Reflection-Asymmetric Thin-Shell Wormhole

• Reflection-asymmetric thin-shell wormholes are constructs connecting two distinct spherically symmetric spacetimes via a thin-shell, resulting in a traversable throat lacking mirror symmetry.
• The configuration relies on junction conditions and localized exotic matter to sustain stability, with the effective potential modified by differing mass and charge parameters.
• Observational predictions include multi-photon ring structures and double shadow features, offering potential differentiation from standard black hole shadows in high-resolution imaging.

A reflection-asymmetric traversable thin-shell wormhole is a gravitational configuration in which two distinct spherically symmetric spacetimes—typically Reissner–Nordström or Schwarzschild geometries with differing @@@@1@@@@ and/or charge parameters—are joined across a thin-shell hypersurface, producing a traversable throat that lacks mirror symmetry under reflection through the throat. The resulting manifold possesses different effective potentials, photon sphere locations, and global properties on each side of the junction. This asymmetry fundamentally influences both the stability and the observational signatures of the wormhole, and it arises naturally in dynamic settings or by construction via generalized junction conditions.

1. Geometric Construction and Junction Formalism

The canonical construction begins by considering two metrics, such as

$$ds_i^2 = -f_i(r) dt_i^2 + f_i^{-1}(r) dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2),\quad f_i(r) = 1 - \frac{2M_i}{r} + \frac{Q_i^2}{r^2}$$

for $i=1,2$, each describing an asymptotically flat or AdS/dS region with respective mass $M_i$ and charge $Q_i$. A thin-shell hypersurface $\Sigma$ at radius $r_0$ is introduced, and the two spacetimes are cut at $r_0$ and glued together along $\Sigma$:

• The induced metric $h_{ab}$ on the shell is continuous ($r_1 = r_2 = r_0$ and $f_1(r_0) = f_2(r_0)$).
• The extrinsic curvature jump $[K_{ab}]$ determines the shell's surface energy-momentum tensor $S_{ab}$ via the (generalized for modified gravity) Darmois–Israel conditions.

The asymmetry arises when $M_1 \neq M_2$ and/or $Q_1 \neq Q_2$, ensuring the lack of reflection symmetry. In dynamical situations, as illustrated by the passage of infalling matter, even initially symmetric wormholes can become reflection-asymmetric after shell collisions that impart different effective mass parameters on either side (Nakao et al., 2013).

2. Physical Support and Stability Criteria

To sustain the wormhole throat, exotic matter violating the null energy condition is generally required. However, the thin-shell formalism allows localization of the energy condition violation to the shell itself, with the exterior (and sometimes even the shell for special parameter choices) obeying the null and weak conditions (Lobo, 25 Aug 2025).

The radial dynamics of the throat are encoded in an effective potential $V(a)$: $\dot{a}^2 + V(a) = 0$ For the Schwarzschild or Reissner–Nordström spacetimes, and in the presence of a cosmological constant $\Lambda$, the potential takes the form

$$V(a) = 1 - \frac{2M}{a} - \frac{\Lambda}{3}a^2 - [2\pi\sigma a]^2$$

Stability against radial perturbations requires $V''(a_0) > 0$. The inclusion of reflection-asymmetry influences the explicit form of $V$, modifying the allowed stability regions: increased asymmetry (difference in $M$, $Q$, or redshift function) often reduces the stability domain (Forghani et al., 2018, Macedo et al., 22 Oct 2025). However, parameters such as a positive cosmological constant can enhance stability (Lobo, 25 Aug 2025).

In dynamic interactions, such as the passage of a matter shell, the reflection asymmetry is further accentuated and can render the wormhole only metastable, with oscillatory persistence depending on a finely tuned parameter window (see the explicit traversability conditions P1–P4 in (Nakao et al., 2013)).

3. Traversability and Physical Constraints

Traversability is determined by both the avoidance of horizons (the throat must satisfy $a > 2M_i$ for regularity on each side) and by imposing tolerable tidal accelerations. The acceleration experienced by a traveler must not exceed $g_\oplus$, and the tidal forces must be less than the dissociation threshold of the traveler's body (Lobo, 25 Aug 2025, Celis et al., 2021).

In reflection-asymmetric geometries, one must compute the tidal acceleration across the shell, which typically depends on the jump in extrinsic curvature and the metric derivatives on either side. The lack of symmetry can lead to asymmetric tidal environments, with the possibility of strong localized stresses at the shell.

