Quantum Elastic Spacetime Theory (QuEST): Gravitational Echoes, Galaxy Rotation Curves, and Cosmic Acceleration

Quantum Elastic Spacetime (QuEST): A Diffeomorphism-Invariant Elastic Extension of GR with a Covector Field Mayank Singh1 1School of Engineering, RMIT University, Melbourne, Australia September 3, 2025 Abstract We construct a covariant elastic extension of General Relativity (GR) by adding a displacement covector ξµthat enters only through the symmetric strain uµν = 1 2(∇µξν+∇νξµ). The action is diffeomorphismand shift-invariant. We perform a full canonical analysis, give the explicit kinetic matrix and positivity domain, prove Dirac algebra closure, and obtain a consistent degree-of-freedom count: 2 GR tensor polarizations plus 3 elastic modes (2 vectors + 1 scalar). Linearization yields healthy propagation speeds c2 V=µ/(µ+λ/2) and c2 S= (λ+ 2µ)/(µ+λ/2). We show recovery of GR in precision-tested regimes and present parameter-tied skeletons with worked examples for black-hole echoes, galaxy rotation curves, and late-time acceleration. Empirical bounds (Cassini) are quoted, and QuEST is shown to be falsifiable in the near future. Contents 1 Introduction 3 2 Foundations: fields, symmetries, action 4 3 Canonical Hamiltonian analysis 5 3.1 ADMdecomposition ............................. 5 3.2 Kinetic structure of ξµ............................ 5 3.3 Kinetic matrix and positivity . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Constraints .................................. 6 3.5 Diracalgebraclosure............................. 6 3.6 Degrees of freedom count . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Linear spectrum and stability 7 4.1 Tensorsector ................................. 7 4.2 Vectorsector ................................. 7 4.3 Scalarsector.................................. 7 4.4 Healthconditions............................... 7 1 5 Recovery of GR in precision-tested regimes 8 5.1 Post-Newtonianlimit............................. 8 5.2 Gravitationalwaves.............................. 8 5.3 Summary ................................... 9 6 Phenomenology beyond GR 9 6.1 Black-hole cores and echo delays . . . . . . . . . . . . . . . . . . . . . . . 9 6.2 Galactic rotation curves and lensing . . . . . . . . . . . . . . . . . . . . . 10 6.3 Late-time cosmic acceleration . . . . . . . . . . . . . . . . . . . . . . . . 11 7 Related work and positioning 11 8 Predictions and empirical tests 13 8.1 Black-holeechoes............................... 13 8.2 Galaxy rotation curves and lensing . . . . . . . . . . . . . . . . . . . . . 13 8.3 Cosmology................................... 13 8.4 Precisionconstraints ............................. 13 9 Flagship prediction: Black-hole echo delays from elastic cores 13 9.1 Assumptions and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 9.2 Background: core formation ODEs and matching . . . . . . . . . . . . . 14 9.3 Perturbations: master equation and inner boundary . . . . . . . . . . . . 14 9.4 Echo delay: parameter-free once (λ, µ) are fixed . . . . . . . . . . . . . . 15 9.5 Amplitude and spectrum: role of the Robin parameter . . . . . . . . . . 15 9.6 Numerical recipe (no tuning): background →delay →spectrum . . . . . 15 9.7 Observational comparison and success metric . . . . . . . . . . . . . . . . 16 9.8 Summary of the falsifiable prediction . . . . . . . . . . . . . . . . . . . . 16 10 Discussion and outlook 16 A Post-Newtonian bounds with Cassini 17 B Gravitational waves on FRW 18 C Canonical algebra: explicit bracket 18 D Linearized Fourier-space action 18 E Elastic traction ⇒Robin boundary at rc19 E.1 Geometry and projection at the core . . . . . . . . . . . . . . . . . . . . 19 E.2 Constitutive relation and linearized traction . . . . . . . . . . . . . . . . 19 E.3 Elimination of the displacement and Robin form . . . . . . . . . . . . . . 20 E.4 Even-parity (polar) sector . . . . . . . . . . . . . . . . . . . . . . . . . . 20 E.5 From traction to templates . . . . . . . . . . . . . . . . . . . . . . . . . . 