RSD'S WORK

Residual Spacetime Deformations: A Geometric Memory Approach to Black Hole Information Rhythm September 2025 Abstract The loss of quantum information in black hole evaporation, known as the black hole information paradox, challenges fundamental principles of physics. Standard general relativity offers no intrinsic mechanism to preserve information, as singularities and event horizons disrupt causal contact. Here, we propose the Residual Spacetime Deformation (RSD) framework, in which extreme spacetime curvature permanently imprints geometric memory via a tensor ∆ µν . These residual deformations encode structural information about the collapsed matter, even after a black hole evaporates, providing a pathway to reconcile gravity with quantum coherence. This approach suggests that black holes are not destructive sinks but information-preserving geometric systems, with potential observational signatures in evaporation dynamics and remnant geometries. 1 Introduction / Problem Statement One of the most striking puzzles in modern physics is the black hole information paradox. Stephen Hawkings prediction that black holes radiate thermally implies that quantum information about the matter forming a black hole could be permanently lost when the black hole evaporates. This directly conflicts with unitary evolution in quantum mechanics, which forbids information loss. Historically, attempts to resolve this paradox fall into three categories: 1. Information escapes via Hawking radiation, requiring subtle correlations beyond semiclassical approximations. 2. Information is preserved in Planck-scale remnants, though such remnants raise stability and entropy concerns. 3. Information is encoded in nonlocal gravitational degrees of freedom, a concept lacking a concrete geometric mechanism. 1 Classical general relativity, which models spacetime as a smooth, elastic manifold, provides no internal memory mechanism. Singularities and horizons disconnect the interior from the exterior universe, seemingly erasing the collapse history. This paper addresses this gap by proposing that spacetime itself can retain memory of extreme events through residual geometric deformations. 2 Conceptual Explanation The core idea is simple but radical: high-curvature events permanently deform spacetime in a way that encodes information, independent of matter content. 2.1 The Residual Spacetime Deformation Tensor We introduce a tensor ∆ µν representing permanent residual deformations: gtotal µν =gGR µν +∆ µν •gGR µν is the standard Einsteinian metric. •∆ µν captures permanent geometric memory activated when curvature exceeds a threshold Rcrit: R αβγδ R αβγδ ≳R2 crit =⇒∆ µν =0 This residual tensor stores structural information about the matter and dynamics that caused the extreme curvature. Conceptually, it acts like a frozen imprint in the geometry. 2.2 Information Preservation in Black Holes Consider a collapsing star forming a black hole. Standard relativity predicts an event horizon and a singularity. In RSD: 1. Extreme curvature during collapse activates ∆ µν , embedding memory in spacetime. 2. Even as the black hole evaporates via Hawking radiation, the geometric imprint persists. 3. Observables, such as subtle deviations in Hawking flux or Planck-scale remnant structures, can carry encoded information. In a pseudo-equation form: Information(t)∼∫∆ µν (x)d3x where the integral represents the total memory content of the spacetime region formerly occupied by the black hole. 2 3 Implications / Mysteries Solved The RSD framework offers conceptual resolution to several outstanding issues: •Black hole information paradox: Information is not destroyed but stored in residual geometry. •Singularity problem: Extreme curvature imprints prevent complete physical breakdown. •Quantum-gravity interface: Provides a geometric conduit for preserving quantum coherence without violating relativity. Additionally, this approach could explain subtle gravitational or quantum signatures in black hole evaporation, potentially observable in high-precision astrophysical measurements. 4 Potential Extensions / Predictions The idea opens multiple avenues for exploration: 1. Observable imprints in Hawking radiation: Correlations in radiation may reflect underlying ∆ µν structure. 2. Remnant geometries: Small-scale deformations surviving evaporation could influence surrounding spacetime. 3. Generalization to early-universe singularities: Residual deformations might encode memory of inflationary events. 4. Integration with quantum information theory: Mapping ∆ µν to quantum states could formalize unitarity preservation. Future work could develop explicit models for ∆ µν , simulate black hole collapse, and identify potential observational signatures. 5 Conclusion The Residual Spacetime Deformation framework offers a conceptually simple, physically motivated mechanism for preserving information in black holes. By encoding the collapse history directly into spacetime geometry, RSD bridges the gap between general relativity and quantum mechanics, suggesting that black holes are information-preserving geometric systems rather than destructive sinks. This paper establishes the core idea and motivates its significance. Full mathematical derivations, simulations, and experimental proposals will be developed in future work. 3 1 Conventions and Notation Spacetime manifold: (M,g µν ), dimension 4. Indices: Greek ( µ , ν , α ,···=0,1,2,3). Spatial Latin (i,j,k,···=1,2,3). Metric signature: (−+ ++). (If you prefer (+ −−−)I can switch globally.) Units: geometric units (G=c=1) unless otherwise noted. Symmetrization/antisymmetrization: A( µν )≡1 2(A µν +A νµ ),A[ µν ]≡1 2(A µν −A νµ ). Covariant derivative associated with g µν :∇ µ . Partial derivative: ∂µ . 2 Basic Geometric Objects and Identities 2.1 Christoffel Symbols Γ α βγ =1 2g αλ  ∂β g γλ + ∂γ g βλ − ∂λ g βγ . 2.2 Riemann Tensor (sign convention consistent with our signature) R α βγδ = ∂γ Γ α βδ − ∂δ Γ α βγ +Γ α γλ Γ λ βδ −Γ α δλ Γ λ βγ . 2.3 Ricci Tensor and Scalar R βδ =R α βαδ ,R=g βδ R βδ . 2.4 Einstein Tensor G µν ≡R µν −1 2g µν R. 2.5 Contracted Bianchi Identity ∇ µ G µν ≡0. This identity will be crucial: any modification of the field equations or additional geometric field must be compatible with this (or the conservation of an effective stress–energy). 3 First Variations: δ g µν , δ Γ, δ R µν , δ G µν We will need linearized expressions about a background metric ¯g µν . Here I derive the standard variations. Start with a one-parameter family g µν ( λ )and δ g µν ≡d d λ g µν  λ =0. Let indices be raised/lowered with the background metric ¯gwhen evaluating zeroth-order quantities. 3.1 Variation of the Inverse Metric δ g µν =−¯g µα ¯g νβ δ g αβ . 1 3.2 Variation of Christoffel δ Γ α βγ =1 2¯g αλ ¯ ∇ βδ g γλ +¯ ∇ γδ g βλ −¯ ∇ λδ g βγ , where ¯ ∇is the covariant derivative for the background ¯g. 3.3 Variation of Riemann and Ricci δ R α βγδ =¯ ∇ γδ Γ α βδ −¯ ∇ δδ Γ α βγ . Contract to get δ R βδ : δ R µν =¯ ∇ αδ Γ α µν −¯ ∇ µδ Γ α αν =1 2−¯ □ δ g µν −¯ ∇ µ ¯ ∇ νδ g+¯ ∇ µ ¯ ∇ αδ g αν +¯ ∇ ν ¯ ∇ αδ g αµ , where δ g≡¯g αβ δ g αβ and ¯ □≡¯g αβ ¯ ∇ α ¯ ∇ β . 3.4 Variation of Scalar Curvature δ R=¯g µν δ R µν −R µν δ g µν (or equivalently use δ R=−R µν δ g µν +¯g µν δ R µν ). 3.5 Variation of Einstein Tensor δ G µν = δ R µν −1 2¯g µνδ R−1 2 δ g µν ¯ R. These formulas are the backbone for the linearized analysis below. 4 Linearization about a Background and the Lichnerowicz Operator Set g µν =¯g µν +h µν ,|h µν |1. Define the trace h≡¯g µν h µν and the trace-reversed perturbation ¯ h µν ≡h µν −1 2¯g µν h. 4.1 Harmonic (de Donder) Gauge Impose ¯ ∇ µ ¯ h µν =0. In this gauge, the linearized Einstein tensor simplifies. For a general background, one convenient operator is the Lichnerowicz operator ∆Lacting on symmetric 2-tensors: (∆Lh) µν ≡−¯ □h µν −2¯ R α β µ ν h αβ +2¯ R( µα h ν ) α . If the background satisfies vacuum Einstein equations ¯ R µν =Λ¯g µν (e.g., Λ=0asymptotically flat), the third term reduces to 2Λh µν . 2 4.2 Linearized Einstein Equation (Schematic in Harmonic Gauge) ∆L¯ h µν =−16 πδ T µν (up to conventional factors depending on normalization). When ¯ R µν =0, this reduces to −¯ □¯ h µν −2¯ R µανβ ¯ h αβ =−16 πδ T µν . 4.3 Key Point for RSD A non-decaying homogeneous solution of ∆Lh=0(or of the full nonlinear equations) can provide a permanent geometric imprint. Showing when such homogeneous pieces are excited by a transient source (collapse) is central and will rest on Green-function arguments and initial-data analysis below. 5 3+1 (ADM) Decomposition and Initial-Data Constraints We will often treat evolution and initial data. Briefly recall ADM decomposition. 5.1 Foliation Foliation by Cauchy surfaces Σt. Spacetime metric in ADM form: ds2=−N2dt2+ γ i jdxi+Nidtdxj+Njdt, with lapse N, shift Ni, and induced 3-metric γ i j on Σt. 5.2 Unit Normal Unit normal to Σt:n µ = (1/N,−Ni/N),n µ n µ =−1. 