EBMT Wormhole: Geometry and Stability

• The EBMT wormhole is a static, spherically symmetric traversable geometry featuring a minimal throat with zero redshift that ensures no tidal forces.
• It is supported by a massless phantom scalar field whose reversed-sign kinetic term violates the null energy condition, a key requirement for traversability.
• Studies show its hyperbolic geodesic structure, unique global energy characteristics, and inherent instability under radial phantom-scalar perturbations.

Searching arXiv for the cited EBMT wormhole papers to ground the article in current literature. arxiv_search(query="Ellis Bronnikov Morris Thorne wormhole", max_results=10, sort_by="relevance") arxiv_search(query="(Culetu, 2014) OR (Cremona et al., 2018) OR (Konoplya, 2018) OR (Blázquez-Salcedo et al., 2022) OR (González et al., 2022) OR (Ahmed et al., 28 Feb 2025) OR (Xu et al., 10 Mar 2025)", max_results=10, sort_by="relevance") The Ellis–Bronnikov–Morris–Thorne wormhole is a static, spherically symmetric traversable wormhole obtained as a particular Morris–Thorne geometry with zero redshift function and shape function $b(r)=a^{2}/r$ or $b(r)=r_{0}^{2}/r$, depending on notation. In curvature coordinates it may be written as

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$

while in proper radial coordinate $\ell=\pm\sqrt{r^{2}-a^{2}}$ it becomes

$$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$

It is asymptotically flat as $|\ell|\to\infty$, possesses a minimal areal radius $a$ at the throat, and is supported by a massless scalar field with reversed-sign kinetic term, so that the null energy condition is violated as required for traversability (Culetu, 2014, Cremona et al., 2018).

1. Geometric definition and coordinate forms

The EBMT wormhole is a special case of the Morris–Thorne ansatz

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$

with the specific choice

$\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$

The condition $\Phi=0$ implies the absence of horizons and vanishing tidal effects in the standard Morris–Thorne sense, while $b(r)=r_{0}^{2}/r$0 gives the flare-out at the throat $b(r)=r_{0}^{2}/r$1 (Culetu, 2014). In the $b(r)=r_{0}^{2}/r$2-coordinate representation,

$b(r)=r_{0}^{2}/r$3

the two asymptotically flat regions are explicit, and the throat radius is $b(r)=r_{0}^{2}/r$4 (Cremona et al., 2018).

The same geometry is often presented with the notation $b(r)=r_{0}^{2}/r$5 for the throat radius, so that

$b(r)=r_{0}^{2}/r$6

In this form the flaring-out condition becomes

$b(r)=r_{0}^{2}/r$7

which ensures that the wormhole is open (Blázquez-Salcedo et al., 2022). The curvature invariants remain finite: $b(r)=r_{0}^{2}/r$8 and

$b(r)=r_{0}^{2}/r$9

so the throat is geometrically regular rather than singular (Blázquez-Salcedo et al., 2022).

For the equatorial spatial slice $$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$0, $$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$1, the induced metric

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$2

admits an embedding in Euclidean $$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$3 with profile

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$4

which yields the familiar flared geometry of two funnels joined at the throat (Resca et al., 2022).

2. Supporting matter and violation of energy conditions

The matter source is a minimally coupled, massless scalar field with a wrong-sign kinetic term. One form of the action is

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$5

with background solution

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$6

which approaches $$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$7 as $$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$8 (Cremona et al., 2018). An equivalent Lagrangian density is

$$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},\qquad r\ge a,$$9

whose sign convention makes explicit the negative kinetic contribution (Culetu, 2014).

For the static EBMT solution, the stress–energy tensor takes the anisotropic form

$\ell=\pm\sqrt{r^{2}-a^{2}}$0

so the energy density is negative, the radial pressure is negative, and the tangential pressure is positive (Culetu, 2014). In proper-radial notation this appears as

$\ell=\pm\sqrt{r^{2}-a^{2}}$1

again showing effective negative energy density everywhere (Blázquez-Salcedo et al., 2022).

The null energy condition is violated directly. For a radial null vector $\ell=\pm\sqrt{r^{2}-a^{2}}$2,

$\ell=\pm\sqrt{r^{2}-a^{2}}$3

and the same conclusion follows from the scalar-field form

$\ell=\pm\sqrt{r^{2}-a^{2}}$4

for a convenient radial null choice (Culetu, 2014, Kang et al., 2019). The integrated physical interpretation in the 2014 analysis identifies the matter content as a massless scalar with negative kinetic term providing an anisotropic fluid with $\ell=\pm\sqrt{r^{2}-a^{2}}$5, $\ell=\pm\sqrt{r^{2}-a^{2}}$6, $\ell=\pm\sqrt{r^{2}-a^{2}}$7 and violating all energy conditions (Culetu, 2014).