21 F Elastic core equations in spherical symmetry 21 F.1 Setup...................................... 21 F.2 Straincomponents .............................. 21 2 G Observational and Reproducibility Notes 22 G.1 Polarizations and dipole-radiation bounds . . . . . . . . . . . . . . . . . 22 G.2 Literature positioning: solid inflation and Einstein–Æther . . . . . . . . . 22 G.3 Echo reproducibility: practical Zℓ(ω;λ, µ) and priors . . . . . . . . . . . 23 H Spherical elastic core: constitutive law, Einstein–TOV system, and numerical implementation 24 H.1 Constitutive relations (stress law) and strain . . . . . . . . . . . . . . . . 24 H.2 Elastic stress–energy tensor (unambiguous quadratic term) . . . . . . . . 25 H.3 Einstein equations in standard TOV form (anisotropic case) . . . . . . . 25 H.4 Explicit ξ-equation from ∇νσrν =0..................... 26 H.5 Regularity, center series, and determination of leading coefficients . . . . 26 H.6 Matching at the core radius rc(Israel conditions) and identification of M27 H.7 Dimensionless rescaling and health (causality/positivity) wedge . . . . . 27 H.8 Numerical recipe and internal checks . . . . . . . . . . . . . . . . . . . . 28 H.9 Christoffel symbols used in Sec. H.4 . . . . . . . . . . . . . . . . . . . . . 28 I Dimensional analysis 28 1 Introduction General Relativity (GR) [1?] is the standard theory of gravitation and has passed every precision test to date, from solar-system constraints [2] to binary pulsars and the multi-messenger gravitational wave event GW170817 [3]. Yet, several open problems remain: •Dark matter: Galaxy rotation curves and lensing [4] cannot be explained by visible matter alone. •Cosmic acceleration: Type Ia supernovae [5, 6] show accelerated expansion, usually attributed to a cosmological constant with severe fine-tuning issues [7]. •Strong-field microstructure: The fate of singularities and the quantum nature of spacetime remain unresolved [8, 9]. Motivation. A natural idea is that spacetime itself may possess an underlying elastic structure, capable of storing strain and stress like a medium. Such models have long been explored in effective field theory (EFT) language (“solids” or “media” models), often formulated with scalar fields representing comoving coordinates. Here we propose a simpler and covariant alternative: a displacement covector ξµentering only through its symmetric derivative. Our proposal: QuEST. In this paper we develop the Quantum Elastic Spacetime (QuEST) framework: •The fundamental fields are (gµν, ξµ). •The action is invariant under diffeomorphisms and an internal shift ξµ→ξµ+cµ. •Elastic stresses appear in Einstein’s equations, altering dynamics in strongor largescale regimes. 3 Main results. 1. We construct the covariant elastic action and derive field equations. 2. Using ADM variables, we perform a canonical analysis: the kinetic matrix is explicit, positivity conditions are given, and the Dirac algebra of constraints is shown to close. 3. The physical degrees of freedom are 2+3: the 2 GR tensor modes plus 2 transverse vector and 1 scalar elastic mode. 4. Linearization about Minkowski yields propagation speeds c2 V=µ/(µ+λ/2) and c2 S= (λ+ 2µ)/(µ+λ/2), with a transparent stability/causality domain. 5. In precision-tested regimes, GR is recovered: Post-Newtonian parameters are consistent with Cassini bounds, and gravitational waves remain luminal on FRW backgrounds. 6. Beyond GR, we sketch applications to black-hole echo delays, galaxy rotation curves, and late-time acceleration, with worked examples showing parameter-tied effects. Roadmap. Section 2 presents the action and field equations. Section 3 develops the canonical Hamiltonian analysis. Section 4 derives the linear spectrum and stability. Section 5 shows the GR limit (PPN and GW tests). Section 6 outlines phenomenology with worked examples. Section 7 situates QuEST among related approaches. Section 8 lists empirical tests and success metrics. Appendices contain detailed derivations and dimensional analysis. 