Black Hole Interiors as a Laboratory for Time-Dependent Classical Double Copy

Abstract: The classical double copy provides a powerful bridge between gravity and gauge theory, but its most explicit realizations remain concentrated in stationary or highly symmetric settings. We show that trapped regions of black-hole geometries furnish an exact setting for time-dependent classical double copy. In the static, spherically symmetric case, each trapped interval admits a local single-copy description on the associated Kantowski--Sachs patch that is intrinsically time dependent, although it can be derived from static Kerr--Schild data and does not require knowledge of any exterior black-hole completion. We prove that this class is characterized intrinsically by a distinguished relation between the Kantowski--Sachs scale factors, equivalently by the longitudinal relation (p_{\parallel}=-ρ), and that the Kerr--Schild scalar and single-copy field are uniquely reconstructible from interior cosmological data. Schwarzschild provides the singular benchmark, for which the single-copy electric field diverges along the interior evolution, while the regular Bardeen solution yields a finite single-copy field throughout the trapped region and a smooth extension into a regular static core. The Bardeen core violates the strong energy condition in a compact region, whereas the corresponding single-copy Maxwell field remains regular and satisfies the standard classical energy conditions. We further show that the Bardeen horizon phase structure is encoded in the single-copy scalar. These results identify trapped Kerr--Schild interiors as an exact local laboratory for time-dependent classical double copy.

• The paper identifies that trapped Kerr–Schild black hole interiors can serve as an exact laboratory for time-dependent classical double copy, linking gravitational and gauge-theory frameworks.
• It employs the Kantowski–Sachs formulation to translate interior gravitational dynamics into gauge field configurations, analyzing both singular Schwarzschild and regular Bardeen solutions.
• The study reveals that horizon structures and energy condition violations are precisely encoded in the single-copy scalar, offering insights into causal transitions and field regularity.

Black Hole Interiors as an Exact Laboratory for Time-Dependent Classical Double Copy

Introduction

The classical double copy, mapping gravitational solutions to gauge theory configurations, has achieved substantial traction in stationary and highly symmetric spacetimes. However, its precise implementation for genuinely time-dependent, anisotropic backgrounds has remained elusive. The paper "Black Hole Interiors as a Laboratory for Time-Dependent Classical Double Copy" (2604.19920) rigorously establishes that trapped regions within black-hole geometries—specifically within static, spherically symmetric Kerr–Schild spacetimes—constitute an exact framework for local, time-dependent classical double copy, realized intrinsically on Kantowski–Sachs patches.

This approach connects the interior cosmological viewpoint, where the areal radius serves as a time parameter, with the Kerr–Schild structure; it directly enables the translation of interior gravitational properties into uniquely defined, time-dependent gauge-theory analogues. The construction applies both to singular solutions (Schwarzschild) and regular black holes (Bardeen), with detailed analysis of interior dynamics, field strength evolution, and energy-conditions duality.

Kantowski–Sachs Formulation of Black Hole Interiors

Upon crossing the event horizon, the causal roles of Schwarzschild coordinates interchange: the areal radius becomes timelike, while the temporal coordinate becomes spacelike. This leads to a cosmological description utilizing the Kantowski–Sachs metric:

$$ds^2 = -d\tau^2 + a(\tau)^2 d\chi^2 + b(\tau)^2 d\Omega_2^2,$$

where $a(\tau)$ and $b(\tau)$ are distinct, time-dependent scale factors. For the Schwarzschild black hole, this anisotropic evolution manifests in divergent shear and expansion scalars near the singularity, resultants of the interior Weyl tensor driving the Raychaudhuri focusing.

The regular Bardeen black hole, parameterized by a mass $M$ and a magnetic charge $g$, possesses a nonstatic interior between two horizons ($r_- < r < r_+$), transitioning to a regular static de Sitter-like core for $r < r_-$. All curvature invariants remain finite in the Bardeen case.

Figure 1: Comparison of the electric single-copy profiles for Schwarzschild and Bardeen, plotting $|F_{tr}(r)| = |\phi'(r)|$. The Schwarzschild solution diverges at $r = 0$, while the Bardeen field remains regular throughout and vanishes at the center.

