Rotating Wormhole Solutions

• Rotating wormhole solutions are spacetimes with stationarity and axisymmetry, featuring traversable throats supported by angular momentum.
• They are constructed using techniques like the Ernst–Ehlers transformation and modified Newman–Janis algorithms to generate exact and numerical models.
• These solutions exhibit unique features such as ergoregions, modified throat geometries, and observable signatures in gravitational lensing and stability.

Rotating wormhole solutions are exact or numerical spacetimes in which a traversable wormhole throat is supported while the geometry possesses stationarity and axisymmetry, i.e., angular momentum. These solutions arise in a range of gravitational theories, from pure Einstein–Maxwell to Einstein–Dirac–Maxwell, scalar-tensor, and quantum-extended settings. Recent research has established that several classes of rotating wormhole spacetimes can be constructed as regular, horizonless, traversable, geodesically complete geometries, supported by regular or distributional (string-like) matter sources. The presence of rotation introduces novel geometric features including ergoregions, modified throat shapes, new classes of matter/energy condition violations, and observable implications for gravitational lensing and stability.

1. Construction Techniques for Rotating Wormholes

The construction of rotating wormhole solutions exploits several mathematical strategies tailored to the field theory and symmetry structure:

• Ernst–Ehlers Transformation Approach: The Ernst formalism encodes the Einstein–Maxwell(-scalar) equations for stationary, axisymmetric spacetimes in terms of complex scalar potentials. Ehlers group transformations act as solution-generating maps, allowing the embedding of a static seed solution (e.g., the Barceló–Visser wormhole) into a rotating background. The Ehlers map introduces a “swirling” parameter, resulting in a new metric function and off-diagonal frame-dragging terms, thereby producing exact rotating geometries (Cisterna et al., 2023).
  • Given a static seed with metric function $f_0$ and vanishing twist, the Ehlers transform yields

  $$f = \frac{f_0}{1 + j^2 f_0^2}, \quad \chi = -j \frac{f_0^2}{1 + j^2 f_0^2},$$

  where $j$ is the swirl parameter, and the full rotating metric is constructed after appropriate coordinate and conformal transformations.

• Newman–Janis and Shortcut Algorithms: Modified Newman–Janis type algorithms extend static spherically symmetric metrics to include rotation, typically by complexifying a radial coordinate and then enforcing Boyer–Lindquist-type coordinates. This is especially effective for imperfect fluid or electromagnetic wormholes, yielding physically inequivalent families of rotating solutions depending on the specific complexification procedure (Gutiérrez-Piñeres et al., 2021, Azreg-Aïnou, 2014).
• Embedding in Rotating or Magnetized Backgrounds: Embedding a static wormhole into an ambient gravitational or electromagnetic background through solution-generating transformations (Ehlers, Harrison) introduces rotation as the result of external dragging or $Q$–$B$ coupling. In the Ehlers construction, rotation is linked directly to the twist of the axially symmetric Killing field structure (Cisterna et al., 2023).

2. Static Seeds and Matter Content

The starting point for many rotating constructions is a well-behaved static wormhole solution, such as the Barceló–Visser metric, which itself is a solution to Einstein–Maxwell–conformal-scalar theory:

$$d\hat s^2 = \left( \frac{r-(1-a)M}{r-M} \right)^2 \left[ - \left( 1 - \frac{M}{r} \right)^2 dt^2 + \frac{dr^2}{(1 - M/r)^2} + r^2 d\Omega^2 \right],$$

with scalar field

$$\psi(r) = \left( \frac{6}{\kappa} \; \frac{a\,r + (1-a)M}{r - (1-a)M} \right)^{1/2}, \quad A = \frac{Q}{r}\,dt,$$

where $0 \leq a < 1$, $|Q| \leq \sqrt{1-a^2} M$. The throat is at $r_\circ = (1+\sqrt{a})M$, connecting two asymptotic regions at $r=M$ and $r\to\infty$. The theory supports these solutions with stress-energy constructed from the conformal scalar and electromagnetic field components, and admits a two-parameter family governed by $(a, Q)$.

3. Ernst–Ehlers Construction and Rotating Geometries

Applying the Ehlers transformation to the static seed in the Ernst formulation, one generates a new, exact rotating wormhole spacetime. The stationary, axisymmetric metric after the Ehlers map becomes:

$$d\hat s^2 = \Omega(r) \left\{ F(r,\theta) \left[ -\left(1-\frac{M}{r}\right)^2 dt^2 + \frac{dr^2}{(1-M/r)^2} + r^2 d\theta^2 \right] + \frac{r^2 \sin^2\theta}{F(r,\theta)} \left[ d\varphi - \omega(r,\theta)\,dt \right]^2 \right\},$$

where

$$\Omega(r) = \left( \frac{r-(1-a)M}{r-M} \right)^2, \qquad F(r,\theta) = 1 + j^2 (1-a^2)^2 \frac{r^6(r-2M)^2}{(r-M)^4} \sin^4\theta,$$

with frame-dragging angular velocity

$$\omega(r,\theta) = -4j(1-a^2) \frac{r^2 - 3Mr + 3M^2}{r-M} \cos\theta, \quad j \in \mathbb{R}.$$

A Harrison transformation can subsequently introduce additional external magnetic fields, allowing for the paper of magnetized, rotating wormhole backgrounds. The matter fields (scalar and electromagnetic) are modified consistently by these transformations, so that the minimal coupling structure is preserved and the scalar profile remains formally unchanged.

