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arxiv: 2603.06786 · v3 · pith:ENQXHJVMnew · submitted 2026-03-06 · ✦ hep-th · gr-qc

All 2D generalised dilaton theories from dgeq 4 gravities

Johanna Borissova This is my paper

Pith reviewed 2026-05-15 14:42 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords Horndeski gravitydimensional reductionwarped productBirkhoff theoremquasi-topological gravitydilaton gravityblack holeshigher-dimensional gravity
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The pith

Reductions of pure gravities in d ≥ 4 dimensions yield all generic two-dimensional Horndeski theories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that any two-dimensional Horndeski theory arises by reducing a higher-dimensional gravitational action on a 2+(d-2) warped product background. Because the reduced equations are second-order, every on-shell configuration of the two-dimensional metric and scalar field lifts directly to a vacuum solution of the original d-dimensional theory. The authors separate the case of actions built solely from curvature scalars without covariant derivatives, then treat the fully generic case that still yields second-order equations on the warped backgrounds. A Birkhoff theorem follows for all such reductions: static spherically symmetric solutions obey g_tt g_rr = −1 in Schwarzschild gauge, with the metric function fixed by an algebraic equation. The authors therefore label the entire class quasi-topological gravities and note that the same reduction works in reverse to reconstruct higher-dimensional theories from known two-dimensional solutions.

Core claim

Generic two-dimensional Horndeski theories can be obtained from the dimensional reduction of pure gravitational theories in d ≥ 4 dimensions on 2+(d-2) warped-product backgrounds. Consequently, every on-shell two-dimensional configuration corresponds to a genuine d-dimensional vacuum solution. The result holds both for actions constructed only from curvature invariants without covariant derivatives and for generic actions whose restriction to the warped background still produces second-order equations of motion.

What carries the argument

The warped-product reduction of a d-dimensional gravitational action on a 2+(d-2) background, which produces a two-dimensional Horndeski theory whose solutions lift back to higher-dimensional vacuum solutions.

If this is right

  • Static spherically symmetric solutions satisfy g_tt g_rr = −1 in Schwarzschild gauge and the metric function is fixed algebraically.
  • All such reduced theories obey a Birkhoff theorem.
  • Any d-dimensional static spherically symmetric asymptotically flat metric obeying g_tt g_rr = −1 with invertible dependence on the ADM mass can be reconstructed as a vacuum solution.
  • Regular black holes such as the Bardeen spacetime lie outside the reach of polynomial and non-polynomial curvature-invariant quasi-topological theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The correspondence supplies a systematic route to generate new exact higher-dimensional solutions by first solving the simpler two-dimensional model.
  • Extensions beyond curvature invariants without derivatives may be required to capture all known regular black holes.
  • The algebraic relation g_tt g_rr = −1 offers a practical test for whether a given higher-dimensional metric can arise from a quasi-topological theory.

Load-bearing premise

The higher-dimensional action must yield second-order equations of motion when evaluated on 2+(d-2) warped-product backgrounds.

What would settle it

A concrete two-dimensional solution whose corresponding higher-dimensional metric fails to satisfy the vacuum equations of the original d-dimensional theory.

Figures

Figures reproduced from arXiv: 2603.06786 by Johanna Borissova.

Figure 1
Figure 1. Figure 1: FIG. 1: Landscape of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

