Constraining superluminal Einstein-\AE{}ther gravity through gravitational memory

Benedetta Rosatello; Guangzi Xu; Jann Zosso; Lavinia Heisenberg

REVIEW 2 major objections 4 minor 2 cited by

Constraining superluminal Einstein-\AE{}ther gravity through gravitational memory

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper computes the first gravitational displacement memory in Einstein-Aether gravity and finds that superluminal scalar or vector aether waves generate a formally divergent tensor memory along the critical directions where their…

desk verdict A careful, honest first computation of gravitational memory in Einstein-Æther gravity with a genuinely new superluminal divergence mechanism, but the advertised exclusion of superluminal parameter space remains a conjecture whose quantitative basis is missing exactly where the effect is largest. read the letter →

arxiv 2505.09544 v2 pith:ZOXHEJAP submitted 2025-05-14 gr-qc

Lavinia Heisenberg , Benedetta Rosatello , Guangzi Xu , Jann Zosso This is my paper · ORCID

classification gr-qc MSC 83C3583D05 PACS 04.30.-w04.50.Kd

keywords Einstein-AethergravitygravitationalmemorydisplacementsuperluminalpropagationLorentzviolationIsaacsonenergy-momentumgauge-invariantperturbationswaves

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 computes the gravitational displacement memory of Einstein-Aether gravity, a theory in which a preferred-frame vector field breaks Lorentz symmetry in the gravitational sector. Using an Isaacson-style average over high-frequency waves, the authors derive a memory formula for a metric theory with a nontrivial asymptotic vector background, and they find that the memory equation diverges in specific directions whenever the scalar or vector aether waves propagate faster than the tensor waves. Because existing observations force tensor waves to travel at the speed of light, the divergent directions are unprotected: any spherically asymmetric source of superluminal aether radiation would produce an a priori unbounded memory. The paper therefore conjectures that the superluminal parameter space of Einstein-Aether gravity is excluded by the absence of such large memory signals in current gravitational-wave data, leaving only the fully luminal theory.

What carries the argument

The argument rides on three pieces: (i) a manifestly gauge-invariant second-order action for the linearized theory written in terms of only the five dynamical degrees of freedom (tensor $h^{TT}_{ij}$, vector $\Sigma_i$, scalar $\Theta$); (ii) a second-variation method that converts this action into a gauge-invariant asymptotic Isaacson energy-momentum tensor; and (iii) the velocity factor $V_\psi = 1/(1 - \beta_\psi/\beta_T \, \vec{n}'\cdot\vec{n})$ appearing in the memory Green's function. The factor $V_\psi$ diverges when the emission direction and the observer direction satisfy $\vec{n}'\cdot\vec{n} = \beta_T/\beta_\psi$, and that divergence is the physical core of the paper.

What would settle it

Compute the displacement memory at the critical direction $\vec{n}_p\cdot\vec{n} = \beta_T/\beta_S$ using a source with finite spatial extent, or a fully nonlinear numerical relativity simulation of an Einstein-Aether binary merger: if the amplitude saturates to a finite value as the source size shrinks, the claimed unbounded memory is an artifact and the exclusion conjecture fails; if it grows without bound as the source is made smaller, the conjecture is supported.

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Extended reading notes

Core claim

The central discovery is that the tensor displacement memory in Einstein-Aether gravity, sourced by the self-stress of scalar and vector aether radiation, is not always finite. When a source emits aether waves with group velocity $\beta_S > \beta_T$, there is a critical emission direction $\vec{n}_p \cdot \vec{n} = \beta_T/\beta_S$ at which the memory Green's function formally diverges, because the wave's causal cone intersects the past null cone of the memory event. In the luminal case this divergence is cancelled by the transverse-traceless projection of the source, but for superluminal sources no such cancellation occurs. The authors conjecture, from this unbounded formal amplitude together with existing constraints on luminal tensor waves, that Einstein-Aether theory is only viable if all modes propagate at the speed of light, $\beta_T = \beta_V = \beta_S = 1$.

Load-bearing premise

The conjecture depends on the assumption that the formally divergent memory amplitude computed with the localized-source approximation ($r'\ll r$) is a physical, unbounded effect rather than a sign that the approximation itself has broken down at the critical direction, where finite source size, backreaction, or nonlinearity could cut off the divergence.

