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REVIEW 2 major objections 5 minor 105 references

Stealth black holes in Aether Scalar Tensor theory

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Pith's one-line read Solving AeST's most general static, spherically symmetric vacuum equations, this paper finds two classes of stealth black holes with exact Reissner-Nordstrom geometry and secondary hair, one joinable to cosmology.

desk verdict New stealth Reissner-Nordstrom black holes in AeST are genuinely derived and worth publishing, but the 'most general' classification rests on an unproven restriction of the scalar ansatz to q=0,1. read the letter →

arxiv 2412.15395 v2 pith:MK3SO4BT submitted 2024-12-19 gr-qc astro-ph.GAastro-ph.HE

Constantinos Skordis , David M. J. Vokrouhlicky This is my paper · ORCID
classification gr-qcastro-ph.GAastro-ph.HE MSC 83C5783D0583C15 PACS 04.70.-s04.50.Kd
keywords AetherScalarTensortheorystealthblackholessecondaryhairReissner-NordstrommetricstaticsphericallysymmetricsolutionsmodifiedNewtoniandynamicsshiftsymmetrywormhole
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to close a gap in the Aether Scalar Tensor (AeST) theory: before the theory can stand as a dark-matter-free explanation of galaxy and cosmological observations, it needs consistent strong-field objects that look like the black holes we observe. The authors solve the most general static, spherically symmetric vacuum equations of AeST in its strong-field regime and find two classes of stealth black holes, both with exact Reissner-Nordstrom geometry and with non-trivial secondary hair. The $q=1$ class has a timelike scalar gradient and can in principle be smoothly joined to the cosmological solution, making it a concrete black-hole candidate for the theory. The $q=0$ class carries non-zero shift charge and a spacelike gradient, so it is not a cosmological-join candidate but completes the classification. A separate $A=0$ branch gives non-black-hole wormhole-type solutions with no horizon.

What carries the argument

The load-bearing ansatz is the static shift-symmetric scalar $\varphi=Q_0(qt+R(r))$ with $q$ restricted to 0 or 1, together with the unit-timelike vector field written in terms of one function $A(r)$ and $\chi=\pm\sqrt{1+A^2e^{-2\Psi}}$. Integrating the shift-symmetry Noether current yields a conserved charge $\varphi_0$, and the constraint $q\varphi_0=0$ splits the analysis into the two families. The central reduction is the combination of the gravitational field equations that, when the $\mu^2$ term is negligible, enforces $\Psi=-\Phi$ and leaves a single ODE for $\Phi$ whose unique solution is $e^{2\Phi}=1-2G_N M/r+q_{\rm BH}^2/r^2$. Once the metric is fixed, the remaining equations determine $E=\chi'+\chi\Phi'$ and the hair fields algebraically, and the asymptotic frame freedom fixes the integration constants. The discrete symmetry $A_\mu\to-A_\mu$ then organizes the two sign branches that appear in the solutions.

What would settle it

Take the field equations (2.14)-(2.23) with $\varphi=Q_0(qt+R)$ and $A\neq0$ for $q=-1$ (or any other real $q$ outside $\{0,1\}$) and look for a regular static, spherically symmetric vacuum solution; one such branch would break the claimed two-class completeness, while a proof that all such $q$ are inconsistent would confirm it.

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Extended reading notes

Core claim

Within AeST's strong-field regime, meaning scales well below the MOND radius where the free function takes the form $F=(2-K_B)\lambda_s Y-2K_2(Q-Q_0)^2$, the paper shows that every static, spherically symmetric vacuum solution in the $q=1$ or $q=0$ branch has the Reissner-Nordstrom metric $e^{2\Phi}=1-2G_N M/r+q_{\rm BH}^2/r^2$. The hair is secondary: the vector charge $q_A$ and scalar charge $\varphi_0$ obey fixed relations to $M$ and $q_{\rm BH}$, for example $q_A^2=\tilde{m}^2\varphi_0^2/Q_0^2+q_{\rm BH}^2/\tilde{n}$ in the $q=0$ branch, so they do not add independent charges. The $q=1$ branch has a timelike scalar gradient and zero shift charge, and in the Schwarzschild limit $Q=Q_0$ exactly, which is why it can in principle connect to the AeST cosmological scalar; the $q=0$ branch has a spacelike gradient and non-zero shift charge, so it is not a cosmology-join candidate. The algebraically special $A=0$ family yields horizonless wormhole-type solutions with a minimum radius $r_0$; some branches are regular and others contain naked singularities. The hair fields are regular at the black-hole horizons once the coordinates are changed to regular null coordinates.

Load-bearing premise

The classification's claim to completeness rests on restricting the scalar ansatz to $q\in\{0,1\}$; the paper mentions $q=-1$ but never analyzes it and gives no proof that other real values of $q$ are excluded by staticity, shift symmetry, or boundary conditions.

