Traversable Wormholes Supported by Entropy-Inspired Effective Matter Sectors

This paper investigates the viability of using entropy-induced density profiles from various modified gravity frameworks (Barrow, Tsallis, Kaniadakis, logarithmic, and exponential) as effective sources for traversable wormholes in the Morris-Thorne spacetime, demonstrating that these entropy-inspired sectors can support such geometries by redistributing exoticity through anisotropic stresses while satisfying necessary equilibrium and energy-condition constraints.

The Gist

Imagine the universe as a giant, flexible sheet of fabric. In standard physics, if you want to punch a hole through this fabric and connect two distant points (creating a "wormhole"), you need something very strange to hold that hole open. Usually, this requires "exotic matter"—stuff that behaves in ways normal matter doesn't, like having negative weight or pushing outward instead of pulling inward.

This paper asks a fascinating question: What if the "exotic stuff" holding the wormhole open isn't a mysterious new particle, but rather a consequence of how we count the microscopic "pixels" of space itself?

Here is a simple breakdown of what the researchers did and found, using everyday analogies.

The Big Idea: Gravity as a Thermometer

For a long time, scientists have suspected that gravity isn't just a force, but a result of thermodynamics (heat and entropy). Think of a black hole not just as a cosmic vacuum cleaner, but as a hot object with a specific temperature and a specific amount of "disorder" (entropy) on its surface.

The researchers started with a theory that says: If you change the rules for how we calculate this "disorder" (entropy), the shape of space itself changes.

Usually, this theory was used to describe black holes. But these authors asked: "Can we use these new, weird rules for entropy to build a wormhole instead?"

The Experiment: Building a Wormhole from "Entropy Recipes"

The team didn't try to build a whole new universe. Instead, they took five different "recipes" for how entropy might behave (inspired by different theories of quantum physics) and asked: "If we use the density of matter predicted by these recipes, can it prop open a wormhole?"

They treated the wormhole like a tunnel. To keep the tunnel from collapsing, you need a specific amount of "push" (negative pressure) at the narrowest point (the throat). They tested five different mathematical "flavors" of entropy to see if they could provide that push.

Here are the five "flavors" they tested, explained simply:

1. The "Fractal" Flavor (Barrow)

• The Analogy: Imagine a coastline. From far away, it looks smooth. But if you zoom in, it gets jagged and complex. This theory suggests space has a similar "jagged" texture at the smallest scales.
• The Result: This creates a wormhole supported by a "negative density" that fades away slowly, like a gentle slope. It works, but the math gets tricky if you try to make the texture perfectly smooth (the standard version).

2. The "Non-Additive" Flavor (Tsallis)

• The Analogy: Imagine a crowd of people. In normal physics, the total energy is just the sum of everyone's energy. In this theory, the crowd interacts so much that the whole is different than the sum of its parts.
• The Result: This creates a wormhole where the "exotic" matter is very concentrated right at the throat and fades away very quickly. It's like a tight knot of support that holds the tunnel open, but the effect dies off fast as you move away.

3. The "Relativistic" Flavor (Kaniadakis)

• The Analogy: This is based on how particles move at near-light speeds. It suggests the "disorder" of space behaves differently when things are moving fast.
• The Result: Unlike the previous two, which fade away gradually, this one creates a "blob" of exotic matter. It's like a compact, localized cushion right at the throat. The support is strongest in a specific zone and then drops off sharply. It's not a smooth slope; it's a distinct, localized bump.

4. The "Logarithmic" Flavor (The Chameleon)

• The Analogy: This is the most flexible one. Imagine a shape-shifter. Depending on the settings, it can be a "negative weight" object OR a "positive weight" object that pushes incredibly hard.
• The Result: This is unique. It can support a wormhole in two ways:
  1. By having negative density (the usual exotic stuff).
  2. By having positive density but a "phantom-like" pressure that pushes outward violently.
     It's the only one that can switch between these two modes, making it very versatile for building a stable tunnel.

