On regular black strings spacetimes in nonlinear electrodynamics

This paper investigates the coupling of General Relativity with Nonlinear Electrodynamics to address axial singularities in four-dimensional black strings, proving that regular purely electric configurations are impossible for Lagrangians recovering the Maxwell limit while simultaneously constructing and validating new exact regular solutions, such as cylindrical analogues of the Bardeen and Hayward metrics, that satisfy causality and unitarity constraints.

The Gist

Imagine the universe as a giant, stretchy fabric called spacetime. In the 1910s, Einstein showed us that massive objects like stars and black holes create deep "dips" or "holes" in this fabric. Usually, if you get too close to the center of a black hole, the fabric tears apart completely. This tear is called a singularity—a point where the math breaks down, the density becomes infinite, and our understanding of physics stops working. It's like hitting a wall that doesn't exist in reality.

For a long time, scientists thought these tears were inevitable. But recently, they've been trying to "patch" these holes using a special kind of glue called Nonlinear Electrodynamics (NED).

This paper is about a specific type of black hole called a Black String.

The Setup: Black Strings vs. Black Holes

Think of a standard black hole as a sphere (like a beach ball). It has a singularity right in the middle.
Now, imagine stretching that beach ball into a long, thin rope or noodle that goes on forever. That's a Black String. Instead of a point singularity, it has a line singularity running down its entire center axis.

The problem? Just like the beach ball, this rope has a tear down the middle. The authors wanted to see if they could use their "NED glue" to smooth out that tear and make the rope solid and safe all the way through.

The Big Discovery: The "No-Go" Rules

The authors did some heavy math to see if this was possible. They discovered some strict "rules of the road" (No-Go Theorems) that act like traffic signs:

1. The Electric Rule: If you try to fix the rope using only electricity (and you want the electricity to act normal when it's weak, like in our everyday world), you cannot fix the tear. It's impossible. The math simply won't allow a smooth, electric-only rope.
2. The Mixed Rule: If you try to use a mix of electricity and magnetism (dyonic), you also cannot fix the tear.
3. The Magnetic Exception: The only way to potentially fix the rope is to use magnetism in a very specific, weird way.

It's like trying to fix a broken bridge. The authors found out that if you try to use "standard" materials (normal electricity), the bridge will always collapse in the middle. You have to use a very exotic, non-standard material (nonlinear magnetism) to even have a chance.

The Experiments: Trying Different Glues

To test their theory, the authors tried two different types of "glue":

1. The "Strong" Glue (Euler-Heisenberg & Logarithmic):
They tried using models based on how electrons behave in strong magnetic fields.

• Result: These glues were too weak. They actually made the tear worse. The center of the rope became even more jagged and dangerous. The singularity remained, just with a different shape.

2. The "Magic" Glue (Bardeen & Hayward):
They then tried using special, pre-designed formulas (named after scientists Bardeen and Hayward) that are known to fix spherical black holes.

• Result: Success! When they applied these specific magnetic formulas to the Black String, the tear disappeared. The center of the rope became a smooth, solid core. The "infinite tear" was replaced by a gentle, finite bump. The rope was now "regular" (safe) all the way through.

The Catch: The Speed Limit Violation

Here is the twist. While they successfully patched the hole, they found a new problem.

In our normal world, nothing can travel faster than the speed of light. It's the universe's speed limit. However, in these "patched" black strings, the authors found that near the very center (the core), the "glue" gets so weird that light itself can break the speed limit.

Imagine driving on a highway where, suddenly, the speed limit signs disappear, and cars start zooming faster than physics allows. This is called superluminal propagation. It means that inside the core of these smooth black strings, the laws of cause and effect get messed up. You could theoretically send a message to the past, which breaks the rules of reality.

The Conclusion

So, what did this paper teach us?

1. You can't fix everything: You can't use standard electricity to fix a Black String; it's mathematically impossible.
2. You can fix the hole, but... You can use special magnetic fields to smooth out the tear and make a "regular" Black String.
3. But there's a cost: To do this, you have to break the universal speed limit near the center. The solution is smooth, but it's physically "sick" because it allows time-travel-like paradoxes.

The Takeaway:
The universe seems to have a trade-off. You can have a black object with no tears (singularities), or you can have one that obeys the speed of light and cause-and-effect. But you probably can't have both at the same time in these cylindrical shapes. The authors have mapped out exactly where these rules apply, giving us a better understanding of how the universe might (or might not) allow for perfect, smooth black objects.

Technical Summary

1. Problem Statement

General Relativity (GR) predicts the existence of spacetime singularities where curvature invariants diverge, rendering the theory non-predictive. While significant progress has been made in resolving singularities in spherically symmetric spacetimes (regular black holes) using Nonlinear Electrodynamics (NED), the cylindrical topology (black strings) remains less explored.

• The Core Issue: Classical 4D black strings possess an intrinsic curvature singularity along their symmetry axis ($r=0$).
• The Challenge: It is unclear whether NED, which successfully regularizes spherical black holes, can similarly resolve the axial singularity of black strings without violating fundamental physical principles like causality and unitarity.
• Specific Gap: Previous "no-go" theorems regarding the impossibility of generating regular, purely electric black holes with a standard Maxwellian weak-field limit had not been rigorously extended to cylindrical symmetries.

2. Methodology

The authors employ a model-independent theoretical framework to analyze the coupling of GR with NED in a 4D spacetime with a negative cosmological constant ($\Lambda = -3/\ell^2$).