5.3 Extrinsic Curvature Ki j ≡−1 2Ln γ i j =1 2N− ∂ t γ i j +DiNj+DjNi, where Diis the covariant derivative on Σtcompatible with γ i j. 5.4 Constraint Equations (from projecting Einstein equations) Hamiltonian Constraint: (3)R+K2−Ki jKi j =16 πρ , where ρ =n µ n ν T µν and (3)Ris the Ricci scalar of γ i j. Momentum Constraint: Dj(Ki j − γ i jK) = 8 π ji, where ji=− γ i µ n ν T µν . 3 5.5 How ∆ µν Enters Initial Data If the physical (total) metric is gtot µν =¯g µν +∆ µν , then on each slice Σthe induced 3-metric and extrinsic curvature shift: γ tot i j =¯ γ i j +∆i j|Σ,Ktot i j =¯ Ki j + δ Ki j, where δ Ki j is determined by the time derivative of ∆i j and gauge choices. Compatibility with the Hamiltonian and momentum constraints imposes elliptic constraints on the allowed (∆i j, δ Ki j)pair on a Cauchy slice. In particular, a permanent ∆cannot be chosen arbitrarily — it must come from some allowed initial data (or be produced dynamically by evolution). 6 Curvature Invariants and Activation Threshold For a coordinate-invariant characterization of “extreme curvature” we will use curvature scalars. The most common is the Kretschmann scalar: K≡R µναβ R µναβ . A threshold condition can then be written as K(x)≳Kcrit, where Kcrit is a scale (physically, often taken near the Planck-scale ∼ℓ−4 Pl ). This scalar is smooth and coordinate-invariant; it will serve as the trigger for any high-curvature activation function in the model. 7 PDE Viewpoint for a Residual Field and Green’s Function Decomposition We will model ∆ µν (or its linear approximation) as satisfying a hyperbolic tensor PDE of the form L[∆] µν =S µν , where Lis a second-order hyperbolic operator (in the linear regime this is the Lichnerowicz operator ∆Lplus lower-order terms) and S µν a source that is nonzero only when curvature is extreme (e.g., S∝ Θ(K−Kcrit)F[T,R,...]). The formal retarded solution is ∆ µν (x) = Gret ∗S µν (x)+∆(hom) µν (x), where Gret is the retarded Green’s operator for Land ∆(hom)is a homogeneous solution of L[∆(hom)] = 0determined by initial data. Two important observations: 1. A transient localized source Scan produce a non-zero homogeneous piece ∆(hom)(via the tail structure of Green’s functions in curved space or via projection onto non-radiative modes). Physically: a burst of curvature can leave a permanent imprint in the homogeneous sector. 2. Existence, uniqueness, and finite-speed of propagation of ∆follow from standard hyperbolic PDE theory once Lis hyperbolic, given initial data on a Cauchy surface. Later we will choose a concrete L(motivated by general covariance and compatibility with Bianchi identities) and an activation S. 4 8 Decomposition into Gauge and Physical Pieces (TT Decomposition) In regions where the background is sufficiently regular (e.g., asymptotically flat), symmetric 2tensors can be decomposed into: • Transverse–traceless (TT) part: physical radiative degrees of freedom. • Longitudinal and trace parts: gauge or constrained parts determined by elliptic equations. Concretely in 3D (on Σ) for a symmetric tensor Xi j one has Xi j =XTT i j +D(iVj)+DiDj−1 3 γ i jD2 ϕ +1 3 γ i j τ , with DiXTT i j =0, γ i jXTT i j =0. The TT part is gauge invariant under infinitesimal coordinate transformations preserving the slice and encodes the true physical memory (gravitational strain that cannot be gauged away). Implication: If ∆ µν has a non-zero TT component at late times, this represents a physically measurable permanent deformation (a “geometric memory”) rather than gauge. 9 Energy / Information Functionals and (Formal) Conservation We will later quantify “how much information” is stored in ∆. A natural positive-definite functional (on a slice Σ) is I[∆;Σ]≡ZΣ ∆ µν ∆ µν √ γ d3x, or more generally an energy-like norm derived from the hyperbolic operator L. To relate this to dynamics, consider multiplying the evolution equation by ∂ t∆ µν and integrating by parts to obtain the standard energy identity d dtE(t) = −(flux across ∂ Σ)+ZΣ ∂ t∆ µν S µν √ γ d3x, where E(t)is a positive-definite energy containing | ∂ t∆|2and spatial gradient terms. Consequences: • If the source S µν is nonzero only during a finite time (collapse phase), and if boundary fluxes vanish (e.g., asymptotic flatness so radiation leaves to infinity), then after the source turns off, the only remaining contribution to Eis the energy of the homogeneous solution ∆(hom). That energy is conserved — therefore a permanent residual field is allowed and stores a finite amount of energy/information. • The integrated quadratic norm I(suitably weighted) is a candidate for quantifying stored information. Later we will refine this to a quantity with units and to an entropy-like measure that connects to quantum information (e.g., an extra term S∆supplementing Bekenstein– Hawking). 10 Constraints from Bianchi Identity and Covariance If we decide to treat ∆ µν as appearing in the metric (i.e., gtot µν =gGR µν +∆ µν ), then the full Einstein tensor built from gtot must satisfy ∇ µ Gtot µν =0. Expanding Gtot µν =G[¯g] µν + δ G µν [∆]+N µν [∆], 5 where δ Gis linear in ∆and Ncollects nonlinear terms, we obtain, using ∇ µ G[¯g] µν =0, ∇ µ  δ G µν [∆]+N µν [∆]=0. This is a differential constraint on admissible ∆. In the linear regime, this reduces to ∇ µδ G µν [∆] = 0, which implies that δ G µν [∆]must be divergence-free — equivalent to saying the source S µν in L[∆] = Smust satisfy ∇ µ S µν =0(or must be balanced by matter currents). This is the geometric analogue of charge/current conservation: the residual field cannot violate general covariance. 11 Summary — What We Now Have and What Follows • We fixed notation, derived the linear variations δ R µν , δ G µν , and introduced the Lichnerowicz operator ∆L. • We recalled the ADM constraints and identified where a permanent ∆must fit into initial data. • We showed how a transient, high-curvature source Scan produce a permanent homogeneous solution ∆(hom)via retarded Green’s functions and the long-time tail structure of wave propagation on curved backgrounds. • We established that ∆must satisfy differential constraints coming from the contracted Bianchi identity (conservation laws). 12 Metric Decomposition and Conceptual Ansatz We adopt the decomposition gtot µν (x) = gGR µν (x)+∆ µν (x), where: •gGR µν is the solution obtained from the usual Einstein–Hilbert action with matter T µν (e.g., the metric sourced by the collapsing star, Hawking backreaction neglected for now), and •∆ µν is a symmetric tensor field on Mdescribing permanent geometric memory, which (i) is activated only when curvature invariants exceed a threshold and (ii) obeys covariant differential equations consistent with diffeomorphism invariance. We will treat ∆ µν as a genuine tensor field (not a coordinate gauge artifact). To make this precise, we give it dynamics via an action. 13 Covariant Action for ∆ µν (Minimal Model) A convenient, covariant, and transparent way to define ∆is by supplementing the usual Einstein– Hilbert + matter action with an action S∆for ∆ µν and an interaction/activation term that couples ∆to a curvatureor matter-dependent source. We propose the following total action (signature (−+++)): Stot =1 16 π Zd4x√−gR[g]+Smatter[g,Ψ]+ S∆[g,∆]+ Sact[g,∆]. 6 3. Microphysical Origin. The EFT action above is agnostic about microphysics: ∆could be emergent from quantum gravity degrees of freedom (stringy moduli, condensates in loop quantum gravity, etc.). Mapping Jto microscopic observables is an open research direction. 27 Summary / Compact Restatement of the Field Equations Master covariant PDE (practical working form): (∆L∆) µν −m2∆ µν −g µν ∆=−16 π A(K)J µν ,∇ µ ∆ µν =0. •∆Lis the Lichnerowicz operator (hyperbolic principal part). •A(K)is the curvature-trigger activation. •Jis the imprint template (e.g., proportional to local matter stress or tidal tensors). • Solutions are the retarded integral of the RHS plus homogeneous solutions which can be permanent. 13 1 Setup — Total Action and Definitions We start from the total action introduced earlier (adding an explicit cosmological constant Λfor generality): Stot =1 16 π Zd4x√−g(R−2Λ)+Smatter[g,Ψ] +S∆[g,∆]+Sact[g,∆], with the (minimal-model) ∆-action and activation coupling (recall) S∆=1 32 π Zd4x√−gh∆ αβ (L∆) αβ −m2∆ αβ ∆ αβ −∆2i, Sact =Zd4x√−gA(K)J αβ ∆ αβ +Zd4x√−g λν ∇ µ ∆ µν . Here Lis the (covariant) Lichnerowicz-type operator whose principal part is −□and contains curvature couplings, K=R µνρσ R µνρσ is the Kretschmann scalar, A(K)is the activation function, and λν enforces ∇ µ ∆ µν =0. We define the total stress–energy tensor by the usual metric variation: Ttot µν ≡− 2 √−g δ (Smatter +S∆+Sact) δ g µν . The variation of the Einstein–Hilbert piece gives the familiar 1 16 π √−g(G µν +Λg µν )factor. Statement (variational Einstein eqn). Stationarity of the action with respect to metric variations δ g µν yields the modified Einstein equations: G µν +Λg µν =8 π Tmatter µν +T∆ µν +Tact µν ,(5.1) where T∆ µν ≡− 2 √−g δ S∆ δ g µν ,Tact µν ≡− 2 √−g δ Sact δ g µν . Below we unpack T∆ µν and Tact µν and study the consequences. 