Image of a wormhole with an arbitrary throat profile

Abstract: We investigate the observable signatures -- the shadow, the throat silhouette, and the image of a thin accretion disk -- for a family of static, spherically symmetric wormholes with an arbitrary throat profile. First, we derive expressions for the shadow radius, the throat silhouette radius, and the photon energy shift for a general static, spherically symmetric metric. Then we apply these results to a specific wormhole metric containing three free parameters: the throat radius~$a$, the throat length~$λ$, and the parameter~$u_0$ that controls the depth of the gravitational well. We numerically obtain the shadow and silhouette radii as functions of $λ$, $u_0$, and $a$, construct accretion disk images for three representative parameter sets, and compare the results with those for a Schwarzschild black hole. We find that there exist sets of parameters $a$, $λ$, $u_0$ such that the wormhole shadow and throat silhouette radii coincide with the shadow and event horizon silhouette of a Schwarzschild black hole of the same mass. Nevertheless, the accretion disk images of these objects differ substantially. In wormhole images, the Doppler effect plays a major role, not the gravitational redshift. As a result, the accreting wormhole images appear brighter.

• The paper derives general formulas for wormhole photon shadows, throat silhouettes, and circular photon orbits, showing that the shadow depends on the outer photon sphere while the silhouette reflects the full throat geometry.
• The long-throat model reveals a strong size degeneracy: by adjusting the throat radius, wormhole shadows and silhouettes can match those of equal-mass Schwarzschild black holes, including shadow and silhouette radii of approximately 5.196m and 4.457m.
• The paper finds that accretion-disk brightness can break this degeneracy because wormholes produce a non-black throat and Doppler-dominated inner emission, with maximum energy shifts of about 1.31–1.45 compared with roughly 0.74 for Schwarzschild black holes.

Overview

This paper by Ishkaeva and Sushkov analyzes the observable signatures of static, spherically symmetric wormholes with an adjustable throat length, focusing on three features: the photon shadow, the throat silhouette, and the image of a thin accretion disk (2605.16413). The work addresses a gap in the literature: while wormhole lensing and shadows have been studied extensively for short-throat geometries, configurations with long throats—motivated both by domain-wall-supported geometries and by semiclassical solutions supported by vacuum polarization—had not previously been characterized observationally.

General framework

The authors work with a general static, spherically symmetric metric in the proper radial coordinate $u \in (-\infty, +\infty)$:

$ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$

with asymptotic flatness on the observer's side and $r(u)$ having a global minimum at the throat $u = 0$. Notably, the far side is not required to be asymptotically flat, in contrast to the Ellis–Bronnikov wormhole.

Using the Hamilton–Jacobi separation, photon trajectories fall into three classes: deflected rays with a turning point, throat-crossing rays without a turning point, and critical rays on circular photon orbits satisfying $U = U' = 0$. The shadow radius for a distant equatorial observer is

$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$

determined by the outermost photon sphere on the observer's side. Inner photon spheres, including one potentially located at the throat, do not affect the shadow boundary but can generate additional relativistic Einstein rings.

A central methodological contribution is the derivation of an integral equation for the throat silhouette radius $\alpha_{\text{sil}}$, following the silhouette formalism of Dokuchaev and Nazarova:

$$\int_0^\infty \frac{du}{r^2(u)\sqrt{N^2(u) - \alpha_{\text{sil}}^2/r^2(u)}} = \frac{\pi}{\alpha_{\text{sil}}}.$$

Unlike the shadow radius, which depends only on the photon sphere location, the silhouette radius depends on the full shape of $N(u)$ and $r(u)$—a distinction the authors exploit for parameter inference.

The paper also derives conditions for circular orbits at the throat itself: such orbits exist only if $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$0, i.e., when the redshift function is (locally) symmetric about the throat. Photon throat orbits are always unstable for $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$1 geometries, and the authors show that a massless wormhole ($ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$2) admits no stable circular orbits whatsoever, hence cannot host a thin accretion disk. This motivates the use of massive wormholes with nontrivial $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$3.

The long-throat wormhole model

The specific model uses

$ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$4

with three free parameters: the throat radius $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$5, the throat length $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$6, and $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$7 controlling the depth of the gravitational well. Embedding diagrams confirm that increasing $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$8 renders the throat nearly cylindrical, while $ds^2 = -N^2(u)\,dt^2 + du^2 + r^2(u)\,d\Omega^2,$9 affects only photon trajectories, not the spatial geometry.