Editorial extensions

If this is right

  • The $q=1$ branch gives AeST black holes that are geometrically indistinguishable from general relativity's Reissner-Nordstrom solutions, so any observational discrimination must come from the hair's imprint on perturbations, quasinormal modes, or thermodynamics.
  • The $q=1$ Schwarzschild subclass has $Q=Q_0$ exactly and can in principle be the strong-field endpoint of the same scalar gradient that drives AeST cosmology; the authors conjecture that a full cosmological embedding is regular but leave the explicit check to future work.
  • For one $q=1$ sign branch ($\epsilon_A=-1$) the requirement that the scalar $Q$ not vanish outside the horizon imposes a lower bound on the black hole mass, equation (3.14), suggesting that such objects may only form above a minimum mass.
  • The $q=0$ branch carries non-zero shift charge and a spacelike scalar gradient, so it is unlikely to join onto the cosmological solution; it nevertheless provides a complete second family that stability analysis must treat separately.
  • The $A=0$ algebraically special family yields horizonless compact objects with a minimum radius $r_0$; some branches reproduce the spherical wormhole solutions of the preceding vector-tensor theory, while other branches contain naked singularities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $q=-1$ case, or any real $q$ outside $\{0,1\}$, turned out to admit a regular solution, the claimed two-class completeness would fail; checking that case is the most direct test of the classification.
  • Because the hair is secondary and tied algebraically to $M$ and $q_{\rm BH}$, the $q=1$ family may predict parameter-free relations between gravitational-wave ringdown frequencies and the hair charge; computing the perturbation equations would make such predictions testable.
  • Embedding the $q=1$ black hole into the actual FLRW AeST background, rather than an asymptotically flat patch, could reveal whether the scalar gradient can pass through both the black-hole and cosmological horizons without acausal features; the paper conjectures this works, but the calculation remains open.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper analyzes static spherically symmetric vacuum solutions in the Aether Scalar Tensor (AeST) theory in the strong-field regime, where the free function F is taken in the quadratic form (2.2) and MOND corrections are neglected. Using the ansatz phi = Q0(q t + R) with q restricted to 0 or 1 and a radial vector component A(r), the authors reduce the field equations (2.14), (2.16), (2.20), (2.21), (2.22), (2.23) to a tractable system and solve it. They find two families of stealth black holes with Reissner-Nordström metrics and nontrivial secondary hair (q = 1 and q = 0), plus algebraically special solutions with A = 0 that include a Schwarzschild-type solution and wormhole-like branches. The paper claims these are the most general static spherically symmetric vacuum solutions in this regime and that the q = 1 branch can be continuously joined to the cosmological AeST solution.

Significance. If the classification is complete, the paper makes a strong contribution by providing explicit, exact stealth black hole solutions in a modified-gravity theory designed to reproduce MOND phenomenology and cosmological observations. The derivations are algebraic rather than numerical, with useful internal consistency checks: regularity of Q and of the Noether current at the horizon, coordinate regularizations in Eddington-Finkelstein and Lemaitre-Novikov forms, and explicit constraints on parameters for avoiding pathologies. The solutions are in principle falsifiable through strong-field tests. However, the significance is moderated by the unproven restriction on the scalar ansatz parameter q, on which the 'most general' claim rests, and by the conjectural status of the cosmological matching advertised in the abstract.

major comments (2)
  1. [2.2, Eq. (2.12)] The claim that the solutions are the 'most general' static spherically symmetric vacuum solutions rests on the assertion in Sec. 2.2 that the constant q in the ansatz phi = Q0(q t + R) takes only the values q = 0 or q = 1. No derivation of this restriction is given. The text promises a discussion of q = -1, but no such discussion appears anywhere in the paper. The field equations (2.16)-(2.20) do not force q to be an integer or to lie in {0,1}; in particular, the constraint (2.20) only forces phi0 = 0 for any q != 0. Unless the authors prove that all other real q either make the field equations inconsistent or are gauge-equivalent to the q = 0 or q = 1 cases, the abstract's 'most general' claim and the enumeration in Table 1 are not established. This is a load-bearing gap because additional allowed q values would correspond to additional solution branches not covered by the paper.
  2. [Abstract; Sec. 5 and 6] The abstract states that one of the solution classes 'can be continuously joined to the cosmological solution of AeST.' The body of the paper does not demonstrate such a join. Section 5 notes that the solutions are asymptotically flat and that 'more checking is necessary' for a continuous extension to cosmology, and the authors only 'conjecture' that this poses no problem. Section 6 repeats that the connection can be made 'in principle.' As written, the abstract overstates what has been shown. Either the matching to an FLRW or asymptotically de Sitter solution should be performed (or at least explicitly constructed at the level of the asymptotic matching), or the abstract should be softened to say the solution 'has properties expected to allow' such a join.
minor comments (5)
  1. [2.2, after Eq. (2.12)] The sentence 'We briefly discuss the possibility q = -1 below' is never followed up; either provide the discussion or remove the promise.
  2. [2.1, Eq. (2.2)] There is a typo: 'withF having the expansion' should read 'with F having the expansion'.
  3. [5, first paragraph] 'the fullJ (Y)' should be 'the full J(Y)'.
  4. [References [33] and [37]] The author name 'Z/suppress lo´ snik' contains a LaTeX artifact and should be corrected to 'Zlosnik'.
  5. [Table 1, q=1 row] The entry for R' contains qA in the numerator of the first term, while Eq. (3.12) has |qA|; please clarify the notation or correct the table if these differ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the solutions are derived from the AeST action and field equations; the q∈{0,1} ansatz restriction is a completeness gap, not a circular input.