5. The "Exponential" Flavor

• The Analogy: Think of a spotlight that is incredibly bright in the center but turns off almost instantly a few inches away.
• The Result: This creates the most "localized" wormhole. The exotic matter is crammed tightly into the throat and disappears almost immediately as you move outward. It's a very sharp, intense support system that doesn't linger.

What They Found

The researchers discovered that all five of these entropy-inspired recipes can theoretically hold a wormhole open.

However, they also found a crucial rule: You can't just pick the "push" (pressure) of the matter arbitrarily. The math forces a specific relationship between the shape of the wormhole and the pressure needed to keep it open. If you try to force the wormhole to be perfectly smooth (like a standard black hole), the pressure required becomes infinite, which breaks the model.

The Key Takeaway:
The paper shows that you don't necessarily need to invent new, undiscovered particles to build a wormhole. Instead, if the microscopic rules of space (entropy) are slightly different than we thought, the geometry of space itself might naturally create the "exotic" conditions needed to keep a wormhole open.

• Some recipes create a gentle, long-lasting support (Barrow).
• Some create a tight, localized support (Kaniadakis, Exponential).
• One recipe is a shape-shifter that can work in two different ways (Logarithmic).

The Bottom Line

This paper is a theoretical "proof of concept." It says: "If the universe's entropy works like these five specific mathematical models, then traversable wormholes are a natural consequence." It doesn't say we can build one tomorrow, but it proves that the math of modified entropy is compatible with the geometry of a wormhole, offering a new way to think about how such structures could exist without breaking the laws of physics.

Technical Summary

Technical Summary: Traversable Wormholes Supported by Entropy-Inspired Effective Matter Sectors

Problem Statement
Traversable wormholes in general relativity require the violation of the null energy condition (NEC) at the throat to satisfy the flare-out condition. While classical matter cannot sustain such geometries, various extensions of gravity and effective field theories suggest that non-classical contributions (quantum, semiclassical, or geometric) can act as effective sources. Recent developments in the "entropy–geometry correspondence" (Refs. [1, 2]) demonstrate that modifications to the Bekenstein–Hawking entropy induce specific effective anisotropic matter sectors and deformed black-hole geometries. However, it remains an open question whether the density profiles derived from these entropy deformations can serve as viable phenomenological sources for traversable wormholes when decoupled from their original black-hole context.

Methodology
The authors adopt a phenomenological approach within the Morris–Thorne framework for static, spherically symmetric traversable wormholes. Instead of importing the full anisotropic effective fluid (including the specific radial equation of state $p_r = -\rho$) generated by the entropy–geometry correspondence in black-hole physics, the authors isolate only the effective density profiles ($\rho(r)$) associated with five specific entropy deformations: Barrow, Tsallis, Kaniadakis, logarithmic, and exponential.

The reconstruction procedure involves:

1. Prescribing the Density: The density profile $\rho(r)$ is taken directly from the entropy–geometry correspondence (Eq. 8 of Ref. [1]).
2. Reconstructing the Shape Function: Using the Einstein equation component $G^t_t = 8\pi\rho$, the shape function $b(r)$ is reconstructed via integration (Eq. 28).
3. Imposing Redshift Regularity: To ensure the absence of event horizons, the redshift function $\Phi(r)$ must be finite. The authors impose a barotropic equation of state $p_r = w\rho$. Crucially, they demonstrate that for $\Phi(r)$ to be regular at the throat ($r=r_0$), the equation-of-state parameter $w$ cannot be arbitrary. It is fixed by the throat condition $b(r_0)=r_0$ and the derivative $b'(r_0)$, yielding $w_{th} = -1/b'(r_0)$.
4. Deriving Pressures: With $w$ fixed and $\rho$ known, the radial pressure $p_r$ is determined. The tangential pressure $p_t$ is then reconstructed using the anisotropic equilibrium condition (Tolman–Oppenheimer–Volkoff equation) rather than the angular Einstein equation, ensuring consistency with the prescribed density and redshift regularity.
5. Diagnostics: The authors analyze the resulting geometries for asymptotic flatness, the flare-out condition ($b'(r_0) < 1$), energy condition violations (NEC, WEC, SEC), and the balance of hydrostatic, gravitational, and anisotropic forces.

