PT-Symmetric Wormhole Model

• The PT-symmetric wormhole model is a bimetric construction that glues two Eddington–Finkelstein metrics via a PT-identification, creating a unified causal structure.
• It employs a null junction supported by an exotic, lightlike shell with negative energy density and positive tangential pressure to sustain traversability.
• The model provides insights into potential closed timelike curves and quantum-field implementations while linking non-Hermitian mechanics with gravitational phenomena.

Searching arXiv for papers on PT-symmetric wormhole models and closely related PT-symmetric wormhole/SYK literature. The PT-symmetric wormhole model is a modified Einstein–Rosen bridge in which traversability is achieved by gluing two regular Eddington–Finkelstein charts across a null throat and interpreting the two sides as parity-time-related sectors of a single bimetric construction. In its four-dimensional realization, the throat lies at $r=\alpha$ and supports a distributional curvature source: a lightlike shell of exotic matter with negative surface energy density and positive tangential pressure. The model is therefore neither a globally smooth single-metric spacetime nor a standard timelike thin-shell wormhole, but a $\mathcal{PT}$-symmetric null-junction geometry whose causal structure has been used to discuss traversability, closed timelike curves, scalar-field quantization, and possible observational signatures (Zejli, 17 Nov 2025).

1. Bimetric construction and $\mathcal{PT}$ identification

The defining geometric input is the use of two regular Eddington–Finkelstein-type metrics, one on each side of the throat,

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$

These metrics are regular at $r=\alpha$. Their only difference is the sign of the mixed term $g_{tr}$, which reverses the causal orientation of the bridge and implements the $\mathcal{PT}$ pairing between the “incoming” and “outgoing” sheets (Zejli, 17 Nov 2025).

This construction developed from earlier work on a modified Einstein–Rosen bridge formulated in ingoing and outgoing Eddington coordinates. In that antecedent form, the bridge is described as traversable in finite Eddington time, nondegenerate at the throat, and operationally one-way: infall is associated with one Eddington patch and emergence with the other. The transformation

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$

was identified as the basic $\mathcal{PT}$-symmetry of the bridge, and the identification of congruent points on the two sheets motivated the later bimetric interpretation (Koiran et al., 2024).

Within the more developed model, the two regions $\mathcal{PT}$0 and $\mathcal{PT}$1 are identified at the throat through

$\mathcal{PT}$2

so the mirror region is not treated as an independent copy of spacetime. Instead, the geometry is presented as a single spacetime sheet assembled from two $\mathcal{PT}$3-related causal presentations. A plausible implication is that the model relocates the usual “two-sheet” interpretation of the Einstein–Rosen bridge from global topology to a discrete symmetry identification across the junction (Zejli, 30 Jul 2025).

2. Null throat geometry and junction structure

The throat is defined by

$\mathcal{PT}$4

with normal

$\mathcal{PT}$5

At $\mathcal{PT}$6, the inverse metrics satisfy

$\mathcal{PT}$7

hence

$\mathcal{PT}$8

The throat $\mathcal{PT}$9 is therefore a null hypersurface rather than a timelike or spacelike matching surface. This null character determines the use of the Barrabès–Israel formalism in Poisson’s reformulation (Zejli, 17 Nov 2025).

The induced degenerate metric on $\mathcal{PT}$0 is continuous,

$\mathcal{PT}$1

which is the required junction condition for a null shell. An adapted basis on $\mathcal{PT}$2 is given by the null tangent generator

$\mathcal{PT}$3

the angular tangents

$\mathcal{PT}$4

and side-dependent transverse null vectors satisfying

$\mathcal{PT}$5

For the incoming side,

$\mathcal{PT}$6

and for the outgoing side,

$\mathcal{PT}$7

The central geometric quantity is the transverse curvature,

$\mathcal{PT}$8

or, when $\mathcal{PT}$9 is constant on $$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$0,

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$1

Its nonvanishing components are

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$2

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$3

with jump

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$4

These jumps are the distributional data that source the shell. In contrast with timelike thin-shell wormholes, the matching surface here is intrinsically lightlike, and the supporting matter is accordingly interpreted as a lightlike membrane rather than an ordinary surface layer (Zejli, 17 Nov 2025).