Kerr–Schild Framework and the Classical Double Copy

Static, spherically symmetric black holes can be recast in the Kerr–Schild ansatz: $$g_{\mu\nu} = \eta_{\mu\nu} + 2\phi(r) k_{\mu} k_{\nu}$$. The map to the gauge sector yields $a(\tau)$0, with the scalar profile $a(\tau)$1 encapsulating both gravitational and gauge properties. For Bardeen,

$a(\tau)$2

In static domains, the single-copy field is electrostatic; in the trapped interior, it acquires intrinsic time dependence due to the coordinate transformation. The field strength and current can be expressed as

$a(\tau)$3

with $a(\tau)$4 the timelike areal radius in the interior.

Interior observers sample this field along their proper-time evolution, yielding profiles $a(\tau)$5 that diverge for Schwarzschild and remain finite for Bardeen.

Intrinsic Characterization and Reconstruction

A key technical result is that trapped Kerr–Schild interiors can be characterized solely via Kantowski–Sachs scale factors; specifically, the class is singled out by the relation $a(\tau)$6. This equivalently implies the longitudinal equation of state $a(\tau)$7 in the effective stress tensor of the interior. Thus, knowledge of the interior cosmological evolution suffices to reconstruct both the Kerr–Schild scalar and the single-copy gauge potential:

$a(\tau)$8

This intrinsic mapping ensures that physical properties such as field regularity and energy conditions admit precise duals within the gauge sector.

Energy-Condition Duality

The regular Bardeen solution is sourced by matter that saturates the null energy condition (radially) and violates the strong energy condition within a compact core, enabling singularity resolution. Its stress tensor is anisotropic but transitions to isotropic (de Sitter-like) behavior at the center. In contrast, the corresponding single-copy Maxwell field remains globally regular and satisfies all classical pointwise energy conditions. This stark duality illustrates how strong energy-condition violation in the gravitational side translates to field regularity and well-behaved energetics in the gauge theory.

Horizon Phase Structure in the Single Copy

While the double copy is local, the paper demonstrates that horizon structure is inherently encoded in the single-copy scalar. Adopting the Chawla–Keeler criterion, trapping surfaces correspond to zeros of the expansion parameter $a(\tau)$9:

$b(\tau)$0

For Bardeen, the number and nature of horizons (non-extremal, extremal, or horizonless) follow directly from the roots of $b(\tau)$1.

Figure 2: $b(\tau)$2 as a function of $b(\tau)$3 for the Bardeen black hole, illustrating transitions between two, one, or zero horizons depending on the mass-to-charge ratio.

The gauge-side scalar thus captures gravitational horizon transitions, providing an algebraic diagnostic of causal structure entirely within the single-copy data.

Implications and Future Directions

Trapped intervals of Kerr–Schild black hole interiors constitute an exact laboratory for probing time-dependent classical double copy. The formalism supports:

• Unique mapping from interior cosmological data to gauge sector configuration.
• Sharp correspondence between energy condition violation and field regularity.
• Encoding of horizon phase transitions within single-copy variables.

These results furnish a self-contained framework for analyzing perturbations, dynamical response, and more complex time-dependent matter-supported examples beyond stationary, vacuum spacetimes. The methodology may inspire systematic studies of interior dynamics and their gauge-theory analogues, unifying gravitational and electromagnetic phenomena in black hole interiors under the double-copy paradigm.

Conclusion

The work establishes that black hole interiors—particularly trapped Kerr–Schild intervals—provide a rigorous, technically robust setting for studying time-dependent classical double copy. The framework identifies a distinguished, intrinsically characterized class of interiors where the full gravitational, cosmological, and gauge-theoretic structures are explicitly and equivalently interrelated. Regular solutions such as Bardeen black holes highlight the interplay between energy condition violation and gauge field regularity, while the encoding of horizon structure in single-copy data elucidates new facets of the double copy's physical reach. These findings open avenues for deeper explorations of nonstationary and anisotropic double-copy correspondences in gravitational physics.