full rationale

The derivation is self-contained: starting from the AeST action (2.1) and the strong-field form F = (2−KB)λsY − 2K2(Q−Q0)^2 (2.2), the authors vary to obtain the scalar, vector, and Einstein equations (2.5)–(2.9), impose the static spherically symmetric ansatz (2.10)–(2.12), and then solve the reduced system (2.14)–(2.23). The q=1 and q=0 branches are obtained by integration; for example, Eq. (3.4) yields the RN metric and Eq. (3.3) then fixes E and the fields. No observed quantity is fitted and then renamed a prediction; the constants M, qA, and φ0 are integration constants, not parameters tuned to the target solutions. The citations to [64] and [65] define the theory and its stability ranges; these are prior independent works, not a self-citation chain forcing the present result. The one genuine concern is a completeness gap rather than circularity: Eq. (2.12) asserts that q is an integer taking only q=0 or q=1 and promises a discussion of q=−1 that does not appear in the paper, so the claimed 'most general' classification is not proven for other real q values. That is a potential overclaim about exhaustiveness, but it does not make any derived solution identical to an input by construction. Therefore the paper receives a circularity score of 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data in this paper; M, qBH, qA, and phi0 are integration constants of the vacuum equations, and KB, lambda_s, K2, Q0, mu, n-tilde, and m-tilde are theory parameters inherited from the AeST action and prior stability analysis. The central solutions rest on the six assumptions listed. No new fields, particles, or conserved quantities are invented; the scalar and vector fields are the theory's existing degrees of freedom.

assumptions (6)
  • domain assumption Strong-field truncation of the free function: F(Y,Q) = (2 - KB) lambda_s Y - 2 K2 (Q - Q0)^2 + ... with higher-order and MOND terms neglected.
    Invoked in Sec. 2.1 (Eq. 2.2) to close the system and reproduce dust-like cosmological scaling; all subsequent equations use this truncated form.
  • domain assumption Scale separation mu r << 1, so the right-hand side of Eq. (3.1) is negligible and Psi = -Phi.
    Used in Sec. 3 (Eqs. 3.1 and 3.2) to reduce the metric to the RN form; restricts validity to radii well below the MOND radius.
  • ad hoc to paper Static scalar ansatz phi = Q0(q t + R) with q taking only values 0 and 1.
    Introduced in Sec. 2.2 (Eq. 2.12) to make the scalar field time-independent in the action; exhaustiveness for "most general" is asserted, not proven, since q = -1 is mentioned but not analyzed.
  • domain assumption Stability parameter ranges 0 < KB < 2, K2 > 0, lambda_s >= 0 from [65].
    These ranges define the regime of interest; imported from the prior linear stability analysis of the theory.
  • domain assumption Asymptotic Lorentz frame freedom lets one set chi0 = +/-1 by boosting to the common frame of A_mu and grad_mu phi at infinity.
    Used in Sec. 3.1 and Appendix A to fix an integration constant and remove a redundant parameter.
  • domain assumption Physical branches are selected by requiring scalar quantities such as Q and S_mu S^mu to be regular at the horizon and non-vanishing at finite r.
    Used in Secs. 3 and 5 to derive mass bounds (3.14) and (3.26) and to exclude some branches.

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Cite this review

Pith. "Pith review of Stealth black holes in Aether Scalar Tensor theory." pith.science (2026). https://pith.science/paper/MK3SO4BT

@misc{pith2026241215395,
  author       = {Pith},
  title        = {Pith review of: Stealth black holes in Aether Scalar Tensor theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MK3SO4BT}},
  note         = {Machine review of arXiv:2412.15395}
}
read the original abstract

The Aether Scalar Tensor (AeST) theory is an extension of general relativity(GR) successful at reproducing galactic rotational curves, gravitational lensing, linear large scale structure and cosmic microwave background power spectrum observations. We solve the most general static spherically symmetric vacuum equations in the strong-field regime of AeST and find two classes of stealth black hole solutions -- those with exact GR geometries -- containing non-trivial secondary hair. In particular, one of these can be continuously joined to the cosmological solution of AeST. We also derive a non-black hole solution with zero spatial component in the vector field. This result proves the existence of mathematically and observationally consistent candidates for black holes in AeST, and creates a basis for testing the theory in the strong-field regime.

Discussion (0). Continue with ORCID to comment.

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