Traversable Wormholes induced by Thomas-Fermi energy density

Abstract: We investigate spherically symmetric and static traversable wormholes supported by exotic matter, focusing on solutions sourced by physically motivated dark matter energy density profiles. Considering the Thomas-Fermi-type distribution, we construct explicit forms of the shape function $b(r)$ and analyze the resulting radial and tangential pressures, carefully addressing the requirements of the flare-out condition at the throat and the absence of horizons. We explore zero-tidal-force configurations as well as inhomogeneous equations of state, demonstrating how appropriate choices of the radial pressure allow for finite and well-behaved redshift functions throughout the spacetime. Boundary conditions at a finite radius are implemented to ensure vanishing energy density and pressures, and asymptotic expansions are derived to characterize the behavior of the metric and matter content near the edge of the dark matter halo. Additionally, we reformulate the Einstein field equations entirely in terms of the energy density, radial and tangential pressures, and their derivatives, providing a framework to analyze the matter distribution independently of the explicit metric functions. Our results offer a systematic methodology to construct physically consistent wormhole geometries supported by realistic dark matter halos, highlighting the intricate interplay between matter profiles, equations of state, and geometric constraints.

• The paper constructs static, spherically symmetric traversable wormholes by using the Thomas–Fermi Bose–Einstein condensate dark-matter density as input and deriving the shape function from Einstein’s equations.
• The analysis shows that flare-out and throat energy-condition requirements constrain the central density, while boundary conditions sharply limit viable redshift profiles, including R ≲ 1.5r₀ for the simplest cored ansatz and R > 3.06r₀ for a logarithmic construction.
• The paper demonstrates that finite-support dark matter can produce horizon-free geometries with smooth pressure matching, but stability, microscopic justification for NEC violation, rotation, and observational signatures remain unresolved.

Overview and motivation

The paper constructs static, spherically symmetric traversable wormholes in general relativity whose matter source is a physically motivated dark matter profile, specifically the Thomas–Fermi (TF) density distribution arising in Bose–Einstein condensate dark matter (BEC-DM) models (2603.12527). The central methodological move is to treat the energy density $\rho(r)$ as the primary input and to reconstruct the geometry — the shape function $b(r)$ and redshift function $\Phi(r)$ — from the Einstein field equations rather than from ad hoc metric ansätze. The authors also reformulate the field equations entirely in terms of the matter variables $\rho$, $p_r$, $p_t$ and their derivatives, providing a framework that is independent of any particular parametrization of the metric functions.

The motivation rests on the standard Morris–Thorne requirements: a throat at $r_0$ with $b(r_0)=r_0$, the flare-out condition $(b - b'r)/b^2 > 0$ together with $b'(r_0)<1$, finiteness of $b(r)$0 everywhere (no horizons), and regular behavior of the stress-energy tensor. The TF profile,

$b(r)$1

is regular at the origin, has finite support at $b(r)$2 (the halo edge), and vanishes smoothly there — properties the authors exploit to impose boundary conditions $b(r)$3 without invoking thin shells.

Shape function and throat constraints

Inserting the TF density into the $b(r)$4 Einstein equation yields an explicit shape function,

$b(r)$5

with $b(r)$6. For $b(r)$7, the boundary value scales as $b(r)$8, dominated by the $b(r)$9 term as expected for a finite-size halo.

The flare-out condition imposes a direct upper bound on the central density:

$\Phi(r)$0

Independently, the null energy condition violation required at the throat, $\Phi(r)$1, gives the bound $\Phi(r)$2 with $\Phi(r)$3. These two inequalities jointly constrain the parameter space; their compatibility becomes a nontrivial filter on each subsequent ansatz for $\Phi(r)$4, and several candidate geometries are eliminated precisely because they violate it.

Zero-tidal-force construction with an inhomogeneous equation of state

With $\Phi(r)$5 in the interior, the radial equation of state $\Phi(r)$6 reproduces the flare-out condition at the throat but diverges as $\Phi(r)$7 where $\Phi(r)$8. To cure this, the authors introduce a transition layer $\Phi(r)$9 and a piecewise $\rho$0 that is driven linearly to zero at $\rho$1, guaranteeing $\rho$2. Integrating the $\rho$3 equation then gives a redshift function that is constant ($\rho$4) throughout the interior and grows linearly only within the transition layer, so $\rho$5 remains finite everywhere and no horizon forms. The tangential pressure obtained from the angular equation vanishes identically at $\rho$6 since $\rho$7. This construction demonstrates that zero-tidal-force traversable wormholes can be sourced consistently by a finite-support BEC-DM halo, provided the equation of state is allowed to be radially inhomogeneous.

