⚛️ general relativity

Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions

This paper establishes that static, regular black holes in Horndeski theories are generically obstructed by instabilities or no-hair theorems, proving that analytic configurations reduce to singular Schwarzschild solutions while the only marginal nonanalytic completion (involving scalar-Gauss-Bonnet couplings) still fails to produce regular black holes.

Original authors: Antonio De Felice, Shinji Tsujikawa

Published 2026-07-10

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CC BY 4.0

Original authors: Antonio De Felice, Shinji Tsujikawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic video game. For decades, players have been trying to build a level where the "game over" screen—a black hole with a crushing, infinite singularity at its center—never appears. Instead, they want a "regular" black hole: a smooth, safe center where the laws of physics still make sense.

In this new study, physicists Antonio De Felice and Shinji Tsujikawa act like the game's chief engineers, checking the code to see if such a level is actually possible. They are looking at a specific type of cosmic engine called "Horndeski theories," which are fancy upgrades to Einstein's gravity that include a mysterious, invisible field (a scalar field) running through space.

Here is the verdict on whether we can build a smooth, singularity-free black hole in this game.

The "Speed Trap" at the Edge

First, the team looked at the edge of the black hole, known as the horizon. They asked: What if the invisible scalar field is moving fast right at the edge?

They found a massive problem. If the field is moving (having a non-zero "kinetic term"), the math says the ripples in this field would travel at infinite speed right at the horizon. Imagine a car trying to drive on a road that suddenly turns into a vertical wall; the car's speed would have to be infinite to stay on it. In physics, infinite speed is a red flag that the whole theory breaks down.

Unless the theory has some very specific, weird "glitches" (mathematical degeneracies) that cancel this out perfectly, this path is blocked. The authors show that for almost all standard versions of these theories, you simply cannot have a moving scalar field at the horizon without breaking the rules of causality or stability.

The "Stillness" Dead End

So, the team tried the other path: What if the scalar field is perfectly still at the horizon? This is the "regular branch."

Here, they applied a clever trick. They treated the math like a recipe that gets simpler the closer you get to the center or the edge. They found that if the theory is "shift-symmetric" (meaning the rules don't change if you shift the value of the field), the math forces the field to be completely constant everywhere.

Think of it like a river that is forced to be perfectly flat and still all the way from the ocean to the source. If the river is still, the water doesn't flow. In this cosmic case, if the field is constant, the black hole loses its "hair" (its unique features) and becomes a boring, standard Schwarzschild black hole.

And here is the kicker: A standard Schwarzschild black hole always has a nasty, crushing singularity at its center. So, by trying to make the center smooth, the math forced the black hole to become the exact kind of object with a singularity that we were trying to avoid. The result? No smooth, singularity-free black holes exist on this path.

The "Perturbative" Trap

What if we try to cheat by adding a tiny bit of extra complexity (a "non-shift-symmetric" theory)? The authors checked the "perturbative branch," which is like starting with the standard Schwarzschild black hole and adding a tiny drop of new physics.

They found that if you start with the standard black hole and try to wiggle it into a smooth one, the math just won't let you. The equations force you to stay on the standard, singular path. It's like trying to turn a square peg into a round one by gently tapping it; the peg just stays square. The only way to get a smooth center would be to start with a completely different, "disconnected" kind of black hole that doesn't look like the standard one at all, but the paper shows that the standard, smooth transition is impossible.

The "Magic Logarithm" Loophole (That Still Fails)

Finally, the team asked: "What if we use a weird, non-smooth math trick, like a square root or a logarithm, to fix the code?"

They found one specific, very narrow escape route: a "scalar–Gauss–Bonnet chain" involving a logarithm. This is a very specific, exotic recipe that avoids the "infinite speed" trap and the "constant field" trap. It's the only mathematically consistent way to have a hairy black hole (one with a scalar field) in this framework.

However, even this "magic" solution has a fatal flaw. While it manages to be smooth at the horizon, the authors prove that it still ends up with a singularity at the very center. It's like building a castle with a perfect, smooth drawbridge, only to find out the dungeon underneath is still a bottomless pit.

The Bottom Line

The paper doesn't just suggest that smooth black holes are hard to find; it proves that in the specific class of theories they studied, you cannot have a static, smooth, singularity-free black hole.

If the field moves at the edge, the physics breaks (infinite speed).
If the field is still, the black hole becomes the standard, singular kind.
Even the most exotic, "non-smooth" math tricks that try to bypass these rules still result in a singularity at the center.

The universe, according to this analysis, seems to insist that if you have a black hole with mass, you must also have a singularity at its heart. The dream of a perfectly smooth, regular black hole in these theories remains just that—a dream, blocked by the hard, unyielding laws of the equations.

