large-$$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$6 behavior is $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$7, with

$$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$8

A notable and candidly stated limitation emerges here: $$f(\mathcal{R},\mathbb{T})=\mathcal{R}+\beta\mathbb{T}$$9 occurs only for $f(\mathcal{R},\mathbb{T})$0 or for $f(\mathcal{R},\mathbb{T})$1, both of which are singular values of the field equations. Consequently, a non-trivial Bekenstein–Hawking-supported wormhole in this model is always non-asymptotically flat, and must be interpreted as a local wormhole interior requiring matching to an exterior vacuum at finite radius. This contrasts sharply with the Rényi and Moradpour cases, which are globally asymptotically flat—a genuine qualitative distinction between the entropy formalisms rather than a mere quantitative difference.

Energy conditions and exotic matter content

Across all three models, the analysis shows that $f(\mathcal{R},\mathbb{T})$2 near the throat while $f(\mathcal{R},\mathbb{T})$3: the radial NEC is violated, as required for traversability, but the tangential NEC is satisfied. The modified gravity sector therefore supplies part of the effective exotic contribution, reducing—but not eliminating—the reliance on exotic matter.

To quantify this reduction, the paper computes the volume-integral quantifier of the averaged null energy condition (ANEC),

$f(\mathcal{R},\mathbb{T})$4

for $f(\mathcal{R},\mathbb{T})$5, $f(\mathcal{R},\mathbb{T})$6, $f(\mathcal{R},\mathbb{T})$7, $f(\mathcal{R},\mathbb{T})$8, and cutoff $f(\mathcal{R},\mathbb{T})$9, comparing against the GR limit $$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$0:

Model	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$1 ($$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$2)	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$3	Reduction
Rényi HDE	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$4	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$5	26.68%
Moradpour HDE	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$6	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$7	28.52%
Bekenstein–Hawking HDE	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$8	$$S = \int d^4x\sqrt{-g}\left[\frac{f(\mathcal{R},\mathbb{T})}{\kappa} + \mathcal{L}_m\right],$$9	32.30%

The Bekenstein–Hawking profile minimizes the amount of exotic matter required, reducing the ANEC integral magnitude by roughly one-third relative to GR. The authors justify the finite cutoff by the rapid decay of the integrand for the asymptotically flat profiles; for the non-asymptotically flat BH case, a finite $\kappa = 8\pi G/c^4$0 is necessary in any case given the local-interior interpretation. One should note this quantifier is cutoff-dependent for the BH model, so its cross-model comparison carries that caveat.

Gravitational lensing

Using the proper radial coordinate $\kappa = 8\pi G/c^4$1, the paper shows that for constant redshift function the throat constitutes an unstable photon sphere: the flaring-out condition implies $\kappa = 8\pi G/c^4$2 and hence $\kappa = 8\pi G/c^4$3. The apparent failure of $\kappa = 8\pi G/c^4$4 in Schwarzschild-like coordinates is identified as a coordinate artifact, consistent with the general characterization of photon surfaces by Claudel et al. Bozza's strong-deflection analysis then yields the universal deflection form $\kappa = 8\pi G/c^4$5, with critical impact parameter $\kappa = 8\pi G/c^4$6 and strong-field observables $\kappa = 8\pi G/c^4$7, $\kappa = 8\pi G/c^4$8, and $\kappa = 8\pi G/c^4$9 determined solely by $\mathcal{L}_m = p$0, where $\mathcal{L}_m = p$1 encodes $\mathcal{L}_m = p$2. All three HDE models share this structure, differing only through the model-dependent second derivative of the shape function at the throat.

Hydrostatic equilibrium

The generalized TOV equation decomposes into hydrostatic ($\mathcal{L}_m = p$3), gravitational ($\mathcal{L}_m = p$4), matter–geometry coupling ($\mathcal{L}_m = p$5), and anisotropic ($\mathcal{L}_m = p$6) forces:

$\mathcal{L}_m = p$7

With constant redshift, $\mathcal{L}_m = p$8 vanishes identically, and the equilibrium residual $\mathcal{L}_m = p$9 is evaluated numerically for each model. In all three cases $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$0 vanishes throughout $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$1 up to numerical precision, confirming exact hydrostatic equilibrium. The anisotropic coefficient $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$2 is positive for $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$3, making the anisotropic force repulsive when $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$4—a condition tied to the NEC requirement. Closed-form expressions for the individual forces are provided for each HDE profile, allowing the balance to be verified analytically as well.

Limitations and open questions

Several caveats qualify the results. First, the restriction to the linear model $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$5 and constant redshift function is made for analytic tractability; nonlinear realizations, whose field equations require numerical treatment or scalar-tensor reformulation, remain unexplored here. Second, the choice $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$6 affects the numerical coefficients and parameter bounds, even if the qualitative NEC violation is robust. Third, the Bekenstein–Hawking solution's generic non-asymptotic flatness means no complete global geometry is constructed for that case—the exterior matching is asserted as necessary but not performed. Fourth, the ANEC comparison uses a fixed cutoff that is well-motivated for two models but structurally required for the third. Finally, the stability analysis establishes equilibrium, not perturbative stability; whether these configurations are stable against radial perturbations is not addressed and remains an open question, as does the observational distinguishability of the lensing signatures from black-hole shadows given that all three models share the same universal strong-deflection structure.

Conclusion

The paper demonstrates that traversable wormholes supported by holographic dark energy arise naturally in linear $$\Theta_{\mu\nu} = -2\mathbb{T}_{\mu\nu} + p\,g_{\mu\nu}$$7 gravity for three distinct entropy-based density profiles. The Rényi and Moradpour models yield globally asymptotically flat geometries satisfying all traversability conditions, while the Bekenstein–Hawking model provides only a local interior solution. In every case the radial NEC is violated near the throat but the tangential NEC holds, and the modified-gravity sector reduces the required exotic matter by roughly 27–32% relative to GR, with the Bekenstein–Hawking profile performing best. Exact TOV equilibrium and a common unstable-photon-sphere lensing signature hold across all models. The results support the view that entropy-corrected holographic sources combined with curvature–matter coupling can substantially alleviate, though not remove, the exotic-matter problem in wormhole physics.