Key Contributions and Results

• Throat-Selected Equation of State: A central theoretical result is that the equation-of-state parameter $w$ is not a free parameter in this construction. Regularity of the redshift derivative at the throat uniquely fixes $w$ in terms of the local geometry (specifically the density at the throat). This resolves the apparent singularity in the redshift derivative, showing it is a removable feature provided the flare-out condition is strictly satisfied.
• Radial NEC Violation: In all regular configurations, the radial NEC ($\rho + p_r \geq 0$) is violated at the throat. The authors show this is a direct geometric consequence of the flare-out condition combined with the redshift regularity constraint, rather than an arbitrary choice of matter properties.
• Sector-Specific Analysis:
  • Barrow and Tsallis Sectors: These generate power-law negative-density sources. The Barrow sector is controlled by a fractal parameter $\Delta$, while the Tsallis sector is controlled by a non-extensive parameter $\delta$. In both cases, the undeformed limits ($\Delta \to 0$ or $\delta \to 1$) suppress the density but require divergent radial tension, making them physically less natural as regular wormhole configurations. The Tsallis sector exhibits a stronger tangential support (positive SEC combination) compared to Barrow.
  • Kaniadakis Sector: This sector produces localized negative-density distributions governed by hyperbolic functions ($\tanh, \sech$). Unlike the algebraic sectors, the source is concentrated in a finite radial region. The behavior is non-monotonic with respect to the deformation parameter $\kappa$, with an intermediate range where the source is most relevant.
  • Logarithmic Sector: This is the most flexible sector. Depending on the sign of the correction parameter $\lambda$, it can support the wormhole via negative effective density ($\lambda < 0$) or via positive density coupled with a phantom-like radial pressure ($w < -1$) when $\lambda > 0$. The redshift function for this sector admits a closed-form analytical solution.
  • Exponential Sector: This generates localized negative-density distributions with rapid exponential suppression away from the throat. The exotic matter is confined to a narrow region near the throat, and the force balance is similarly localized.
• Anisotropy and Energy Conditions: While the radial NEC is universally violated (as required), the tangential NEC and the Strong Energy Condition (SEC) combination vary significantly between sectors. The SEC combination helps distinguish how each entropy deformation redistributes the required "exoticity" through anisotropic stresses. For instance, the Tsallis and Logarithmic sectors show positive SEC combinations in certain regimes, indicating that tangential pressures can partially compensate for radial exoticity.

Significance and Claims
The paper claims that modified entropy profiles provide viable effective sources for traversable wormholes, but the supporting mechanism is highly sensitive to the specific entropy deformation. The study establishes that:

1. Entropy-inspired density profiles can sustain traversable throats without importing the full black-hole effective fluid.
2. The requirement of redshift regularity imposes a strict constraint on the equation of state, linking the macroscopic wormhole geometry directly to the microscopic entropy deformation parameters.
3. Different entropy corrections lead to qualitatively distinct matter distributions: algebraic power-law tails (Barrow, Tsallis), compact hyperbolic localization (Kaniadakis), dual regimes of negative density and phantom pressure (Logarithmic), and sharply localized exponential suppression (Exponential).

The authors position this work as a "controlled phenomenological step" rather than a fundamental derivation. They do not claim to have solved the problem of exotic matter generation from first principles but demonstrate that the density profiles emerging from entropy–geometry correspondence are mathematically consistent with traversable wormhole geometries under specific reconstruction rules. Future extensions suggested by the authors include relaxing the constant equation of state, analyzing stability, and investigating observational signatures like lensing and shadows.