Dymnikova Black Holes in Unimodular Gravity: Maxwell Sources and Vacuum Contributions

This paper demonstrates that the Dymnikova regular black hole can be consistently generated within unimodular gravity using standard Maxwell electrodynamics, where a dynamically emerging, radial-dependent vacuum contribution regularizes the geometry and yields a localized charge distribution with vanishing asymptotic charge.

The Gist

The Big Picture: Fixing a "Broken" Black Hole

Imagine a black hole as a cosmic whirlpool. In the classic version of this story (the Schwarzschild black hole), the water spins faster and faster until it hits a point of infinite density—a "singularity." At this point, the laws of physics break down, like a map that suddenly says "Here be dragons" and then stops making sense.

Physicists have long wanted to fix this. They want a "regular" black hole: one that has a smooth, safe center (like a calm eye of the storm) instead of a broken singularity, but still looks like a normal black hole from far away.

One famous model for this is the Dymnikova black hole. It has a smooth, de Sitter core (a tiny, expanding universe inside) that prevents the singularity. The question this paper asks is: What kind of "stuff" creates this smooth center?

The Two Ways to Build a Smooth Black Hole

The authors explore two different ways to explain the "stuff" holding this black hole together.

1. The "Exotic Battery" Approach (Nonlinear Electrodynamics)

First, they look at the black hole as if it were powered by a very strange, custom-made battery.

• The Analogy: Imagine a standard lightbulb (Maxwell's electricity) that works great everywhere, but if you try to turn it up too high, it burns out. To get a smooth black hole, you need a "super-bulb" that changes its rules depending on how close you are to the center. This is called Nonlinear Electrodynamics (NED).
• The Result: The authors calculated exactly what this "super-battery" would look like. They found that to create the smooth center, you need either a purely magnetic charge or a purely electric charge, but the rules of the electricity have to be twisted and warped near the center. It's like a river that flows normally far away but turns into a thick, slow-moving gel right in the middle to prevent a crash.

2. The "Magic Vacuum" Approach (Unimodular Gravity)

This is the main focus of the paper. The authors ask: Can we use a normal, standard lightbulb (standard Maxwell electricity) and still get a smooth black hole?

In standard physics, the answer is usually "no." Standard electricity follows strict rules that would create a singularity. However, the authors use a different theory of gravity called Unimodular Gravity.

• The Analogy: Imagine the universe is a balloon. In standard gravity, the air inside (matter) and the rubber of the balloon (space) are tightly linked. If you squeeze the air, the rubber stretches exactly as expected.
  In Unimolar Gravity, the total size of the balloon is fixed by a rule. This allows the "air" (matter) and the "rubber" (space) to behave a little differently. Specifically, it allows the "vacuum" (empty space) to act like a dynamic, shifting force that changes depending on where you are.
• The Magic Trick: The authors show that if you use standard, normal electricity (no weird "super-batteries" needed) inside this "fixed-size balloon" universe, the vacuum itself steps in to do the heavy lifting.
  • Near the center, the vacuum acts like a cushion, pushing back to keep the geometry smooth.
  • Far away, the vacuum fades away, and the black hole looks normal again.
• The "Cosmological Function": They found that the "vacuum energy" isn't a constant number (like a fixed cosmological constant). Instead, it's a radial function, $\Lambda(r)$. Think of it as a "smart cushion" that is thick and strong right in the middle of the black hole to prevent a singularity, but gets thinner and thinner until it disappears completely as you move away.

Key Findings in Plain English

1. No "Super-Charges" Needed: Usually, to make a smooth black hole, you need exotic, weird forms of electricity that break standard rules. This paper shows that in Unimodular Gravity, you can use standard, boring electricity and let the "vacuum" of space do the exotic work.
2. The Charge Disappears: In this new model, the electric field is perfectly smooth everywhere. It starts at zero at the very center, grows a little bit, and then fades back to zero as you go far away.
   • The Analogy: It's like a localized storm. You have a burst of wind in the middle, but if you stand far enough away, the wind is gone. The total amount of "charge" you measure from the outside is actually zero. The black hole isn't "charged" in the traditional sense; the charge is just a temporary, localized ripple that helps smooth out the center.
3. The Vacuum is the Hero: The paper concludes that the "regularity" (the smoothness) of the black hole comes entirely from the effective vacuum sector. The ordinary matter (the electric field) is well-behaved and follows standard rules. The "magic" that fixes the singularity is provided by the geometry of space itself, which acts like a dynamic, shape-shifting cushion.

Summary

The paper takes a famous model of a smooth black hole and re-imagines it. Instead of needing a weird, custom-made "exotic battery" to keep the center smooth, they show that if you change the rules of gravity slightly (Unimodular Gravity), you can use standard electricity. The "smoothness" is then provided by the vacuum of space itself, which acts like a smart, adjustable cushion that is strong in the middle and vanishes at the edges. This separates the "ordinary matter" from the "vacuum effects," giving a clearer picture of how a regular black hole could theoretically exist.