3. Surface stress-energy and exotic matter content

The shell stress tensor is decomposed as

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$5

where $$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$6 is the surface energy density along the null generators, $$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$7 is the surface current, and $$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$8 is the isotropic tangential pressure. The defining relations are

$$\mathrm{d}s_+^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 - \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right),$$9

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$0

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$1

with

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$2

For the induced 2-metric

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$3

one obtains

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$4

and therefore

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$5

Spherical symmetry and the absence of mixed angular curvature terms give

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$6

Since

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$7

the tangential pressure is

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$8

with angular components

$$\mathrm{d}s_-^2 = \left(1 - \frac{\alpha}{r}\right) \mathrm{d}t^2 - \left(1 + \frac{\alpha}{r}\right) \mathrm{d}r^2 + \frac{2\alpha}{r} \mathrm{d}r \, \mathrm{d}t - r^2 \left(\mathrm{d}\theta^2 + \sin^2\theta \, \mathrm{d}\phi^2\right).$$9

The shell is thus characterized by negative surface energy density, vanishing surface current, and positive tangential pressure (Zejli, 17 Nov 2025).

On a null shell, the null energy condition reduces to $r=\alpha$0. Since

$r=\alpha$1

the value

$r=\alpha$2

implies explicit NEC violation at the throat. This identifies the junction matter as exotic. The positive tangential pressure is interpreted as a repulsive source that helps stabilize the throat and keep the bridge open. In the model’s formulation, the exotic lightlike membrane supplies the surface stress-energy required for the Einstein field equations to hold globally, while the bulk regions satisfy

$r=\alpha$3

away from the throat (Zejli, 17 Nov 2025).

The distributional Bianchi identities further imply

$r=\alpha$4

In the stationary spherically symmetric configuration this reduces to identities because $r=\alpha$5, $r=\alpha$6 and $r=\alpha$7 are constant on the sphere, and the induced 2-geometry is metric-compatible. The intrinsic projection

$r=\alpha$8

ensures consistency with Einstein’s equations in distributional form.

4. Traversability, one-way propagation, and closed timelike curves

The wormhole is traversable because the metric remains regular at $r=\alpha$9 in Eddington–Finkelstein coordinates and the null junction permits passage in finite coordinate time. Earlier formulations emphasized the advanced or retarded Eddington coordinates

$g_{tr}$0

or equivalently

$g_{tr}$1

to exhibit the absence of a coordinate singularity at the throat and the finiteness of radial null propagation there (Zejli, 30 Jul 2025).

The model is not bidirectionally traversable within a single metric patch. The sign of the cross term $g_{tr}$2 fixes a preferred direction: $g_{tr}$3 describes an incoming one-way bridge and $g_{tr}$4 an outgoing one-way bridge. In this sense, the PT-symmetric wormhole is a one-way traversable geometry assembled from two causally opposed Eddington–Finkelstein sectors. This distinguishes it from conventional traversable wormholes in which a single smooth geometry supports motion in both directions (Koiran et al., 2024).

A further extension couples two such one-way bridges to generate closed timelike curves. The mechanism combines opposite causal orientations with the time inversion induced by the $g_{tr}$5 identification. A traveler can traverse one throat from $g_{tr}$6 to $g_{tr}$7 and a second throat from $g_{tr}$8 back to $g_{tr}$9, with return time satisfying

$\mathcal{PT}$0

The composite worldline is

$\mathcal{PT}$1

with

$\mathcal{PT}$2

The curve remains timelike everywhere; the nontrivial causal effect is global rather than local. The construction is described as consistent with Novikov’s self-consistency principle, so only globally self-consistent histories are admissible (Zejli, 30 Jul 2025).

A common misconception is that the model claims paradox-generating “free” time travel. The published formulation does not do so. It instead ties CTC formation to a specific spacetime arrangement built from two traversable one-way throats and explicitly invokes Novikov self-consistency as the interpretive framework (Zejli, 30 Jul 2025).

5. Quantum-field and phenomenological aspects

An effective scalar-field theory has been constructed on the two $\mathcal{PT}$3-related sectors by introducing fields $\mathcal{PT}$4 on $\mathcal{PT}$5 and $\mathcal{PT}$6 on $\mathcal{PT}$7 with

$\mathcal{PT}$8

and throat matching

$\mathcal{PT}$9

The action is

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$0

with

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$1

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$2

Variation yields

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$3

Under mode decomposition $\eta \mapsto -\eta, \qquad t' \mapsto -t'$4, the junction condition implies

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$5

or, at operator level,

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$6

This frequency pairing has been used to motivate PT-organized spectral structures and possible QNM doublets (Zejli, 30 Jul 2025).