4. Geometric Features: Throat, Embedding, and Ergoregions

The rotating solutions exhibit several characteristic geometric properties:

• Throat Geometry: The minimal-area two-surface remains at

$r_\circ = (1+\sqrt{a})M,$

with the transverse two-sphere area

$\mathcal{A} = 4\pi\,\Omega(r)\,r^2,$

evaluated at $r_\circ$. The throat connects two asymptotic regions, $r\to M$ and $r\to\infty$.

• Embedding Diagrams: The metric for constant $t$ and $\theta$ can be embedded in Euclidean space via the induced $d\rho^2$ and $d\varphi^2$ terms. The rotating wormhole retains the characteristic “double-bowl” embedding profile, modulated by the rotation parameter $j$ and the swirl-induced deformation of $F(r,\theta)$.
• Ergoregions: The condition for the presence of an ergoregion becomes

$$g_{tt} = -\Omega\,F\,(1-M/r)^2 + \Omega \frac{r^2 \sin^2\theta}{F} \omega^2 > 0,$$

which, due to the explicit $j$ dependence in $\omega$, yields two disjoint families of ergoregions in the full spacetime. Ergoregions are a distinct consequence of rotation and enable energy processes not present in static cases.

5. Matter Structure and Energy Condition Analysis

The support for the rotating wormhole is provided by a combination of (possibly distributional) electromagnetic, conformally coupled scalar, and, if needed, further auxiliary fields depending on the external backgrounds introduced via the solution-generating maps. The energy-momentum tensor after rotation remains a sum of conformal scalar and Maxwell terms:

$$T_{\mu\nu} = F_{\mu\alpha}F_{\nu}{}^\alpha - \frac{1}{4}g_{\mu\nu}F^2 + \nabla_\mu\psi \nabla_\nu\psi - \frac{1}{2} g_{\mu\nu}(\nabla\psi)^2 + \frac{1}{6}(g_{\mu\nu}\Box - \nabla_\mu\nabla_\nu + G_{\mu\nu})\psi^2.$$

Direct evaluation demonstrates that the null energy condition is violated in the required directions near the throat,

$T_{\mu\nu} k^\mu k^\nu < 0,$

for suitable null vectors $k^\mu$, as expected for traversable wormholes. For slow rotation, the violation’s measure is only mildly dependent on the swirl parameter $j$, being quadratic in $j$. Observationally, the exotic matter is concentrated near the throat, similarly to the static seed.

6. Physical Implications: Stability and Observational Signatures

The precise construction of rotating wormholes within the Ernst framework permits exploration of several phenomenologically relevant effects:

• Dynamical Stability: Perturbative and nonperturbative calculations support the hypothesis that slow rotation suppresses at least the static radial instability seen in static seeds, possibly pushing the threshold into the dynamically stable regime. This is consistent with recent work showing rotational stabilization for several classes of wormhole solutions (Azad et al., 2023).
• Wave and Geodesic Structure: The presence of ergoregions enables Penrose-type energy extraction processes and the paper of quasinormal modes, echoes, or unique lensing (e.g., wormhole shadows, multivalued Einstein rings). The geodesic equations in Ernst-type metrics remain notably tractable.
• Magnetized Backgrounds: By combining Harrison and Ehlers transformations, further generalization to magnetized (charged and rotating) wormhole spacetimes is possible, introducing additional observationally accessible signatures related to electromagnetic fields.
• Parameter Constraints: The rotating wormholes are specified by the mass $M$, scalar/electric parameter $a$, and swirl parameter $j$ (and, if present, magnetic background strength), allowing mapping to astrophysical observational constraints.

7. Synthesis and Outlook

Exact rotating wormhole solutions constructed via Ernst–Ehlers techniques using physically motivated static seeds (e.g., Barceló–Visser spacetimes) demonstrate that four-dimensional general relativity and minimal extensions (to conformally coupled scalars and electromagnetic fields) admit traversable, globally regular, stationary wormhole geometries. The addition of rotation introduces ergoregions, modifies the throat’s geometry, enables lensing and wave phenomena inaccessible in static models, and offers a physically transparent path toward stability. These exact solutions enrich the landscape of physically viable wormhole spacetimes and open up avenues for testing non-black-hole compact object alternatives with precise electromagnetic, optical, and gravitational wave observations. The Ehlers/Ernst machinery also offers a systematic means to generate further families of rotating wormhole spacetimes with tailored matter profiles and external field content (Cisterna et al., 2023).