Regular Black Strings and BTZ Black Hole in Unimodular Gravity Supported by Maxwell Fields

This paper demonstrates that within the framework of unimodular gravity, Maxwell electrodynamics can serve as a source for regular black strings and BTZ black holes, where the cosmological constant emerges as a radial-dependent vacuum contribution derived from the non-conservation of the energy-momentum tensor.

The Gist

Imagine the universe as a giant, stretchy fabric called spacetime. For decades, physicists have used a rulebook called General Relativity to describe how heavy objects (like stars and black holes) warp this fabric. But there's a problem: when you zoom in on the very center of a black hole, the math breaks down. The fabric tears, creating a "singularity"—a point of infinite density where our laws of physics stop working. It's like a hole in the fabric that the universe just can't explain.

This paper proposes a clever workaround using a slightly different rulebook called Unimodular Gravity. Here's how it works, explained simply:

1. The "Fixed Volume" Rule

In standard physics, the universe is like a flexible balloon that can be stretched or squished in any way. In Unimodular Gravity, there's a new rule: the total volume of the balloon must stay exactly the same, no matter how you stretch it.

This small change has a huge side effect. In the old rulebook, the "Cosmological Constant" (a mysterious energy that pushes the universe apart) had to be put in by hand at the start. In this new rulebook, that constant isn't a fixed number; it's a variable that can change depending on where you are. Think of it like a thermostat that automatically adjusts the temperature based on the room, rather than a heater stuck on one setting.

2. The "Leaky" Energy Tank

Usually, physicists assume that energy and momentum are perfectly conserved (nothing is ever lost or gained). But because of the "fixed volume" rule, this paper suggests that energy can "leak" a little bit.

This leak allows the universe to create a dynamic vacuum energy (let's call it $\Lambda$) that changes from place to place.

• The Analogy: Imagine a black hole is a whirlpool in a river. In standard physics, the water just spins forever. In this new theory, the whirlpool can "suck" energy from the riverbank (the vacuum) to change its own shape. This extra energy acts like a cushion.

3. Fixing the Black Holes

The authors applied this idea to two types of black holes:

• The Black String: Imagine a black hole that isn't a ball, but a long, infinite cylinder (like a cosmic spaghetti noodle).
• The BTZ Black Hole: A black hole in a universe with only two dimensions (like a flat sheet of paper).

In standard physics, these objects have a sharp, infinite point in the middle. But in this paper, the authors show that if you use Maxwell's Electromagnetism (the same physics that governs light and magnets) combined with this "leaky" vacuum energy, you can build a Regular Black Hole.

What does "Regular" mean?
Instead of a sharp, infinite point, the center becomes a smooth, finite core.

• The Metaphor: Imagine a storm. In a normal black hole, the wind speed at the center goes to infinity (a singularity). In this "Regular" black hole, the wind speeds up as it gets closer to the center, but then it hits a "ceiling" and slows down, creating a calm, smooth eye. The singularity is gone.

4. The Magic Ingredient: The "Geometric Function"

To prove this works, the authors used a mathematical tool they call $H(r)$.

• Think of $H(r)$ as a health checkup for the black hole.
• If $H(r)$ is positive, the black hole is healthy and can be supported by normal electromagnetic fields (like light).
• If $H(r)$ is negative, the math breaks down, and the "cushion" isn't strong enough to stop the singularity.

They found that for the Black String and the Hayward-type black hole, the health check is perfect everywhere. The electromagnetic field and the changing vacuum energy work together perfectly to keep the center smooth.

However, for the BTZ black hole, there's a catch. The health check is good far away, but close to the center, it dips into the "danger zone" (negative values). This means that for this specific type of black hole, normal electromagnetism alone can't fix the singularity all the way to the very center; it needs a little extra help from the vacuum energy to stay smooth.

The Big Takeaway

This paper suggests that we might not need "exotic" or imaginary matter to fix black holes. Instead, by slightly tweaking the rules of gravity (fixing the volume) and letting the vacuum energy adapt to the situation, standard physics (Maxwell's equations) might be enough to turn a terrifying, infinite singularity into a smooth, manageable core.

It's like realizing that to fix a pothole in the road, you don't need a new type of asphalt; you just need to let the existing road material settle and adapt to the shape of the hole.

Technical Summary

1. Problem Statement

General Relativity (GR) predicts that black holes and black strings generically possess spacetime singularities where curvature invariants diverge, causing the breakdown of classical physics. While "regular" black hole solutions (those without singularities) have been proposed, they typically require nonlinear electrodynamics (NED) as a source. These NED sources are often highly complex, obscuring physical interpretation.

The authors aim to address two specific issues:

1. Can regular black string (3+1 dimensions) and BTZ black hole (2+1 dimensions) solutions be supported by standard Maxwell electrodynamics (linear theory) rather than complex nonlinear models?
2. How does Unimodular Gravity (UG) facilitate this, specifically regarding the role of the cosmological constant and energy-momentum conservation?

2. Methodology and Theoretical Framework

Unimodular Gravity (UG)

The authors utilize UG, an alternative to GR where the spacetime volume element is fixed to a constant ($\det(g_{\mu\nu}) = g_0$).