Estimates for the minimum throat and junction interface radii can be obtained by requiring sub-gravitational accelerations: $$a \geq \frac{|\zeta-1|c^2}{g_\oplus},\quad r_0 = \left(\frac{|b'-1||\zeta-1|^2 c^4}{g_\oplus^2}\right)^{1/3}$$ where $b(r)$ is the shape function and $\zeta$ parameterizes the redshift profile (Lobo, 25 Aug 2025).

4. Optical Appearance: Multi-Ring and Double Shadow Structure

A defining feature of reflection-asymmetric thin-shell wormholes is the presence of two photon spheres at distinct radii and with distinct effective potentials. When illuminated by surrounding thin accretion disks, the image observed from one side can exhibit multiple photon rings, a strong reduction in the central brightness depression ("shadow"), and sometimes double (or even multi) shadow structures (Macedo et al., 22 Oct 2025, Wielgus et al., 2020).

• Light rays with appropriate impact parameters can penetrate the throat, orbit near the opposite photon sphere, and return to produce additional rings inaccessible to canonical black holes.
• The number, location, and luminosity of these rings depend sensitively on the matching parameters (mass, charge, shell radius), the presence of emitting material on both sides (single-disk vs. double-disk scenarios), and the geometry-induced demagnification.
• The size of the central depression is markedly reduced compared to black hole shadows with similar mass (Macedo et al., 22 Oct 2025), and in two-disk scenarios, inner rings become more numerous and luminous.

The multi-ring structure provides an unequivocal observational distinction from Kerr or charged black holes, especially in high-resolution interferometric imaging (Wielgus et al., 2020, Macedo et al., 22 Oct 2025).

5. Mathematical Expressions and Parameter Matching

The matching at the shell imposes precise algebraic relations among the parameters: $A_+(r_0) = A_-(r_0),\qquad r_0^2 = \text{(shared)}$ and

$$\xi = \frac{M_+}{M_-} = 1 - \frac{y}{2x_0}(1-\eta),\quad \eta = \frac{Q_+^2}{Q_-^2},\ y = \frac{Q_-^2}{M_-^2},\ x_0 = \frac{r_0}{M_-}$$

These ensure the continuity of the induced metric and allow the explicit calculation of effective potentials $V_+(r)$ and $V_-(r)$, photon sphere radii $r_{ps}^{\pm}$, and the set of impact parameters $b$ yielding distinct photon trajectories.

For null geodesics on each side: $$\left(\frac{dr}{d\phi}\right) = \mp \frac{b}{r^2}\left[1 - b^2 V_\pm(r)\right]^{-1/2}$$ and the photon sphere conditions are: $\frac{d}{dr}V_\pm(r)\bigg|_{r_{ps}^\pm} = 0$ These equations underpin the full ray-tracing simulations yielding observable images (Macedo et al., 22 Oct 2025).

6. Observational and Experimental Prospects

Reflection-asymmetric thin-shell wormholes present unique, testable optical and dynamical signatures:

• Multi-photon ring structures observable in high-resolution astronomical imaging (e.g., EHT-class VLBI arrays), with enhanced ring contrast in two-disk configurations.
• Shrunk or multi-component shadows—deviations from the universal shadow radius-mass relation of black holes.
• Potential for distinctive lightcurve behavior in time-domain observations, due to round-trip trajectories ("delayed echoes") or non-exponential luminosity decay for infalling matter (Chen et al., 2022).

The detection of such features would constitute strong evidence for horizonless ultra-compact objects and probe both modified gravity and the microphysical structure of spacetime beyond standard general relativity (Macedo et al., 22 Oct 2025, Wielgus et al., 2020).

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Table 1: Key Features of Reflection-Asymmetric Thin-Shell Wormholes

Feature	Physical Origin	Observational Consequence
Asymmetric metric across throat	Distinct $M_\pm$, $Q_\pm$	Different photon spheres; multi-rings
Discontinuous photon impact param	Junction at shell	Non-universal shadow size
Tunable exotic matter localization	Junction choice, gravity	Exoticity minimized at shell
Multi-disk configurations	Accretion disk placement	Enhanced multi-ring visibility

Reflection-asymmetric thin-shell wormholes provide a rigorous theoretical testbed for the interplay of junction conditions, matter sources, gravitational stability, and potentially observable astrophysical phenomena. High-resolution imaging and precise lightcurve analysis offer promising avenues for empirical discrimination from canonical black holes.