Damour–Solodukhin Wormholes

• Damour–Solodukhin wormholes are traversable structures derived by minimally deforming black-hole metrics, replacing event horizons with throats connecting asymptotically flat regions.
• The deformation parameter (λ) governs deviations from Schwarzschild or Kerr geometries, affecting geodesic dynamics, light-ring stability, and accretion profiles.
• Distinct observational features, including unique lensing patterns, non-logarithmic strong deflection, and late-time echoes, set these wormholes apart as black-hole mimickers.

Damour–Solodukhin wormholes are Schwarzschild-like or Kerr-like traversable Lorentzian wormholes obtained by a small deformation of black-hole geometries that removes the event horizon and replaces it with a throat connecting two asymptotically flat regions. In the static case the deformation is commonly encoded by a dimensionless parameter written as $\lambda$, $\Lambda$, or $\omega$ depending on convention; when it vanishes the geometry reduces to Schwarzschild, while rotating generalizations reduce to Kerr when $\lambda=0$ (Matyjasek, 2020, Karimov et al., 2019). Because the exterior geometry can remain very close to the corresponding black-hole metric, these spacetimes are studied as black-hole mimickers in geodesic dynamics, lensing, accretion, perturbation theory, and semiclassical stress-energy analyses (Nandi et al., 2018, Qian et al., 2024).

1. Metric structure and parameterizations

The recent literature uses several closely related static and rotating forms of the Damour–Solodukhin geometry. A recurrent static form modifies only the redshift factor,

$$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$

so that the Schwarzschild horizon is replaced by a throat at $r=2M$ (Bermúdez-Cárdenas et al., 14 Apr 2025, Yusupova et al., 1 Jul 2026). A second common form, obtained after a rescaling of time and mass, is written as

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$

with

$$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$

for which the throat is at

$r_{\rm th}=2M(1+\lambda^2),$

and the ADM mass is

$M_{\rm ADM}=M(1+\lambda^2)$

(Tsukamoto, 2020). In a further static convention used in galactic embeddings, the isolated wormhole is written as

$\Lambda$0

with throat at $\Lambda$1 (Biswas et al., 2023).

The rotating generalization, often called a Kerr-like wormhole or rotating Damour–Solodukhin wormhole, is written in Boyer–Lindquist-like coordinates as

$\Lambda$2

with

$\Lambda$3

Here the throat is determined by $\Lambda$4, and $\Lambda$5 enters only through $\Lambda$6, i.e. only the $\Lambda$7 component is modified relative to Kerr (Karimov et al., 2019). A related dimensional form used in electromagnetic-flux calculations employs

$\Lambda$8

with ergoregion existence limited by

$\Lambda$9

(Urtubey et al., 2024).

Setting	Metric functions as written	Throat / limit
Static $\omega$0-shift form	$\omega$1, $\omega$2, $\omega$3	throat at $\omega$4; Schwarzschild at $\omega$5
Rescaled static form	$\omega$6, $\omega$7, $\omega$8	throat at $\omega$9
Rotating Kerr-like form	$\lambda=0$0	Kerr recovered at $\lambda=0$1

This multiplicity of conventions matters for direct comparison of orbit radii and critical parameters. Several papers therefore state results in the notation natural to the chosen parameterization rather than in a single universal form (Bermúdez-Cárdenas et al., 14 Apr 2025, Tsukamoto, 2020).

2. Light rings, photon spheres, and timelike circular orbits

A major development is the intrinsic-curvature treatment of circular geodesics in traversable wormholes. For the static form

$\lambda=0$2

the projected Jacobi metric on the equatorial plane is

$\lambda=0$3

Circular orbits occur where the geodesic curvature $\lambda=0$4 vanishes, and stability follows from the sign of the Gaussian curvature $\lambda=0$5 (Bermúdez-Cárdenas et al., 14 Apr 2025). In this approach the massive-particle circular-orbit condition becomes

$\lambda=0$6

while for null orbits it reduces to

$\lambda=0$7

For $\lambda=0$8 this yields

$\lambda=0$9

The throat itself also satisfies $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$0, so the throat supports a circular orbit; in the null case this gives a light ring at $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$1 (Bermúdez-Cárdenas et al., 14 Apr 2025).

The stability of the throat light ring is controlled by the near-throat Gaussian curvature. For photons,

$$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$2

so the throat light ring is stable if $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$3, unstable if $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$4, and marginal at $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$5. For the additional outer light ring, the stability is complementary: one of the two light rings is stable and the other unstable, depending on $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$6. In the same framework the innermost stable circular orbit follows from $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$7,

$$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$8

which reduces to the Schwarzschild value when $$ds^2=-\bigl(f(r)+\lambda^2\bigr)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2, \qquad f(r)=1-\frac{2M}{r},$$9 (Bermúdez-Cárdenas et al., 14 Apr 2025).

