Quasitopological gravity and double-copy formalism

Technical Summary: Quasitopological Gravity and Double-Copy Formalism

Problem Statement
General Relativity (GR) successfully describes gravity in weak and strong fields but predicts curvature singularities in vacuum black-hole solutions. To address this, quasitopological theories of gravity (QTG) have been proposed, extending the Einstein-Hilbert action with an infinite series of polynomial scalar curvature invariants. While these theories admit regular (non-singular) black-hole solutions in $D \geq 5$ dimensions, the field equations are generally highly non-linear and involve higher-order derivatives. However, for spherically symmetric metrics, these equations reduce to second-order differential equations. The challenge lies in finding a systematic method to generate these solutions and understand the underlying structure of the higher-curvature interactions without solving complex differential equations directly.

Methodology
The paper proposes a modified classical double-copy formalism to solve the vacuum field equations of $D$-dimensional quasitopological gravity. The approach relies on the Kerr-Schild representation of the metric, $g_{\mu\nu} = \eta_{\mu\nu} + 2\Phi k_\mu k_\nu$, where $\eta_{\mu\nu}$ is a flat metric and $k_\mu$ is a null vector.

The core methodology involves the following steps:

1. Dimensional Shift: Instead of working in the physical $D$-dimensional spacetime ($M^D$), the author introduces an auxiliary flat spacetime of dimension $D+1$ ($M^{D+1}$).
2. Auxiliary Non-Linear Electrodynamics: The gravitational problem is mapped to a static, spherically symmetric solution of a non-linear electrodynamics (NLED) theory defined in $M^{D+1}$. The action for this auxiliary theory is $W_A = \int d^{D+1}X \mathcal{L}(E) + \int d^{D+1}X J^a A_a$, where $E$ is the electric field strength.
3. Reduction and Projection: The NLED equations in $M^{D+1}$ reduce to a Gauss-law constraint for a point-like charge. The solution yields a function $h(E) = d\mathcal{L}/dE$. By projecting the solution from $M^{D+1}$ onto a $D$-dimensional hyperplane ($X^D=0$), the electric field $E$ is identified with a primary curvature invariant $p$ of the gravitational metric.
4. Algebraic Reconstruction: The gravitational field equation is shown to be equivalent to an algebraic relation $h(p) = \mu/r^{D-1}$, where $\mu$ is related to the mass. Solving this algebraic relation for $p(r)$ allows the reconstruction of the metric function $f(r)$ via the relation $f = 1 - r^2 p(r)$.

Key Contributions

• Unified Framework: The paper demonstrates that all vacuum, static, spherically symmetric solutions of $D$-dimensional quasitopological gravity can be generated by solving a specific non-linear electrodynamics problem in a flat $(D+1)$-dimensional spacetime.
• Reduction of Dynamics: It establishes that the complex gravitational dynamics, encoded in an infinite series of curvature invariants, map directly to the constitutive relation $h(E)$ of the auxiliary gauge theory. The gravitational field equations reduce to an algebraic relation between the primary curvature invariant and the electric field, while the remaining dynamics are governed by a simple Gauss-law constraint.
• Explanation of Regularity: The formalism provides a transparent explanation for the absence of higher-derivative terms in the reduced gravitational equations. In the gauge-theory side, this corresponds to the purely algebraic nature of the constitutive relation between $E$ and $h(E)$.
• Explicit Solutions: The author applies this formalism to specific NLED models to derive new gravitational solutions:
  • Born-Infeld Model: Yields a higher-dimensional black hole solution with a metric function $f = 1 - \frac{\mu r^2}{\sqrt{r^{2(D-1)} + \mu^2 \ell^4}}$.
  • Hayward-Type Model: Yields a higher-dimensional version of the Hayward metric, $f = 1 - \frac{\mu r^2}{r^{D-1} + \mu \ell^2}$.
    Both examples result in regular black holes with a de Sitter-like core and bounded curvature invariants.

Results
The study confirms that the modified double-copy approach successfully generates the known Schwarzschild-Tangherlini solution (in the linear limit) and extends it to non-linear regimes. By choosing NLED Lagrangians that regularize the electric field at the origin, the corresponding gravitational solutions are shown to possess finite curvature invariants everywhere, effectively curing the singularity problem within the quasitopological framework. The scaling properties of the solutions are analyzed, showing that if the auxiliary function $h(p)$ is well-behaved, the resulting spacetime satisfies the Markov limiting curvature condition.

Significance
The paper claims that this work extends the scope of the double-copy paradigm beyond Einstein gravity and linear theories. It suggests a unifying framework where higher-curvature gravitational interactions are interpreted as non-linear gauge dynamics. The author posits that this correspondence clarifies why quasitopological theories admit regular black-hole solutions without introducing higher-derivative instabilities in the spherically symmetric sector. The results imply that the double-copy is not restricted to Einstein gravity but represents a more general organizing principle linking gravity and gauge theories, even in the presence of higher-curvature interactions. The paper concludes by noting that future work could explore rotating solutions, time-dependent backgrounds, and the stability of these regular black holes.