Torsional black holes and wormholes in Einstein-Cartan-Maxwell gravity with a conformal scalar field

This paper introduces a one-parameter extension of Weyl transformations in first-order gravity that yields a conformally coupled scalar sector with dynamical torsion, leading to exact static solutions in Einstein-Cartan-Maxwell theory that describe scalar-dressed black holes, regular black holes, and traversable wormholes, where torsion plays a crucial role in regularizing both scalar and geometric singularities.

The Gist

Imagine the universe as a giant, stretchy trampoline. In the standard view of gravity (Einstein's General Relativity), this trampoline is smooth and perfect. If you place a heavy bowling ball (a star or black hole) on it, the fabric curves down, creating a deep well. This theory works incredibly well, but it has a strict rule: the fabric must be smooth everywhere. If you try to add certain types of "hair" (like a specific kind of energy field) to the bowling ball, the fabric usually rips or creates a "naked singularity"—a point where the math breaks down and the rules of physics stop working.

This paper introduces a new way to think about that trampoline. The authors suggest that the fabric isn't just smooth; it can also have a subtle, hidden "twist" or "torsion" built into its structure. Think of it like a trampoline made of a special fabric that can be twisted like a screw, not just bent.

Here is the breakdown of their discovery using simple analogies:

1. The New "Twist" in the Rules

The authors propose a new mathematical "dial" (a parameter they call $\lambda$) that controls how this twisting happens.

• The Old Way ($\lambda = 1$): If you turn the dial to the standard setting, the fabric is smooth, and the twisting disappears. This is the familiar Einstein gravity we know.
• The New Way ($\lambda \neq 1$): If you turn the dial, the fabric gains this hidden "torsion." This twist is generated by a "scalar field" (a type of energy field) that wraps around the black hole.

2. Taming the "Hair"

In standard physics, black holes are boring; they are described only by their mass, spin, and electric charge. This is called the "no-hair" theorem. If you try to give them "hair" (extra fields), the hair usually causes the black hole to explode or become a singularity.

The authors found that by using their new "twisted" fabric:

• The Hair Stays Neat: The scalar field (the "hair") can wrap around the black hole without tearing the fabric or creating a singularity. The twist in the fabric acts like a safety net, keeping the energy field smooth and regular everywhere, even right at the edge of the black hole.
• The Result: They created exact mathematical models of "dressed" black holes—black holes that have this extra, smooth hair without breaking the laws of physics.

3. Two New Types of Cosmic Objects

Depending on how they set the "dial" and the values of the constants, they found two fascinating types of solutions:

• Regular Black Holes: Imagine a black hole that has a center, but instead of a point where physics breaks down (a singularity), the center is smooth and finite. The "twist" in the fabric smooths out the sharp edge that usually exists in these models.
• Traversable Wormholes: Think of a wormhole as a tunnel connecting two distant points in the universe. Usually, these tunnels are unstable or require "exotic" matter (stuff with negative energy) to stay open. The authors found that in their twisted universe, the torsion itself acts like the glue holding the tunnel open. They found a solution where a wormhole connects two flat regions of space, and you could theoretically travel through it without hitting a singularity or being crushed.

4. The Role of Electric Charge

The paper highlights a specific rule for these new objects:

• In a "Flat" Universe: You can have these smooth black holes or wormholes without needing electric charge.
• In an "AdS" Universe (a universe with a specific type of curvature): To have these black holes, you must have an electric charge. It's as if the electric charge is the key that unlocks the door to these twisted, smooth black holes in that specific environment.

Summary

The authors didn't just tweak the math; they found a new "gear" in the engine of gravity. By allowing space to have a hidden "twist" (torsion) that interacts with energy fields, they showed that:

1. Black holes can have extra "hair" without breaking.
2. The sharp, dangerous centers of black holes can be smoothed out.
3. Stable wormholes can exist naturally, held open by the geometry of space itself rather than exotic matter.

They proved that if we allow gravity to be a bit more flexible (by including this twist), the universe can support a much richer variety of stable, smooth, and fascinating structures than we previously thought possible.

Technical Summary

Technical Summary: Torsional Black Holes and Wormholes in Einstein–Cartan–Maxwell Gravity with a Conformal Scalar Field

Problem Statement
The paper addresses the limitations of the classical "no-hair" theorems in General Relativity, which assert that stationary black holes in Einstein–Maxwell theory are uniquely characterized by mass, charge, and angular momentum. These results rely on assumptions of minimal coupling and a purely Riemannian geometry (vanishing torsion). The authors investigate how relaxing these assumptions—specifically by introducing non-minimal coupling in a first-order gravity framework with non-vanishing torsion—can generate new families of black hole and wormhole solutions. A central motivation is the observation that while conformally coupled scalars in Riemannian geometry often lead to naked singularities or divergent fields at horizons, the inclusion of torsion may regularize these fields and alter the geometric singularity structure.