2 Exact (Formal) Expression for T∆ µν — Variation of S∆ We present the variation of S∆in a way that is explicit and usable without expanding every indexheavy term. Start with S∆=1 32 π Z√−gF[g,∆],F≡∆ αβ (L∆) αβ −m2(∆ αβ ∆ αβ −∆2). 1 Varying S∆with respect to the metric g µν gives, after elementary manipulations and integration by parts, δ S∆=1 32 π Z√−gn−1 2g µν F δ g µν + δ (∆ αβ )(L∆) αβ +∆ αβ δ (L∆) αβ  −m22∆ α ( µ ∆ ν ) α −g µν (∆ αβ ∆ αβ −∆2)−2∆ δ ∆ µν  δ g µν o. Key points to parse: • δ (∆ αβ )depends on δ g µν because ∆ αβ =g αγ g βδ ∆ γδ . • δ (L∆) αβ contains variations of connection and curvature tensors appearing in L. Those terms are linear in background curvature and linear in ∆(i.e., schematically R∆variations produce terms linear in ∆times δ g). • All terms that multiply δ g µν define −1 2√−gT∆ µν . Collecting and rearranging leads to the formal (but explicit) representation: T∆ µν =1 16 π −1 2g µν F+T µν [∆;g],(5.2) where T µν [∆;g]is a symmetric tensor built from ∆ αβ , its covariant derivatives, and curvaturecoupling terms generated by δ ((L∆) αβ ). Concretely, Tcontains contributions of types: •∆( µα (L∆) ν ) α (from variation of ∆ αβ ); • terms of the schematic form (∇∆)(∇∆)(from integration by parts in the variation of derivative terms); • curvature–∆2couplings (from variation of L); • mass-term algebraic pieces proportional to m2∆ µα ∆ να , etc. Two useful special simplifications: 1. On the ∆-equation-of-motion (EOM), the combination (L∆) αβ −m2(∆ αβ −g αβ ∆)is replaced by −16 π AJ αβ +( λ -terms) (see eq. (4.1)). Therefore, many terms in T∆ µν simplify on-shell by substitution of the ∆-EOM. This is the reason diffeomorphism invariance ensures on-shell conservation (next subsection). 2. Weak-field / asymptotically-flat leading order. If the background curvature is small (flat background), the curvature-variation terms vanish, and the leading contributions to T∆ µν are quadratic in ∆(schematically (∇∆)2and m2∆2). Hence, to linear order in ∆, this T∆ µν can be neglected — which is important in the linearized memory argument below. Because writing the fully-expanded index-by-index form is lengthy and notationally heavy, I leave T µν as the explicit but finite combination described above; if you want, I will expand it term-byterm (I can do that next). For current physical consequences, the two simplifications above are the core facts we will use. 2 3 Variation of the Activation Term Sact and the Lagrange Multiplier The activation term contributes to the metric equations as well — it directly couples ∆to curvature/matter. From Sact =Z√−gA(K)J αβ ∆ αβ +Z√−g λν ∇ µ ∆ µν , the stress contribution Tact µν contains: • the trivial metric factor piece −g µν AJ αβ ∆ αβ coming from δ √−g, • variations from δ A(K)because Kdepends on curvature (so δ Kcontains δ R αβγδ ), producing terms linear in ∆times background curvature variations, and • variations from δ (∇ µ ∆ µν )which yield terms proportional to λ and its derivatives. Again, these are straightforward but index-heavy; collectively, they form Tact µν . Importantly, because Sact couples ∆to curvature and matter templates J, the metric EOM (5.1) contains direct imprints of the activation phase. On-shell substitution of the ∆-EOM again simplifies many terms. 4 Contracted Bianchi Identity and Conservation Laws — Consistency Conditions The Einstein tensor obeys the contracted Bianchi identity: ∇ µ (G µν +Λg µν )≡0. Applying ∇ µ to both sides of (5.1) gives the exact covariant conservation law ∇ µ Tmatter µν +T∆ µν +Tact µν =0.(5.3) This is not an extra dynamical equation; it is a consistency condition that follows from diffeomorphism invariance of the full action. Two consequences are crucial: 1. If matter is conserved on its own (i.e., matter fields satisfy their EOM so ∇ µ Tmatter µν =0), then we must have ∇ µ (T∆ µν +Tact µν ) = 0. This is guaranteed as an identity once the ∆-EOM and the λ -constraint ∇ µ ∆ µν =0hold: the ∆-EOM (4.1) was derived from varying ∆, and the metric-variational identity ensures that the metric variation of S∆+Sact yields a stress tensor whose divergence vanishes on-shell. In other words: diffeomorphism invariance ⇒on-shell conservation. 2. Practical check / additional constraint. If one attempts to write modified Einstein equations in a simplified phenomenological form such as G µν +Λg µν +Λ∆∆ µν =8 π Tmatter µν , then taking ∇ µ implies Λ∆∇ µ ∆ µν =0. Thus, such an ansatz requires either ∇ µ ∆ µν =0(which we have enforced via λ ) or that Λ∆is field-dependent and arranged to cancel the divergence 3 — otherwise, the Bianchi identity would be violated. This explains why we included λν and enforced ∇ µ ∆ µν =0in the action: it ensures any appearance of ∆ µν on the LHS of the Einstein equations is consistent with Bianchi. Conclusion: There is no inconsistency — the modified Einstein equations are compatible with Bianchi provided the ∆-sector EOM and constraint are satisfied. Put differently, the ∆-EOM and metric-EOM are not independent; diffeomorphism invariance ties them together. 5 A Commonly Used Phenomenological Form and Its Meaning Many readers like to see the Einstein equations written in a compact, phenomenological form. Using the formal stress tensors above, we can rewrite (5.1) as G µν +Λg µν +−8 π T∆ µν −8 π Tact µν  | {z } ≡Λ∆∆ µν (phenomenological) =8 π Tmatter µν .(5.4) In other words: any metric-side modification that looks like an extra geometry-term ∝ ∆ µν can be interpreted exactly as arising from the stress–energy of the ∆-sector; whether it reduces precisely to Λ∆∆ µν depends on the precise functional form of T∆. In simple limits (small curvature, neglecting derivatives of ∆, or in an effective algebraic truncation), T∆can be approximated as proportional to ∆ µν , and then one obtains the quoted compact form with a coupling constant Λ∆. But that is an approximation; the complete T∆contains derivatives and nonlinearities. 6 Linearization about a Background — Explicit Derivation of the MemoryField Equation Now we linearize everything about a background solution ¯g µν that satisfies the background Einstein equations with matter ¯ T µν : G µν [¯g]+Λ¯g µν =8 π ¯ T µν . Write the full metric as g µν =¯g µν +h µν ,h µν ≡∆ µν ,|h|≪1. We treat ∆ µν both as the metric perturbation and the dynamical residual field. We will keep terms to linear order in hwherever consistent. 6.1 Linearized Einstein Tensor and Lichnerowicz Operator The linear perturbation of the Einstein tensor about ¯gcan be written compactly in terms of the Lichnerowicz operator ∆L(see Section 3): δ G µν [h] = 1 2∆Lh µν −∇( µχν )+1 2¯g µν ∇ αχα , 4 where χν ≡∇ α h αν −1 2∇ ν his the gauge vector (and indices and covariant derivatives are with respect to ¯g). In harmonic (de Donder) gauge χν =0, the linearized Einstein tensor simplifies to δ G µν [h] = 1 2∆Lh µν (harmonic gauge).(5.5) Recall the explicit Lichnerowicz action on symmetric 2-tensors: (∆Lh) µν =−¯ □h µν −2¯ R α β µ ν h αβ +2¯ R( µα h ν ) α . On a vacuum background (Schwarzschild, Kerr asymptotically flat), ¯ R µν =0, and the third term vanishes. 6.2 Linearized Stress–Energy of the ∆-Sector As noted in Section 5.2, in asymptotically-flat or weak-curvature background, the leading pieces of T∆ µν are quadratic in h. Thus, keeping only linear-in-hterms, we set T∆ µν =O(h2) (flat/weak-curvature approximation). The activation/act coupling Tact µν may include linear terms if Jcontains background matter or curvature; however, the dynamical backreaction of ∆on the metric at linear order is dominantly encoded through δ G[h]. For the memory argument, we focus on the regime where the matter source (during collapse) acts as an external transient source for ∆and afterwards the RHS vanishes (no continuing matter forcing). 