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REVIEW 3 major objections 5 minor 106 references

Perturbative black-hole and horizon solutions in gravity with explicit spacetime-symmetry breaking

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Pith's one-line read This paper derives four static, spherically symmetric black-hole metric families to linear order in explicit spacetime-symmetry-breaking coefficients, and shows that combining two nonzero coefficients produces a three-horizon spacetime.

desk verdict New explicit first-order black hole metrics for a two-tensor SME coefficient in explicit-breaking gravity, but the central consistency claim with the full field equations is asserted, not shown, and the perturbative expansion is not uniform at the horizon. read the letter →

arxiv 2411.18255 v2 pith:TNXTEG3P submitted 2024-11-27 gr-qc

Samy Aoulad Lafkih , Nils A. Nilsson , Marie-Christine Angonin , Christophe Le Poncin-Lafitte This is my paper · ORCID
classification gr-qc
keywords explicitspacetime-symmetrybreakingblackholesolutionseffectivefieldtheorySchwarzschildperturbationreduced-actionmethodhorizonstructureperiastronprecessionphotongeodesics
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 show that vacuum, static, spherically symmetric black holes can exist when gravity's diffeomorphism symmetry is broken explicitly by a fixed background two-tensor $s^{\mu\nu}$ rather than by a dynamical field. Working to first order in the symmetry-breaking coefficients around a Schwarzschild seed, the authors obtain four metric families — one for $s^{00}\neq 0$, one for its trace subset, one for $s^{11}\neq 0$, and one for both — and argue that these metrics solve the modified Einstein equations and the relevant Bianchi constraint at the same order. If true, the result matters because it turns a class of explicit symmetry-breaking effective field theories into concrete spacetimes whose horizons, thermodynamics, photon rings, and orbital precession can be compared with observations. The most striking output is that turning on both coefficients produces a solution with three horizons, resembling a Reissner-Nordström-de Sitter structure but with all horizons finite.

What carries the argument

The engine is the reduced-action method (Palais symmetric criticality): insert the static spherically symmetric ansatz $ds^2=-N^2(r)dt^2+f^{-1}(r)dr^2+r^2d\Omega^2$ with a diagonal constant $s^{\mu\nu}$ into the Lagrangian $S=\frac{1}{2\kappa}\int d^4x\sqrt{-g}(R+s_{\mu\nu}R^{\mu\nu})$, vary the reduced one-dimensional action with respect to $N$ and $f$, and solve order by order around the Schwarzschild seed $f_0=N_0^2=1+k/r$. Spherical symmetry forces $s^{22}=s^{33}=0$, leaving $s^{00}$ and $s^{11}$ as the switches that generate the four families. The method relies on the perturbative identity $N^2=f+O(s^2)$ at first order, and on an exact Lambert-$W$ solution $f(r)=1+2W(C/r)+W(C/r)^2$ over $s_{00}$ which contains the perturbative Case 1 metric as its linear subset.

What would settle it

Substitute the explicit metrics (17), (24), (26), and (27) into the full modified Einstein equations (4) and the Bianchi identity (5), keeping all terms linear in the coefficients and using $\partial_\mu s^{\alpha\beta}=0$; if any component of (5) fails to vanish at first order, the solutions are not consistent vacuum solutions of this theory. A concrete first check is to evaluate $\nabla_\nu(s^{\mu\nu}R^{\beta}{}_{\mu})$ for $\beta=t$ and $\beta=r$ on the Case 1 metric near $r=-k$ and see whether the constraint is identically satisfied.

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Extended reading notes

Core claim

The paper claims that, in an effective field theory where particle diffeomorphisms are broken explicitly by a fixed, constant, diagonal background tensor $s^{\mu\nu}$ coupled to the Ricci tensor, static and spherically symmetric vacuum solutions exist as first-order perturbations around Schwarzschild. With only $s^{00}\neq 0$, the metric functions are $N(r)=\sqrt{1+k/r}-\frac{s_{00}k^2}{8r^2}(1+k/r)^{-1/2}$ and $f(r)=1+k/r-\frac{s_{00}k^2}{4r^2}$ (Eq. 17); the trace-subset version (Eq. 24), the $s^{11}$ case (Eq. 26), and the combined case (Eq. 27) follow the same order-by-order scheme. The paper reports that these linearized metrics satisfy the modified Einstein equations and the traced Bianchi identities at the same order, that Case 1 has an inner and an outer horizon similar to Reissner-Nordström, and that Case 3, with both coefficients nonzero, develops three horizons. It then computes the consequences a reader would test: horizon radii, Hawking temperatures and entropies obtained from the first law, curvature scalars, radial photon geodesics, massive-particle effective potentials, periastron precession, and the unstable light-ring radius, which shifts to $-\frac{3k}{2}(1+s_{00}/27)$. The paper therefore supplies a family of perturbative, observationally addressable spacetimes for explicit diffeomorphism breaking.