Technical Summary: Static Regular Black Holes in Horndeski Theories

Problem Statement
The existence of regular black holes (BHs)—spacetimes with event horizons but without central curvature singularities—remains an open question in classical gravity. While metric-level constructions (e.g., Bardeen, de Sitter cores) provide explicit regular centers, they often lack a consistent classical field theory realization. In the context of four-dimensional Horndeski scalar-tensor theories, previous studies have established no-hair theorems for static, shift-symmetric BHs or identified pathologies (ghosts, gradient instabilities) in non-minimal derivative couplings. However, a comprehensive analysis of whether static, spherically symmetric, asymptotically flat configurations can simultaneously satisfy horizon stability and center regularity within the full Horndeski framework has been incomplete. Specifically, it was unclear if the analytic branch where the scalar kinetic term $X$ vanishes at the horizon ($X(rs)=0$) could reach a regular center, or if a controlled non-analytic completion could evade existing obstructions.

Methodology
The authors analyze the most general four-dimensional Horndeski action with a static, time-independent scalar profile $\phi = \phi(r)$ on a spherically symmetric background. The study proceeds by classifying near-horizon behaviors and applying a "finite-jet" reduction to the field equations at critical endpoints ( $r \to \infty$ , $r = r_s$ , $r=0$ ).

Horizon Branch Analysis: The authors first examine the branch where the scalar kinetic term $X = -\nabla_\mu \phi \nabla^\mu \phi / 2$ is finite and non-zero at the horizon ( $X_s \neq 0$ ). They expand the metric and scalar fields near the horizon radius $r_s$ and evaluate the propagation speeds of odd-parity (gravitational) and even-parity (scalar) perturbations.
Analytic Branch Reduction: For the branch where $X(rs) = 0$, the authors assume the coupling functions $G_i(\phi, X)$ are analytic at $X=0$ . They demonstrate that at asymptotic infinity, the horizon, and the center, the scalar equation of motion is determined by a finite set of Taylor coefficients (a "finite jet") of the coupling functions. This allows for a non-perturbative reduction of the scalar equation to a current conservation law.
Non-Analytic Completion: The authors investigate whether non-analytic dependencies on $X$ (specifically fractional powers, square roots, and logarithms) can evade the analytic exclusion. They impose the requirement of covariant regularity, demanding that logarithmic and inverse- $X$ singularities in the equations of motion cancel out.

Key Results

Obstruction of the $X_s \neq 0$ Branch: The branch with a finite, non-zero kinetic term at the horizon is generically obstructed. Unless specific degeneracies occur (simultaneous vanishing of $G_{5,X}$ , $\kappa_r$ , and $\kappa$ ), this branch suffers from divergent propagation speeds or ghost/gradient instabilities. Even if degeneracies are tuned, the pathology typically reappears at higher orders, and known non-analytic examples (like 4D Einstein-Gauss-Bonnet) are ruled out by strong coupling or Laplacian instabilities.
Non-Perturbative No-Hair Theorem (Shift-Symmetric): On the regular branch where $X(rs) = 0$, for non-degenerate shift-symmetric analytic theories, the authors derive a non-perturbative no-hair theorem. The radial current $J_r$ must vanish to satisfy regularity at both the horizon and the center. This forces $\phi' = 0$ throughout the spacetime. Consequently, the metric reduces to the Schwarzschild solution. For any non-zero ADM mass, this implies a central singularity; thus, no regular BH exists in this class.
Perturbative Exclusion (Non-Shift-Symmetric): For non-shift-symmetric theories with positive-power couplings (analytic in $X$ ), the authors show that on the perturbative branch continuously connected to General Relativity (small coupling limit), the solution is again forced to be Schwarzschild. The horizon regularity and asymptotic flatness fix the integration constants to the Schwarzschild values, leading to a central singularity.
Marginal Non-Analytic Completion: The authors identify that the only marginal non-analytic completion capable of evading the analytic current reduction is the scalar-Gauss-Bonnet (sGB) chain, specifically involving a quintic logarithmic term in $G_5$ (e.g., $G_5 \sim \ln|X|$ ). While this specific structure allows for hairy black hole solutions that evade the current reduction step, the authors find that these solutions remain centrally singular.

Significance and Claims
The paper claims to close significant gaps in the static-scalar sector of Horndeski theories regarding regular black holes. Its primary contributions are:

Establishing that the generic $X_s \neq 0$ horizon branch is unstable or ill-posed.
Proving a non-perturbative no-hair theorem for analytic, shift-symmetric theories on the $X(rs)=0$ branch, demonstrating that regularity at the center is incompatible with a non-zero mass.
Extending the exclusion to perturbative non-shift-symmetric branches connected to General Relativity.
Identifying the scalar-Gauss-Bonnet chain as the unique marginal non-analytic completion that satisfies covariant regularity, while noting that even this completion fails to produce a regular black hole, as the resulting hairy solutions retain a central singularity.

The work concludes that within the static, spherically symmetric, asymptotically flat sector of Horndeski theories, there are no known mechanisms to construct a black hole that is simultaneously stable, has a regular horizon, and possesses a regular center, unless one moves beyond the specific analytic or marginally non-analytic classes analyzed here.