3. Energetics, geodesics, and traversability

A notable feature of the EBMT spacetime is the separation between different global energy notions. The Komar energy is

$\ell=\pm\sqrt{r^{2}-a^{2}}$8

because the positive pressure contribution cancels the negative density contribution. By contrast, the ADM energy is

$\ell=\pm\sqrt{r^{2}-a^{2}}$9

with the factor of two arising from the two copies of the throat; for $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$0 this is minus the Planck energy (Culetu, 2014). The same integrated picture assigns a quasilocal Misner–Sharp mass

$$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$1

to the geometry (Culetu, 2014).

Because $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$2 and $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$3 are Killing fields, timelike geodesics admit conserved energy $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$4 and angular momentum $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$5, with effective potential

$$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$6

For radial timelike motion, $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$7, one obtains

$$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$8

which is a hyperbola in the $$ds^{2}=-dt^{2}+d\ell^{2}+(a^{2}+\ell^{2})\,d\Omega^{2}.$$9 plane (Culetu, 2014). Static observers are inertial: for fixed $|\ell|\to\infty$0, $|\ell|\to\infty$1, $|\ell|\to\infty$2, and the four-acceleration $|\ell|\to\infty$3 vanishes (Culetu, 2014).

Null radial trajectories also have a hyperbolic form. From $|\ell|\to\infty$4 and $|\ell|\to\infty$5,

$|\ell|\to\infty$6

which integrates to

$|\ell|\to\infty$7

The 2014 treatment emphasizes that this is a Lorentz-invariant hyperbola directly analogous to the $|\ell|\to\infty$8-symmetric Coleman–de Luccia bubble wall or Ipser–Sikivie domain wall (Culetu, 2014).

On the embedded equatorial surface, geodesics separate into two classes. “Regular geodesics” have turning points $|\ell|\to\infty$9 and remain confined to one asymptotic region, whereas “singular geodesics” have $a$0 and pass through the throat, connecting both halves of the wormhole. The orbit equations can be written in terms of elliptic integrals and Jacobi functions, making the throat-crossing versus scattering distinction explicit (Resca et al., 2022). This geometric picture is consistent with the broader traversability statement that the absence of horizons and finite tidal forces permit passage through the throat (Culetu, 2014).

4. Dynamical throat models and instability

A thin-shell dynamical version of the EBMT geometry can be constructed by identifying two flat Minkowski regions at a throat radius $a$1 with surface stress tensor $a$2. In the cut-and-paste formalism,

$a$3

With the domain-wall equation of state

$a$4

the resulting equation of motion is

$a$5

with solution

$a$6

for $a$7, $a$8. The surface energy density is negative,

$a$9

and the proper acceleration is

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$0

again matching the hyperbolic motion characteristic of the Ipser–Sikivie wall (Culetu, 2014).

The static EBMT background is nevertheless linearly unstable under radial phantom-scalar perturbations. In the perturbative formulation

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$1

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$2

the linearized Einstein–scalar system reduces to a master equation

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$3

The associated Schrödinger-type operator has exactly one discrete negative eigenvalue $$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$4, with $$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$5, so the unstable mode behaves as

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$6

with $$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$7 and e-folding time

$$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$8

The physical interpretation given in the perturbative analysis is a pinching of the throat, leading to collapse in a time of order the throat size (Cremona et al., 2018).

Fully nonlinear double-null simulations exhibit two distinct instability scenarios. A normal scalar pulse drives gravitational collapse into a black hole, with a spacelike curvature singularity $$ds^{2}=-e^{2\Phi(r)}dt^{2}+\left[1-\frac{b(r)}{r}\right]^{-1}dr^{2}+r^{2}d\Omega^{2},$$9 hidden by the event horizon, while a phantom pulse drives inflationary expansion, separates the two asymptotic regions, and produces cosmological horizons. In both cases the process accelerates as the pulse amplitude increases and is delayed as the wormhole mass increases. The same simulations also show that a suitably timed outgoing phantom pulse colliding with an ingoing normal scalar pulse can temporarily restore the coincidence of $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$0 and $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$1, giving a transient re-stabilization of the throat before the instability resumes (Xu et al., 10 Mar 2025).

5. Wave propagation, quasinormal structure, and field probes

The EBMT wormhole has also been studied as a scattering background. For a spherically symmetric traversable wormhole in Morris–Thorne form, the eikonal quasinormal spectrum can be used to reconstruct the near-throat geometry. With the Taylor expansions

$\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$2

the large-$\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$3 WKB spectrum determines the coefficients $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$4, $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$5, and $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$6 in

$\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$7

For the Bronnikov–Ellis case, $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$8 and $\Phi(r)=0,\qquad b(r)=\frac{a^{2}}{r}.$9; setting $\Phi=0$0 gives

$\Phi=0$1

so $\Phi=0$2, and the high-$\Phi=0$3 quasinormal expansion reproduces $\Phi=0$4, $\Phi=0$5, $\Phi=0$6 exactly (Konoplya, 2018).

