Slowly rotating traversable wormholes supported by radially varying string-fluid matter: From regular geometries to photon trajectories

Abstract: This work investigates slowly rotating traversable wormholes supported by string fluids whose properties vary with distance from the throat. This radial variation allows the matter to transition smoothly from a de Sitter-like core near the center to a string-dominated environment further out, producing a regular, horizon-free, and asymptotically flat spacetime. By letting the transverse pressure depend on radius, the fluid naturally adapts to the surrounding geometry, resulting in a well-behaved energy density and shape function. Even modest rotation introduces frame-dragging effects that gently twist photon paths, creating subtle differences between co-rotating and counter-rotating trajectories. These effects are strongest near the throat, while at larger distances the spacetime is largely governed by the static gravitational potentials. Circular photon orbits reveal that the interplay of the redshift function, wormhole shape, and rotation shapes the photon-sphere structure. Different radial profiles of the string fluid generate distinctive photon-ring patterns, offering potential observational signatures of both the rotation and the internal matter distribution. Overall, radially varying string fluids provide a flexible and physically consistent source for traversable wormholes, bridging smoothly between vacuum-like and string-dominated regions while maintaining regularity and supporting slow rotation. This study highlights how anisotropic matter can influence both curvature and light propagation, providing a realistic framework for horizonless exotic spacetimes and suggesting new avenues to explore subtle observational effects around traversable wormholes.

• The paper constructs a fully analytic, regular, horizonless, and asymptotically flat slowly rotating wormhole supported by radially varying string-fluid matter, with flare-out satisfied for ε < 1 and moderate r₀/a.
• The model derives frame dragging as ω = 2J/r³, confines exoticity mainly to tangential NEC violation while the radial volume integral vanishes, and remains free of singularities, ergoregions, and closed timelike curves in the tested slow-spin regime.
• The paper shows that rotation splits photon spheres into prograde and retrograde branches, producing predicted shadow diameters of 36–45 μas and sub-Kerr distortions of δ ≈ 0.03–0.09 that could be tested with next-generation EHT observations.

Motivation and scope

This paper constructs a fully analytic, slowly rotating traversable wormhole sourced by an anisotropic string fluid whose transverse equation of state (EoS) depends on the radial coordinate, $W_t(r) = 1/\tilde{\sigma}(r)$. The stated novelty is the combination of three ingredients previously treated only separately: rotating wormholes with constant string-fluid parameters, radially varying EoS wormholes without rotation, and slow-rotation constructions with other matter sources. The work traces a single causal chain from the matter content through frame dragging to photon trajectories and shadow observables, within a geometry that is regular, horizonless, and asymptotically flat (2606.11261).

The broader context is that exact rotating wormhole solutions with specified matter sources are rare; most analyses rely on perturbative slow-rotation expansions or numerical methods, or on Teo-type metrics where the supporting matter is left implicit. The authors also emphasize that Geroch-Hansen multipole moments of rotating wormholes remain largely unexplored beyond the lowest orders.

The radially varying string fluid

The source builds on Letelier's string cloud formalism, in which a surface bivector $\Sigma^{\mu\nu}$ on the timelike worldsheet replaces the $u^\mu u^\nu$ structure of a dust energy-momentum tensor. For static spherical symmetry, symmetry restricts $\Sigma_{\mu\nu}$ to two components, giving $T^t{}_t = T^r{}_r$ and $T^\theta{}_\theta = T^\varphi{}_\varphi = p$. Extending Soleng's constant-tension-ratio model, the authors take

$$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$

while retaining the radial condition $p_r = -\rho$, interpreted as a de Sitter-like radial EoS rather than as a horizon signature. Conservation of the energy-momentum tensor then fixes $\tilde{\sigma}(r)$ once a density profile is chosen:

$$\tilde{\sigma}(r) = -\frac{2\rho(r)}{r\rho'(r) + 2\rho(r)},$$

which remains positive for the adopted profile, ensuring transverse pressure of the same sign as the density.

The smoothed density profile,

$\Sigma^{\mu\nu}$0

is borrowed from regular black hole constructions: the parameter $\Sigma^{\mu\nu}$1 sets the smoothing scale, $\Sigma^{\mu\nu}$2 controls the string-fluid strength, and $\Sigma^{\mu\nu}$3 coincides with the throat radius $\Sigma^{\mu\nu}$4. The authors are explicit that this profile is not fundamental but chosen for analytic tractability, and they verify robustness against power-law and Gaussian-core alternatives (see below).

