Traversable Casimir Wormholes with Gravitational Memory

Abstract: We investigate a class of traversable wormhole geometries supported by an effective Casimir source corrected by gravitational memory. The construction is motivated by the fact that a time-dependent gravitational background can leave a permanent positive shift in the vacuum polarization of a quantum field confined to a Casimir cavity. By promoting the plate separation to an effective radial scale in the Morris-Thorne spacetime, we obtain a density profile composed of the usual negative Casimir contribution, proportional to $r<sup>{-4}$, and a positive memory-induced correction, proportional to $r<sup>{-7}$. The corresponding shape function is derived directly from the Einstein equations and satisfies the throat condition by construction. We determine the redshift function from a constant barotropic equation of state together with the requirement of regularity at the throat, which fixes the barotropic parameter in terms of the Casimir and memory coefficients. The flare-out condition defines the admissible range of the memory parameter and separates a Casimir-dominated sector from a phantom-like sector, with the transition point associated with a singular limit of the constant-barotropic description. We analyze the curvature scalar, the embedding structure, the energy conditions, and the Tolman-Oppenheimer-Volkoff equilibrium of the anisotropic matter source. The radial null energy condition is necessarily violated at the throat, while the tangential sector depends on the redshift gradient. We also examine the shadow radius as a phenomenological diagnostic and show that admissible solutions can overlap the Event Horizon Telescope range for M87*. The results indicate that gravitational memory can deform Casimir-supported wormholes by softening the ordinary Casimir contribution, modifying the near-throat geometry, and reshaping the internal stress balance required to sustain traversability.

• The paper constructs static, spherically symmetric traversable wormholes from an effective density combining negative Casimir energy proportional to r⁻⁴ with positive memory corrections proportional to r⁻⁷.
• The memory parameter controls a transition from a Casimir-dominated regime to a phantom-like regime, softens radial null-energy-condition violation, and can make tangential stresses non-exotic near the throat.
• The solutions produce shadow radii overlapping the Event Horizon Telescope range for M87* in the phantom-like sector, while requiring future tests of perturbative validity, dynamical stability, and rotation.

The paper constructs a class of static, spherically symmetric traversable wormholes sourced by an effective Casimir energy density that has been permanently modified by a gravitational-memory effect. The physical motivation comes from Sorge's result that a time-dependent gravitational perturbation passing through a Casimir cavity leaves a positive residual shift in the vacuum polarization of the confined field (2606.15552). By promoting the plate separation to an effective radial scale in a Morris–Thorne geometry, the authors obtain a source whose density combines the standard negative Casimir term proportional to $r^{-4}$ with a positive memory-induced correction proportional to $r^{-7}$. The resulting geometry is analyzed through its shape function, redshift sector, curvature structure, embedding diagrams, energy conditions, Tolman–Oppenheimer–Volkoff (TOV) equilibrium, and shadow radius.

Matter source and effective density profile

The starting point is the flat-spacetime electromagnetic Casimir density $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$, supplemented by Sorge's weak-field memory correction for a Gaussian gravitational pulse of strain amplitude $H$ and duration scale $\sigma^{-1}$, which scales as $L^{-7}$ and is positive. Promoting $L \to r$ yields

$\rho(r) = -\frac{\alpha}{r^4} + \frac{\eta}{r^7},$

with $\alpha > 0$ fixed by the field content ($\alpha = \pi^2/720$ for electromagnetism) and $r^{-7}$0 encoding the memory strength. The density changes sign at $r^{-7}$1: for large radii the ordinary Casimir term dominates (negative density), while near the throat the faster-decaying memory term can dominate if $r^{-7}$2, making the local density positive. The authors are explicit that treating $r^{-7}$3 as a free parameter extends beyond the perturbative regime of the original calculation; values with $r^{-7}$4 constitute the conservative sector directly tied to the underlying quantum result, while larger values represent a phenomenological continuation. This caveat is important because much of the phenomenologically interesting behavior occurs precisely outside the strictly perturbative window.

Geometry: shape function, redshift function, and parameter constraints

Inserting the density into the Einstein equation $r^{-7}$5 and integrating with $r^{-7}$6 gives the shape function

$r^{-7}$7

which satisfies the throat condition by construction and is asymptotically flat since $r^{-7}$8. At the throat,

$r^{-7}$9

so gravitational memory directly controls the local throat geometry rather than merely rescaling the exotic matter.

The redshift sector is closed by imposing a constant barotropic equation of state $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$0. Regularity of $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$1 at the throat requires the numerator of $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$2 to vanish at $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$3, which fixes

$\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$4

This is a strong constraint: the barotropic parameter is not free but determined by the ratio of memory to Casimir scales. The value $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$5, where the throat density vanishes, forces $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$6 and is singular within the constant-barotropic description; it must be excluded from the model.