• Symmetry Restriction: This condition restricts diffeomorphism invariance to volume-preserving transformations.
• Cosmological Constant: In UG, the cosmological constant ($\Lambda$) does not appear in the action but emerges as an integration constant of the field equations.
• Non-Conservation: Crucially, the restriction on symmetry implies that the energy-momentum tensor ($T_{\mu\nu}$) is not automatically conserved ($\nabla_\mu T^{\mu\nu} \neq 0$). Instead, its non-conservation is balanced by a position-dependent cosmological term:
  $$\nabla_\mu T^{\mu\nu} = -\nabla^\nu \Lambda(x)$$
  This allows $\Lambda$ to become a dynamical function $\Lambda(r)$, acting as an effective source that can absorb matter contributions.

Field Equations

The authors derive the field equations for $n$ dimensions under UG:
$$R_{\mu\nu} - \frac{1}{n}R g_{\mu\nu} = T_{\mu\nu} - \frac{1}{n}T g_{\mu\nu}$$
By imposing the non-conservation condition, they relate the divergence of the energy-momentum tensor to the gradient of the vacuum contribution $\Lambda(r)$.

The Geometric Function $H(r)$

To test if a solution can be supported by Maxwell electrodynamics, the authors define a geometric function $H(r)$ derived from the mass function $m(r)$ (for black strings) or $M(r)$ (for BTZ).

• In the Maxwell limit (linear electrodynamics), the electric field $E(r)$ is related to $H(r)$ by $H(r) = 2E^2(r)$.
• Validity Condition: For the solution to be physically supported by Maxwell fields, the electric field must be real, requiring $H(r) \geq 0$ everywhere.

3. Key Contributions and Results

The paper constructs and analyzes three specific regular solutions within the UG framework:

A. Regular Black Strings (3+1 Dimensions)

The authors analyze two types of regular black strings sourced by Maxwell fields:

1. Bardeen-type Black String:

   • Mass Function: $m(r) = \frac{\mu r^3}{(q^2\ell^2 + r^2)^{3/2}}$.
   • Result: The geometric function $H(r)$ is strictly positive for all $r$.
   • Conclusion: The solution is globally supported by Maxwell electrodynamics. The electric field behaves as $E \sim r$ near the origin and $E \sim r^{-3/2}$ asymptotically. The dynamical vacuum $\Lambda(r)$ is regular at the origin and recovers the standard AdS constant ($\Lambda_0$) at infinity.

2. Hayward-type Black String:

   • Mass Function: $m(r) = \frac{\mu r^3}{r^3 + \lambda^3}$.
   • Result: Similar to the Bardeen case, $H(r)$ remains positive for all $r$.
   • Conclusion: The solution is globally supported by Maxwell electrodynamics. The vacuum contribution $\Lambda(r)$ acts as an effective cosmological constant near the core, preventing singularity formation.

B. Regular Charged BTZ Black Hole (2+1 Dimensions)

The authors analyze a regular BTZ solution constructed by Cataldo and Garcia:

• Mass Function: $M(r) = m + q^2 \ln(r^2 + a^2)$.
• Result: The geometric function $H(r)$ is not positive everywhere.
  • There exists a critical radius $r_c$ (approx. 0.643 for specific parameters) where $H(r) < 0$.
  • Implication: Maxwell electrodynamics cannot support the solution globally. It is only valid for $r > r_c$. Inside the critical radius ($0 \leq r < r_c$), the solution requires a different source (likely nonlinear electrodynamics) or the vacuum contribution $\Lambda(r)$ must compensate for the failure of the Maxwell source.
• Vacuum Behavior: $\Lambda(r)$ shifts near the origin ($\Lambda \approx q^2/a^2 + \Lambda_0$) and recovers the AdS constant asymptotically.

4. Significance and Conclusions

1. Simplification of Sources: The paper demonstrates that the complex nonlinear electrodynamics usually required for regular black objects can be replaced by standard Maxwell electrodynamics in the context of Unimodular Gravity. The "complexity" is effectively absorbed by the dynamical cosmological function $\Lambda(r)$.
2. Role of $\Lambda(r)$: The integration constant $\Lambda$ is reinterpreted as a position-dependent vacuum energy. It exerts negative pressure near the center ($r \to 0$), counteracting gravitational collapse and eliminating the singularity, while asymptotically recovering the standard AdS cosmological constant.
3. Dimensional Dependence:
   • In (3+1) dimensions (Black Strings), standard Maxwell fields are sufficient to support regular Bardeen and Hayward geometries globally.
   • In (2+1) dimensions (BTZ), Maxwell fields are insufficient globally; a critical radius exists where the linear theory fails, highlighting a dimensional distinction in how UG regularizes these solutions.
4. Theoretical Consistency: The work reinforces the compatibility of the Einstein Equivalence Principle with Unimodular Gravity, showing that the condition $\det(g_{\mu\nu}) = g_0$ naturally leads to a relaxed conservation law that facilitates regular geometries without exotic matter sources in higher dimensions.

In summary, this work provides a novel mechanism to resolve black hole singularities using linear Maxwell theory by leveraging the unique properties of Unimodular Gravity, specifically the dynamical nature of the cosmological constant.