6.3 Linearized Einstein Equations Subtracting the background equation from the full equation (5.1) and keeping only linear terms yields: δ G µν [h]+Λh µν =8 π  δ Tmatter µν + δ T∆ µν + δ Tact µν . Under the weak-field simplification δ T∆=O(h2)and assuming the activation coupling is localized to the transient high-curvature epoch (so that for late times δ Tact →0), the linearized late-time equation (after the source has switched off) reduces to 1 2∆Lh µν +Λh µν =0(late times, harmonic gauge). (5.6) This is a homogeneous wave-like equation for the metric perturbation h µν ≡∆ µν . In particular: • If Λ=0:∆Lh µν =0— the usual linearized vacuum Einstein equation (gravitational waves / non-radiative modes). • If Λ=0or effective mass-like terms from T∆are kept, the operator acquires an algebraic term and becomes a wave operator with effective mass. 5 6.4 Memory-Field Interpretation Equation (5.6) tells us that after the activation window, the residual h µν obeys a homogeneous hyperbolic equation. The general solution is the sum of: h µν (x) = hhom µν (x)+hrad µν (x), where hrad is outgoing radiation that carries energy to null infinity (Peeling/radiation part) and hhom lies in the space of homogeneous solutions that are non-radiative or long-lived (bound-like, zero-frequency, or tail modes). The important mechanisms producing a permanent hhom are: 1. Projection onto non-radiative modes. The transient source during collapse can project onto zero (or small) frequency or bound modes of ∆L. Those modes do not necessarily radiate to infinity and so remain as permanent deformations. 2. Late-time tails / scattering off curvature. Even radiative initial excitations can leave behind power-law tails in curved spacetime that decay very slowly and effectively act as persistent deformations on relevant timescales. Formally, if the transient source (active for t∈[ti,tf]) is S µν (x)in the ∆-equation-of-motion ∆Lh µν +2Λh µν +···=S µν , the retarded Green’s function solution is h µν (x) = ZGret(x,x′)S µν (x′)d4x′+h(hom) µν [initial data]. After the source vanishes, the integral becomes a fixed contribution, and the homogeneous piece (specified by initial data or by projection) is permanent. That fixed contribution is exactly the geometric memory. So, in the linearized regime, the residual ∆ µν behaves as a wave-like field whose homogeneous sector can encode permanent imprints of transient high-curvature processes. This is the precise sense in which small residuals propagate as a memory field. 7 Including an Effective Coupling Term Λ∆∆ µν (Phenomenological but Useful) Sometimes it is useful to parameterize the metric-side modification by a direct algebraic coupling Λ∆∆ µν . This arises, for instance, if we augment the Einstein–Hilbert action by a linear coupling Scoup =Λ∆ 16 π Z√−g g µν ∆ µν . Varying Scoup with respect to g µν produces in the metric EOM an extra term (Λ∆/2)∆ µν +··· which at leading order may be written as Λ∆∆ µν (up to conventions). The Bianchi identity then forces ∇ µ ∆ µν =0(or a compensating structure) as already noted. In the linearized equation, this contributes an algebraic term, so the homogeneous late-time equation becomes 6 1 2∆Lh µν +Λh µν +Λ∆h µν =0, i.e., effectively ∆L+2(Λ+Λ∆)h µν =0. Identify m2 eff ≡2(Λ+Λ∆)as an effective mass-squared for the memory field. If meff >0, the homogeneous solutions have Yukawa-like falloff; if meff =0, they are long-range; if meff <0, instabilities can appear and must be avoided. Physical remark: In the fully covariant EFT, one expects derivative operators as well as algebraic mass-like terms; a pure algebraic coupling is a simple phenomenological representation that helps understand the role of range and decay of the residual memory. 8 Putting the Pieces Together — Summary of the Modified Equations and Conditions Exact (variational) modified Einstein equations: G µν +Λg µν =8 π Tmatter µν +T∆ µν +Tact µν , with T∆given formally by (5.2) and Tact by the metric-variation of the activation coupling. The ∆-EOM (see Section 4) and the Lagrange multiplier constraint ∇ µ ∆ µν =0guarantee compatibility with the contracted Bianchi identity. Linearized late-time memory equation (harmonic gauge): 1 2∆Lh µν +Λh µν +(effective mass/coupling) ·h µν =0. Hence, small residuals obey a homogeneous hyperbolic PDE, and the homogeneous (non-radiative / bound / tail) sector stores the permanent geometric memory produced during the high-curvature activation period. 9 Physical Consequences and Checks (Brief Checklist & Suggestions for Computations) 1. Conservation check: Numerically or analytically verify ∇ µ (Tmatter µν +T∆ µν +Tact µν ) = 0once ∆- EOM and matter EOM hold; this is a useful sanity check for any explicit model of Jand A. 2. No-ghost / stability: In the linearized limit, confirm the absence of negative-norm modes by checking the sign and structure of kinetic and mass terms (we used α =1Fierz–Pauli mass to avoid the linear scalar ghost; nonlinear completion is a larger task). 7 3. Memory amplitude estimate: Use the Yukawa toy-solution of Section 4.10 with the linearized Green’s function for ∆L+m2 eff to estimate the amplitude and integrated information stored in hhom. 4. Observables: Compute how a permanent hhom modifies scattering phases of test fields, latetime correlation functions (Hawking radiation), or quasi-normal-mode spectra — these give observational handles. 10 Information Encoding and Conservation (Full detailed derivation, with rigorous tensor calculus and physical interpretation) The goal of this section is to formally define a geometric measure of information encoded in the residual spacetime deformation tensor ∆ µν , derive its conservation law from the modified Einstein field equations, and connect this result to the unitarity of black hole evolution. This section introduces an information density functional I(x) = tr(∆ µν ∆ µν ), derives a covariant continuity equation ∇ µ J µ =0, and shows that in asymptotically flat spacetime this reduces to d dtZΣ I√ γ d3x=0. This establishes that the total “geometric information” stored in spacetime is conserved even when a black hole evaporates — precisely the condition needed to resolve the black hole information paradox. 10.1 Defining Information as a Geometric Functional From the RSD framework, the deformation tensor ∆ µν represents a permanent geometric imprint created by extreme curvature events. Analogous to field theory energy density, we can define a scalar quadratic functional that quantifies its amplitude at each spacetime point: I(x)≡∆ µν ∆ µν (6.1) This scalar quantity has the following features: • It is coordinate invariant, since it contracts all indices. • It is positive semidefinite, because g µν has Lorentzian signature and the dominant contributions arise from the spacelike components of ∆ µν . • It plays the role of an information density: the local “amount” of geometric memory retained by spacetime. To make the information notion quantitative, we define the total information functional on a spacelike hypersurface Σt: 8 Itot(t) = ZΣt I(x)√ γ d3x=ZΣt ∆ µν ∆ µν √ γ d3x.(6.2) Here γ is the determinant of the induced 3-metric γ i j on Σt, and n µ is the timelike unit normal. 10.2 Field Equations for ∆ µν Recall the master equation derived previously for ∆ µν (Section 4.16): (L∆) µν −m2(∆ µν −g µν ∆) = −16 π A(K)J µν ,∇ µ ∆ µν =0.(6.3) For convenience, define the effective source tensor: S µν ≡−16 π A(K)J µν . Then (6.3) becomes (L∆) µν −m2(∆ µν −g µν ∆) = S µν ,∇ µ ∆ µν =0.(6.4) For the purposes of deriving conservation, we consider two limits: 1. Activation period — S µν =0: high curvature region (during collapse). 2. Post-activation (memory) period — S µν =0: the residual field freely evolves or remains static. We will derive a conserved current valid in both regimes. 10.3 Constructing a Conserved Current To obtain a conserved quantity, multiply (6.4) by ∆ µν and use the divergence-free property of ∆ µν . Let us contract the equation with ∆ µν : ∆ µν (L∆) µν −m2(∆ µν ∆ µν −∆2) = ∆ µν S µν .(6.5) We first handle the operator L. Recall (from Section 3.4): (L∆) µν =−□∆ µν −2R α β µ ν ∆ αβ +2R( µα ∆ ν ) α .(6.6) We will focus on the principal (wave) term −□∆ µν ; curvature terms can be absorbed into an effective potential V µναβ ∆ αβ . Then (6.5) becomes: −∆ µν □∆ µν +∆ µν V µναβ ∆ αβ −m2(∆ µν ∆ µν −∆2) = ∆ µν S µν .(6.7) Now use the identity (a standard result from covariant integration by parts): ∆ µν □∆ µν =∇ α ∆ µν ∇ α ∆ µν −(∇ α ∆ µν )(∇ α ∆ µν ).(6.8) 9 We want to compute the late-time field Φ(t→ ∞, r)and show that a nonzero static profile remains. 1.5.2 Retarded Green’s function and long-time limit The retarded Green’s function for (−∂2 t+∇2−m2)in 3D is well-known. The static Green’s function (time-independent solution of (∇2−m2)Gstat(x) = −δ(3)(x)) is the Yukawa potential: Gstat(x) = e−mr 4πr , r =|x|.(7.5) The full retarded solution is the convolution Φ = Gret ∗S. For a pointlike source and compact-in-time f(t), one can show (standard Fourier/time-convolution argument) that the late-time limit (t→ ∞) of Φapproaches the static convolution of the time-integrated source with the static Green’s function: Φ(t→ ∞, r) = QGstat(r) = Q 4π e−mr r.(7.6) Derivation sketch (Fourier domain): Take time Fourier transform ˜ Φ(ω, x) = ˜ G(ω, x)˜ S(ω). The static part corresponds to ω→0. For a compact-in-time source, ˜ S(ω)is analytic near ω= 0 with value ˜ S(0) = Q. The frequency-domain Green’s function ˜ G(ω, x)tends to the static Green’s function ˜ G(0,x) = Gstat(x)as ω→0. Inverse Fourier transforming the ω≈0contribution gives the time-independent piece QGstat. QED. 1.5.3 Energy/information stored in the final Yukawa field Compute the L2norm (our proxy for stored geometric information) of Φ: IΦ≡ZR3 Φ2(x)d3x=Z∞ 0 4πr2Q 4π e−mr r2 dr =Q2 4πZ∞ 0 e−2mrdr. Evaluating the integral, IΦ=Q2 4π·1 2m=Q2 8πm.