Novel higher-dimensional regular black holes in general relativity coupled to nonlinear electrodynamics

Technical Summary: Novel Higher-Dimensional Regular Black Holes in General Relativity Coupled to Nonlinear Electrodynamics

Problem Statement
The paper addresses the singularity problem in general relativity, where gravitational collapse inevitably leads to spacetime singularities with infinite curvature, as dictated by the Penrose and Hawking singularity theorems. While quantum gravity is anticipated to resolve this, researchers pursue "regular" black hole models as semi-classical alternatives. A specific theoretical difficulty arises in constructing electrically charged regular black holes: standard approaches often define the matter sector (Lagrangian) in terms of the solution's mass ($M$) and charge ($Q$), rendering the theory dependent on the solution itself. Furthermore, for electric solutions, the electromagnetic invariant $F$ vanishes at both the center and infinity, forcing the Lagrangian $L(F)$ to split into two branches joined at a cusp. Recent work has shown that for such regular black holes, mass and charge cannot be independent. The authors aim to resolve these issues by constructing a higher-dimensional regular black hole family where the matter sector is a fixed theory with independent couplings, and the mass and charge emerge strictly as integration constants.

Methodology
The authors employ a formalism coupling general relativity to nonlinear electrodynamics (NED) in $D$ dimensions ($D \geq 4$).

1. Metric Ansatz: They propose a generalized Schwarzschild-Tangherlini-like metric with a radial-dependent mass function $m(r)$ designed to satisfy regularity conditions (finite curvature invariants at $r \to 0$).
2. Dual Hamiltonian Formulation: To avoid the branch ambiguity inherent in the Lagrangian $L(F)$ for electric fields, the authors utilize the dual $P$-framework. They define the auxiliary tensor $P_{\mu\nu} = L_F F_{\mu\nu}$ and its invariant $P$. Unlike $F(r)$, the dual invariant $P(r)$ is strictly monotonic, allowing for a single-valued, globally defined Hamiltonian density $\mathcal{H}(P)$.
3. Theory Construction: By expressing the Hamiltonian density in terms of fixed coupling constants ($\sigma, \beta$) and a structural parameter $\gamma$, rather than solution parameters, they define a fixed theory. The mass and charge are then derived as integration constants of the field equations.
4. Thermodynamic Analysis: The authors analyze the thermodynamics by distinguishing between variations that change the theory's couplings versus variations that keep the theory fixed. They derive the horizon potential, the extended first law (treating couplings as thermodynamic variables), and a generalized Smarr formula.

Key Contributions and Results

• Unified Regular Black Hole Family: The authors propose a new class of electrically charged regular black holes in $D$ dimensions, parameterized by a constant $\gamma > 0$. The metric function is given by:
  $$f(r) = 1 - \frac{Mr^2}{(r^\gamma + Q^\gamma)^{(D-1)/\gamma}}$$
  This family unifies the Bardeen ($\gamma=2$ in $D=4$) and Hayward ($\gamma=3$ in $D=4$) models into a single analytic form valid for any $D \geq 4$.

• Fixed Theory and Integration Constants: By working in the dual Hamiltonian framework, the authors demonstrate that the matter sector can be written as a Hamiltonian density $\mathcal{H}(P)$ containing only fixed coupling constants. Consequently, the mass $M$ and charge $Q$ are strictly integration constants. This resolves the issue of the theory changing when a black hole absorbs a particle; the theory remains fixed, while the integration constants evolve.

  • The theory admits a two-parameter family of solutions. Regularity at the center requires one integration constant to vanish, leaving a one-parameter family of regular solutions where mass is tied to charge ($M \propto Q^{(D-1)/(D-2)}$).

• Maxwellian Weak-Field Limits: The paper analyzes the weak-field limit ($r \to \infty$). It is shown that the Lagrangian recovers the standard Maxwellian behavior ($L \sim F$) if and only if $\gamma = D - 3$.

  • In $D=4$, only the $\gamma=1$ model recovers the Maxwell limit; the Bardeen ($\gamma=2$) and Hayward ($\gamma=3$) models do not.
  • In higher dimensions ($D \geq 5$), models with higher $\gamma$ values recover the Maxwell limit. Specifically, the 5D Bardeen model ($\gamma=2$) and the 6D Hayward model ($\gamma=3$) possess correct Maxwellian limits in their respective dimensions.

• Branch Structure and Causal Location: The authors locate the radius $r_{crit}$ where the Lagrangian $L(F)$ splits into two branches (the cusp).

  • For $D=4$, $r_{crit}$ is always hidden inside the event horizon.
  • For $D \geq 5$, $r_{crit}$ can lie outside the event horizon for near-extremal charges (specifically when $Q_{crit} < Q \leq Q_{max}$). This distinction was previously unnoted.

• Energy Conditions:

  • The Null Energy Condition (NEC) and Weak Energy Condition (WEC) are satisfied everywhere.
  • The Strong Energy Condition (SEC) is violated near the core ($r < r_{SEC}$), consistent with the de Sitter-like behavior required to avoid singularities.
  • The Dominant Energy Condition (DEC) is satisfied everywhere if $\gamma \leq D-3$. If $\gamma > D-3$, the DEC is violated in the region $r > r_0$.

• Thermodynamics and the First Law: The paper addresses the apparent failure of the first law for regular black holes, where the derivative of mass with respect to entropy often differs from the Hawking temperature by a factor $\Xi$.

  • The authors clarify that this discrepancy arises because varying the mass and charge independently changes the theory's couplings.
  • When the theory is held fixed (varying only integration constants), the first law holds in its standard form: $\delta M_{ADM} = T_h \delta S + \Phi_h \delta Q_e$.
  • They derive a closed-form expression for the horizon potential $\Phi_h$ and an extended first law where the couplings are treated as thermodynamic variables, leading to a generalized Smarr formula.

Significance and Claims
The paper claims to provide a rigorous formalism for constructing higher-dimensional regular black holes that resolves the theoretical inconsistency of defining the matter sector via solution parameters. By utilizing the dual Hamiltonian formulation, the authors establish a fixed theory where mass and charge are proper integration constants. This construction clarifies the conditions under which regular black holes recover Maxwellian electrodynamics in higher dimensions and resolves the thermodynamic "failure" of the first law by distinguishing between variations within a fixed theory versus variations across the space of theories. The work also highlights a dimensional dependence in the causal location of the Lagrangian branch point, a feature specific to higher dimensions ($D \geq 5$) that was previously overlooked. The authors conclude that while the construction is consistent, regularity is a special property of a subfamily of solutions and is not preserved under accretion, as the final state generally becomes singular.