Rotating Global Monopole Solutions

• The paper establishes a no-go theorem that forbids rotation in self-consistent global monopole spacetimes based on the coupled Einstein–scalar equations.
• Methodologies like the Newman–Janis algorithm and multipole expansion reveal that introducing rotation results in irreconcilable angular dependencies in the field equations.
• The findings imply that astrophysical models using rotating monopole metrics must be reexamined, suggesting the need for alternative theories beyond general relativity.

A rotating global monopole solution refers to a hypothetical stationary, axially symmetric generalization of the well-known static global monopole spacetime sourced by a triplet of scalar fields with spontaneously broken global O(3) symmetry. Interest in such configurations has persisted due to their potential astrophysical and cosmological signatures, especially regarding gravitational lensing, cosmic defects, and black hole physics. However, the existence of exact rotating global monopole solutions within Einstein’s general relativity is now precluded by a definitive no-go theorem (Lu et al., 2 Oct 2025), which rigorously establishes the incompatibility of rotation and nontrivial monopole charge in the framework of the coupled Einstein–scalar field equations.

1. Static Global Monopole: Field Content and Spacetime Structure

A static global monopole arises in theories with an O(3) → U(1) pattern of spontaneous symmetry breaking, typically realized with a triplet of scalar fields χᵢ (i = 1,2,3) subject to the Lagrangian

$$\mathcal{L}_\text{GM} = \frac{1}{2} \partial_\mu \chi^i \partial^\mu \chi^i + \frac{\lambda}{4} \left( \chi^i \chi^i - \eta^2 \right)^2.$$

The “hedgehog” ansatz

$$\chi^1 = \eta h(r)\sin\theta\cos\phi, \qquad \chi^2 = \eta h(r)\sin\theta\sin\phi, \qquad \chi^3 = \eta h(r)\cos\theta$$

with h(r) → 1 outside the monopole core, implies a spherically symmetric, static metric solution. The quintessential geometry is the Barriola–Vilenkin metric,

$$ds^2 = -f(r)\,dt^2 + \frac{1}{f(r)}\,dr^2 + r^2(d\theta^2 + \sin^2\theta\, d\phi^2), \quad f(r) = 1 - 8\pi G\eta^2 - \frac{2GM}{r},$$

where the deficit solid angle 8\pi G\eta^2 is a hallmark of the global monopole.

2. Previous Efforts at Rotating Generalizations and Algorithmic Approaches

Rotating generalizations—including metrics constructed via the Newman–Janis algorithm (NJA)—sought to extend static monopole solutions to stationary, axially symmetric cases reminiscent of the Kerr metric. The NJA typically replaces the radial coordinate with a complexified form and applies coordinate transformations to generate a rotating metric, resulting in

$$ds^2 = -\frac{\Psi(r,\theta)}{\rho^2} \left[\frac{\Delta}{\rho^2} (dt - a\sin^2\theta\,d\phi)^2 - \frac{\rho^2}{\Delta} dr^2 - \rho^2 d\theta^2 - \frac{\sin^2\theta}{\rho^2}(a\,dt - (K(r)+a^2)\,d\phi)^2 \right],$$

with Δ(r), ρ^2(r,θ), and Ψ(r,θ) defined according to the algorithm.

However, such algorithmically constructed metrics do not guarantee that the full set of source field equations—specifically those derived from the scalar field dynamics—remain satisfied.

3. Consistency Analysis: Scalar Field Equations in Axially Symmetric Backgrounds

A complete, self-consistent rotating global monopole solution requires that both the Einstein equations (with the monopole stress tensor) and the scalar field Euler–Lagrange equations are fulfilled. The coupled system,

$$R_{\mu\nu} - \frac{1}{2} R\,g_{\mu\nu} = 8\pi G\,T_{\mu\nu}, \qquad \nabla^2 \chi^i = \lambda \chi^i (\chi^j \chi^j - \eta^2),$$

imposes nontrivial constraints on the functional dependence of h(r) and the metric functions. Upon inserting a rotating metric ansatz (such as the NJA-rotated form) and the standard hedgehog profile, the scalar field equations split into two classes (due to angular dependence). The resulting system requires, for nontrivial solutions, that certain angular dependencies must identically vanish or match. However, detailed analysis (Lu et al., 2 Oct 2025) reveals that,

• The scalar equations produce conflicting terms: the equations for χ^{1,2} and for χ^3 involve different structures in cos^2θ and higher-order powers,
• Requiring the validity of the equations for all θ demands trivialization (h(r) ≡ 0) or reduction to the static (a = 0) case.

This demonstrates that the NJA-rotated metric does not support a self-consistent nontrivial global monopole profile for any nonzero rotation parameter.

4. General Axially Symmetric Metric and Asymptotic Analysis

The proof is extended to the most general stationary, axially symmetric metric, parameterized as

$$ds^2 = -\frac{f(r,\theta) - \omega^2(r,\theta)\sin^2\theta}{\kappa^2(r,\theta)} dt^2 - 2r\,\omega(r,\theta) \sin^2\theta\, dt\,d\phi + r^2 \kappa^2(r,\theta) \sin^2\theta\, d\phi^2 + \sigma(r,\theta)\left(\frac{\beta^2(r,\theta)}{f(r,\theta)} r^2\, dr^2 + r^2 d\theta^2\right).$$

A multipole expansion of all metric and scalar field functions, together with the requirement that all Einstein equations and scalar field equations (especially the vanishing of the off-diagonal Einstein tensor components like G_{rθ}) are satisfied at each order in 1/r, enforces:

• All leading angular variations must vanish at large r,
• The rotation function must be identically zero,
• The only allowed solution is a spherically symmetric, static global monopole.

This result is robust, independent of the specific ansatz, and holds for any stationary, axisymmetric asymptotic expansion.

5. Implications and Consequences for Phenomenology and Modeling

The no-go theorem definitively excludes the existence of physically consistent, rotating global monopole solutions in Einstein’s general relativity: no choice of metric functions or generalized “hedgehog” profile—including those with a r-dependent azimuthal deformation—can produce a stationary, axially symmetric monopole spacetime.

This has direct implications:

• Any purported “rotating global monopole” metric in general relativity—whether constructed via the Newman–Janis algorithm, ansatz-based methods, or perturbative techniques—does not represent a solution to the full coupled Einstein–scalar system unless it is static.
• Astrophysical and cosmological models employing such metrics to characterize lensing, shadow, or accretion signatures of “spinning monopoles” must be revisited or reinterpreted. All consistent global monopole geometries are strictly static (with a possible solid angle deficit) as in the Barriola–Vilenkin solution.

6. Outlook for Extensions Beyond General Relativity

While the no-go theorem precludes rotating global monopole solutions in GR, avenues remain for possible rotating defects in extended frameworks:

• Modified gravity: The conclusion is specific to Einstein’s equations with the canonical scalar field; in f(R), f(Q), or higher-curvature theories, or with nonminimal couplings, additional degrees of freedom could relax the constraints.
• Multi-field or gauge-coupled models: Additional scalar fields, non-Abelian structures, or gauge interactions may support more exotic rotational configurations.
• Numerical and dynamical simulations: Although no stationary, asymptotically flat solutions with rotation exist, transiently rotating global monopole configurations may still arise in numerical evolutions or unstable regimes.

A complete understanding of rotating topological defects will thus require investigation beyond the classic Einstein–scalar paradigm, informed by the analytical structure of the no-go theorem (Lu et al., 2 Oct 2025).