Regular black holes from pure gravity in four dimensions

We derive static spherically symmetric regular black holes as vacuum solutions to purely gravitational theories in four dimensions. To that end, we construct four-dimensional non-polynomial gravities starting from subclasses of two-dimensional Horndeski actions. By construction, these theories possess second-order equations of motion on spherically symmetric backgrounds. We show that a subset of these non-polynomial gravities, referred to as non-polynomial quasi-topological gravities, admit single-function static spherically symmetric solutions whereby the metric function is determined by an algebraic equation. Solutions to these theories include the Hayward regular black-hole spacetime, for which a corresponding gravitational action is stated explicitly.

💡 Research Summary

The paper tackles the long‑standing problem of obtaining regular (singularity‑free) black‑hole solutions as vacuum configurations of a purely gravitational theory in four dimensions. Traditional approaches rely on coupling to exotic matter fields (e.g. nonlinear electrodynamics) or on higher‑dimensional polynomial quasi‑topological gravities, which do not admit non‑trivial static spherically symmetric vacua in four dimensions. The authors overcome this limitation by constructing a new class of four‑dimensional “non‑polynomial” gravities whose Lagrangians are arbitrary (not necessarily analytic) functions of curvature invariants such as R, R_{μν}R^{μν}, and R_{μνρσ}R^{μνρσ}, without involving covariant derivatives of curvature.

The construction starts from the most general two‑dimensional Horndeski action for a metric q_{ab}(y) and a scalar φ(y). This action depends on two free functions h_2(φ,χ) and h_4(φ,χ) (with χ≡∇aφ∇^aφ) and yields second‑order field equations. By embedding the two‑dimensional system into a four‑dimensional warped product spacetime g{μν}=q_{ab}(y)dy^a dy^b+φ(y)^2 dΩ_2^2, the authors show that any four‑dimensional Lagrangian built from curvature invariants reduces, on such backgrounds, to a two‑dimensional Horndeski theory. Conversely, given a desired Horndeski sector (specified by functions α(φ,χ) and β(φ,χ)), one can reconstruct a covariant four‑dimensional density that reproduces it upon reduction.

Within this framework the authors identify a subclass they call “non‑polynomial quasi‑topological gravities”. These theories satisfy two crucial properties: (i) on any static spherically symmetric ansatz the equations of motion remain second order, and (ii) the solution space collapses to a single metric function f(r) governed by an algebraic equation. In contrast to polynomial quasi‑topological gravities (which exist only in dimensions higher than four), the algebraic equation here involves two theory‑dependent functions of the primary curvature invariant, reflecting the non‑polynomial nature of the action.

The paper provides explicit examples. For the Hayward regular black hole, f(r)=1−2Mr^2/(r^3+2Ml^2), the authors exhibit the corresponding four‑dimensional Lagrangian, which contains non‑polynomial combinations such as √(1+χ) and rational functions of φ. They also treat the Dymnikova regular black hole, whose metric function involves an exponential decay term, and present its associated action. Both examples demonstrate that the regular spacetimes, previously obtained only with matter sources, arise here as pure‑gravity vacuum solutions.

A detailed analysis of the theoretical structure shows that the “single‑function” condition imposes specific relations between α and β, effectively reducing the Horndeski sector to a one‑parameter family. The authors discuss how additional symmetries (e.g., φ→−φ) or constraints on the functional dependence can further simplify the algebraic equation, making it analogous to the polynomial case while retaining genuine non‑polynomial features.

Finally, the authors compare their construction with earlier four‑dimensional non‑polynomial quasi‑topological gravities (e.g., those in references