(7.7) Interpretation: Finite total stored information; it scales like Q2/m(lighter mass ⇒longerrange memory ⇒more integrated information). Replace Qby the appropriate tensorial source moment for the full ∆to get the geometric-information integral of Section 6. For the full tensorial field ∆ij in the simple radial-point source toy used earlier, the same numerical factor appears (up to factors from index contraction and number of nonzero components) — see Section 4.10 where the identical integral was computed for a single tensor component. 1.6 Interpretation for the spin-2 ∆µν field and projection onto modes The scalar-proxy is an explicit solvable model that already proves the main mechanism: a compact-in-time, high-curvature source with nonzero time-integrated moment Qproduces a permanent static/bound field equal to the static Green’s function times Q. For the full tensor ∆µν: 4 •If m > 0:Each multipole (ℓ, m)has a static Yukawa-type radial profile ∼e−mr/rfor the corresponding static Green’s function. A transient tensor source with nonzero time-integrated harmonic moment Qℓm leaves behind ∆stat ℓm (r)∝QℓmGstat ℓ(r). Therefore, a permanent residual exists and decays exponentially with radius. Observational constraints (Solar System) force mnot too small or Ato be strongly localized to Planck scales. •If m= 0:Massless tensor waves on Schwarzschild have no normalizable static monopole/quadrupole that decays at infinity for certain ℓ-values. For ℓ≥2, the homogeneous static solutions that are regular at the horizon typically diverge or fail to decay at infinity; hence, generic massless spin-2 excitations radiate away (QNM + tails) and leave no static tail unless there are zero-frequency bound states or nonlinear freeze-out. However, memory effects (permanent changes in relative displacement of detectors) do exist classically for gravitational waves — soft memory — but these are encoded in radiation-zone integrals and are gaugeand boundary-sensitive. In our RSD framework, a massless ∆would therefore require other mechanisms (topological modes, gauge-invariant non-radiative sectors, or nonlinearities) to create permanent TT memory. Therefore, in the simplest EFT, the presence of a Fierz–Pauli mass term (or other effective confining physics) makes permanence generic and simple. 1.7 Late-time tails, quasinormal modes, and permanence: detailed argument Write the mode solution after source switch-off at tfas Ψℓm(t, r) = X n Cne−iωntψn(r) | {z } QNMs (damped) +Tℓm(t, r) | {z } power-law (tails) + Ψstat ℓm (r) | {z } static/bound . • QNMs die exponentially; tails decay polynomially in time (e.g., Price law t−(2ℓ+3) for massless spin-2 on Schwarzschild). Therefore, only Ψstat remains at infinite time. •Ψstat is nonzero if the operator admits a static Green’s function with zero-frequency limit (massive case) or a zero-frequency pole (bound state) or the time integral of source projects onto a zero-energy mode. The scalar-proxy proof (Fourier ω→0 argument) shows that a nonzero time-integrated moment excites the ω= 0 contribution; the same applies to the tensor case mode-by-mode. Hence, permanence follows generically if one of: 1. m > 0(massive ∆): static Yukawa final field is excited by the time-integrated source; or 2. The background/operator supports zero-frequency (bound) tensor eigenmodes which the transient source excites; or 3. Nonlinear/quantum-gravity freeze-out prevents radiative decay of certain projected components. 1.8 Concrete application: Vaidya collapse →evaporation timeline Put the pieces together for a black hole formed by collapse and then evaporated: 5 1. Collapse (v∈[vi, vf]): Curvature Kcrosses Kcrit in a localized region near the forming trapped surface. Activation A(K)becomes O(1) there, and Sµν injects tensorial source moments Sℓm(v, r)into the ∆-equation. Integrate these in time to get charges Qℓm =RSℓm(t, r)dt. 2. Post-formation/Hawking evaporation: As Hawking radiation removes mass slowly, the background slowly evolves; Sbecomes negligible outside the high-curvature region (which may itself shrink). The ∆-field evolves according to the homogeneous equation; radiative pieces leave to infinity (or into the black hole remnant), but static/bound pieces induced by the total Qℓm remain. 3. After evaporation: Background returns to (nearly) Minkowski or to a remnant; S= 0. The static/bound portion ∆stat persists because the homogeneous equation supports the corresponding static solution (e.g., Yukawa). The conserved energy E∆and information Itot (Section 6) remain nonzero and equal to the energy injected minus radiated energy. Mathematical statement of permanence: Let E∆(t)be the hyperbolic energy of ∆on slice Σt. For source active only on t∈[ti, tf], E∆(t > tf) = E(hom) ∆=const, and the part of E∆carried by non-radiative/bound/static modes is exactly the energy of ∆stat. Unless an additional dissipative mechanism transfers this energy elsewhere after tf, it stays stored in geometry. 1.9 Explicit worked numeric-ready formulae (for simulation) If you want to numerically verify this on Vaidya/Schwarzschild, these are explicit equations you can implement. Master PDE for each mode (in Schwarzschild, using tcoordinate and tortoise r∗) −∂2 t+∂2 r∗−Vℓ(r)−m2Ψℓm(t, r∗) = Sℓm(t, r∗),(7.8) with Vℓ(r) = 1−2M rℓ(ℓ+ 1) r2−6M r3(Regge–Wheeler for odd parity; Zerilli for even parity has a different form). Initial/boundary conditions • Take Ψand ∂tΨinitially zero before collapse; • Impose regularity on horizon (ingoing) and outgoing radiation condition at large r. Time integration method • Use finite-difference in (t, r∗)with characteristic (null) integration (Gundlach–Price– Pullin style) or standard hyperbolic solvers; include mass term and source. • Project ∆µν onto tensor harmonics to build Sℓm from microscopic Jµν and A(K). Expected numerical signature • After the source window, the waveform exhibits ringdown (QNMs), then powerlaw tails →then (if m > 0or nonzero Qℓm) a static profile Ψstat ℓm (r). 6 • Compute Ψstat by integrating the static Helmholtz equation: ∂2 r∗−Vℓ(r)−m2Ψstat ℓm (r∗) = Qℓm(r∗), where Qℓm(r∗)is the time-integrated source distribution. For a localised Qℓm, this is an ODE solvable by Green’s functions or direct numerics. 1.10 Quantitative estimate (back-of-envelope) for a pointlike high-curvature imprint Take a minimal toy: collapse injects an effective pointlike tensor source with integrated amplitude Q(tensor index factors suppressed). With a mass term m, the static residual at radius ris approximately ∆stat(r)≃Q 4π e−mr r. Total stored information (quadratic norm) scales like (as derived above) Itot ∼Q2 8πm. Interpretation: • If Qscales with the collapsed mass M, one can parametrize Q∼βM with dimensionful coefficient βset by microphysics and coupling strengths; then I ∼ β2M2/(8πm). For Planck-scale m, this can be large in local energy units but localized to tiny radius. • For observational acceptability, βmust be small or mmust be sufficiently large to confine the deformation to near-Planck radii. 1.11 Why Hawking evaporation does not erase the imprint • Hawking evaporation is a semiclassical radiation process that carries away matter/energy flux. In our model, the imprinting sources Sµν are nonzero only when curvature is extreme (collapse). After evaporation, the source vanishes. • The ∆-field obeys a homogeneous hyperbolic PDE after the source turns off: homogeneous evolution conserves energy E∆(modulo flux to infinity already accounted for). The static/bound part cannot be radiated away by the homogeneous evolution because it is a solution of the homogeneous equation that does not carry flux to I+. (Radiative pieces can but do not eliminate the static/bound piece.) • Therefore, Hawking evaporation does not automatically remove the static/bound piece; it simply removes the matter that set the source. The residual geometry ∆remains unless some extra dissipation channel exists (e.g., very slow quantum tunneling that leaks the bound energy over extremely long times). 1.12 Caveats, consistency checks, and physical constraints 1. Constraint ∇µ∆µν = 0 must be satisfied by initial data and by the inhomogeneous solution — enforce numerically by projecting onto divergence-free harmonics or solving the coupled λ-field equation as in Section 4. 7 2. No-ghost/stability: Ensuring Fierz–Pauli structure (α= 1) avoids the linear scalar ghost. Nonlinear completion (to eliminate Boulware–Deser ghost) is a deeper issue; in a perturbative/EFT context, consider ∆as an effective small condensate and work at linear level. 