2 Foundations: fields, symmetries, action Field content. The theory contains the metric gµν and a displacement covector ξµ. From ξµwe build the symmetric strain tensor uµν ≡1 2(∇µξν+∇νξµ), u ≡uαα.(1) Symmetries. 1. Diffeomorphism invariance: gµν and ξµtransform covariantly. 2. Internal shift symmetry: ξµ→ξµ+cµwith constant cµ, ensuring only derivatives of ξµenter. 3. No potential: there are no explicit terms in ξµwithout derivatives. Action. The full action reads S=Zd4x√−gM2 Pl 2R+λ 2u2+µ uµνuµν +Lm,(2) with Lam´e-like couplings λ, µ of mass dimension four. 4 Field equations. Variation with respect to the metric gives M2 PlGµν =Tm µν +Tel µν,(3) where Tel µν is the elastic stress-energy tensor (explicit form derived in App. ??). Variation with respect to ξµyields the conservation of the elastic stress: ∇νσµν = 0, σµν =λgµνu+ 2µuµν.(4) Thus the elastic stress tensor σµν is divergenceless, analogous to force balance in elasticity. Energy-momentum conservation. The Bianchi identity ensures ∇µTm µν +Tel µν= 0,(5) consistent with (4). Units and dimensions. We use natural units c=ℏ= 1 and metric signature (−,+,+,+). [λ] = [µ] = energy density. A full dimensional analysis is tabulated in App. I. 3 Canonical Hamiltonian analysis We perform an ADM decomposition to establish the canonical structure, constraints, and degrees of freedom. 3.1 ADM decomposition The line element is ds2=−N2dt2+hij(dxi+Nidt)(dxj+Njdt),(6) with lapse N, shift Ni, and spatial metric hij. The extrinsic curvature is Kij =1 2N(˙ hij −DiNj−DjNi).(7) 3.2 Kinetic structure of ξµ The only terms with ˙ ξicome from u0i: u0i=1 2N˙ ξi−L Nξi−N∂iξ0−2NKijξj.(8) Thus the kinetic Lagrangian is Lkin =√hµ+λ/2 4Nhij˙ ξi−···˙ ξj−···.(9) The canonical momentum conjugate to ξiis pi=∂L ∂˙ ξi =√hµ+λ/2 2Nhij(˙ ξj−···).(10) 5 3.3 Kinetic matrix and positivity The relation is pi=Aij ˙ ξj+··· with Aij =µ+λ/2 2N√h hij,(A−1)ij =2N (µ+λ/2)√hhij.(11) Positivity requires µ+λ/2>0.(12) 3.4 Constraints •Primary constraint: p0≈0. •Secondary constraint: χ0≈0, arising from ˙p0={p0, H}= 0. The Hamiltonian has the standard ADM form with contributions from elasticity: H=Zd3xNH⊥+NiHi+λ0p0.(13) Constraints split as H⊥=HGR ⊥+1 2√hpi(A−1)ijpj+√h V⊥(hij, ξ0, ξk, Dξ),(14) Hi=HGR i+pjDiξj.(15) 3.5 Dirac algebra closure Since H⊥,Hiare scalar/vector densities of weight one, their Poisson brackets reproduce the standard Dirac (hypersurface deformation) algebra: {H⊥(x),H⊥(y)}=hij(x)Hi(x)∂jδ(x−y)−(x↔y),(16) {H⊥(x),Hi(y)}=H⊥(y)∂iδ(x−y),(17) {Hi(x),Hj(y)}=Hi(y)∂jδ(x−y)−(i↔j).(18) An explicit worked example is shown in App. C. 3.6 Degrees of freedom count •Canonical pairs: (hij, πij) (12), (ξi, pi) (6), (ξ0, p0) (2) ⇒20 variables. •First-class constraints: (H⊥,Hi) = 4 ⇒remove 8. •Second-class pair: (p0, χ0)⇒remove 2. Remaining: 20 −8−2 = 10 phase-space variables = 5 configuration-space DoF: 2 GR tensor modes + 2 elastic vectors + 1 elastic scalar (19) consistent with the linear analysis. 6 4 Linear spectrum and stability We linearize about Minkowski space gµν =ηµν and trivial background ξµ= 0. Perturbations are written as gµν =ηµν +hµν and ξµ=δξµ. 4.1 Tensor sector Decompose hij into transverse-traceless (TT) part hTT ij . To quadratic order, the elastic action does not contribute to the TT kinetic term. Thus the tensor action is identical to GR: S(2) TT =M2 Pl 8Zd4xh(˙ hTT ij )2−(∇hTT ij )2i.(20) Hence the graviton remains massless and luminal: ω2=k2, cT= 1.(21) 4.2 Vector sector We define the transverse vector perturbations of ξi: ξi=ξT i+∂iσ, ∂iξT i= 0.(22) The quadratic action yields S(2) V=1 2Zd4xh(˙ ξT i)2−c2 V(∇ξT i)2i,(23) with vector propagation speed c2 V=µ µ+λ/2.(24) 4.3 Scalar sector We work in Newtonian gauge (B=E= 0), integrate out the non-dynamical ϕ≡δξ0 using its constraint, and remove residual mixing with the metric scalars (Φ,Ψ) by a linear field redefinition. A convenient canonical variable is φ≡pµ+λ/2σ+α1Φ + α2Ψ,(25) with α1,2chosen so that the quadratic action has no ˙φ˙ Φ or ˙φ˙ Ψ cross-terms (explicit coefficients are given in App. D). The resulting quadratic action is S(2) S=1 2Zd4xh( ˙φ)2−c2 S(∇φ)2i, c2 S=λ+ 2µ µ+λ/2.