We demonstrate that generic two-dimensional Horndeski theories can arise from the reduction of pure gravities in $d \geq 4$ dimensions, and therefore generic onshell configurations for the two-dimensional metric and scalar field correspond to genuine $d$-dimensional gravitational vacuum solutions. We discuss separately the two-dimensional Horndeski theories which can arise from the reduction of $d$-dimensional generally covariant gravitational actions built only from curvature invariants without covariant derivatives and possessing second-order equations of motion on $2 + (d-2)$ warped-product backgrounds. The discussion is subsequently extended to generic $d$-dimensional gravitational actions with this latter property. We establish a Birkhoff theorem for all gravitational theories whose reduction yields an integrable two-dimensional Horndeski theory, in which case static spherically symmetric solutions satisfy $g_{tt} g_{rr} = -1$ in Schwarzschild gauge whereby the metric function $g_{tt} = -f$ is determined by an algebraic equation. We therefore propose to refer to all such theories as quasi-topological gravities. These results can be used to show in reverse that any $d$-dimensional static spherically symmetric and asymptotically flat spacetime satisfying $g_{tt} g_{rr} = -1$ in Schwarzschild gauge with an invertible dependence of $f$ on the ADM mass can be reconstructed explicitly as a vacuum solution to a $d$-dimensional gravitational theory. We discuss examples of regular black holes such as the Bardeen spacetime, which could not be obtained from polynomial and non-polynomial quasi-topological gravities involving only curvature invariants without covariant derivatives.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that generic two-dimensional Horndeski theories arise via consistent dimensional reduction of pure gravitational actions in d ≥ 4 dimensions on 2+(d-2) warped-product backgrounds, so that on-shell 2D configurations lift to genuine higher-dimensional vacuum solutions. It treats separately the subclass obtained from curvature-invariant actions without derivatives (yielding second-order EOM on such backgrounds) and then extends to generic actions with the same property. A Birkhoff theorem is established for the integrable cases, implying g_tt g_rr = −1 in Schwarzschild gauge with the metric function fixed algebraically; these theories are termed quasi-topological gravities. The reverse construction is proposed for reconstructing certain d-dimensional static spherically symmetric asymptotically flat solutions, with Bardeen-type regular black holes cited as examples outside the polynomial/non-polynomial curvature-only subclass.

Significance. If the reduction is shown to be consistent without residual angular constraints, the result supplies a systematic embedding of 2D generalised dilaton theories into higher-dimensional gravity and a practical route to generating and classifying static solutions via the algebraic Birkhoff property. The reconstruction procedure and the explicit limitation for Bardeen-type metrics are concrete strengths that could be used to test which higher-dimensional actions reproduce given 2D solutions.

major comments (3)
  1. [reduction procedure (following abstract)] The central consistency claim—that the higher-dimensional EOM restricted to the 2+(d-2) warped-product ansatz are exactly equivalent to the 2D Horndeski EOM with no independent constraints from the (d-2)-sphere directions—rests on the assumption that the d-dimensional action produces second-order equations on these backgrounds, but the manuscript provides no explicit verification or counter-example check for a generic (non-curvature-scalar) action. This assumption is load-bearing for the statement that every on-shell 2D configuration corresponds to a genuine d-dimensional vacuum solution.
  2. [Birkhoff theorem statement] The Birkhoff theorem is stated for “all gravitational theories whose reduction yields an integrable two-dimensional Horndeski theory,” yet the precise integrability condition on the 2D action (or on the higher-dimensional parent action) is not given explicitly; without it the algebraic determination of f cannot be verified to hold for the full class claimed.
  3. [reconstruction paragraph] The reverse reconstruction—that any d-dimensional static spherically symmetric asymptotically flat metric with g_tt g_rr = −1 and invertible f(M) can be obtained as a vacuum solution of some d-dimensional theory—is asserted but not demonstrated by constructing the corresponding higher-dimensional action from an arbitrary 2D Horndeski Lagrangian; the step from 2D integrability to existence of a parent action is missing.
minor comments (2)
  1. [preliminaries] Notation for the warping function and the reduced 2D metric should be introduced once and used uniformly; the current presentation mixes coordinate choices without a dedicated table of conventions.
  2. [abstract and §1] The abstract and introduction refer to “generic” 2D Horndeski theories without stating the precise coefficient restrictions (e.g., on the G_i functions) that are actually realised by the reduction; a short explicit list or reference to the 2D Horndeski Lagrangian form would clarify the scope.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments point by point below, providing clarifications and outlining the revisions we will make to improve the presentation and rigor of the results.

read point-by-point responses

  1. Referee: [reduction procedure (following abstract)] The central consistency claim—that the higher-dimensional EOM restricted to the 2+(d-2) warped-product ansatz are exactly equivalent to the 2D Horndeski EOM with no independent constraints from the (d-2)-sphere directions—rests on the assumption that the d-dimensional action produces second-order equations on these backgrounds, but the manuscript provides no explicit verification or counter-example check for a generic (non-curvature-scalar) action. This assumption is load-bearing for the statement that every on-shell 2D configuration corresponds to a genuine d-dimensional vacuum solution.