Massive scalar perturbations reveal a different sectoral behavior. In the generalized Bronnikov–Ellis wormhole, an anomalous decay rate appears for certain parameter ranges, whereas in the tideless Morris–Thorne wormhole the fundamental mode $\Phi=0$7 shows no anomalous decay and $\Phi=0$8 falls off with $\Phi=0$9 for all masses studied. The same analysis states that both wormholes are stable against the massive scalar perturbations considered, since $b(r)=r_{0}^{2}/r$00 for all modes studied (González et al., 2022). A plausible implication is that the instability structure is strongly perturbation-sector dependent rather than exhausted by a single master equation.

For a free massive complex scalar field in the EB background,

$b(r)=r_{0}^{2}/r$01

the radial equation reduces to a confluent Heun equation, and the general solution can be expressed in terms of $b(r)=r_{0}^{2}/r$02 functions (Blázquez-Salcedo et al., 2022). In the free case, decaying solutions on the two sides of the throat can be continuous while having a discontinuous radial derivative at $b(r)=r_{0}^{2}/r$03,

$b(r)=r_{0}^{2}/r$04

except in the spherical case. With a quartic-plus-sextic self-interaction,

$b(r)=r_{0}^{2}/r$05

the pathology is removed and smooth, finite-energy Q-balls exist, including spherically symmetric and spinning configurations (Blázquez-Salcedo et al., 2022).

Gravitational perturbations have been analyzed with the Newman–Penrose formalism and the Teukolsky equation. For the static wormhole with $b(r)=r_{0}^{2}/r$06, $b(r)=r_{0}^{2}/r$07, the only nonzero background Weyl scalar is

$b(r)=r_{0}^{2}/r$08

and the perturbed outgoing scalar $b(r)=r_{0}^{2}/r$09 satisfies a Teukolsky-type master equation with source term. In the treatment based on a Gaussian pulse of pressureless dust, the real part of the effective potential is positive everywhere and is interpreted there as implying no growing modes in that perturbative setup (Kang et al., 2019). Taken together with the radial phantom-scalar instability result, these studies indicate that “stability of the EBMT wormhole” is not a single statement but depends on the perturbation channel and matter content being considered.

6. Optical signatures and later extensions

Weak-field lensing provides a direct optical probe of the EBMT geometry and of deformations of it. For the pure EBMT metric

$b(r)=r_{0}^{2}/r$10

dressing the geometry with a global monopole and a cosmic string introduces parameters

$b(r)=r_{0}^{2}/r$11

For one commonly used dressed metric, the total angular change is

$b(r)=r_{0}^{2}/r$12

so the deflection angle is

$b(r)=r_{0}^{2}/r$13

In the weak-field limit $b(r)=r_{0}^{2}/r$14,

$b(r)=r_{0}^{2}/r$15

and in the no-throat limit $b(r)=r_{0}^{2}/r$16 one recovers the pure conical-deficit contribution exactly (Ahmed et al., 28 Feb 2025).

The EBMT geometry also serves as a seed for multi-throat deformations. In a string-cloud background, a localized perturbation

$b(r)=r_{0}^{2}/r$17

can convert the single-throat EB structure into a double-throat geometry once

$b(r)=r_{0}^{2}/r$18

for which $b(r)=r_{0}^{2}/r$19 becomes a local maximum and two symmetric minima appear. The resulting energy density and pressures decay asymptotically as $b(r)=r_{0}^{2}/r$20, as in a Letelier string cloud, while the null energy condition violations become confined to narrow bands around the two throats and the inter-throat “belly” is supported by ordinary matter (Amaral et al., 7 Apr 2026).

A rotating extension exists in an exact Teo-type form with the same Morris–Thorne shape function $b(r)=r_{0}^{2}/r$21 and unit lapse. In proper radial distance $b(r)=r_{0}^{2}/r$22, the frame-dragging function is

$b(r)=r_{0}^{2}/r$23

with $b(r)=r_{0}^{2}/r$24 given in closed form, producing a two-parameter family labeled by $b(r)=r_{0}^{2}/r$25 and total angular momentum $b(r)=r_{0}^{2}/r$26. This spinning geometry remains regular at the throat, violates all standard energy conditions, develops an ergoregion only for sufficiently large $b(r)=r_{0}^{2}/r$27, yet remains stably causal because $b(r)=r_{0}^{2}/r$28 is a global time function. Its shadow is smaller than Kerr’s, and its Geroch–Hansen multipoles show a massless but spinning configuration with higher moments depending explicitly on the throat scale (Batic et al., 25 Feb 2026).

The EBMT wormhole therefore occupies a distinctive place in wormhole physics: it is simultaneously an exactly solvable traversable geometry, a canonical example of phantom-scalar support and null-energy-condition violation, a system with hyperbolic geodesic structure and unusual global energetics, a linearly unstable background in the radial Einstein–phantom sector, and a continuing seed metric for inverse spectral problems, scalar bound states, lensing analyses, multi-throat constructions, and exact spinning generalizations (Culetu, 2014, Cremona et al., 2018).