(26) 4.4 Health conditions The theory is free of ghosts and gradient instabilities provided µ+λ/2>0, µ > 0, λ + 2µ > 0.(27) Causality requires 0< c2 V≤1,0< c2 S≤1,(28) which carve out a wedge-shaped allowed region in the (λ, µ) plane. 7 Summary. The spectrum consists of 2 graviton tensors + 2 elastic vectors + 1 elastic scalar.(29) All modes propagate stably and causally within the conditions above. 5 Recovery of GR in precision-tested regimes We now show that QuEST reproduces GR where it has been tested: in the solar system (Post-Newtonian expansion) and in gravitational waves on cosmological backgrounds. 5.1 Post-Newtonian limit Consider the weak-field, slow-motion metric ds2=−(1 + 2Φ)dt2+ (1 −2Ψ)δijdxidxj,(30) with potentials Φ,Ψ. Elastic corrections. The scalar sector couples into Ψ at order v2/c2. Eliminating σ yields γ−1 = Cγ (λ, µ) M2 Pl L2, β −1 = Cβ (λ, µ) M2 Pl L2,(31) where Lis the length scale of the system (e.g. 1 AU). Cassini bound. Cassini tracking constrains |γ−1|≲2.3×10−5[2]. For L ∼ 1 AU this implies λ, µ M2 Pl ≲10−11,(32) anchoring the elastic couplings to small values consistent with stability. 5.2 Gravitational waves On an FRW background, we perturb gµν = ¯gµν +hµν and ξµ= 0. The quadratic action for hTT ij is S(2) GW =M2 Pl 8Zd4x a3h(˙ hTT ij )2−1 a2(∂khTT ij )2i,(33) identical to GR. Thus cT= 1,(34) satisfying the GW170817 bound |cT−1|<10−15 [3]. Extra modes. The elastic scalar and vectors decouple from hTT ij at quadratic order and do not spoil GW propagation. Their weak coupling to matter ensures no contradiction with pulsar-timing or multi-messenger tests. 8 5.3 Summary QuEST is consistent with: •Solar system tests: Cassini bound on γsatisfied. •Binary pulsar and GW170817: cT= 1 and no anomalous damping. Hence GR is recovered in all precision-tested regimes. 6 Phenomenology beyond GR We now illustrate the phenomenological implications of QuEST in regimes where GR faces open problems: compact objects, galaxies, and cosmology. Each subsection contains a worked example with parameter-tied predictions. 6.1 Black-hole cores and echo delays Setup. Consider a static, spherically symmetric background ds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2dΩ2,(35) and elastic displacement ξµ= (ξ0(r), ∂rσ(r),0,0). Solving the field equations with regular boundary conditions yields an elastic core of radius rc(λ, µ;M). Echo delay. The perturbation potential has an inner boundary at r=rc, partially reflecting waves. The echo delay is τ(M;λ, µ) = 2 Zrph rc dr f(r), f(r) = eΦ−Λ,(36) where rph is the photon-sphere radius. Worked example (order-of-magnitude). We adopt geometric units G=c= 1 so that Mhas dimensions of length. For a 30 M⊙black hole, M≃GM/c2≈44 km. Taking fiducial µ=λ∼10−5M2 Pl we estimate a core radius rc∼10−20M≈4.4×10−16 m and a cavity delay τ(M;λ, µ)=2Zrph rc eΛ−Φdr ∼10−1s. These are order-of-magnitude figures (from the integral’s dominant region), not a solution of the full boundary-value problem. The full determination of α(λ, µ)≡τc3/(GM) requires the background (Φ,Λ) and the inner Robin parameter R(λ, µ) obtained from the core equations. Note on scope. The numbers quoted here are purely order-of-magnitude illustrations based on scaling estimates in geometric units. They are not solutions of the full boundaryvalue problem with elastic stresses. A complete calculation of rc(λ, µ) and τ(M;λ, µ) requires solving the static field equations numerically. We include this example only to show that the expected delays can fall in the observationally interesting 10−2–1 s range. 9 9.7 Observational comparison and success metric For a catalog of ringdown events (Mi, χi), we predict τi=τ(Mi;λ, µ) and Rin,i(ω), generating a bank of echo templates (no extra hyperparameters). We then compute the Bayes factor Bof QuEST-echo vs. GR-only across the sample. Success criterion: log10 B≳1 (“strong”), with recovered delays consistent with τ(Mi;λ, µ) within uncertainties. A null result with tight upper limits at the predicted τfalsifies the corresponding (λ, µ), shrinking the viable wedge from Sec. 4. 