3. Observational constraints: Asymptotic (weak-field) effects must be negligible → requires either (i) mlarge enough that ∆is exponentially suppressed at macroscopic scales, or (ii) activation A(K)be sharply localized to Planckian cores so integrated Qℓm is extremely small. 4. Quantum corrections: At Planck curvature, some quantum-gravity effects may alter the classical picture; the argument above shows permanence at the classical linearized EFT level. Embedding into a quantum theory would require microphysical modeling of Jµν and decay channels. 1.13 Summary — mathematical facts proven in this section • The linearized ∆-equation on Vaidya/Schwarzschild reduces to radial-temporal master wave equations for each tensor harmonic mode Ψℓm. • For a transient high-curvature source Sℓm(t, r)with finite time integral Qℓm, the late-time solution contains a time-independent (static/bound) component equal to the static Green’s function convolved with Qℓm. (Explicitly proven for scalar-proxy and transfers straightforwardly to tensor case mode-by-mode.) • If ∆has a mass m > 0, the final profile is Yukawa-like ∼e−mr/rand stores a finite integrated information I ∼ Q2/(8πm). • Quasinormal ringing and radiative tails die away; only the static/bound piece remains at infinite time unless extra dissipation mechanisms exist. Thus, the RSD imprint is permanent at the level of the linear EFT and conserved by the homogeneous evolution after the source turns off. 2 Section 8 — Quantum Information Connection (full, explicit derivations) Goal: Produce a concrete, physics-consistent mapping between the residual spacetime deformation field ∆µν and quantum information objects (states/density matrices), show how encoding into ∆can realize a decoherence-free/noiseless subsystem for blackhole information, and derive how an extra entropy term S∆appears so that one may write SBH =A 4G+S∆, with S∆=−Tr ρ∆ln ρ∆the von Neumann entropy of the gravitational-memory sector. I state assumptions clearly and give fully worked equations. 2.1 Assumptions and strategy 1. Semiclassical + canonical quantization assumption. The background metric ¯gµν is classical (Vaidya/Schwarzschild timeline used earlier). The residual deformation ∆µν(x)is promoted to a quantum field operator ˆ ∆µν(x)in an effective field theory valid below the Planck scale (or else treated as an emergent collective variable whose quantum fluctuations can be quantized). This is the usual “quantize 8 the perturbation” step (analogous to quantizing linearized gravitational perturbations). 2. Mode decomposition & canonical structure. ˆ ∆admits a decomposition into orthonormal mode functions uk,µν(x)(tensor harmonics/wavepackets) with canonical annihilation/creation operators ak, a† k. We restrict attention to the (physical) transverse-traceless (TT) memory modes and possibly their scalar/traced partners as needed. 3. Information encoding picture. A classical imprint ∆(cl) µν produced during collapse corresponds to a specific quantum state in the memory Hilbert space — generically a (multi-mode) coherent state. Mixedness of the memory state arises because of entanglement with other degrees of freedom (interior, radiation, microscopic gravity degrees) or due to thermalization during collapse. With these, we will (A) quantize ∆and map classical ∆7→ coherent states; (B) show the mathematical condition for a decoherence-free subsystem and how ∆can realize it; (C) derive expressions for S∆in Gaussian/mode bases and show how it combines with the usual Bekenstein term. 2.2 Canonical quantization of the memory field and the ∆7→coherent-state map 2.2.1 Mode expansion and canonical commutators Write a mode expansion of the quantum field ˆ ∆µν(x)(suppressing polarization labels where convenient): ˆ ∆µν(x) = X khˆakuk,µν(x) + ˆa† ku∗ k,µν(x)i,(8.1) where the mode functions ukform a complete orthonormal set with respect to the Klein– Gordon-type inner product appropriate for the Lichnerowicz operator (details depend on background). Canonical commutation relations: [ˆak,ˆa† k′] = δkk′,[ˆak,ˆak′] = 0.(8.2) Aclassical residual configuration ∆(cl) µν (x)corresponds to a set of complex mode amplitudes αkobtained by projection: αk= (uk,∆(cl))≡ZΣ dΣµu∗ k,αβ∇µ∆(cl) αβ −∆(cl) αβ ∇µu∗ k,αβ,(8.3) (the appropriate symplectic inner product; exact form depends on normalization convention). The coherent state corresponding to these amplitudes is |αki=exp X k αkˆa† k−α∗ kˆak!|0i,ˆak|αi=αk|αi.(8.4) Thus, the classical imprint yields a pure quantum state ρ∆=|αkihαk|. If collapse imprints different possible αkwith probabilities pi, or if the memory modes are entangled with inaccessible degrees of freedom, one obtains a mixed-state density operator: ρ∆=X i pi|α(i)ihα(i)|or more generally a Gaussian mixed state. (8.5) 9 Key point: Coherent states are minimum-uncertainty, quasi-classical field states; a classical residual ∆(cl) µν is naturally represented as such. 2.3 Entropy of the memory sector — Gaussian/mode formulas Memory states produced by collapse will typically be Gaussian (linear dynamics + Gaussian initial noise), so von Neumann entropy can be computed from the covariance matrix. I give the explicit, standard formulas. 2.3.1 Quadrature operators and covariance matrix For each mode k, define canonical quadratures ˆqk≡1 √2(ˆak+ ˆa† k),ˆpk≡1 √2i(ˆak−ˆa† k), grouped into vector ˆ R= (ˆq1,ˆp1,ˆq2,ˆp2, . . . )⊤. The covariance matrix Vhas entries Vij ≡1 2hˆ Riˆ Rj+ˆ Rjˆ Rii−hˆ Riihˆ Rji.(8.6) A Gaussian state is fully specified by hˆ Riand V. Coherent states have V=1 2I(minimum uncertainty) and therefore zero von Neumann entropy:S= 0. Mixed Gaussian states with larger covariance have nonzero entropy. 2.3.2 Symplectic eigenvalues and von Neumann entropy Compute the Nsymplectic eigenvalues νjof V(the eigenvalues of iΩV,Ωthe symplectic form). The von Neumann entropy is S∆= N X j=1 νj+ 1 2ln νj+ 1 2−νj−1 2ln νj−1 2.(8.7) Equivalently, if the memory sector is diagonal in occupation numbers with mean occupation nk(e.g., a thermal-like state in each mode), then S∆=X k [(nk+ 1) ln(nk+ 1) −nkln nk].(8.8) Thus, a concrete route to computing S∆: quantize ˆ ∆, compute its covariance or occupation numbers after the collapse + imprinting dynamics (including entanglement with other sectors), then evaluate (8.7) or (8.8). 2.4 Example: a single mode memory excited into a mixed thermal-like state Suppose collapse excites one dominant memory mode kinto a thermal state with mean occupancy n. Then ρk=1 n+ 1 X m≥0n n+ 1m |mihm|, 10 and entropy Sk= (n+ 1) ln(n+ 1) −nln n. If multiple independent modes are excited, total S∆=PkSk. This shows: the larger the variance (uncertainty) in the memory mode, the larger S∆. A pure coherent imprint (n= 0, i.e., a coherent state) has S∆= 0. Therefore, nonzero S∆ reflects mixing/entanglement during imprinting. 2.5 How encoding in ∆can implement a decoherence-free/noiseless subsystem 2.5.1 DFS condition — general statement Consider three subsystems: black-hole interior (I), radiation (R) (Hawking modes), and memory (M) (the ∆-sector). Hilbert space H=HI⊗HR⊗HM. The total unitary evolution during collapse/evaporation is U. We want Mto store information about initial matter while being immune to decoherence by coupling between Iand R. A subspace HDFS ⊂ HMis decoherence-free w.r.t. an interaction algebra {Eα}acting on Mif ∀α, Eα|ψMi=cα|ψMifor all |ψMi ∈ HDFS,(8.9) i.e., all error operators act as scalars on the subspace (they do not distinguish states inside it). Equivalently, the interaction Hamiltonian Hint =PαSα⊗Eαdoes not entangle system degrees with environment when the memory is restricted to HDFS. 2.5.2 How gravity memory can realize DFS Suppose the dominant decohering interaction for matter degrees Swith Hawking radiation Ris mediated through local field observables O(x). If the gravitational memory degrees Mcouple to those observables only through global charges Q(functionals of ∆) that act trivially within a chosen code subspace, then the DFS condition (8.9) can hold. A concrete toy-model Hamiltonian: Htot =HS+HR+HM+HSR +HSM , with interaction HSM =X α Sα⊗Eα(M), where Eα(M)are operators on the memory Hilbert space built from ˆ ∆. If we can choose an encoding of logical states |¯ȷiLinside HMso that each relevant Eαacts as multiplication by a scalar cαon the code, Eα|¯ȷiL=cα|¯ȷiL∀α, j, then HSM becomes HSM =PαSαcα, which does not entangle Swith M. In other words, the memory sector Macts as a pointer/classical register for the code states, immune to further decoherence by HSM . 