Technical Summary

Technical Summary: Dymnikova Black Holes in Unimodular Gravity: Maxwell Sources and Vacuum Contributions

Problem Statement
The paper addresses the theoretical challenge of constructing regular black hole solutions that avoid the curvature singularities inherent in classical General Relativity (GR), specifically focusing on the Dymnikova spacetime. While regular black holes have traditionally been modeled using nonlinear electrodynamics (NED) to support anisotropic matter distributions with a de Sitter core, the authors investigate whether the same geometry can be generated by standard Maxwell electrodynamics within the framework of unimodular gravity. The central problem is to determine if the regularization of the Dymnikova geometry can be attributed to an effective vacuum contribution arising from the specific constraints of unimodular gravity, rather than requiring exotic matter sources that violate standard energy conditions.

Methodology
The authors employ a two-pronged analytical approach:

1. Nonlinear Electrodynamics (NED) Reconstruction:
   The Dymnikova metric is first reinterpreted as a solution sourced by NED. The authors derive the corresponding Lagrangian $L(F)$ for both purely magnetic and purely electric configurations by solving the Einstein field equations coupled to the NED energy-momentum tensor. They verify the behavior of the Lagrangian near the origin ($r \to 0$) and at asymptotic infinity ($r \to \infty$) to ensure consistency with Bronnikov's criteria for regular black holes.

2. Unimodular Gravity Extension:
   The authors then apply the unimodular gravity framework, where the determinant of the metric is fixed ($\sqrt{-g} = g_0$). In this theory, the covariant conservation of the energy-momentum tensor is relaxed, allowing for a dynamical cosmological term $\Lambda(x)$ that emerges as an integration function.

   • They utilize the trace-reversed field equations of unimodular gravity: $R_{\mu\nu} - \frac{1}{4}R g_{\mu\nu} = T_{\mu\nu} - \frac{1}{4}g_{\mu\nu}T$.
   • By assuming a standard Maxwell source ($T_{\mu\nu}$ is traceless), they derive a radial-dependent cosmological function $\Lambda(r)$ that satisfies the modified conservation law $\nabla_\mu T^{\mu\nu} = -F^{\nu\alpha}J_\alpha$.
   • They define a geometric function $H(r)$ to ensure the electric field remains real and physically valid throughout the spacetime.

Key Contributions and Results

• NED Sources: The authors successfully reconstruct the Dymnikova geometry using NED.

  • For a purely magnetic source, they derive a Lagrangian $L(F)$ that is regular at the origin and vanishes asymptotically.
  • For a purely electric source, they derive a corresponding Lagrangian and electric field. The electric field is shown to be regular everywhere, vanishing at both the origin and infinity, consistent with Bronnikov's theorem that regular NED black holes cannot exhibit a standard Maxwell limit at infinity.
  • The analysis confirms that a dyonic (mixed electric and magnetic) source cannot generate the Dymnikova spacetime, consistent with existing no-go theorems.

• Maxwell Source in Unimodular Gravity: The primary contribution is demonstrating that the Dymnikova geometry can be generated by standard Maxwell electrodynamics coupled to unimodular gravity.

  • The regularization is not achieved by modifying the matter sector (Maxwell electrodynamics satisfies standard energy conditions) but by an effective vacuum contribution encoded in a radial-dependent cosmological function, $\Lambda(r)$.
  • The derived electric field is $E(r) \propto r^{3/2} e^{-r^3/(2q^3)}$, which is everywhere regular and vanishes asymptotically.
  • Crucially, the total electric charge measured at spatial infinity is zero ($Q_\infty = 0$), indicating the spacetime behaves as a localized charge distribution rather than an asymptotically charged object.

• The Cosmological Function $\Lambda(r)$:

  • The function $\Lambda(r)$ is explicitly derived as $\Lambda(r) = \frac{3m e^{-r^3/q^3}(4q^3 - 3r^3)}{2q^6}$.
  • Near the origin ($r \to 0$): $\Lambda(r)$ approaches a constant positive value ($\approx 6m/q^3$), reproducing the de Sitter core responsible for the geometry's regularity.
  • Asymptotically ($r \to \infty$): $\Lambda(r)$ vanishes exponentially, recovering asymptotic flatness and the Schwarzschild behavior.

Significance and Claims
The paper claims that this construction provides a "clean separation" between ordinary matter and the mechanism of regularization. Unlike standard NED models where the same matter sector generates the geometry and violates energy conditions to ensure regularity, the unimodular framework attributes the necessary violations entirely to the effective vacuum sector ($\Lambda(r)$).

The authors assert that this result:

1. Reinforces the interpretation that the Dymnikova regular geometry is associated with vacuum effects (gravitational vacuum polarization).
2. Demonstrates that standard Maxwell electrodynamics is sufficient to support regular black holes if the gravitational sector is modified via unimodular gravity.
3. Supports the view that in unimodular gravity, the cosmological term is not a fundamental constant but an emergent geometric quantity determined by the spacetime geometry and matter distribution.

The work does not propose new experimental tests or applications but offers a theoretical reinterpretation of the Dymnikova solution, aligning it with the geometric constraints of unimodular gravity and the equivalence principle.