In the rescaled static convention, null-orbit analyses emphasize a regime structure. There are two photon spheres at $r=2M$0 and an antiphoton sphere at the throat for $r=2M$1; at $r=2M$2 the photon spheres and antiphoton sphere degenerate into a marginally unstable photon sphere; and for $r=2M$3 there is one photon sphere located at the throat (Tsukamoto, 2020). This is consistent with the broader statement that the light-ring structure of wormholes contains an odd number of light rings and that the throat is always one of them (Bermúdez-Cárdenas et al., 14 Apr 2025).

Timelike geodesics have also been solved analytically in the rescaled static geometry

$r=2M$4

The throat radius is

$r=2M$5

and the radial potential has four roots because the throat itself is a root: $r=2M$6 The throat can merge with other roots and generate double-, triple-, and quartic-root degeneracies. The triple root pinned at the throat determines the ISCO when $r=2M$7, whereas if $r=2M$8 the ISCO remains at $r=2M$9. Using Mino time, bound and unbound trajectories are obtained in closed form in terms of incomplete elliptic integrals. When the throat is a simple root, particles traverse smoothly between the two asymptotically flat regions; when the throat is a double or triple root, the azimuthal angle and coordinate time develop logarithmic or power-law divergences as the particle approaches the throat (Ho et al., 21 May 2026).

3. Lensing, strong deflection, and shadow phenomenology

Weak deflection in static Damour–Solodukhin wormholes has been computed by applying the Gauss–Bonnet theorem to the optical metric. For the non-rotating case the deflection angle is

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$0

so the Schwarzschild result is recovered at $ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$1 and the deformation parameter increases the bending angle (Övgün, 2018). For the rotating Kerr-like wormhole, the Ono–Ishihara–Asada extension yields

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$2

with the plus sign for retrograde rays and the minus sign for prograde rays (Övgün, 2018).

Strong-field lensing in the static case has been analyzed with Bozza’s method. In one commonly used form of the metric, the photon sphere remains at

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$3

which is a central reason why the lensing observables remain close to those of Schwarzschild. The 2018 strong-lensing study concluded that for

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$4

the strong-field coefficients and observables are already very close to their Schwarzschild values, and for

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$5

they coincide with Schwarzschild to the displayed precision for the Sgr A* benchmark used there. In that sense strong lensing suggests an observational upper bound

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$6

if one insists on practical indistinguishability from Schwarzschild by those observables alone (Nandi et al., 2018).

A distinct strong-deflection phenomenon arises at the critical value

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$7

At this point the throat becomes a marginally unstable photon sphere and the usual logarithmic divergence of the deflection angle breaks down. Instead,

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$8

with

$ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,$9

This nonlogarithmic divergence is the characteristic signature of the degenerate marginally unstable photon sphere in the Damour–Solodukhin geometry (Tsukamoto, 2020).

The lensing literature also contains an explicit methodological dispute. A comment argued that the static strong-deflection coefficients in one Gauss–Bonnet analysis were algebraically inconsistent and that the rotating weak-deflection result missed the direct/retrograde asymmetry required by frame dragging (Bhattacharya et al., 2018). The reply defended the original use of the Gibbons–Werner and Ono–Ishihara–Asada frameworks as a leading-order weak-deflection method and maintained that

$$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$0

is the correct asymptotic leading-order behavior for the rotating case in that setting (Övgün, 2018). The controversy is therefore primarily about the scope of the approximation and the interpretation of leading versus higher-order terms.

Ray-tracing studies treat the Damour–Solodukhin wormhole as the $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$1 limit of an RS-like wormhole family,

$$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$2

with $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$3 for the DS case. In that framework the photon sphere exists, the photon-ring radius shrinks as $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$4 increases, the shadow radius likewise decreases as $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$5 increases, and a secondary lensed ring appears for small positive $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$6 before becoming less visible and eventually disappearing for larger $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$7 (Sokoliuk et al., 2022). By contrast, once a galactic dark-matter halo is included, the environmental corrections can push the photon sphere and shadow outward: $$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$8

$$A(r)=1-\frac{2M}{r}, \qquad B(r)=\left[1-\frac{2M(1+\lambda^2)}{r}\right]^{-1}, \qquad C(r)=r^2,$$9

This indicates that photon-ring and shadow diagnostics are sensitive not only to the throat deformation but also to the surrounding matter distribution (Biswas et al., 2023).