Editorial extensions

If this is right

If the conjecture holds, the viable parameter space of Einstein-Aether gravity collapses to the luminal surface $\beta_T = \beta_V = \beta_S = 1$, eliminating all superluminal aether propagation.
In the luminal theory, vector polarizations disappear from the detector response while the scalar longitudinal and breathing polarizations remain, so gravitational-wave polarimetry can directly test the theory.
The absence of large memory signals in current LIGO/Virgo/KAGRA binary coalescence events becomes a direct observational check of the conjecture.
The same divergent-memory mechanism applies to vector aether waves, so the constraint does not depend on scalar radiation alone.
The critical-angle relation $\vec{n}_p\cdot\vec{n} = \beta_T/\beta_S$ gives a concrete angular signature that future memory searches could target.

Reading between the lines

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

The divergence mechanism is structurally analogous to the Cherenkov angle, suggesting that finite-size or nonlinear effects might regulate the memory amplitude to a large but finite value rather than removing it entirely, which would still leave observable signals.
The mechanism should extend beyond Einstein-Aether: any metric theory of gravity whose additional degrees of freedom propagate faster than the tensor modes is likely to exhibit the same unprotected memory directions.
A fully nonlinear numerical-relativity computation of an Einstein-Aether binary merger, with finite source sizes and metric backreaction, would settle whether the divergence is physical or an artifact of the localized-source approximation.
If superluminal aether modes existed, memory signals would be statistically enhanced toward the critical cone, producing a characteristic angular bias across multiple observed events.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is the first direct displacement-memory calculation in Einstein-Æther gravity, and it is done with care and unusual candor about where the computation stops being valid. The advertised claim—exclusion of all superluminal EÆ parameter space—is a well-motivated conjecture, not a proven result.

What is actually new: the manifestly gauge-invariant second-order action reduced to the five dynamical degrees of freedom; the second-variation derivation of the asymptotic Isaacson energy-momentum tensor; the polarization content, checked against earlier gauge-fixed literature in Appendix F; and the luminal memory formula (127) with theory-dependent coefficients. The superluminal mechanism is also new: the Green's function enhancement when the source direction satisfies n_p·n = 1/β_S, where the source cone intersects the observer's past null cone. The geometry in Figures 2 and 3 makes the mechanism clear. The paper also does the reader the service of stating its own limitations plainly in Sec. IV B 4.

Soft spots: the central exclusion conjecture rests on a formal divergence computed under the localized-source assumption r' ≪ r and a point-pulse model. At the critical direction those assumptions fail, and the authors say so. No regulated amplitude is computed for a finite-size source, and no comparison is made to actual memory searches. The divergence may be a coherent superposition over a finite source volume rather than an unbounded signal; the Cherenkov analogy the authors themselves cite points to finite-size effects taming the divergence. So the step from formal divergence to incompatibility with GW data is not established. That is the load-bearing step, not a small blemish. However, this is a conjecture the authors label as such; the derivation up to that point is solid and cross-checked. The vector-sector argument is presented by analogy rather than worked out, which is a minor gap.

The citation pattern looks appropriate: they cross-check against Foster and Saffer/Yunes/Yagi, note a typo in Ref. [74] confirmed by its authors, and import the speed constraints from the right places. The self-citations to the Isaacson framework are justified because that framework is the direct input.

Who is this for: people working on modified gravity, gravitational memory, and Lorentz violation. It deserves a serious referee. I would send it out rather than desk reject, with the expectation that the referee asks for either a regulated finite-size estimate or a softening of the abstract's exclusion language. I would also cite it if I wrote on memory in beyond-GR theories.