Geometry and regularity

The spacetime uses the stationary axisymmetric form with potentials

$\Sigma^{\mu\nu}$5

where the shape function follows from the density via $\Sigma^{\mu\nu}$6, integrating to an expression involving incomplete gamma functions. Two consistency checks are reported explicitly. First, the flare-out condition $\Sigma^{\mu\nu}$7 holds analytically for $\Sigma^{\mu\nu}$8 and moderate $\Sigma^{\mu\nu}$9, with the paper stating it holds for $u^\mu u^\nu$0. Second, both Ricci and Kretschmann scalars are finite for all $u^\mu u^\nu$1 across the examined parameter space, peaking at the throat and decaying monotonically — confirming a horizon-free, singularity-free geometry.

The frame-dragging function is derived rather than assumed: at first order in spin, $u^\mu u^\nu$2 with $u^\mu u^\nu$3 yields $u^\mu u^\nu$4, and imposing $u^\mu u^\nu$5 at infinity gives $u^\mu u^\nu$6, with $u^\mu u^\nu$7 identified through the asymptotic $u^\mu u^\nu$8. In Appendix B, the Geroch-Hansen moments are computed to this order: the mass monopole equals the Komar mass, $u^\mu u^\nu$9, approximately $\Sigma_{\mu\nu}$0 for fiducial parameters, and the current dipole equals $\Sigma_{\mu\nu}$1. Higher multipoles would require second-order terms and are deferred.

The slow-rotation expansion is controlled by $\Sigma_{\mu\nu}$2 for all spins used ($\Sigma_{\mu\nu}$3), so neglected corrections stay below the percent level. The paper also verifies that no ergoregion forms: $\Sigma_{\mu\nu}$4 would require $\Sigma_{\mu\nu}$5, which fails for the adopted spins and radii, and no closed timelike curves appear.

Photon trajectories and shadows

In the equatorial plane, radial photon motion reduces to $\Sigma_{\mu\nu}$6, with the circular-orbit condition splitting the impact parameter into prograde and retrograde branches,

$\Sigma_{\mu\nu}$7

so rotation opens a "photon band" between the two circular orbits. The Lense-Thirring precession frequency scales as $\Sigma_{\mu\nu}$8, maximal at the throat, with dimensionless values $\Sigma_{\mu\nu}$9–$T^t{}_t = T^r{}_r$0 for representative parameters — small but non-negligible near $T^t{}_t = T^r{}_r$1.

To probe how the redshift gradient imprints on optics, five phenomenological logarithmic lapse functions $T^t{}_t = T^r{}_r$2 are introduced (linear, quadratic, exponentially damped, tanh-regulated, saturating). The rotational displacement of the photon sphere depends directly on the local logarithmic slope,

$T^t{}_t = T^r{}_r$3

so steeper lapse gradients amplify the spin-induced shift. A caveat is stated plainly: these lapse profiles are not derived from the field equations for the string-fluid source; they are regularity-motivated phenomenological choices, and solving the fully coupled system is left open.

Representative quantitative results for $T^t{}_t = T^r{}_r$4 and $T^t{}_t = T^r{}_r$5:

Lapse	$T^t{}_t = T^r{}_r$6	$T^t{}_t = T^r{}_r$7	$T^t{}_t = T^r{}_r$8 ($T^t{}_t = T^r{}_r$9as)	$T^\theta{}_\theta = T^\varphi{}_\varphi = p$0	$T^\theta{}_\theta = T^\varphi{}_\varphi = p$1
$T^\theta{}_\theta = T^\varphi{}_\varphi = p$2 (linear)	1.52	1.38	41.2	0.048	0.14
$T^\theta{}_\theta = T^\varphi{}_\varphi = p$3 (quadratic)	1.65	1.42	43.8	0.075	0.23
$T^\theta{}_\theta = T^\varphi{}_\varphi = p$4 (exponential)	1.31	1.24	36.4	0.027	0.07
$T^\theta{}_\theta = T^\varphi{}_\varphi = p$5 (tanh)	1.58	1.40	42.5	0.060	0.18
$T^\theta{}_\theta = T^\varphi{}_\varphi = p$6 (saturating)	1.74	1.44	45.3	0.094	0.30
Kerr BH	2.60	2.20	42.0	0.167	0.40

