Noncommutative geometry-inspired wormholes supported by quasi-de Sitter and Chaplygin-like equations of state

Abstract: We construct static, spherically symmetric wormhole solutions with a nontrivial redshift function, inspired by noncommutative geometry, in which point sources are replaced by Gaussian smearing of minimal length, yielding a regular shape function. Within this framework, we derive model-independent relations that isolate the role of the redshift function in controlling the stress-energy components and the violation of the null energy condition (NEC). Negative or suitably tuned redshifts confine the exotic matter to a thin neighborhood of the throat. We then reformulate this redshift engineering in matter terms through a quasi-de Sitter equation of state (EOS) with localized Gaussian or Lorentzian perturbations, obtaining minimally exotic wormholes that are regular, horizon-free, and asymptotically flat. Finally, we extend the analysis to a Chaplygin-like EOS, introducing a nonlinear coupling between pressure and density that yields redshift wells with possible local blueshift regions and tunable anisotropies governed by a certain nonlinearity parameter. Together, these results provide a unified and physically transparent framework for constructing traversable noncommutative-geometry-inspired wormholes with controlled, spatially localized exotic matter content.

• The paper constructs regular, horizon-free, asymptotically flat wormholes from Gaussian-smeared matter, identifying an extremal mass ratio of μₑ ≈ 1.9041 and a throat radius of xₑ ≈ 3.0224.
• The paper shows that every regular redshift function leaves an NEC-violating layer at the throat, while increasing its near-throat gradient systematically reduces the layer’s thickness.
• The paper finds that Chaplygin-like models satisfy subluminal radial sound-speed constraints only for small nonlinearities, with the allowed α range narrowing from 0.35 near extremality to 0.016 at μ = 2.5.

Overview

This paper constructs static, spherically symmetric traversable wormhole solutions within general relativity, sourced by a noncommutative-geometry-inspired Gaussian energy density in which point-like matter is smeared over a minimal length scale $\sqrt{\theta}$. The central contribution is a set of model-independent relations that elevate the redshift function $\Phi(r)$ from an arbitrary ansatz to a quantitative control parameter governing the magnitude and spatial extent of the null energy condition (NEC) violation. The authors then translate this "redshift engineering" into matter language via two families of equations of state: a quasi–de Sitter EOS with localized Gaussian or Lorentzian perturbations, and a Chaplygin-like EOS with a tunable nonlinearity parameter $\alpha$. All resulting configurations are regular, horizon-free, and asymptotically flat.

Geometry and throat structure

The metric is the standard Morris–Thorne form $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$, with the shape function fixed by integrating the noncommutative density $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$ through the $tt$ field equation, yielding $b(r)$ in terms of the lower incomplete Gamma function. Working with dimensionless variables $x = r/\sqrt{\theta}$ and $\mu = M/\sqrt{\theta}$, the function $f(x)$ admits an extremal configuration at $\Phi(r)$0 where two coincident zeroes appear at $\Phi(r)$1; this degenerate case saturates the flare-out condition ($\Phi(r)$2) and is not traversable. For $\Phi(r)$3 two distinct throats exist, and the outer root $\Phi(r)$4—which grows monotonically with $\Phi(r)$5—is identified as the physical throat satisfying $\Phi(r)$6. A key physical consequence follows immediately: since the mass scale is set by $\Phi(r)$7, these wormholes are intrinsically microscopic; for $\Phi(r)$8 the throat approaches the Schwarzschild radius, recovering classical behavior only at masses far above extremality.

The embedding analysis shows that the noncommutative profile produces a less flared geometry than the corresponding Morris–Thorne wormhole, reflecting the larger curvature radius of the equatorial embedding surface near the throat induced by the Gaussian smearing.

Model-independent NEC control by the redshift function

At the throat, the radial NEC combination takes the redshift-independent value

$\Phi(r)$9

so exoticity at the throat is fixed entirely by the shape function. Away from the throat, however, imposing $\alpha$0 requires $\alpha$1 to satisfy a lower bound that diverges as $\alpha$2 near $\alpha$3. Consequently, any regular redshift function necessarily leaves a nonzero NEC-violating layer adjacent to the throat; only its thickness can be minimized. The paper quantifies this via the near-throat slope of $\alpha$4, which depends linearly on $\alpha$5, and a first-order estimate for the width $\alpha$6 of the violating shell whose derivative with respect to $\alpha$7 is strictly negative whenever the denominator is positive. Thus increasing $\alpha$8 monotonically shrinks the exotic layer, independent of the sign of $\alpha$9. These relations are exact consequences of the field equations and hold for any choice of redshift profile.

Four representative redshift families (exponential, rational power-law, tanh-sigmoidal, and Gaussian) are analyzed numerically. Across all families, negative $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$0 (equivalently positive near-throat gradient) weakens and localizes the violation: e.g., for the exponential profile with $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$1 and $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$2, the first NEC zero occurs at $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$3, with the NEC remaining positive up to $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$4. Positive $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$5 deepens the well and extends the violating region. For Type II profiles with $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$6, $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$7 identically, so the leading influence enters through $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$8 at quadratic order—a subtlety the authors use to explain why feature locations shift slightly outward relative to Type I. The most economical configurations combine negative $$ds^2 = -e^{2\Phi(r)}dt^2 + dr^2/f(r) + r^2 d\Omega^2$$9 with rapidly descending, asymptotically flat profiles (Type III), all without horizon formation.