4. Accretion, disk physics, and electromagnetic output

Steady spherical Bondi accretion onto the static Damour–Solodukhin wormhole has been developed in the metric

$r_{\rm th}=2M(1+\lambda^2),$0

with

$r_{\rm th}=2M(1+\lambda^2),$1

When $r_{\rm th}=2M(1+\lambda^2),$2, the event horizon disappears and the spacetime instead has a throat at

$r_{\rm th}=2M(1+\lambda^2),$3

The standard mimicker argument based on light travel time is

$r_{\rm th}=2M(1+\lambda^2),$4

which is large for tiny $r_{\rm th}=2M(1+\lambda^2),$5, but the Bondi analysis shows that mimicry in accretion profiles is less restrictive: the radial velocity, density, and mass accretion rate near $r_{\rm th}=2M(1+\lambda^2),$6 can remain close to the Schwarzschild black-hole case even for $r_{\rm th}=2M(1+\lambda^2),$7. The exact behavior at the inner radius differs, however: for Schwarzschild the radial velocity remains finite at the horizon, whereas for the Damour–Solodukhin wormhole

$r_{\rm th}=2M(1+\lambda^2),$8

(Yusupova et al., 1 Jul 2026).

Thin-disk accretion around the rotating Damour–Solodukhin wormhole has been studied within the Novikov–Thorne/Page–Thorne formalism. Because $r_{\rm th}=2M(1+\lambda^2),$9 enters only through $M_{\rm ADM}=M(1+\lambda^2)$0, i.e. only through $M_{\rm ADM}=M(1+\lambda^2)$1, the quantities $M_{\rm ADM}=M(1+\lambda^2)$2, $M_{\rm ADM}=M(1+\lambda^2)$3, $M_{\rm ADM}=M(1+\lambda^2)$4, $M_{\rm ADM}=M(1+\lambda^2)$5, $M_{\rm ADM}=M(1+\lambda^2)$6, and the radiative efficiency

$M_{\rm ADM}=M(1+\lambda^2)$7

are independent of $M_{\rm ADM}=M(1+\lambda^2)$8 and are identical to their Kerr values for a given spin (Karimov et al., 2019). The emissivity sector is different: increasing $M_{\rm ADM}=M(1+\lambda^2)$9 shifts the peaks of $\Lambda$00 and $\Lambda$01 to larger radius, decreases the maximum flux and temperature, and lowers the peak frequency of the emitted spectrum. Nevertheless, for the strong-lensing-motivated regime

$\Lambda$02

the flux differences are too small to be observable with present precision (Karimov et al., 2019).

Neutrino-pair annihilation above thin disks has been analyzed in the static wormhole background. The metric used there modifies only the time-time component,

$\Lambda$03

and the study finds that the Damour–Solodukhin disk is about $\Lambda$04 hotter than the Schwarzschild disk in the immediate vicinity of the compact object. The ISCO radius decreases as $\Lambda$05 increases, the neutrino-antineutrino energy deposition is enhanced over the Newtonian estimate near the compact object, and the emitted power remains large enough that the black-hole-like wormhole can trigger a gamma-ray burst for approximately

$\Lambda$06

(Shi et al., 2023).

Rotating Damour–Solodukhin wormholes can also emit an outward Poynting flux when threaded by magnetized accreting matter. In a force-free magnetosphere with imposed field geometry, the local radial energy flux density is

$\Lambda$07

and the total extracted power is

$\Lambda$08

A Blandford–Znajek-like mechanism is possible when the ergosphere exists and the disk inner edge lies inside it; the relevant high-spin regime is

$\Lambda$09

with

$\Lambda$10

in geometrized units. The resulting power is reported to be of the same order as for Kerr black holes, with a sample maximum near $\Lambda$11, $\Lambda$12 (Urtubey et al., 2024).

5. High-energy collisions and neutrino oscillation probes

The static Damour–Solodukhin wormhole can act as a particle accelerator because the throat permits head-on collisions between ingoing and outgoing trajectories. In the metric

$\Lambda$13

the throat is at $\Lambda$14, and the center-of-mass energy for two particles is

$\Lambda$15

For a head-on collision at the throat,

$\Lambda$16

so the collision energy grows without bound as the wormhole approaches the black-hole limit. Rear-end collisions do not show this enhancement. The throat also admits a circular orbit with

$\Lambda$17

and for small $\Lambda$18 this orbit is stable, again allowing collisions with the same $\Lambda$19 scaling (Tsukamoto et al., 2019).