Referee Report

2 major / 4 minor

Summary. This paper computes gravitational displacement memory in Einstein-Æther (EÆ) gravity using an Isaacson-based, gauge-invariant framework. The authors derive a manifestly gauge-invariant second-order action (Eq. (46)), identify the propagating scalar, vector, and tensor degrees of freedom with their respective speeds (Eqs. (25), (33), (39)), determine the gravitational polarization content (Eq. (59)), and construct the asymptotic Isaacson energy-momentum tensor. The central technical result is the luminal tensor memory formula (Eq. (127)), which is cross-checked against prior literature in Appendices E and F. The paper then considers superluminal scalar and vector aether waves in the simplified context of individual energy pulses (Sec. IV B 2) and finds that the memory amplitude contains a factor V_p^S = 1/(1 - beta_S n_p·n) that diverges at the critical direction n_p·n = 1/beta_S (Eq. (114)). Based on this formal divergence, the authors conjecture that the superluminal parameter space of EÆ gravity is excluded by current gravitational wave data (Sec. IV B 4).

Significance. If the technical derivation stands, the paper makes several solid contributions: a new gauge-invariant second-order action for EÆ gravity, a complete analysis of its gravitational polarizations in gauge-invariant language, and a luminal memory formula that is explicitly cross-checked against earlier results in Refs. [16,74]. These results are reproducible from the provided equations and are valuable for future work on Lorentz-violating metric theories. The identification of a potential mechanism for large memory along critical directions is an interesting and falsifiable idea in principle. However, the paper's headline claim—the exclusion of superluminal EÆ gravity—is a conjecture that is not established by the presented computation, because the divergence occurs precisely where the authors' own approximations break down and no regulated amplitude or observational comparison is provided. The manuscript is therefore more convincing as a derivation of the memory framework and of a formal critical-direction enhancement than as a constraint on the theory's parameter space.

major comments (2)

[Sec. IV B 2, Eqs. (112)–(114) and (118), (123)] The exclusion conjecture rests on treating the divergence of V_p^S at n_p·n = 1/beta_S as a physical unbounded memory. However, the divergence is an artifact of a singular change of variables: in deriving Eq. (95), the delta-function identity δ(g(r')) = δ(r' - r0)/|g'(r0)| is invoked, but at the critical direction the coefficient of r' in g(r') vanishes, so the identity is not a valid distributional statement. For a finite-size source, the r' integral is bounded by the source extent, and the contribution along the critical ray is a coherent superposition over a finite region, not an infinite amplitude. The authors explicitly acknowledge in Sec. IV B 4 that 'strictly speaking not valid at the diverging point,' that perturbation theory may break down, and that finite-size effects are neglected. Since no regulated amplitude is computed, the step from the formal divergence in Eqs. (118) and (123) to the conclusion that the theory is 'incompatible with current gravitational wave data' is not established. A finite-source calculation, or at least an explicit regularization scheme, is required to support the conjecture.
[Sec. IV B 4] The paper moves from 'a priori unbound memory' to a statement about current observational data without a quantitative bridge. Even if the formal divergence were physical, the authors do not estimate the expected memory amplitude for realistic sources (e.g., neutron star binaries), do not compare it to LIGO/Virgo detector noise or to existing memory upper limits, and do not quantify how 'large' the effect must be to be excluded. The text says 'we do not provide a quantitative calculation of how big the memory offset will truly be' and that a full waveform knowledge would be required. In the absence of such estimates, the assertion that 'already the statement ... is, from our point of view, a notable new mechanism' is a reasonable qualitative conclusion, but the title and abstract's 'stringent exclusion of the superluminal parameter space' overstates what the computation supports. The authors should either provide a concrete observational bound or explicitly reframe the result as a conjectural mechanism that requires further work before any exclusion can be claimed.

minor comments (4)

[Sec. II B 2 vs. Appendix G, Eq. (G1b)] The main text (after Eq. (33)) assumes c14 ≠ 0, but Appendix G, Eq. (G1b) states a condition 'c14 ≠ 1.' Please reconcile the notation; if both conditions are intended, state them explicitly.
[Sec. IV B 2, Fig. 2 caption] The caption refers to a 'source regarded time'; this appears to be a typo for 'source retarded time.' Please correct.
[Sec. IV B 4, footnote 17] The text relies on scalar and vector emission from compact objects, but notes that 'the precise nature of EÆ black holes... is not yet completely settled.' This caveat weakens the claim that binary black hole mergers would necessarily source the predicted large memory. Please state more explicitly how the final conjecture depends on this unsettled issue.
[Sec. III C 2] The claim that gauge invariance of the linear equations 'automatically carries over' to the Isaacson energy-momentum tensor is stated without a proof; given the known gauge ambiguities in earlier literature (Ref. [74]), a brief argument or a pointer to the second-variation theorem would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the superluminal memory divergence follows from the Einstein-Æther action and the retarded Green's function, not from fitted inputs or self-defined quantities.