Quasi-de Sitter equations of state

A strict de Sitter EOS $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$0 imposed up to the throat forces $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$1, saturating flare-out and yielding a marginal, non-traversable geometry. Traversability therefore demands a localized positive departure $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$2 with $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$3. This multiplicative form preserves the scale-free character of the de Sitter fluid and ties the flare-out condition directly to the deformation amplitude:

$\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$4

Notably, $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$5 is not free—it is fixed by $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$6—so the construction has no adjustable exoticity parameter beyond the mass. The resulting master ODE for $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$7 has a removable singularity at the throat; a Taylor expansion gives finite $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$8, and asymptotic integration yields $\rho(r) = M(4\pi\theta)^{-3/2}e^{-r^2/4\theta}$9, guaranteeing asymptotic flatness after the harmless time rescaling absorbing $tt$0.

Numerically (RKF4/5), the Gaussian-bump model yields everywhere non-negative tangential pressure with a single peak just outside the throat that grows taller and narrower with increasing $tt$1. The Lorentzian bump, differing from the Gaussian only at fourth order near the throat, produces nearly identical redshift profiles but qualitatively different $tt$2: for $tt$3 the tangential pressure is negative everywhere, a consequence of the heavier tail extending the region where $tt$4. In both cases the NEC is approached asymptotically ($tt$5) rather than restored pointwise at finite radius—the authors note that strict pointwise recovery would require compact-support bumps, which they deliberately do not pursue.

Chaplygin-like EOS and causality bounds

The third matter model interpolates between a generalized Chaplygin law $tt$6 at the throat and the de Sitter relation $tt$7 at large radii, via an exponential localization factor. The constant $tt$8 is chosen so the throat NEC identity holds automatically, and $tt$9 at infinity requires $b(r)$0.

The radial adiabatic sound speed provides a stringent causality constraint on $b(r)$1:

$b(r)$2

with admissible ranges shrinking rapidly with mass: $b(r)$3 for $b(r)$4, $b(r)$5 for $b(r)$6, and $b(r)$7 for $b(r)$8. This tightening window means the Chaplygin nonlinearity becomes progressively less influential for heavier configurations, and the redshift profiles become nearly insensitive to $b(r)$9 at $x = r/\sqrt{\theta}$0.

Qualitatively, the Chaplygin sector differs sharply from the quasi-de Sitter models: the redshift can be positive at the throat, signaling local blueshift regions where proper time runs faster than at infinity, while still avoiding horizons. Increasing $x = r/\sqrt{\theta}$1 simultaneously softens the redshift well and enhances the positive $x = r/\sqrt{\theta}$2 peak, spreading the anisotropic stresses outward. The tangential pressure remains negative in a narrow interval adjacent to the throat across all cases, with the width of this tension region growing modestly as $x = r/\sqrt{\theta}$3 departs from extremality.

Limitations and open questions

Several caveats are stated explicitly. First, the configurations are intrinsically microscopic: with $x = r/\sqrt{\theta}$4 cm, throat radii and masses lie far below astrophysical scales, so no direct gravitational-wave or lensing signatures are expected; the models serve as controlled probes of minimal-length effects rather than astrophysical candidates. Second, the quasi-de Sitter constructions violate the NEC asymptotically rather than restoring it at finite radius, so "minimal exoticity" here means localization in magnitude, not compact support. Third, stability under perturbations is not addressed—radial perturbation spectra and tidal constraints remain open diagnostics within the framework. Fourth, the causality bounds on $x = r/\sqrt{\theta}$5 are derived from the radial sound speed alone; since $x = r/\sqrt{\theta}$6 is determined indirectly by the metric rather than by an independent constitutive law, a full causal characterization of the anisotropic fluid is not available. Finally, the analysis is restricted to static, non-rotating geometries; how rotation modifies the throat structure and near-throat NEC behavior at order $x = r/\sqrt{\theta}$7 remains unresolved.

Conclusion

The paper establishes a unified, largely analytic framework for noncommutative-geometry-inspired traversable wormholes in which the redshift function acts as a quantitative dial on the thickness and depth of the NEC-violating layer, and in which the same physics can equivalently be encoded in matter terms through quasi–de Sitter or Chaplygin-like equations of state. The main structural results—the redshift-independent throat NEC, the unavoidable violating layer for any regular $x = r/\sqrt{\theta}$8, and the monotonic shrinkage of that layer with $x = r/\sqrt{\theta}$9—are model-independent and transferable to other wormhole constructions. The Chaplygin extension demonstrates that nonlinear pressure-density coupling can generate horizon-free local blueshift regions, bounded tightly by subluminal-sound-speed requirements. Remaining questions concern dynamical stability, rotation, and whether compact-support deformations achieving strict pointwise NEC restoration can be constructed without sacrificing the minimality and analytic tractability of the present profiles.