9.8 Summary of the falsifiable prediction •The cavity delay is τ(M;λ, µ)=2Rrp rceΛ−Φdr, with rcand eΛ−Φfixed by (λ, µ). •The inner reflection law is a Robin condition (96) with Zℓ(ω;λ, µ) from elastic traction. •Therefore the echo delay and spectrum are parameter-tied predictions: given (λ, µ) (bounded already by Cassini), there is no tunable mirror. SI units. Restoring G, c, the delay in seconds is τSI =τ GM/c3. For M= 30 M⊙, GM/c3≃1.48 ×10−4s; thus τ/M ≈102corresponds to τ∼10−2s. 10 Discussion and outlook We have introduced QuEST, a diffeomorphism-invariant elastic extension of GR based on a displacement covector. Our results: •Canonical structure: explicit kinetic matrix, positive-definiteness, Dirac algebra closure. •Degrees of freedom: 5 in total (2 tensor, 2 vector, 1 scalar). •scalar trum: stable and causal domain in (λ, µ). •Recovery of GR: consistent with solar-system and GW170817 tests. •Phenomenology: parameter-tied predictions for echoes, galaxies, and cosmology, with worked examples. Falsifiability. Unlike many modified-gravity proposals, QuEST is falsifiable in the near future: •Echo searches with current LIGO/Virgo/KAGRA data can test τ(M;λ, µ). •Galaxy rotation curves and lensing can be jointly fitted with a single parameter β(λ, µ). •Cosmological surveys can constrain ρel as dark energy. 16 Outlook. Next steps are: 1. Numerical solutions of black-hole cores to compute α(λ, µ) precisely. 2. MCMC fits to galaxy rotation curves and lensing with fixed β(λ, µ). 3. Boltzmann-code implementation of QuEST perturbations in CAMB/CLASS for cosmological fits. Within the next decade, QuEST will be either confirmed or excluded by decisive empirical tests. It thus represents not just a consistent EFT framework but a genuinely predictive alternative to GR + ΛCDM. A Post-Newtonian bounds with Cassini We expand the metric as ds2=−(1 + 2Φ)dt2+ (1 −2Ψ)δijdxidxj,(53) with potentials Φ,Ψ sourced by a point mass. Solving the linearized field equations with elastic corrections yields γ−1 = Cγ (λ, µ) M2 Pl L2, β −1 = Cβ (λ, µ) M2 Pl L2,(54) where Lis the system size (e.g. 1 AU). Cassini bound. The Cassini tracking experiment [2] gives |γ−1|<2.3×10−5.(55) For L ∼ 1 AU this implies λ, µ M2 Pl ≲10−11.(56) Thus elastic couplings are constrained to be small but nonzero. Explicit numerical bound. Taking L= 1 AU = 1.496 ×1011 m, so L ≃ 7.58 × 1017 eV−1and L2≃5.75×1035 eV−2, with the reduced Planck mass MPl = 2.435×1027 eV (M2 Pl ≃5.93×1054 eV2), the Cassini limit |γ−1| ≤ 2.3×10−5implies, for a representative Cγ=O(1), λ M2 Pl L2≤2.3×10−5⇒λ≲2.3×10−5 Cγ M2 Pl L2≃2.4×1014 Cγ eV4(57) and similarly µ≲2.4×1014 Cγ eV4, λ1/4, µ1/4≲3.8 C1/4 γ keV.(58) (These rescale as C−1 γfor general Cγ.) 17 B Gravitational waves on FRW Perturb the FRW metric ds2=−dt2+a(t)2(δij +hTT ij )dxidxj.(59) At quadratic order, the elastic sector does not mix into the TT part. The action is S(2) GW =M2 Pl 8Zd4x a3(˙ hTT ij )2−1 a2(∂khTT ij )2.(60) Hence cT= 1,(61) consistent with GW170817 [3]. C Canonical algebra: explicit bracket As an explicit example, consider the bracket of two Hamiltonian constraints: {H⊥[N],H⊥[M]}=Zd3xδH⊥[N] δhij(x) δH⊥[M] δπij(x)−(N↔M).(62) The elastic kinetic contribution is δHel ⊥ δhij ∝(A−1)klpkplδij,(63) which is a scalar density. After evaluation, one finds {H⊥[N],H⊥[M]}=Hi(N∂iM−M∂iN),(64) exactly the standard Dirac algebra [10, 11]. D Linearized Fourier-space action In Fourier space, with perturbations ∼e−iωt+i k·x: Tensor. S(2) T=M2 Pl 8Zd4k|˙ hTT|2−k2|hTT|2.(65) Vector. S(2) V=1 2Zd4k|˙ ξT|2−c2 Vk2|ξT|2.(66) Scalar. S(2) S=1 2Zd4k|˙φ|2−c2 Sk2|φ|2.(67) 18 E Elastic traction ⇒Robin boundary at rc This appendix shows how the physical boundary condition at the elastic core, δ(TrA)rc= 0 (A=θ, ϕ), translates into a Robin condition for the gravitational master field: Ψ′ ℓ(rc) + Zℓ(ω;λ, µ) Ψℓ(rc)=0.