full rationale

The paper's central result — the formally divergent tensor displacement memory at n_p·n = 1/β_S for β_S > β_T — is obtained by solving the Isaacson memory equation (84) with the retarded Green's function of the tensor wave operator, Eqs. (85)-(95), and the explicit point-pulse energy-momentum tensor constructed in Appendix D (Eqs. (104)-(110)). No step equates the output to an input: the divergence is not put in by hand, and the pulse amplitude E_p is a bookkeeping parameter, not fitted to the memory signal. The external constraints β_T = 1 from GW170817 and β_ψ ≥ 1 from Cherenkov bounds are imported as observational inputs, not derived from or calibrated by the memory computation. The framework of Refs. [7,30,55] is cited for the general Isaacson/Lemma-1 reduction, but those citations are parameter-free and their assumptions do not include the EÆ superluminal divergence, so they are independent support rather than load-bearing circularity. The paper itself flags the breakdown of the localized-source and perturbative assumptions at the critical direction (Sec. IV B 4) and labels the exclusion statement a conjecture; that is a validity caveat, not a circularity. I find no equation or fitted parameter that reduces by construction.

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

The paper introduces no new fields or particles and fits no parameters to data. It imports the Einstein-Aether action, asymptotic flatness, the Isaacson averaging scheme, and external speed constraints from GW170817 and Cherenkov arguments. The main load-bearing choice is treating the formal divergence as physical despite the breakdown of the source-localization approximation.

assumptions (6)

domain assumption The Einstein-Aether action in Eq. (1) is the correct classical theory for the gravitational sector, with a fixed-norm aether vector field.
The paper studies only this action; as noted in App. A, higher-order derivative operators could be added without changing the dof count, so the conclusions are theory-specific.
domain assumption The asymptotic background is Minkowski with a purely temporal, constant aether vector and the source-centered frame aligned with the preferred frame.
Section II B assumes Eq. (10). A background with spatial aether components or a relative boost would modify the SVT decomposition and the critical-angle condition.
domain assumption High-frequency and low-frequency scales are cleanly separated and the Isaacson averaging rules hold.
Section III B imports these from Refs. [30,56,57]; the memory equation is the averaged backreaction Eq. (69), so the conclusion depends on this scale separation.
domain assumption The source is localized with r' << r at all points where the memory source is nonzero.
Eq. (93) enables the expansion in Eq. (94). The paper itself notes this condition breaks down along the critical directions where the memory diverges.
domain assumption Tensor waves propagate luminally (beta_T = 1 from GW170817) and Cherenkov constraints rule out subluminal aether modes (beta_psi >= 1).
Section IV B 1 imports these empirical constraints to restrict the parameter space before applying the memory argument; the exclusion conjecture inherits them.
domain assumption Nondispersive propagation makes phase velocity equal to group velocity for each Einstein-Aether mode.
Eqs. (26), (33) and (39); the retarded time in the energy fluxes uses group velocity, which is essential to the critical-angle condition.

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Pith. "Pith review of Constraining superluminal Einstein-\AE{}ther gravity through gravitational memory." pith.science (2026). https://pith.science/paper/ZOXHEJAP

@misc{pith2026250509544,
  author       = {Pith},
  title        = {Pith review of: Constraining superluminal Einstein-\AEther gravity through gravitational memory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOXHEJAP}},
  note         = {Machine review of arXiv:2505.09544}
}

read the original abstract

Every emission of radiation in gravity also includes a nonwavelike component that leaves a permanent change in proper distances of the spacetime it travels through. This phenomenon is known as gravitational displacement memory. Building up on a recently developed computation framework that harnesses Isaacson's insights on a fundamental definition of gravitational waves, we compute the leading displacement memory formula in Einstein-Aether gravity. Our analysis represents the first direct calculation of gravitational memory in a metric theory with nontrivial asymptotic vector field value. We find that an emission of scalar and vector aether waves at a propagation speed greater than the speed of tensor radiation features unprotected causal directions with a priori unbound memory build-up. Based on the results and the existing constraint of luminally propagating tensor waves, we conjecture a stringent exclusion of the superluminal parameter space of Einstein-Aether gravity.