Methodology
The authors formulate a gravitational theory in the first-order formalism (Einstein–Cartan gravity) where the metric (vielbein) and the affine connection (spin connection) are treated as independent dynamical variables.

1. Generalized Conformal Symmetry: They introduce a one-parameter extension of Weyl transformations ($\lambda$) that acts on both the vielbein and the spin connection. This parameter interpolates between the canonical Weyl scaling ($\lambda=1$, where torsion remains invariant) and a case where the spin connection is fixed ($\lambda=0$).
2. Field Content: The theory couples the Einstein–Cartan gravitational sector to a Maxwell field and a conformally invariant scalar field $\phi$. The scalar field is coupled non-minimally to the curvature and acts as a source for torsion.
3. Field Equations: The total action is varied with respect to the vierbein, spin connection, scalar field, and Maxwell potential. Crucially, the equation of motion for the spin connection is solved algebraically, expressing the torsion tensor $T^a$ directly in terms of the scalar field gradient:
   $$T^a = \frac{\kappa(1-\lambda)}{6 - \kappa\phi^2} \phi d\phi \wedge e^a$$
   This demonstrates that torsion is purely of the vector-trace type and vanishes if $\lambda \to 1$ or if $\phi$ is constant.
4. Ansatz: The authors seek exact static solutions in four dimensions. They assume a metric with a codimension-2 hypersurface of constant curvature $k$ (spherical $k=1$ or hyperbolic $k=-1$) and assume radial dependence for the scalar and electromagnetic fields.

Key Contributions and Results
The study derives exact static solutions in two distinct asymptotic regimes: asymptotically flat ($\Lambda=0$) and asymptotically locally Anti-de Sitter ($\Lambda < 0$).

1. Asymptotically Flat Solutions ($\Lambda = 0, V(\phi) = 0$):

• Scalar-Dressed Black Holes: For positive mass ($M>0$), the theory admits black hole solutions. Depending on the deformation parameter $a$ (related to $\lambda$) and the charge $q$, the scalar field can be regular everywhere or singular at the horizon.
  • For specific integer values of $a$, the solutions represent torsional extensions of the Bekenstein black hole.
  • In the absence of electric charge ($q=0$) and for specific parameters, the authors find regular black holes where both the scalar field and the geometric/torsional invariants are finite everywhere, including the origin.
• Traversable Wormholes: For negative mass ($M<0$), the solutions describe wormhole geometries.
  • The scalar field remains regular throughout the spacetime.
  • The geometry possesses a throat connecting two asymptotic regions with no event horizons.
  • For specific parameter ranges ($Q=1, a \geq 5$), the curvature and torsional singularities at the throat are removed, yielding a fully regular traversable wormhole supported by torsion.

2. Asymptotically AdS Solutions ($\Lambda < 0, V(\phi) \neq 0$):

• Topological Black Holes: The authors construct solutions with a negative cosmological constant and a specific scalar potential $V(\phi)$ that preserves conformal invariance.
• Charge Dependence: A critical finding is that in the AdS sector, the existence of these scalar-dressed black hole solutions is intrinsically linked to the presence of a non-vanishing electric charge ($q \neq 0$). If $q=0$, the scalar field vanishes, and the solution reduces to a topological black hole without scalar hair.
• Regularity: For even integer values of the parameter $a$, the scalar field is regular everywhere. The solutions describe charged topological black holes where the scalar field is regular, and singularities (if they exist for odd $a$) are hidden behind the event horizon.

Significance and Claims
The paper claims to provide new exact examples of regular and torsional configurations in four-dimensional gravity. The primary significance lies in demonstrating that:

1. Torsion as a Regularizer: Torsion, sourced dynamically by the conformal scalar field, can regularize the scalar field itself and, for suitable branches, improve the geometric singularity structure (e.g., removing curvature singularities to create regular black holes).
2. Unified Mechanism: The one-parameter generalization of conformal symmetry in Riemann–Cartan geometry provides a unified framework to generate exact solutions with nontrivial torsion, continuously recovering standard metric configurations in the limit $\lambda \to 1$.
3. Counterexamples to No-Hair: These results serve as counterexamples to traditional no-hair paradigms by exhibiting black holes and wormholes with nontrivial "torsional hair" and regular scalar fields.
4. Distinction of Regularity: The authors emphasize a distinction between the regularity of the scalar field and the regularity of the spacetime geometry, reserving the term "regular black hole" specifically for configurations where curvature and torsional invariants are also finite.

The work does not propose experimental tests or specific phenomenological applications but establishes a theoretical foundation for further study of torsional effects in black hole thermodynamics, singularity theorems, and wave propagation.