The photon sphere structure is rich: a circular orbit always exists at the throat (since $r(u)$0 is symmetric), and for sufficiently large $r(u)$1 or small $r(u)$2, an additional pair of unstable circular orbits appears symmetrically on either side. The shadow boundary is set by the outer orbit on the observer's side; increasing $r(u)$3 pushes it outward, while increasing $r(u)$4 pulls it back toward the throat.

Degeneracy of shadow and silhouette sizes

Both $r(u)$5 and $r(u)$6 increase with the throat radius $r(u)$7 and decrease with $r(u)$8 and $r(u)$9. Because $u = 0$0 is a free parameter, the paper finds a strong degeneracy: parameter sets exist for which the wormhole shadow and silhouette radii exactly match those of a Schwarzschild black hole of equal mass. For $u = 0$1, $u = 0$2, the Schwarzschild shadow radius $u = 0$3 is reproduced at $u = 0$4, and the Schwarzschild silhouette radius $u = 0$5 at $u = 0$6. The implication is that shadow and silhouette sizes alone cannot distinguish this wormhole family from a Schwarzschild black hole.

Accretion disk images: the decisive discriminant

The authors construct thin-disk images (emission from $u = 0$7, observer at $u = 0$8, inclination $u = 0$9 close to that of Sgr A$U = U' = 0$0) for $U = U' = 0$1, $U = U' = 0$2 (equal to the Schwarzschild horizon radius) and three parameter sets. Despite matched masses and throat/horizon radii, the images differ substantially:

Object	$U = U' = 0$3 at throat/horizon	$U = U' = 0$4 at ISCO
Schwarzschild BH	$U = U' = 0$5	$U = U' = 0$6
WH ($U = U' = 0$7)	$U = U' = 0$8	$U = U' = 0$9
WH ($$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$0)	$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$1	$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$2
WH ($$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$3)	$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$4	$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$5

Two results stand out. First, photons emitted at the wormhole throat are only moderately redshifted ($$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$6), whereas horizon-emitted photons in the Schwarzschild case have $$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$7, rendering the event horizon silhouette black. Second, the maximum energy shift at the ISCO is roughly twice the Schwarzschild value ($$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$8–$$\alpha_{\text{sh}} = \left|\frac{r(u_{\text{ph}})}{N(u_{\text{ph}})}\right|,$$9 versus $\alpha_{\text{sil}}$0), because the Doppler effect, not gravitational redshift, dominates in the wormhole images. The consequence is that accreting wormholes appear systematically brighter in their inner disk regions than equal-mass black holes.

The parameters $\alpha_{\text{sil}}$1 and $\alpha_{\text{sil}}$2 provide independent control over the observable features: a deeper well (smaller $\alpha_{\text{sil}}$3) suppresses $\alpha_{\text{sil}}$4 at the throat while leaving ISCO shifts nearly unchanged, whereas a longer throat homogenizes the energy-shift distribution. The silhouette size responds oppositely to the two parameters, so combined measurements of silhouette size and brightness distribution can in principle break the degeneracy left by size measurements alone.

Limitations and open questions

The paper concedes several restrictions. The analysis is confined to static, spherically symmetric geometries; rotation, which is astrophysically expected, is not treated, and the degeneracy-breaking strategy via Doppler-dominated brightness may be altered in rotating counterparts. The accretion model is deliberately simplified: emission is restricted to the region inside the ISCO, only gravitational redshift and Doppler shift are included, and radiative transfer, disk thickness, and emission profiles are neglected—so quantitative flux comparisons with EHT-style observations would require more realistic modeling. The far-side asymptotics are left unconstrained, and the paper does not examine whether the additional photon spheres and Einstein rings predicted for long-throat geometries are observationally resolvable. Whether the parameter degeneracy in shadow/silhouette size persists across other wormhole families, and whether the Doppler-brightness signature survives realistic accretion physics, remain open questions raised by this work.

Conclusion

The paper provides general formulas for the shadow radius, throat silhouette radius, and photon energy shift in arbitrary static, spherically symmetric wormhole spacetimes, and applies them to a three-parameter long-throat model. Its principal findings are that shadow and silhouette sizes are degenerate with Schwarzschild black holes due to the free throat radius, but that accretion disk images are not: wormhole disks exhibit a Doppler-dominated, systematically brighter inner region and a non-black throat silhouette. These results identify the energy-shift distribution of the inner accretion flow, rather than shadow geometry, as the more robust observable for distinguishing traversable wormholes from black holes in strong-field imaging.