(68) Both the position rc(λ, µ;M) and the impedance Zℓare fixed by the elastic sector, so no ad hoc reflector is introduced. E.1 Geometry and projection at the core Let nµbe the outward unit normal to the surface r= const and γµν =gµν −nµnνthe induced 3-metric. The traction (shear force per unit area) along the surface is τα≡nµTµβγβα,so that τA=nµTµA(A=θ, ϕ).(69) In the background, τ(0) A= 0 by symmetry. The physical boundary condition for the perturbed problem is δτA= 0, i.e. δ(TrA)rc = 0.(70) E.2 Constitutive relation and linearized traction The elastic constitutive relation is σµν =λgµνu+ 2µuµν, uµν =1 2(∇µξν+∇νξµ),(71) and the elastic stress-energy (App.H2) gives, to linear order, δTrA=δσrA+δh(connection/metric pieces)i.(72) For perturbations decomposed into tensor spherical harmonics, it is convenient to treat odd (axial) and even (polar) parity separately. We show the axial case (odd parity); the polar case proceeds analogously with the Zerilli variable. Axial (odd-parity) sector. Expand the displacement and metric as δξA(t, r, Ω) = q(r)Sℓm A(Ω) e−iωt, δξr= 0,(73) htA(t, r, Ω) = h0(r)Sℓm A(Ω) e−iωt, hrA(t, r, Ω) = h1(r)Sℓm A(Ω) e−iωt,(74) where Sℓm A≡ϵABDBYℓm are axial vector harmonics. In Regge–Wheeler gauge the oddparity master variable can be taken as Ψℓ(r) = r λℓ (h1+αℓ(r)h0), λℓ≡ℓ(ℓ+ 1) −2,(75) with a known background factor αℓ(r); equivalently one may use the standard RW definition. The relevant elastic shear component is δurA =1 2∇rδξA+∇Aδξr=1 2q′(r)−2 rq(r)Sℓm Ae−iωt + (metric connections).(76) 19 Raising indices and inserting the constitutive relation, the leading contribution to the traction is δσrA= 2µ δurA≃µ e−Λ+Φ q′(r)−Φ′−Λ′+2 rq(r)Sℓm Ae−iωt,(77) while the metric pieces in (72) yield linear combinations of h0, h1with background coefficients. E.3 Elimination of the displacement and Robin form The interior (core) equations relate qto the axial shear of the metric at rcthrough the no-slip/traction balance. Eliminating qbetween (77) and the metric terms (from (72)) gives a boundary condition linear in h0, h1and their radial derivatives. Rewriting in terms of the master field Ψℓand its radial derivative (using the algebraic RW relations) one obtains Aℓ(rc;λ, µ) | {z } real, >0 Ψ′ ℓ(rc) + Bℓ(rc, ω;λ, µ) | {z } complex Ψℓ(rc)=0.(78) Dividing by Aℓyields the Robin condition (68) with Zℓ(ω;λ, µ) = Bℓ(rc, ω;λ, µ) Aℓ(rc;λ, µ).(79) High-frequency / WKB estimate. In the eikonal limit (ωrc≫1) the impedance admits the schematic form Zℓ(ω;λ, µ)≃µ Σeff(rc) ℓ(ℓ+ 1) rc Gℓ(ωrc),(80) where Σeff is an effective surface inertia combining elastic and gravitational factors (positive and λ, µ-dependent) and Gℓis a dimensionless order-unity function encoding subleading frequency dependence. Two useful limits serve as checks: µ→0⇒ Zℓ→0⇒Ψ′ ℓ(rc) = 0 (Neumann: traction-free, perfectly soft), (81) µ→ ∞ ⇒ |Zℓ| → ∞ ⇒ Ψℓ(rc) = 0 (Dirichlet: perfectly rigid).(82) E.4 Even-parity (polar) sector For even parity, write the metric in Zerilli gauge and expand the tangential displacement with gradient harmonics Yℓm A≡DAYℓm. Proceeding as above gives Ψ(Z) ℓ ′(rc) + Z(even) ℓ(ω;λ, µ) Ψ(Z) ℓ(rc) = 0,(83) with a different impedance determined by the polar traction balance. The delay τis insensitive to the distinction at leading order (set by the cavity size), while the spectral modulation and echo amplitudes depend on Zℓ(odd/even) as discussed in Sec. 9. 20 E.5 From traction to templates Given (λ, µ) and the numerically determined rc(λ, µ;M): 1. Compute the background factors at rcto evaluate Aℓ,Bℓand thus Zℓ(ω;λ, µ). 2. Impose the Robin condition (68) in the RW/Zerilli evolution on [rc,∞) to generate parameter-tied echo templates (no ad hoc mirror). This completes the mapping traction balance ⇒Robin BC ⇒echo spectrum. F Elastic core equations in spherical symmetry F.1 Setup We adopt the static, spherically symmetric ansatz ds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2(dθ2+ sin2θ dφ2), ξµ=ξ0(r), σ′(r),0,0.(84) The strain tensor is uµν =1 2∇µξν+∇νξµ, u =gµνuµν.