Load-bearing premise

The load-bearing premise is that the reduced-action, order-by-order solutions with a fixed constant diagonal background tensor satisfy the full modified Einstein equations and the full Bianchi identity (5) at first order — not merely the traced version — so that the metrics are genuine solutions of the explicitly broken theory.

Editorial extensions

If this is right

  • The Case 1 metric predicts horizons at $r_{\star,\pm}=-\frac{k}{2}(1\pm\sqrt{1+s_{00}})$, with the outer horizon close to Schwarzschild and the inner horizon near the singularity.
  • With both $s^{00}$ and $s^{11}$ nonzero, the horizon condition is cubic and admits three real roots for $s^{11}<0$, so a single static spherically symmetric spacetime can carry three horizons.
  • Periastron precession acquires a symmetry-breaking term $\Delta\phi\approx -3\pi k/p + s_{00}\pi k^2/(4L_0^2 p)$; matching the S2 orbit data yields $s_{00}=0.62^{+1.14}_{-1.16}$ at 1$\sigma$, a weak but direct constraint.
  • Case 2 and Case 3 are not asymptotically Minkowski: $f\to 1-s_{11}$ as $r\to\infty$, leaving an angular-deficit signature in radial photon geodesics with $ct/r\to 1+s_{11}/2$.
  • The light ring remains unstable and its radius shifts to $r_{\text{lr}}=-\frac{3k}{2}(1+s_{00}/27)$ relative to Schwarzschild.

Reading between the lines

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

  • If the three-horizon root $r_{\star,3}=k s_{00}/4$ is physical rather than a linearization artifact, multi-horizon structure becomes a generic signature of multi-component explicit symmetry breaking; the paper's own discontinuity analysis indicates that quadratic terms are needed to settle the question within the same EFT.
  • A natural extension the paper leaves open is the black-hole shadow: because the light-ring radius shifts at first order in $s_{00}$, shadow-size measurements could yield competitive constraints once the photon orbits are integrated to asymptotic infinity.
  • The S2-precession bound on $s_{00}$ is far weaker than the gravitational-wave speed bound on the same coefficient, suggesting that the coefficient is better probed by strong-field geometry than by weak-field orbits; computing ringdown frequencies for these metrics would be a sharper test.
  • If the full Bianchi identity (5) is enforced beyond the traced version, the constant-diagonal ansatz may impose additional conditions that select among the four cases; a direct check would either confirm the paper's consistency claim or point to spontaneous breaking as the viable completion.
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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

3 major / 5 minor

Summary. The paper studies a gravitational effective field theory with explicit diffeomorphism breaking by a fixed background two-tensor s^mu nu coupled to the Ricci tensor, plus a scalar trace subset. Using the reduced-action method with a Schwarzschild seed, the authors derive static, spherically symmetric vacuum solutions to linear order in the symmetry-breaking coefficients. They present solutions for a single nonzero component s00 (Case 1), a trace model, a single s11 (Case 2), and the combined s00 and s11 case (Case 3), and they analyze horizons, Hawking temperature, entropy, curvature singularities, photon radial geodesics, timelike orbits, periastron precession (including an application to the S2 star), and the light ring. The central claim is that the metrics in Eqs. (17), (24), (26), and (27) satisfy the modified Einstein equations (4) and the traced Bianchi identity (5) at linear order in the symmetry-breaking coefficients.

Significance. If the central claim is correct, the paper supplies new, explicit, and testable black-hole spacetimes for a class of explicit diffeomorphism-breaking EFTs, with concrete observables such as modified periastron precession, horizon structure, and light-ring radii. The authors include many explicit computations, a numerical integration of the reduced equations, an exact Lambert-function solution, and coordinate transformations for Killing horizons in the Appendix, which are valuable for follow-up work. However, the main consistency statement is asserted rather than demonstrated, and the perturbative expansion appears not to be uniformly valid at the horizon, so the significance of the results depends on whether these gaps can be closed in a revision.