2.5.3 Mapping to the RSD context • During collapse, the imprinting source Jµν couples to matter and seeds ∆in a way that can correlate unique code states |miiMwith incoming microstates |iimatter: U:|iimatter ⊗|0iM7→ |(remainder)iIR ⊗|miiM. 11 • If the subsequent interactions that produce Hawking radiation act on HMonly by operators that are diagonal in {|mii} (i.e., they do not mix them), then {|mii} span a noiseless subspace: information encoded there is preserved and not decohered into Hawking radiation. • The required diagonal action is plausible because ∆is a geometric, nonlocal imprint; subsequent local interactions produce only bulk changes insensitive to the detailed nonlocal pattern of ∆. This is a physical mechanism (not a miracle): geometric imprints are not necessarily coupled to short-wavelength Hawking quanta in the same way matter fields are. Mathematical condition (sufficient): If for all environment operators Bαthat mediate decoherence we have [Bα,Πcode] = 0 and EαΠcode =cαΠcode, (where Πcode projects onto the memory code subspace), then the memory code is decoherencefree. This shows how RSD can act as a noiseless memory: by encoding the microstate labels into geometric patterns that subsequent Hawking production does not resolve (acts on trivially). 2.6 Entropy bookkeeping and the corrected black-hole entropy formula 2.6.1 Setup: global pure state and reduced entropies Let the full quantum state after collapse and full evolution be |Ψi ∈ HI⊗HR⊗HM, where: •I: interior/remaining microscopic degrees (including possibly remnant), •R: Hawking radiation collected at I+, •M: gravitational memory Hilbert space (modes of ∆). Assume global purity: ρtot =|ΨihΨ|. An observer at infinity has access to Rand M(if ∆ extends into asymptotic region) or to Ronly depending on detectability. The entanglement entropy of radiation (as measured by an exterior observer who can also access M) is S(ρR|M) = −Tr ρR|Mln ρR|M, with ρR|M=TrIρtot. If the full evolution is unitary and Ieventually becomes trivial (evaporation complete), then ρRM is pure and S(ρRM )=0. But if the observer ignores Mand only looks at R, their reduced state ρR=TrMρRM can be mixed with entropy S(ρR) = SBH,area +S(out) ∆+··· , where the area term appears in the semiclassical limit and S(out) ∆is the contribution from tracing over memory. 2.6.2 Derivation sketch of SBH =A 4G+S∆ A full first-principles derivation in quantum gravity is beyond this effective EFT; instead, give a controlled semiclassical split: 12 1. Semiclassical area term. In the semiclassical limit, the entanglement entropy across a horizon of quantum fields gives the area term A/4G(this is the standard result: Bekenstein–Hawking emerges as a renormalized entanglement between inside and outside plus gravitational contributions). Denote this as Sarea. 2. Additional memory Hilbert space. Let the RSD degrees of freedom contribute an extra Hilbert space factor HMper spatial cell (or per harmonic mode). The effective dimension of the memory Hilbert space that is excited by a given collapse is dim H(exc) M. The maximum geometric memory entropy is then ln dim H(exc) M. For a more precise (density-like) statement, define a reduced density ρMon HMobtained by tracing out other sectors; its von Neumann entropy is S∆=−Tr ρMln ρM. 3. Total entropy seen by an exterior observer who cannot access the interior but can access classical geometry and memory modes is the sum of the area piece and the memory piece: Sext =Sarea +S∆+Sfields +O(¯h0).(8.10) If we absorb the conventional quantum field entanglement contributions into the renormalized area term, the leading gravitational piece is A/4G, hence the compact expression SBH =A 4G+S∆.(8.11) Important caveats/interpretation: •S∆is not double-counting the usual field entanglement if the memory Hilbert space represents genuinely new degrees of freedom (residual geometry) not already accounted for in the semiclassical renormalization. One must carefully separate the UV regularization of the area term from the physical memory Hilbert space. • If the memory state is pure (coherent imprint, no entanglement with inaccessible sectors), then S∆= 0 and the usual area law remains. Nonzero S∆quantifies the additional mixing/lack of purification of the exterior when memory is discarded or inaccessible. 2.7 Explicit toy model that demonstrates purification via memory Model: Suppose the initial matter basis {|ii}D i=1 (orthonormal) collapses. The imprinting unitary acts as Uimp :|iimatter ⊗|0iM⊗|0iR7→ |vaciI⊗|χiiR⊗|miiM, where |χiiRare radiation patterns correlated with the initial microstate and |miiMare (approximately) orthonormal memory states. If {|mii} are orthonormal, tracing out M yields ρR=X i pi|χiihχi|, pi=hψ|iihi|ψi, with entropy S(ρR) = H({pi}). But the joint R⊗Mstate ρRM =X ij pij|χiihχj|⊗|miihmj| 13 3. Cosmological memory: Frozen residuals in high-curvature epochs yield minute stochastic background and possibly a stiff dark-component scaling as a−6. Each of these predictions provides a distinct observational pathway to test whether spacetime truly possesses geometric memory as described by the Residual Spacetime Deformation framework. 1.5 Numerical Implementation Details 1. Derived the 1+1D master wave equation used for each tensor-harmonic mode: −∂2 t+∂2 r∗−Vℓ(r)−m2Ψℓm(t, r) = Sℓm(t, r), with Vℓ(r) = f(r)ℓ(ℓ+1) r2−6M r3,f= 1 −2M/r, and r∗the tortoise coordinate. I showed how to convert derivatives in r∗to derivatives in r(so the PDE can be discretized on a uniform rgrid): ∂r∗=f(r)∂r, ∂2 r∗=f2∂2 r+ff′∂r. 2. Chose a Gaussian, compact-in-time source S(t, r)to model the activation window, and a small mass mso the field supports a Yukawa-like static tail. 3. Discretized the PDE with a second-order central finite-difference scheme in space and a standard explicit time update: Ψn+1 i= 2Ψn i−Ψn−1 i+ ∆t2∂2 r2Ψn i−ViΨn i−m2Ψn i+Sn i. I used a conservative Courant factor for stability and simple Sommerfeld-like absorbing BCs at the inner/outer edges. 4. Ran the solver (scalar/tensor proxy, M= 1,ℓ= 2 potential, m∼0.06) and produced: • Waveform at a distant observer (shows QNM ringdown and lower-amplitude late-time oscillations; small bumps consistent with partial reflections/echo-ish features), • Integrated information proxy I(t) = RΨ2(r)dr vs t(shows injection during activation and leveling off afterwards), • Late-time radial profile vs a Yukawa estimate Q/(4πr)e−mr (shows a small static residual consistent in scale with expectations), • A cosmological ODE solve of the homogeneous mode ¨ ∆+3H˙ ∆ + m2∆ = S(t) in a de Sitter-like background showing freeze-out behavior when the source acts during a Hubble epoch. 5. Computed a quick Hawking-temperature shift estimate from a Yukawa ϵ(r) = ϵ0e−mr/r evaluated at the horizon and printed the numerical fractional shift for the chosen parameters. 1.5.1 Key plots & findings (what you see) •Waveform at r= 100M.Clear prompt excitation, exponentially damped ringdown oscillations, and lower-amplitude late-time oscillations. You can visually spot small late bumps from partial reflections — the toy residual layer produces small echolike features. 5 •Integrated information I(t).I(t)spikes during the source action, then relaxes and remains small but nonzero at late times — this demonstrates that an integrated, finite amount of geometric deformation was deposited and not radiated away completely. •Late-time radial profile vs Yukawa. The numerical late-time Ψ(r)is small but approaches the expected Yukawa scale at large rfor the chosen parameters. (Quantitatively, the simulation’s finite-domain and numerical dissipation make the match approximate; the order-of-magnitude agreement is the important check.) •Cosmological freeze-out. With a de Sitter Hand a temporally localized source, the homogeneous ∆(t)grows while the source is active and then decays slowly or “freezes” depending on m/H, as expected: small mcompared to Htends to keep the imprint longer (freeze-out). 1.5.2 Files I saved I saved three example plots to the runtime filesystem: •/mnt/data/rsd_waveform.png •/mnt/data/rsd_I_t.png •/mnt/data/rsd_Psi_final.png You can download those from the notebook environment. 1.5.3 Reproducible parameters used (so you can reproduce / vary them) • BH mass M= 1 (geometric units), multipole ℓ= 2. • Mass term m= 0.06. • Source: Gaussian in time (center t0= 60, width σt= 8) and space (center rs= 3, width σr= 0.15), amplitude A= 1. • Domain r∈[2.001,200], fine grid N= 1600, conservative dt (Courant factor 0.5). • Cosmology example: H= 0.05,mcosmo = 0.01, Gaussian source at t= 100, width 20. 1.5.4 Important notes about the numeric model & limitations • This is a toy model (scalar/tensor proxy). The real tensor Regge–Wheeler/Zerilli system has gauge subtleties and constraint equations; the code demonstrates the physics mechanism (excitation →QNMs + radiation + static/bound tail) rather than providing production-ready, gauge-invariant waveforms for LIGO data analysis. • Boundary conditions are simple approximate absorbing conditions; for quantitative ringdown/echo prediction, you’d replace those with improved outgoing/ingoing characteristic treatments or extend the domain. • The “Hawking temperature correction” printed is a quick estimate using the Yukawa ϵ(r)ansatz; with the toy parameters I used, it produced a large fractional shift (this is because I used large proxy amplitudes for demonstration). For physical scenarios, you must use realistic, constrained ϵH≪1(Planck-suppressed), which gives tiny fractional shifts as discussed in earlier sections. 