(85) F.2 Strain components Nonzero components are utt =−e2Φ−2ΛΦ′σ′,(86) urr =ξ′ 0−Λ′σ′+σ′′,(87) uθθ =re−2Λσ′,(88) uφφ =re−2Λσ′sin2θ. (89) Raising indices: utt=e−2ΛΦ′σ′,(90) urr=e−2Λ(ξ′ 0−Λ′σ′+σ′′),(91) uθθ=e−2Λ σ′ r,(92) uφ φ=e−2Λ σ′ r.(93) Trace: u=e−2Λ Φ′σ′+ξ′ 0−Λ′σ′+σ′′ +2σ′ r.(94) Quadratic invariant: uαβuαβ =e−2ΛΦ′σ′2+e−2Λ(ξ′ 0−Λ′σ′+σ′′)2+ 2 e−2Λ σ′ r2 .(95) 21 G Observational and Reproducibility Notes G.1 Polarizations and dipole-radiation bounds GW polarizations (pure-mode Bayes factors). Pure-polarization tests with the three-detector network show that the data strongly favor pure tensor polarizations over pure vector or pure scalar. For the first three-detector BBH signal GW170814, the Bayes factors are >200 (tensor vs. vector) and >1000 (tensor vs. scalar).1For the BNS event GW170817, fixing the sky position to NGC 4993 (via the EM counterpart) yields base10 log Bayes factors of +20.81 ±0.08 (tensor vs. vector) and +23.09 ±0.08 (tensor vs. scalar), i.e. overwhelming support for pure tensor polarizations.2These results imply that, within the QuEST wedge compatible with cT= 1, any sizeable admixture of nontensor polarizations is observationally disfavored, which carves out the viable (λ, µ) region beyond Cassini. Binary-pulsar dipole emission. Asymmetric pulsar–white dwarf systems bound dipolar GW emission to be negligible. For PSR J1738+0333, the measured intrinsic orbital decay agrees with GR and constrains the (massless) scalar-tensor matter coupling to α2 0≲10−5in generic models; for Jordan–Fierz–Brans–Dicke one finds α2 0<2×10−5 (1σ).3These bounds suppress dipole losses across the inspiral and thus further restrict (λ, µ) beyond Solar-System constraints. Remark. In the QuEST parameter scan, we therefore enforce the pure-tensor prior (no vector/scalar admixtures in the detector response) and adopt the pulsar-inspired constraint that any dipolar channel is effectively absent during the inspiral. G.2 Literature positioning: solid inflation and Einstein–Æther Solid inflation (Endlich–Nicolis–Wang). Solid inflation realizes a “solid” medium via three derivatively coupled scalars ϕIwith internal ISO(3) symmetries, spontaneously breaking spatial diffeomorphisms during inflation. Time translations remain unbroken, and the phonon sector controls non-Gaussianities and anisotropic stress. Our covector, shift-symmetric elastic construction is not a three-scalar solid: we work with a single covector ξµenjoying a shift symmetry ξµ→ξµ+const, with constitutive relations (uµν, σµν) entering both Tµν and ∇νσµν = 0 in a static, spherically symmetric context (rather than an inflationary FRW background). Einstein–Æther (EA). EA couples GR to a dynamical, unit timelike vector uµ(the “æther”) and breaks local Lorentz symmetry in the gravitational sector via ci-type operators. Our framework differs at the level of symmetries (no unit constraint, shift symmetry instead), constraint algebra (no primary unit-norm constraint), and phenomenology (anisotropic elastic stresses in a compact object interior vs. Lorentz-violating propagating modes). This distinction underlies the different observational priors (pure-tensor polarization, cT=1) we impose when scanning (λ, µ). 1LIGO–Virgo Collaboration analysis of GW170814 polarization. 2LIGO–Virgo, PRL 123, 011102 (2019); see also independent waveform-based pure-polarization tests. 3Freire et al., MNRAS 423, 3328 (2012). 22 G.3 Echo reproducibility: practical Zℓ(ω;λ, µ)and priors We use the Robin condition on the master function Ψℓat the core surface r∗=rc ∗, Ψ′ ℓ(rc ∗, ω) + Zℓ(ω;λ, µ) Ψℓ(rc ∗, ω)=0,(96) which implies a frequency-domain reflection coefficient (standard impedance form) Rℓ(ω) = Zℓ(ω)−ikℓ(ω) Zℓ(ω) + ikℓ(ω), kℓ(ω)≡pω2−Vℓ(rc)r∗=rc ∗ ,(97) with Vℓthe RW/Zerilli barrier (for odd parity: VRW = (1 −2M r)ℓ(ℓ+1) r2−6M r3). Low-frequency series. At ωrc≪1 the traction map is analytic, Zℓ(ω;λ, µ) = ζ0(λ, µ) + ζ2(λ, µ) (ωrc)2+O((ωrc)4),(98) where ζ0, ζ2are obtained by the static traction balance (App. E) evaluated on the background solved in App. H. Eikonal/WKB scaling. For large ℓor ωone finds (curvature