Neutrino flavor oscillations provide a distinct probe of the static geometry. In the lensing setup

$\Lambda$20

the wormhole factor enters the neutrino phase through the shifted redshift function and modifies both radial and nonradial propagation (Shi et al., 2024). The resulting flavor-transition probabilities depend not only on the mass-squared differences but also on the individual neutrino masses and on the sign of the mass ordering. The closest-approach radius is approximated by

$\Lambda$21

and the paper reports that the contribution of $\Lambda$22 can change the shapes of the probability-versus-angle curves greatly, particularly in the three-flavor case (Shi et al., 2024). This suggests that neutrino lensing and oscillation data could, in principle, be used to estimate $\Lambda$23 and to discriminate a black-hole-like wormhole from a Schwarzschild black hole.

6. Perturbations, quasinormal structure, and echoes

Wave propagation on Damour–Solodukhin wormholes is commonly formulated as a one-dimensional master equation,

$\Lambda$24

with the effective potential constructed by gluing two black-hole-like exteriors across a throat. In a symmetric toy model this is represented by

$\Lambda$25

The early waveform reproduces ordinary black-hole ringdown closely, but the Green’s function has additional singular structure that generates late-time tails and echoes (Qian et al., 2024).

Late-time tails arise from branch cuts of the frequency-domain Green’s function. In the wormhole case the standard black-hole recipe must be generalized because singularities are present in both ingoing and outgoing waveforms. The late-time behavior is found to scale as

$\Lambda$26

so the tail is governed by the slower-decaying side of the two-sided geometry when the asymptotic indices differ (Qian et al., 2024). Echoes, by contrast, arise from a new family of poles supplementing the usual black-hole quasinormal spectrum. The throat reflection amplitude satisfies

$\Lambda$27

which leads directly to the echo period

$\Lambda$28

The original black-hole poles remain largely intact, but an additional branch appears because repeated reflections occur between the effective barrier and the throat (Qian et al., 2024).

Asymmetric extensions sharpen this picture. Modeling the effective potential as

$\Lambda$29

one finds two closely related high-frequency spectra: echo modes and reflectionless modes. In the symmetric case the reflectionless modes lie precisely on the real axis and are approximately

$\Lambda$30

In the asymmetric case the reflectionless modes acquire nonzero imaginary parts, while the asymptotic echo modes still form a ladder nearly parallel to the real axis with spacing

$\Lambda$31

At

$\Lambda$32

the real parts of echo modes coincide with those of reflectionless modes, and the imaginary part of a reflectionless mode becomes a measure of asymmetry (Qian et al., 1 Nov 2025).

Environmental effects modify this spectrum. For a Damour–Solodukhin wormhole embedded in a Hernquist halo, scalar perturbations lead to a double-barrier potential, one barrier for each asymptotic universe, and hence to echoes with delay

$\Lambda$33

The halo increases the throat length $\Lambda$34 and therefore increases the echo spacing. The same study finds that the galactic Damour–Solodukhin wormhole is more stable than the isolated one under linear scalar perturbations because the magnitude of the imaginary part of the quasinormal frequency is slightly larger (Biswas et al., 2023).

7. Exotic matter, semiclassical support, and energy conditions

The matter supporting a traversable Damour–Solodukhin wormhole is nonstandard. In one explicit classical description the isolated geometry is supported by an anisotropic exotic source

$\Lambda$35

with

$\Lambda$36

and negative radial pressure responsible for NEC/WEC violation (Biswas et al., 2023). When the wormhole is embedded in a galactic halo with Hernquist density profile

$\Lambda$37

the positive halo density softens the exoticity: the WEC is satisfied outside the throat region, and the NEC violation is confined to

$\Lambda$38

This does not remove exoticity, but it confines it to a smaller neighborhood around the throat (Biswas et al., 2023).

A complementary semiclassical analysis uses the Schwinger–DeWitt expansion for massive scalar, spinor, and vector fields in the static background

$\Lambda$39

At the throat $\Lambda$40, the Morris–Thorne conditions are

$\Lambda$41

Among the three field types studied, only the massive scalar field with arbitrary curvature coupling $\Lambda$42 has a region in parameter space where both conditions are satisfied. The minimally coupled case $\Lambda$43 and the conformally coupled case $\Lambda$44 both lie outside that allowed region. The massive spinor and vector fields do not support the wormhole in this approximation (Matyjasek, 2020).

This combination of results defines the present technical picture. Damour–Solodukhin wormholes can mimic Schwarzschild or Kerr remarkably well in some observables, especially external lensing and disk kinematics, yet they retain characteristic signatures: a throat-supported light ring, nontrivial circular-orbit root structure, possible nonlogarithmic strong deflection at the critical deformation, head-on throat collisions with $\Lambda$45, late-time echoes from a double-barrier cavity, and explicit reliance on exotic or semiclassically tuned stress-energy rather than ordinary classical matter (Bermúdez-Cárdenas et al., 14 Apr 2025, Tsukamoto et al., 2019, Qian et al., 2024).