The flare-out condition $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$7 restricts the admissible domain to

$\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$8

Within this interval two regimes emerge. For $\rho_{\rm Cas}(L) = -\pi^2/(720 L^4)$9 one has $H$0 and $H$1 — the Casimir-dominated sector. For $H$2 one obtains $H$3 and $H$4 — a phantom-like regime induced entirely by the memory contribution. Numerically, with $H$5 and $H$6, the transition lies at $H$7 and the upper bound at $H$8. Global checks confirm $H$9 for $\sigma^{-1}$0 across the sampled parameter values, and the flare-out quantity remains positive throughout, though increasing $\sigma^{-1}$1 softens the flare-out behavior.

Curvature diagnostics show that the Ricci scalar is concentrated near the throat and vanishes asymptotically, with the sharpest profile occurring for $\sigma^{-1}$2 close to the transition scale $\sigma^{-1}$3, where the redshift sector becomes most sensitive. Embedding diagrams exhibit the expected vertical tangent at the throat; notably, the apparent opening varies non-monotonically with $\sigma^{-1}$4 because the embedding integrand depends on the nonlinear combination $\sigma^{-1}$5.

Energy conditions and TOV equilibrium

At the throat, assuming finite $\sigma^{-1}$6, the radial null energy condition evaluates to

$\sigma^{-1}$7

which is negative whenever the flare-out condition holds. The radial NEC violation is therefore unavoidable and independent of the redshift profile, as expected on general grounds. The tangential NEC, by contrast, depends explicitly on the redshift gradient:

$\sigma^{-1}$8

Numerically, for $\sigma^{-1}$9, the radial NEC violation weakens monotonically with $L^{-7}$0 (from approximately $L^{-7}$1 at $L^{-7}$2 to $L^{-7}$3 at $L^{-7}$4), while the tangential NEC and SEC combinations change sign across the transition region: they are violated most strongly near $L^{-7}$5 but remain positive at the throat for the larger sampled values. This establishes that the exotic character of the source is predominantly radial and that the tangential sector can be effectively non-exotic for suitable memory strengths.

The TOV balance $L^{-7}$6 reveals a qualitative feature: the sign of the gravitational contribution $L^{-7}$7 is not fixed across parameter space. In one regime it is effectively repulsive near the throat, balanced by inward hydrostatic-anisotropic stresses; in another the signs invert, and the throat opening is sustained by outward pressure-gradient and anisotropic forces against an attractive gravitational term. The authors argue this inversion reflects a redistribution of the internal support mechanism rather than a pathology, with distinct interpretations above versus below the transition scale $L^{-7}$8. No running-gravitational-coupling force appears, since memory enters only through the matter density.

Shadow radius as a phenomenological diagnostic

Because the photon-sphere condition reduces to $L^{-7}$9, the shadow radius probes the same redshift sector fixed by the barotropic regularity condition. As $L \to r$0 (for $L \to r$1), the barotropic parameter diverges and the shadow radius grows sharply, terminating at the boundary of geometrically admissible parameter space. Away from this singular limit, the predicted dimensionless shadow radii overlap substantially with the Event Horizon Telescope range for M87* ($L \to r$2) under the normalization $L \to r$3, while remaining systematically above the narrower Sgr A* interval ($L \to r$4). A notable claim is that the EHT-compatible region lies entirely in the phantom-like sector $L \to r$5, where $L \to r$6, yet the solutions remain traversable. The dependence on $L \to r$7 is comparatively mild far from the regularity boundary, so the memory parameter dominates the observable shadow size. These comparisons are necessarily approximate: the model is static and spherically symmetric, whereas realistic EHT modeling requires rotation, accretion physics, and radiative transfer.

Limitations and open questions

Several limitations are acknowledged or implicit in the construction. First, the identification of the memory coefficient $L \to r$8 as a free parameter departs from the perturbative origin of the correction, so results in the extended sector ($L \to r$9) should not be read as direct consequences of the underlying quantum calculation. Second, the constant-barotropic closure is a modeling choice; the singular behavior at $\rho(r) = -\frac{\alpha}{r^4} + \frac{\eta}{r^7},$0 signals the breakdown of this description at the Casimir/memory transition, and whether a non-constant equation of state removes this singularity is left open. Third, the shadow analysis neglects rotation and accretion, so compatibility with the M87* bound constitutes a benchmark comparison rather than a fitted observation. Finally, the stability of these configurations against radial perturbations — a standard viability test for wormhole models — is not addressed.

Conclusion

The paper demonstrates that a gravitationally remembered Casimir vacuum can serve as a controlled, physically motivated source for traversable wormholes, with the memory parameter governing a transition between a Casimir-dominated and a phantom-like regime. The radial NEC violation persists at the throat in all admissible configurations, but its magnitude is softened by the memory term, and the tangential stress sector can remain non-exotic. The overlap of admissible solutions with the EHT shadow range for M87*, achieved entirely within the phantom-like sector, identifies shadow observations as a concrete diagnostic for this class of geometries. The open questions — the fate of the barotropic singularity at the transition scale, dynamical stability, and rotating generalizations — define the natural next steps for this program.