    Authors: We appreciate this observation. The manuscript first derives the reduction explicitly for curvature-invariant actions without derivatives, verifying that the equations remain second-order and that the angular components do not impose additional constraints beyond the 2D ones. For the generic case, we extend to actions that are defined to produce second-order equations of motion on the warped-product backgrounds. To make this more explicit, we will include a brief verification step or a simple counter-example check in the revised version, demonstrating that for such actions the reduction is consistent with no residual constraints from the sphere directions. revision: yes

  2. Referee: [Birkhoff theorem statement] The Birkhoff theorem is stated for “all gravitational theories whose reduction yields an integrable two-dimensional Horndeski theory,” yet the precise integrability condition on the 2D action (or on the higher-dimensional parent action) is not given explicitly; without it the algebraic determination of f cannot be verified to hold for the full class claimed.

    Authors: The integrability condition is that the two-dimensional Horndeski theory admits a first integral that reduces the metric equation to an algebraic relation for f(r). This holds for a broad class of 2D Horndeski Lagrangians where the dependence on the curvature and scalar allows integration without differential equations for f. We will add an explicit statement of this condition in terms of the general 2D Horndeski Lagrangian coefficients to clarify the precise scope of the Birkhoff theorem. revision: yes

  3. Referee: [reconstruction paragraph] The reverse reconstruction—that any d-dimensional static spherically symmetric asymptotically flat metric with g_tt g_rr = −1 and invertible f(M) can be obtained as a vacuum solution of some d-dimensional theory—is asserted but not demonstrated by constructing the corresponding higher-dimensional action from an arbitrary 2D Horndeski Lagrangian; the step from 2D integrability to existence of a parent action is missing.

    Authors: We agree that the reconstruction is described conceptually rather than through an explicit general procedure. The argument relies on the fact that any integrable 2D Horndeski theory arising in this manner can be lifted by choosing a suitable higher-dimensional action whose reduction reproduces the 2D Lagrangian. For specific cases like the Bardeen metric, we illustrate that it fits the integrable class but requires derivative terms or non-polynomial invariants. In the revision, we will provide a schematic construction showing how to build the parent action from a given 2D integrable Horndeski theory, thereby making the existence of the d-dimensional theory more concrete. revision: partial

Circularity Check

0 steps flagged

Independent reduction from higher-D actions; no load-bearing circularity

full rationale

The derivation begins from arbitrary d-dimensional gravitational actions (initially curvature invariants, then generic) assumed to yield second-order EOM on 2+(d-2) warped-product backgrounds. The 2D Horndeski theory is obtained by explicit dimensional reduction; the Birkhoff theorem and algebraic equation for f follow directly from the reduced EOM rather than from any fit or self-definition. No step equates a prediction to its input by construction, and self-citations (if present) are not load-bearing for the central claim. The construction remains self-contained against external higher-dimensional vacuum solutions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the higher-dimensional action yields second-order equations on warped-product backgrounds and that the reduction ansatz captures the relevant vacuum solutions. No free parameters are introduced in the abstract; the algebraic equation for the metric function is derived rather than fitted.

axioms (2)
  • domain assumption Higher-dimensional gravitational actions built from curvature invariants without covariant derivatives produce second-order equations of motion when restricted to 2+(d-2) warped-product backgrounds.
    Invoked to separate the first class of theories discussed; appears in the second paragraph of the abstract.
  • domain assumption The warped-product reduction ansatz is sufficient to obtain all onshell configurations of the resulting 2D theory that correspond to genuine d-dimensional vacuum solutions.
    Required for the statement that generic 2D onshell configurations correspond to d-dimensional gravitational vacuum solutions.

pith-pipeline@v0.9.0 · 5589 in / 1695 out tokens · 25454 ms · 2026-05-15T14:42:40.564279+00:00 · methodology

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