major comments (3)
  1. [Section III, Eqs. (12), (17), (19)] The claim that the metric functions (17), (24), (26), and (27) satisfy the full covariant field equations (4) and the Bianchi identity (5) is not demonstrated. The authors solve only the reduced Euler-Lagrange equations (12) and (23) for the two functions N(r) and f(r), with g_{theta theta}=r^2 fixed in the ansatz (9). In a theory with a fixed background tensor s^{mu nu} that is constant in these coordinates, radial reparametrizations are not symmetries, so the theta-theta component of (4) is an independent constraint that the reduced-action method with this ansatz does not enforce. Moreover, the exact N^2=f solution (19) of the reduced equations is explicitly admitted not to satisfy the non-linearised Einstein equations (text after Eq. (19)); this shows that a stationary point of the reduced action is not automatically a solution of the full action. I request an explicit component-by-component verification of (4) and the full (non-traced) Bianchi identity for each of the four metrics at O(s), or a clear argument that Palais symmetric criticality applies to this non-covariant theory with the fixed background tensor and fixes the theta-theta equation.
  2. [Section IV, Eqs. (16), (29)-(32)] The linearized solution (17) is not uniformly valid near the Schwarzschild horizon. The first-order correction N1 in Eq. (16) contains the factor (1+k/r)^{-1/2}, which diverges as r approaches -k, so the ratio N1/N0 grows without bound in that region. The horizon radii computed in Eqs. (29)-(30) and the Hawking temperatures in Eqs. (32)-(33) are evaluated at r_star approximately -k, precisely where the perturbative expansion breaks down. The numerical comparison in Figure 2 integrates the reduced equations (12) rather than the full covariant equations, so it does not justify the near-horizon results. The authors should either establish the validity of the linearized approximation near r=-k with a more careful expansion, or restrict the claims about horizons and thermodynamics to regions where the expansion is controlled.
  3. [Section II.A and III, Eqs. (6)-(8), (24)] The trace-subset model is distinguished from the model of ref. [73] by which components of s^{mu nu} are held fixed, as the authors note. Nevertheless, the paper does not apply the no-go results for explicit diffeomorphism breaking reported in [73] to the ansatz (9). These no-go constraints are directly relevant to whether consistent solutions with explicit breaking can exist, and the trace solution (24) should be checked against the full equations (7) and (8), not only the reduced Euler-Lagrange equations. A direct substitution of (24) into the full covariant equations would resolve whether this solution is physically admissible.
minor comments (5)
  1. [Section VII, Eq. (87) and Discussion] The statement that the S2 data 'constrain' s00 to -0.54 < s00 < 1.76 is overstated, because the 1-sigma posterior in Eq. (87) is s00 = 0.62^{+1.14}_{-1.16}, which includes zero and is consistent with no symmetry breaking. The wording should be softened to an 'estimated' value with large uncertainty.
  2. [Section V, Eq. (51)] In the photon geodesic section, the metric is written as ds^2 = -c^2 N^2 dt^2 + (1/f) dr^2, but the angular part is omitted; for clarity, the full metric (9) should be stated, or the section should explicitly restrict to radial motion.
  3. [Section IV, Eq. (31)] The Hawking temperature formula (31) is written as a square root of a product of derivatives; the notation should clarify that the expression is evaluated at the outer horizon and that absolute values are taken as in Eq. (32), to avoid ambiguity.
  4. [Throughout] The spelling of 'Kretschmann' is inconsistent (it appears as 'Kretchmann' in several places, including the captions of Figures 3 and 5); please correct this.
  5. [Section III, after Eq. (25)] The condition (25) for the existence of a solution (f', N') to the system (12) is stated without derivation; adding a brief derivation or a reference would help the reader assess the numerical integration.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the metrics are obtained by solving the stated reduced-action equations, and the self-citations are non-load-bearing.

full rationale

The derivation chain is self-contained: the central results (Eqs. (17), (24), (26), (27)) are obtained by substituting the stated metric ansatz (9) into the action (3)/(6), deriving the reduced Euler-Lagrange equations (12)/(23), expanding around the Schwarzschild seed, and solving for the first-order corrections in the EFT coefficients. The coefficients s00 and s11 are free inputs, not fitted parameters used to produce the metrics. The later S2 precession analysis fits s00 to GRAVITY data via Eqs. (83)-(87), which is parameter estimation from an externally measured precession, not a prediction constructed from the metric, and it does not feed back into the derivation of the solutions. Self-citations are present ([56] and [73]) but are not load-bearing: [73] is invoked to note a vDVZ-type discontinuity and to distinguish the trace model as a 'different model', while the actual boundary condition c2 -> 0 is independently motivated by the smooth-GR-limit requirement; [56] is cited only as an FLRW consistency example and is not used in the black-hole derivation. The manuscript itself flags a limitation after Eq. (19), stating that the exact reduced solution with N^2 = f 'does not respect the (non-linearised) Einstein equations', and the verification mentions only the traced Bianchi identity (5), not the full set of Bianchi constraints. These are correctness/consistency risks, not circular reductions: no equation is equivalent by construction to its input, and no fitted parameter is renamed as a prediction. The central derivation therefore does not reduce to its own assumptions or to a self-citation chain.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central construction depends on the EFT action (3), the constant diagonal ansatz for s^{mu nu}, the validity of the reduced-action method in a non-diffeomorphism-invariant theory, and the linear-order truncation. No new particles or forces are introduced. The entropy result also depends on the assumed first law and an undetermined integration constant.