• Theoretical Setup:
  • Action: Einstein-Hilbert action coupled to a general NED Lagrangian density $L(F)$, where $F = F_{\mu\nu}F^{\mu\nu}$ is the electromagnetic invariant.
  • Metric: A static, cylindrically symmetric ansatz (Lemos metric):
    $$ds^2 = -f(r)dt^2 + \frac{1}{f(r)}dr^2 + r^2d\phi^2 + \frac{r^2}{\ell^2}dz^2$$
  • Field Equations: The Einstein equations are coupled with generalized Maxwell equations derived from varying the action with respect to the gauge potential.
• Analytical Approach:
  1. Regularity Analysis: The authors derive necessary conditions for the mass function $m(r)$ to ensure the finiteness of curvature invariants (Kretschmann scalar $K$ and Ricci scalar $R$) at $r=0$. Specifically, $m(r)$ must behave as $O(r^3)$ near the origin.
  2. No-Go Theorems: They extend Bronnikov's spherical no-go theorems to cylindrical symmetry, analyzing purely electric, magnetic, and dyonic configurations.
  3. Exact Solutions: They construct specific exact solutions using:
     • Singular Models: Euler-Heisenberg and Logarithmic electrodynamics.
     • Regular Models: Bardeen and Hayward classes (adapted for cylindrical symmetry).
  4. Physical Consistency Checks: The solutions are tested against causality (no superluminal propagation) and unitarity constraints using the effective metric for photon propagation.

3. Key Contributions

A. Extension of No-Go Theorems to Cylindrical Topology

The paper provides a rigorous mathematical proof extending established no-go theorems to black strings:

• Purely Electric Configurations: It is proven that no purely electric black string can be regular at the axis ($r=0$) if the NED Lagrangian recovers the standard Maxwell limit ($L \approx -F$) in the weak-field regime. To achieve regularity, the theory must violate the Maxwell limit at the core.
• Dyonic Configurations: It is demonstrated that static, cylindrically symmetric dyonic solutions (simultaneous electric and magnetic charges) cannot be regular. The conditions required for the magnetic sector ($L_F \to 0$) and the electric sector ($L_F \to \infty$) at the origin are mutually exclusive.
• Magnetic Configurations: Regularity is possible for purely magnetic strings, provided the Lagrangian approaches a finite constant as $F \to \infty$ (strong field limit).

B. Construction of Exact Regular Black String Solutions

The authors successfully generate new exact solutions for regular black strings by adapting the Bardeen and Hayward models:

• Bardeen Class: A purely magnetic solution where the core geometry transitions between Anti-de Sitter (AdS) and de Sitter (dS) depending on the charge-to-mass ratio. The curvature invariants remain finite everywhere.
• Hayward Class: Another purely magnetic regular solution exhibiting a de Sitter-like core.
• Electric Counterparts: By "reverse-engineering" the Lagrangian from the mass function of the magnetic solutions, they derived the corresponding purely electric sources. These electric solutions are regular but fail to recover the Maxwell limit at spatial infinity (the electric field decays faster than $1/r^2$).

C. Causality and Unitarity Analysis

The paper investigates the physical viability of these regular solutions regarding signal propagation:

• Theorem: Regular purely magnetic black strings necessarily violate causality in the deep core region ($r \to 0$).
• Mechanism: For a regular Lagrangian, $L_F$ must vanish at the origin. This leads to a violation of the condition $\Phi = L_F + 2F L_{FF} \leq 0$, which guarantees subluminal propagation.
• Result: While the Euler-Heisenberg model (singular) preserves causality, the regular Bardeen and Hayward models exhibit superluminal photon propagation near the axis. This implies a breakdown of classical predictability in the innermost region, despite the absence of a curvature singularity.

4. Results

Feature	Singular Models (Euler-Heisenberg, Logarithmic)	Regular Models (Bardeen, Hayward)
Curvature Singularity	Present at $r=0$ (diverges as $r^{-16}$ or similar).	Removed. Invariants are finite everywhere.
Maxwell Limit	Recovered at $r \to \infty$.	Violated for electric solutions; Magnetic solutions require specific strong-field behavior.
Causality	Generally preserved (weak field), though Logarithmic model violates it near the singularity.	Violated near the axis ($r \to 0$) due to superluminal propagation.
Core Geometry	Singular.	Regular (AdS or dS core).

• Asymptotic Behavior: All constructed solutions approach the standard vacuum black string (AdS) geometry at infinity.
• Horizon Structure: The existence of event horizons depends on the relationship between the mass parameter $\mu$ and the charge parameter. Extremal cases exist where the horizon coincides with the regular core.

5. Significance and Implications

• Topological Constraints: The work establishes that the topological difference between spherical and cylindrical symmetries does not relax the fundamental constraints of NED. The "no-go" theorems for regular electric and dyonic configurations are universal across these topologies.
• Trade-off between Regularity and Causality: The paper highlights a fundamental physical tension: regularizing the spacetime singularity via NED often necessitates a violation of causality in the core. This suggests that while NED can smooth out curvature divergences, it may introduce other pathologies (superluminality) that prevent a fully "healthy" regular black string.
• Holographic Applications: Since black strings are relevant in Anti-de Sitter (AdS) backgrounds for the AdS/CFT correspondence, these regular solutions offer new candidates for studying holographic fluid dynamics and phase transitions without the complications of singularities, provided the causal violations are managed or interpreted within a quantum gravity context.
• Future Directions: The authors suggest extending these static solutions to rotating black strings and investigating their thermodynamic properties and stability against perturbations.

In conclusion, this paper provides a comprehensive classification of NED-coupled black strings, proving that while regular cores are mathematically achievable for magnetic configurations, they come at the cost of causality in the deep interior, and purely electric regular configurations are strictly forbidden if the theory must recover standard electrodynamics at infinity.