Charged topological geons with a self-gravitating scalar field

Problem Statement
This paper investigates the theoretical existence and properties of "topological geons"—spacetime configurations with nontrivial topology that possess particle-like characteristics. Specifically, the study focuses on static, spherically symmetric, charged topological geons supported by a self-gravitating, minimally coupled scalar field with negative kinetic energy (a phantom scalar field) and an arbitrary self-interaction potential. The work addresses the gap in analytical models for such geons, particularly those with the topology $\mathbb{R} \times \mathbb{RP}^3\setminus\{p\}$, and explores their viability as classical models for elementary particles beyond the Standard Model, including dark matter candidates.

Methodology
The author employs the Einstein-Maxwell-Klein-Gordon framework using the signature $(+, -, -, -)$ and geometrized units ($G=c=1$). The methodology proceeds as follows:

1. Topological Construction: The paper establishes a one-to-one correspondence between traversable wormholes with topology $\mathbb{R} \times (\mathbb{R}^3 \# \mathbb{R}^3) \approx \mathbb{R}^2 \times S^2$ and topological geons with topology $\mathbb{R} \times (\mathbb{RP}^3\setminus\{p\})$. The geon is constructed as the quotient manifold of a spherically symmetric wormhole spatial slice by the isometric action of the group $\mathbb{Z}_2$.
2. Metric and Field Equations: The study utilizes "quasiglobal" coordinates where the metric is static and spherically symmetric. The action includes a scalar curvature term, a phantom scalar field with an arbitrary potential $V(\phi)$, and an electromagnetic field. The resulting field equations are the coupled Einstein-Klein-Gordon-Maxwell equations.
3. Reconstructed Potential Method: To handle the unknown form of the scalar field potential $V(\phi)$, the author applies the "reconstructed potential method" (inverse problem method). Instead of solving for the fields given a potential, the author selects a specific, physically reasonable form for the area metric function $C(r)$ (representing the throat geometry) and derives the scalar field $\phi(r)$, the metric function $A(r)$, and the potential $V(\phi)$ via explicit quadratures (integrals).
4. Specific Case Analysis: The general analytical results are applied to a specific ansatz, $C(r) = \sqrt{r^2 + \delta^2}$, where $\delta$ represents the throat size. This allows for the analytical derivation of the Ellis-Bronnikov-Sorkin geon solution.

Key Contributions and Results

• General Analytical Solution: The paper derives a set of quadratures (Eqs. 8–11) that provide a general solution for charged wormholes and their corresponding geons for any physically reasonable scalar field potential. This establishes a direct link between the geometric parameters of the throat and the field parameters.
• Mass-Charge-Size Relation: A primary result is the demonstration that for a given scalar field configuration, the gravitational mass $M$ of a topological geon is completely determined by its electric charge $q$ and its size (throat radius $\delta$). Specifically, the mass is expressed as $M = Q(0)/3 + b/3$, where $Q(0)$ is an integral dependent on the charge and throat geometry, and $b$ is a geometric constant.
• The Ellis-Bronnikov-Sorkin Geon: By choosing the specific ansatz $C(r) = \sqrt{r^2 + \delta^2}$, the author constructs an explicit solution. Key findings for this specific geon include:
  • The mass is given by $M = \pi q^2 / (3\delta)$.
  • The scalar field is $\phi = \arctan(r/\delta)$, and the potential $V(\phi)$ is explicitly derived (Eq. 20).
  • The solution is regular everywhere provided the charge and mass satisfy specific inequalities relative to the size $\delta$ (Eq. 18). If these inequalities are violated, the solution describes a regular black hole with the same topology hidden behind an event horizon.
  • In the limit of critical parameters, the center of the geon behaves like an event horizon, but the Kretschmann invariant remains regular, indicating the singularity is purely coordinate-based.
• Particle Modeling: The paper calculates the parameters for an Ellis-Bronnikov-Sorkin geon with the charge and mass of an electron. The resulting geon size is $\delta_e \approx 2.6 \times 10^{-15}$ cm, a value comparable to nuclear scales. Similarly, for Planck-scale parameters, the geon size and charge are close to the Planck length and charge.

Significance and Claims
The paper argues that topological geons supported by phantom scalar fields offer a viable classical model for elementary particles that are not described by the Standard Model.

• Beyond Standard Model: The author posits that these geons could serve as classical models for dark matter particles. Unlike standard black hole models (Reissner-Nordström or Kerr-Newman), which cannot represent elementary particles due to the requirement $q \ll M$, the geon model allows for a relationship where the size is reasonable for particle physics scales even with significant charge-to-mass ratios.
• Dark Matter Implications: The study suggests that if dark matter consists of charged massive particles (a possibility noted in the literature), the Ellis-Bronnikov-Sorkin geon provides a theoretical framework where such particles have a nontrivial spacetime topology.
• Limitations: The author explicitly states that the stability of these configurations is not analyzed, noting the lack of a conceptual framework for stability analysis near the Planck scale or at the quantum level. The paper acknowledges that while classical scalar field wormholes are generally unstable, the instability growth for geons with critical parameters might appear slow to a distant observer.

In summary, the work provides a rigorous analytical framework for charged topological geons, demonstrating that their mass is a function of charge and size, and proposes them as a potential classical explanation for the nature of elementary and dark matter particles.