Figures

Figures reproduced from arXiv: 2505.09544 by the authors.

**Figure 1.** , where the emission angle ϕp is varied at a fixed evaluation point Ωx = { π 2 , 0} and θp = π 2 of the memory, the transition between the two regimes at βS = 1/⃗np · ⃗n involves the expected divergence of the memory signal. In the following, we want to understand the origin of this apparent divergence and draw associated conclusions [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗

**Figure 2.** Penrose diagrams of conformally compactified asymptotically flat spacetimes in the source-centered coordinates, in which the null direction is chosen to be the one set by the gravitational tensor modes that are assumed to propagate luminally βT = 1 with asymptotic retarded time u ≡ t−r. Only the temporal and radial coordinates {t, r} are represented and one may read the diagram as depicting one particular angular d… view at source ↗

**Figure 3.** Causal cone (blue) of emission of superluminal scalar waves at a speed βS = 1.5. Shown are only two spatial directions x and y and the time direction t. The cone intersects a past null cone of a memory event (orange) admitting critical directions of emission that give rise to a formally unbound memory buildup. Shown are also the spatial projections of the cone (dashed blue) and the intersection line (dashed red),… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Penrose diagrams of conformally compactified [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Scalar memory from compact binary coalescences
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In Ricci-coupled scalar-Gauss-Bonnet gravity, the change in scalar charge during binary black hole mergers generates a scalar memory contribution that modifies the total memory signal on observable timescales.
Gravitational Memory in Generalized Proca Gravity
gr-qc 2025-08 conditional novelty 6.0 of 10

The displacement memory formula for Generalized Proca gravity is derived for a massive Lorentz-invariant branch and a massless Lorentz-violating branch, with the dispersive branch requiring a frequency-integrated treatment.

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Equations of motion Constraint equation. If we directly vary the action with respect to the coefficient λ, we will obtain a constraint equation in terms of the gauge-invariant quantities −2 ¯AΩ + 2 ¯A2Φ = 0, (F7) which in the gauge in Eq. (F6) reads 2 ¯Aω0 + ¯A2h00 = 0. (F8) This equation agrees with the corresponding Eq.(96) in Ref. [74] when choosing ¯A...

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Gravitational polarizations Expanding the polarization modes [Eq. (59)] in terms of the SO(3)-invariant components [Eq. (18)] of the perturbation fields, then switching the notation using Eq. (F1) and (F3) we have P+ = eij + 2 ETT ij = eij + 2 ϕTT ij , (F21a) P× = eij × 2 ETT ij = eij + 2 ϕTT ij , (F21b) Pu =− ¯Ac13 βV (aT i + 1 2 ¯AET i )ui =− ¯Ac13 βV (...

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Understanding the directions of large memory buildup We are now in a place to physically understand the divergence in the memory computation that relates the two regimes Vp S > 0 (I) and Vp S < 0 (II) identified above, and deduce a conjecture on a severe constraint on super- luminal EÆ theory. Through Eq. (112), the magnitude of this factor determines the...

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Conjecture of new constraint on EÆ Gravity Given the physical origin of this divergence in the memory formula, it represents first of all an interesting feature of Lorentz-breaking theories that might provide a characterizing signature in GW data. Our computa- tion is strictly speaking not valid at the diverging point, since basic assumptions are not sati...

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Asymtotic energy-momentum tensor To compare the self-stress-energy tensor, we perform gauge fixing on Eq. (74) using Eq. (F6), as follows (2)tT ij = 1 4κ0 D ∂iϕTT ab ∂jϕab TT E = ninj 4κ0β2 T D ˙ϕTT ab ˙ϕab TT E , (F23a) (2)tV ij = ¯CV 4κ0 D ∂iνa∂jνa E = ¯CVninj 4κ0β2 V D ˙νa ˙νa E , (F23b) (2)tS ij = ¯CS 4κ0 1 4 D ∂iF∂jF E = ¯CSninj 4κ0β2 S D ˙F ˙F E , (...