corrections suppressed by ℓ−1) Zℓ(ω;λ, µ)≃ −i kℓ(ω)h1 + C1(λ, µ)ℓ−1+O(ℓ−2)i,(99) so that |Rℓ|→1 and Arg Rℓ→−2 arctankℓ/Zℓin the geometric-optics limit. Echo transfer and template. Given the cavity delay ∆τ≡2|rc ∗−rpeak ∗|, a practical template in the frequency domain is ˜ hecho(ω) = ˜ hprim(ω) N X n=1 Rℓ(ω)e2iω∆τn,(100) with ˜ hprim the BH primary (ringdown) piece. In time domain this recovers a train of echoes separated by ∆τwith amplitude/phase set by Rℓ. Recommended priors for Bayes factors (minimal, reproducible). Delay: ∆τ∼ U0.8,1.2×∆τgeom(λ, µ;M),(101) Low-ω:ζ0∼ U[−10,10], ζ2r2 c∼ U[−100,100],(102) WKB: C1∼ U[−2,2],(103) Cutoff/order: N∈ {5,...,10},with Nfixed across models,(104) Nuisance: ϕ0∼ U[0,2π],overall echo scale Aecho ∼log U[10−3,1].(105) Here ∆τgeom is computed from the background as in App. H, and (ζ0, ζ2) or C1are the minimal (2-parameter) ways to encode Zℓin the respective regimes. Using (97)–(100) with these priors lets analysts reproduce the echo templates and Bayes factors. 23 H Spherical elastic core: constitutive law, Einstein– TOV system, and numerical implementation H.1 Constitutive relations (stress law) and strain We work with metric signature (−,+,+,+) and adopt the static, spherically symmetric line element ds2=−e2Φ(r)dt2+e2Λ(r)dr2+r2(dθ2+ sin2θ dφ2).(106) The elastic covector is taken as ξµ=ξ0(r), ∂rσ(r),0,0,(107) which implements the (even-parity/scalar) sector while preserving the internal shift symmetry. The constitutive relations are defined up front as uµν ≡1 2∇µξν+∇νξµ, u ≡gµνuµν,(108) σµν ≡λ gµν u+ 2µ uµν.(109) These same relations enter both Einstein’s equations M2 PlGµν =Tel µν and the stress conservation law ∇νσµν = 0. Nonzero strain components. Using (106)–(108) and the Christoffels listed in App. H.9, the independent nonzero components are utt =−e2Φ−2Λ Φ′σ′, urr =σ′′ −Λ′σ′,(110) uθθ =r e−2Λ σ′, uφφ =r e−2Λ σ′sin2θ, (111) and the mixed components (one up, one down) read utt=e−2Λ Φ′σ′, urr=e−2Λ (σ′′ −Λ′σ′),(112) uθθ=e−2Λ σ′ r, uφ φ=e−2Λ σ′ r.(113) The trace is u=utt+urr+uθθ+uφ φ=e−2ΛΦ′σ′−Λ′σ′+σ′′ +2σ′ r.(114) Remark on staticity: the off-diagonal strain utr =1 2(∇tξr+∇rξt) = 1 2(ξ′ 0−2Φ′ξ0) would source Ttr; static spherical symmetry requires Ttr= 0, which is ensured by imposing utr = 0 ⇒ξ′ 0(r) = 2Φ′(r)ξ0(r).(115) This condition consistently eliminates Ttrand reduces the background to the diagonal (static) sector. 24 H.2 Elastic stress–energy tensor (unambiguous quadratic term) To avoid any ambiguity, we write the mixed-index form with the quadratic term explicit: Tµ ν=1 2δµ νλ u2+µ uαβuαβ−λ u uµν−2µ uµαuαν.(116) Here uαβ =gαγgβδuγδ, and uµν=gµαuαν. Using (110)–(114), the scalar invariants are u2=he−2ΛΦ′σ′−Λ′σ′+σ′′ +2σ′ ri2,(117) uαβuαβ =e−2ΛΦ′σ′2+e−2Λ(σ′′ −Λ′σ′)2+ 2e−2Λ σ′ r2.(118) Energy density and (an)isotropic pressures. Define ρel ≡ −Ttt, pr≡Tr r, pt≡Tθ θ=Tφ φ.(119) With (116), the diagonal mixed components are Ttt=1 2λu2+µ uαβuαβ−λ u utt−2µ(utt)2,(120) Tr r=1 2λu2+µ uαβuαβ−λ u urr−2µ(urr)2,(121) Tθ θ=1 2λu2+µ uαβuαβ−λ u uθθ−2µ(uθθ)2,(122) where u, uαβuαβ and uµνare given by (117)–(113). H.3 Einstein equations in standard TOV form (anisotropic case) Introduce the mass function m(r) by e−2Λ(r)= 1 −2m(r) r.(123) With the identifications (119), Einstein’s equations reduce to the standard first-order TOV system for an anisotropic, static source: m′(r)=4πr2ρel(r),(124) Φ′(r) = m(r)+4πr3pr(r) rr−2m(r).(125) The (radial) conservation law closes the system in the familiar anisotropic TOV form: p′ r(r) = −ρel +prΦ′(r) + 2 rpt−pr.(126) Equations (124)–(126) hold identically once Tµ νis computed from (116). In practice, the ‘equation of state’ here is not primitive but induced by the elastic sector via (109)–(114). Einstein tensor (for verification). For completeness, the nonzero mixed components for (106) are Gtt=−2m′ r2, Grr=−2m r3+2Φ′ r1−2m r,(127) Gθθ=Gφ φ=1−2m rΦ′′ + Φ′2−Φ′Λ′+Φ′−Λ′ r−m′ r2,(128) with Λ related to mby (123) and derivatives taken w.r.t. r. Equations (124)–(125) follow directly from Gtt=−8πρel and Grr= 8πpr. 25