A sensitivity analysis replacing the nominal density with power-law ($T^\theta{}_\theta = T^\varphi{}_\varphi = p$7) or Gaussian-core profiles shifts photon-sphere radii by roughly 5–15% while preserving the prograde-retrograde split, the photon band, and the spin scaling — indicating the qualitative conclusions are not tied to the specific density choice.

Energy conditions and exotic matter

With $T^\theta{}_\theta = T^\varphi{}_\varphi = p$8 and $T^\theta{}_\theta = T^\varphi{}_\varphi = p$9, the NEC holds marginally in the radial direction ($$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$0 exactly) but is violated tangentially near the throat, where $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$1 region behavior drives $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$2; the SEC fails in the asymptotic string-dominated regime where $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$3. Notably, because $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$4 identically, the Visser-Kar-Dadhich volume integral vanishes:

$$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$5

The authors stress this does not mean absence of exotic matter — the violation lives in the tangential sector, which this integral does not capture — and note that a small deviation $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$6 would make $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$7 arbitrarily tunable. Compared with phantom-scalar wormholes (delocalized violation) and Casimir-supported models (thin-shell violation), the exotic region here is controlled by the smoothing scale $$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$8, offering a direct handle on its localization.

Observational prospects

Scaled to M87* parameters ($$\rho(r) = \tilde{\sigma}(r)\, p(r), \qquad p_\theta = p_\varphi = W_t(r)\,\rho(r),$$9, $p_r = -\rho$0 Mpc), predicted shadow diameters of roughly 36–45 $p_r = -\rho$1as fall within current EHT sensitivity (~20 $p_r = -\rho$2as at 230 GHz) and near the measured M87* value (~42 $p_r = -\rho$3as). However, the distortion parameters $p_r = -\rho$4–$p_r = -\rho$5 are systematically smaller than Kerr values (0.1–0.3), and the paper identifies two practical obstacles: degeneracy with low-spin Kerr shadows absent an independent spin measurement, and the EHT's limited ability to constrain shape distortions below $p_r = -\rho$6. The next-generation EHT, at ~5 $p_r = -\rho$7as resolution, could measure photon-ring thicknesses of 2–8 $p_r = -\rho$8as implied by $p_r = -\rho$9–$\tilde{\sigma}(r)$0, and configurations with $\tilde{\sigma}(r)$1 would enter its reach; a convincing discrimination would likely additionally require a time-domain horizon-absence signature such as gravitational-wave echoes.

Limitations and open questions

Several limitations are conceded explicitly. The five lapse profiles are phenomenological, not derived from the coupled Einstein-string-fluid system; deriving them self-consistently is an open task. No stability analysis is performed — linear perturbations of rotating wormholes are acknowledged as technically demanding — though the authors argue heuristically that anisotropic stresses and the smooth throat profile may favor stability relative to thin-shell constructions. Higher Geroch-Hansen multipoles, which would encode throat structure and deviations from the Kerr pattern $\tilde{\sigma}(r)$2, require second-order slow-rotation terms and remain uncomputed. Ergoregions and closed timelike curves are excluded only within the slow-rotation regime; faster spin would demand separate analysis. Finally, the flare-out constraint restricts parameter space to $\tilde{\sigma}(r)$3 and $\tilde{\sigma}(r)$4.

Conclusion

The paper delivers a self-consistent, fully analytic slowly rotating wormhole supported by a radially varying string fluid, connecting the matter EoS through frame dragging to photon dynamics and shadow observables. Its distinctive results are the tangential localization of the NEC violation with vanishing radial volume-integral quantifier, the demonstration that the lapse-function gradient controls the rotational splitting of photon spheres, and quantitative shadow predictions that lie within EHT reach while exhibiting sub-Kerr distortion. The framework provides a concrete template for how anisotropic matter distributions leave observable imprints on horizonless compact objects, with stability, self-consistent lapse derivation, and higher multipoles constituting the principal unresolved questions.