The same line of work argues that pseudo-unitary evolution can be maintained despite non-Hermiticity in the naive sense. The relevant condition is

$\eta \mapsto -\eta, \qquad t' \mapsto -t'$7

for a positive metric operator $\eta \mapsto -\eta, \qquad t' \mapsto -t'$8, allowing a real energy spectrum in the unbroken $\eta \mapsto -\eta, \qquad t' \mapsto -t'$9-symmetric regime. This is presented as the quantum-mechanical framework supporting scalar-field propagation on the wormhole background (Zejli, 30 Jul 2025).

The phenomenology proposed for the null-shell wormhole is explicitly multi-channel. First, the throat and photon-sphere barrier form a cavity. With tortoise coordinate

$\mathcal{PT}$0

the cavity length is

$\mathcal{PT}$1

and for $\mathcal{PT}$2,

$\mathcal{PT}$3

This yields late-time gravitational-wave echoes from repeated partial reflections between the light ring and the throat (Zejli, 17 Nov 2025).

Second, horizon-scale imaging may show duplicated or through-throat photon rings and non-Kerr asymmetric brightness patterns because the geometry is two-sided and lacks an event horizon. Third, in a 1+1-dimensional toy model using DFU stress tensors,

$\mathcal{PT}$4

with

$\mathcal{PT}$5

the $\mathcal{PT}$6 identification can cancel the net outward flux across the throat by pairing $\mathcal{PT}$7 and $\mathcal{PT}$8 contributions. Fourth, a relic population of such wormholes has been speculatively connected to an effective $\mathcal{PT}$9 and the seeding of cosmic voids. These cosmological points are presented as phenomenological speculation rather than derivation (Zejli, 17 Nov 2025).

6. Relation to PT-symmetric SYK and non-Hermitian wormhole analogues

The expression “PT-symmetric wormhole model” also appears in holography-inspired many-body settings, especially coupled SYK systems. In that literature, a two-site PT-symmetric SYK model functions as a toy model for different wormhole configurations in nearly $\mathcal{PT}$00 gravity. The dictionary given there identifies

$\mathcal{PT}$01

with traversable wormholes,

$\mathcal{PT}$02

with Euclidean wormholes, and

$\mathcal{PT}$03

with Keldysh wormholes in the Lindbladian regime (García-García et al., 2023).

In that framework, PT symmetry reduces the 38 Bernard-LeClair non-Hermitian universality classes to 24 admissible PT-symmetric classes, of which 14 are realized in the coupled two-site SYK model. Four classes,

$\mathcal{PT}$04

are singled out as topological because their Hamiltonians admit rectangular block structures with integer index $\mathcal{PT}$05, guaranteeing $\mathcal{PT}$06 robust purely real eigenvalues. This is presented as a link among non-Hermitian PT symmetry, topology, and wormhole physics (García-García et al., 2023).

A distinct development deforms the standard two-sided SYK traversable-wormhole teleportation setup by adding a PT-symmetric non-Hermitian boundary term,

$\mathcal{PT}$07

so that

$\mathcal{PT}$08

In a two-level reduction,

$\mathcal{PT}$09

with critical threshold

$\mathcal{PT}$10

At $\mathcal{PT}$11, the model reaches an exceptional point; for $\mathcal{PT}$12, eigenvalues become complex-conjugate pairs and the PT-broken phase acts as a causal amplifier. The transmitted amplitude is enhanced without shifting the traversability window, which remains near

$\mathcal{PT}$13

in the reported numerics. In that work, $\mathcal{PT}$14 Majorana modes per side are used, 100 disorder realizations are analyzed, and the threshold $\mathcal{PT}$15 is reported to follow a log-normal distribution with

$\mathcal{PT}$16

This SYK-based program does not describe the same spacetime geometry as the null-shell Einstein–Rosen construction, but it shows that PT symmetry has become a unifying concept across both geometric wormhole models and non-Hermitian many-body wormhole analogues (Joshi et al., 25 Jan 2026).

Taken together, these strands support a narrow but technically coherent usage of the term: a PT-symmetric wormhole model is either a bimetric spacetime construction in which parity-time symmetry organizes the gluing, causal orientation, and matter content of a traversable throat, or a quantum many-body analogue in which PT symmetry organizes the spectral and dynamical structure associated with wormhole-like transport. The four-dimensional null-shell model is the most explicit realization of the term in a general-relativistic setting, and its characteristic signature is the conjunction of a null junction, a lightlike exotic membrane, and a $\mathcal{PT}$17-related two-sheet causal structure (Zejli, 17 Nov 2025).