Cored-inspired redshift function proposals

The paper then explores four families of redshift profiles motivated by the cored structure of the TF density:

• Proposal I: $\rho$8, mimicking the density itself. Fixing $\rho$9 determines $p_r$0 analytically; imposing additionally $p_r$1 forces the boundary radius to satisfy $p_r$2, which requires $p_r$3. Compatibility with the NEC bound restricts $p_r$4 to the interval $p_r$5, and more stringently $p_r$6. The consequence is severe: the halo edge must lie very close to the throat, so this proposal admits only extremely compact configurations and motivates the alternatives below.
• Proposal II: $p_r$7, i.e., Proposal I plus a constant phase. The two boundary conditions $p_r$8 fix both coefficients algebraically. A convenient gauge choice $p_r$9 simplifies the coefficients to closed forms and requires $p_t$0, enlarging the viable geometry relative to Proposal I at the cost of considerably more involved expressions.
• Proposal III: the cored structure is imposed on the derivative, $p_t$1, so that $p_t$2 involves the sine and cosine integrals $p_t$3 and $p_t$4. The coefficients $p_t$5 are fixed by the same boundary conditions and share the validity domain of Proposal II.
• Proposal IV: $p_t$6, anchoring the redshift gradient at the throat scale. The two free constants are again uniquely determined by $p_t$7 and $p_t$8, offering finer control of the intermediate region between throat and boundary while preserving the cored character.

Across all four proposals, the pattern is consistent: the boundary conditions close the system algebraically, leaving no residual freedom except in special gauges such as the fixed-$p_t$9 choice of Proposal II.

Redshift functions extracted from the third field equation

A distinct strategy treats the transverse-pressure equation as a differential equation for $r_0$0 by isolating its substructures. Three cases are analyzed:

Logarithmic sector $r_0$1, giving $r_0$2. The condition $r_0$3 fixes $r_0$4, and $r_0$5 then determines the density parameter uniquely as $r_0$6. Positivity requires $r_0$7, and compatibility with the NEC upper bound strengthens this to $r_0$8. This is the cleanest result of the section: a self-consistent, extended wormhole geometry with the matter content fully fixed by geometry and boundary conditions.

First-order sector $r_0$9, giving $b(r_0)=r_0$0. Here $b(r_0)=r_0$1 vanishes automatically at the throat, and $b(r_0)=r_0$2 fixes $b(r_0)=r_0$3 consistently. However, the radial pressure evaluates to $b(r_0)=r_0$4, strictly negative and never zero. This configuration is therefore discarded — a concrete illustration that an apparently natural redshift ansatz fails once all physical boundary conditions are enforced.

Radial-logarithmic sector $b(r_0)=r_0$5, giving $b(r_0)=r_0$6. Requiring $b(r_0)=r_0$7 forces $b(r_0)=r_0$8 (a constant redshift), but substituting back yields $b(r_0)=r_0$9, which is unphysical. This case is likewise discarded.

The authors note explicitly that the logarithmic profiles are unbounded functions of $(b - b'r)/b^2 > 0$0; the finite halo radius $(b - b'r)/b^2 > 0$1 acts as the necessary infrared cutoff that keeps them under control.

Limitations and open questions

Several caveats qualify the results. First, the entire construction is classical: the exoticity problem is not resolved but relocated — the NEC is still violated at the throat, now attributed to the effective dark matter sector, and the paper does not demonstrate that BEC-DM microphysics actually produces the required negative $(b - b'r)/b^2 > 0$2. Second, the transition-layer construction of the zero-tidal-force model introduces the arbitrary thickness $(b - b'r)/b^2 > 0$3 and a piecewise equation of state whose dynamical justification within a BEC model is not established. Third, stability is not analyzed anywhere in the paper; all solutions are static equilibria, and perturbative or thermodynamic stability remains open. Fourth, the analysis assumes exact spherical symmetry and a sharp halo edge at $(b - b'r)/b^2 > 0$4; rotating generalizations, which the authors themselves flag, may modify the throat structure qualitatively. Finally, the compatibility windows derived for Proposals I–IV depend on the small-ratio expansions in $(b - b'r)/b^2 > 0$5, so the stated bounds on $(b - b'r)/b^2 > 0$6 inherit those approximation errors.

Conclusion

The paper provides a systematic, matter-first methodology for building traversable wormholes from the Thomas–Fermi BEC dark matter profile, with explicit shape functions, algebraically fixed redshift-function coefficients, and smooth matching to vacuum at the finite halo radius. Its most useful contributions are the quantitative viability windows — notably the restriction $(b - b'r)/b^2 > 0$7 for the simplest cored ansatz versus $(b - b'r)/b^2 > 0$8 for the logarithmic construction extracted from the third field equation — and the demonstration, via two discarded ansätze, that the full set of boundary conditions is a stringent filter on candidate geometries. Whether such geometries are dynamically attainable and observationally distinguishable from black holes in galactic environments remains unresolved by this work.