free parameters (5)
  • s00 = 0.62+1.14/-1.16 (1 sigma) from S2 precession
    EFT coefficient that sources the Case 1, trace, and Case 3 solutions; constrained in Section VII from GRAVITY S2 data.
  • s11
    EFT coefficient for Case 2 and Case 3; left free in the paper, with some branches requiring s11 < 0 for real horizons.
  • k = -2M for Schwarzschild comparison
    Mass parameter in the Schwarzschild seed solution; kept generic and then set to -2M. It is a label of the solution family, not fitted to data.
  • Integration constants c1, c2 = 0
    Chosen by hand in Section III to ensure a smooth GR limit and remove a scaling degeneracy.
  • S0
    Undetermined integration constant in the entropy, to be fixed by physical considerations; the entropy is therefore not a unique prediction.
assumptions (5)
  • domain assumption Palais symmetric criticality and the Fels-Torre principle apply to the explicitly broken action, so reduced-action solutions solve the covariant field equations.
    Invoked in Section III to justify the reduced-action method; the paper notes 'some caution' and possible boundary terms, but no explicit check is shown.
  • domain assumption s^{mu nu} is a fixed, constant, diagonal background tensor in the chosen coordinate system.
    The paper restricts to this ansatz in Section III; it is a modeling choice, not derived from the theory.
  • domain assumption The modified Einstein equations (4) and the traced Bianchi identities (5) are the correct field equations for the action (3).
    Taken from the SME gravity EFT literature [36,37,54] and assumed throughout.
  • ad hoc to paper The linear-order perturbative expansion around Schwarzschild is a reliable approximation, including near the horizon.
    The paper computes horizons and singularities at linear order; the non-uniform validity of the expansion at the horizon is not addressed and is contradicted by the N(r*) inconsistency.
  • domain assumption The first law dk = TH dS and the Hawking temperature formula (31) apply to these solutions.
    Used in Section IV to derive entropy; the paper notes the entropy constant S0 is undetermined.

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Pith. "Pith review of Perturbative black-hole and horizon solutions in gravity with explicit spacetime-symmetry breaking." pith.science (2026). https://pith.science/paper/TNXTEG3P

@misc{pith2026241118255,
  author       = {Pith},
  title        = {Pith review of: Perturbative black-hole and horizon solutions in gravity with explicit spacetime-symmetry breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TNXTEG3P}},
  note         = {Machine review of arXiv:2411.18255}
}
abstract

In this paper, we present static and spherically symmetric vacuum solutions to the mass-dimension $d\leq 4$ action of an effective-field theory, choosing the diffeomorphism symmetry to be broken explicitly. By using the reduced-action method with a Schwarzschild seed-solution, we find static and spherically symmetric black hole solutions to the field equations to linear order in the symmetry-breaking coefficients, which are consistent solutions to the modified Einstein equations at the same order. Using several ans\"atze for the symmetry-breaking coefficient we classify the allowed solutions, and we compute standard consequences and observables, including horizons, thermodynamics, photon geodesics, and perihelion precession. We find that the horizon structure of some of our solutions are similar to the Reissner-Nordstr\"om case, and that several of them exhibit physical singularities at $r=2M$. We note in particular that introducing more than one non-zero coefficient for spacetime-symmetry breaking coefficient leads to a solution with three horizons; the aim is to obtain observables that can be confronted to black holes observational data.

Figures

Figures reproduced from arXiv: 2411.18255 by the authors.

Figure 1
Figure 1. shows the difference between this numerical integration and the Schwarzschild solution, for an exag￾gerated value of s 00 = ±0.1. We remark that when r reaches a few hundred Schwarzschild radii, the metric functions become very close to the GR limit. Even with an exaggeratedly high value of s 00 like 0.1, the different cases diverge only slightly, even close to the event hori￾zon. We also compare the analytical firs… view at source ↗
Figure 2
Figure 2. FIG. 2: Comparison between integrated metric (solid [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Hawking temperature (left) and Kretchmann [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The Kretschmann scalar for the trace subset of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The Hawking temperature for Case 2 as a [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Ricci scalar (left) and Kretchmann scalar [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Hawking temperature (left) and Kretchmann [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Effective potential for massive particles. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Estimated posterior distribution function for [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

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    Trace subset solution Plugging in the same metric ansatz as in the previous sections into the Lagrangian (6) and setting the matter sector to zero, we arrive at the following reduced action Strace = 1 2κ Z dr 1√f h 2 s00N 2 − 1 (2N (f + rf ′ − 1) + r(rf ′N ′ + 2f (2N ′ + rN ′′))) i , (22) from which we find the Euler-Lagrange equations for N (r) and f (r)...

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