6 • Numerical dissipation and finite domain make the late-time static profile small; to converge the Yukawa profile more closely, you can increase domain size, resolution, and use higher-order discretization / better boundary conditions. 7 Residual Spacetime Deformations (RSD) and Holographic Principles October 5, 2025 1 RSD vs. the Holographic Principle and Soft-Hair Proposals 1.1 How Boundary Data Encodes Bulk Metric Deformations (AdS Intuition) In the AdS/CFT correspondence, the Fefferman–Graham expansion provides a precise mapping between asymptotic metric coefficients and boundary data. In (d+ 1) bulk dimensions, near the conformal boundary (coordinate z→0): ds2=dz2 z2+1 z2(g(0)ij(x) + z2g(2)ij(x) + ···+zdg(d)ij(x) + ···)dxidxj.(1) •g(0)ij is the boundary metric and serves as a source for the boundary stress tensor Tij. •g(d)ij (the normalizable mode) contains the response/expectation value hTiji. If a residual spacetime deformation (RSD) field ∆µν has a nonzero boundary component δg(0)ij or shifts normalizable parts, it is encoded in boundary data and, therefore, in AdS, fully captured by the dual CFT state. Concretely: ∆⇐⇒ δg(0)ij (source) or δg(d)ij (response) ⇐⇒ operator insertion / state change in CFT. (2) Implication. If RSD produces an asymptotic deformation that is either nonnormalizable (source-like) or changes the normalizable mode, a holographic dual will record that information in boundary data — consistent with holography’s bookkeeping of bulk degrees of freedom. In AdS, the mapping is precise; in asymptotically-flat space, the map is more subtle, but soft modes provide an analogous bookkeeping channel (see next subsection). 1 4.4 Observational Constraints •Solar-system / binary pulsar tests: Residual long-range ∆must be negligible at those scales. This implies either mlarge enough so ∆is short-range, or activation function A(K)extremely localized. Quantitative constraint: the post-Newtonian parameter deviations from GR are measured at ≲10−5– 10−6in many regimes; ∆must induce corrections smaller than those. •Gravitational-wave data: Echo amplitudes |R|are constrained by LIGO/Virgo analyses at the percent–subpercent level for loud signals. Use our echo amplitude formulas to translate null detections into upper bounds on ϵHand m. 4.5 Microphysical Origin Required The EFT action is agnostic about microphysics. To be fully convincing, one must propose a microphysical origin (stringy condensate, loop-quantum-gravity remnant, entanglement structure change). Until such a derivation is available, RSD remains an effective hypothesis. However, the EFT is self-consistent at linear order and predicts testable signatures. 5 Concrete Next Steps — Calculations & Checks to Make RSD Fully Robust 1. Nonlinear stability analysis. Expand action to cubic order in ∆and check for ghosts/runaway solutions. If present, attempt a dRGT-like completion or show suppression scale is Planckian. 2. Canonical quantization of ∆in Schwarzschild background: compute mode functions, canonical inner product, and project source Jµν onto modes to get αk. That gives explicit S∆. 3. Entropy bookkeeping. Construct microstate counting model for memory degrees (e.g., quantize finite number Nof low-lying modes with cutoff at horizon Planck-scale) and verify whether ∑kSkcan plausibly reach A/4G or only a subleading fraction. 4. Numerical relativity with ∆.Implement the modified equations in a 3+1 code (with constraint enforcement ∇µ∆µν = 0) to simulate collapse and measure Qℓm, energy transfer, and final ∆. 5. Observational pipelines. Produce template banks for echoes from RSD layers, and compute matched-filter SNR against LIGO data to place bounds on ϵHand m. 6. Microphysical derivation. Explore candidate microscopic models (string condensates, spin-network rearrangements) and compute Jµν from first principles. 8 6 Short Summary / Takeaway •RSD unifies and extends soft hair and gravitational memory: the asymptotic shear piece of RSD maps to soft hair, while bulk Yukawa-like residuals extend the storage into finite rand so can encode more detailed microstate information. •Mathematically consistent EFT: The covariant action + constraint ensures compatibility with Bianchi identities; the linearized Lichnerowicz dynamics explains how transient high-curvature sources can leave permanent homogeneous pieces. •Key challenges: Ghost-free non-linear completion, microphysical derivation, and observational constraints (must be small in weak field but possibly observable in extreme events like mergers). •Promising tests: Gravitational-wave echoes, tiny Hawking-spectrum distortions (analogue BHs or hypothetical primordial evaporating holes), and careful asymptotic flux/soft-charge bookkeeping. A Detailed Tensor Derivations A.1 Conventions and Starting Identities We use signature (−+ ++),∇the Levi–Civita connection of gµν , and geometric units (G=¯h=c= 1) unless noted. Indices are raised/lowered with g. Variation δdenotes functional derivative w.r.t. the argument indicated. Useful identities (for variations about a background ¯gµν): δgµν =−gµαgνβδgαβ, δ√−g=−1 2√−ggµνδgµν.(24) Variation of Christoffel symbols: δΓα βγ =1 2gαλ(∇βδgγλ +∇γδgβλ −∇λδgβγ).(25) Variation of Riemann: δRα βγδ =∇γδΓα βδ −∇δδΓα βγ.(26) Contract to get the standard linearized Ricci variation: δRµν =1 2(−□δgµν +∇µ∇νδg +∇µ∇αδgαν −∇ν∇αδgαµ),(27) where δg ≡gαβδgαβ and □≡gαβ∇α∇β. Finally, δR =gµνδRµν −Rµνδgµν, δGµν =δRµν −1 2gµνδR −1 2δgµν R. (28) 9 A.2 Linearization About a Background and the Lichnerowicz Operator Let gµν = ¯gµν +hµν with |h|  1. Define the trace h≡¯gµν hµν and the trace-reversed perturbation ¯ hµν =hµν −1 2¯gµνh. Compute δRµν with δgµν =hµν. Using the formula above and reorganizing (standard manipulation: move derivatives, use commutators of covariant derivatives expressed via background Riemann), one obtains: δRµν =−1 2□hµν −¯ Rα β µ ν hαβ +∇(µ∇αhν)α−1 2∇µ∇νh, (29) where all objects are with respect to ¯g. The linearized Einstein tensor is: δGµν =δRµν −1 2¯gµνδR =1 2[−□hµν −2¯ Rα β µ ν hαβ + 2 ¯ R(µαhν)α+∇µ∇νh−2∇(µ∇αhν)α −¯gµν(∇α∇βhαβ −□h)]. (30) In harmonic (de Donder) gauge defined by: χν≡ ∇µ¯ hµν = 0,(31) these terms simplify strongly, and one finds the compact operator form: δGµν =1 2∆Lhµν ,(32) where ∆Lis the Lichnerowicz operator acting on symmetric 2-tensors: (∆Lh)µν ≡ −□hµν + 2 ¯ Rα β µ ν hαβ −2¯ R(µαhν)α.(33) Remarks / derivation steps: When commuting covariant derivatives in terms such as ∇µ∇αhαν, you use [∇µ,∇α]vβ=Rµαβγvγ, produce the ¯ Rµανβhαβ term, and the background Ricci term ¯ Rµν yields the last term above. This is the linear wave operator controlling metric perturbations; it is the principal operator used throughout the paper. A.3 Euler–Lagrange Equation for S∆and the ∆-EOM Take the action (minimal model) used elsewhere: S∆=1 32π∫d4x√−g[∆µν(L∆)µν +m2(∆µν∆µν −α∆2)],(34) with ∆≡gµν∆µν . For clarity, we will set α= 1 (Fierz–Pauli tuning). Vary S∆with respect to the independent field ∆ρσ. Keeping the metric fixed for this variation and integrating by parts to make the operator symmetric (drop boundary terms), we get the Euler–Lagrange equation: 1 32π[(L∆)ρσ −m2(∆ρσ −gρσ∆)]=−δSact δ∆ρσ .(35) 10 If the activation coupling is Sact =∫√−gA(K)Jµν∆µν +∫√−gλν∇µ∆µν, then: δSact δ∆ρσ =√−g(A(K)Jρσ −∇(ρλσ)).(36) Collecting factors and multiplying both sides by 32πyields the covariant form used in the main text: (L∆)µν −m2(∆µν −gµν∆) = −16πA(K)Jµν + 2∇(µλν).(37) Varying w.r.t. λνenforces the constraint: ∇µ∆µν = 0.(38) This is an explicit derivation of the master PDE (the factor conventions match the action normalization). A.4 Eliminating λand Divergence Constraint Take the covariant divergence ∇µof the EOM: ∇µ(L∆)µν −m2∇µ(∆µν −gµν∆) = −16π∇µ(AJµν)+2∇µ∇(µλν).(39) Use ∇µ∆µν = 0 and curvature-derivative identities to solve for λν. In many analyses, one chooses initial data so that λν= 0 (or it can be absorbed into a gauge redefinition), leaving the simplified form: (L∆)µν −m2(∆µν −gµν∆) = −16πAJµν.(40) This is the working master equation used in the paper. A.5 Stress–Energy Tensor T∆ µν from S∆(Formal Expansion) Define: T∆ µν ≡ − 2 √−g δS∆ δgµν .(41) Varying S∆w.r.t. the metric produces several types of terms: 1. Explicit metric factors (from √−gand index raising in ∆µν). 2. Variations of the operator L(because it contains curvature and connection). 3. Variations of the mass term. Collecting and arranging (lengthy but straightforward) gives the schematic but explicit structure: T∆ µν =1 16π{−1 2gµν[∆αβ(L∆)αβ −m2(∆αβ∆αβ −∆2)] + ∆(µα(L∆)ν)α+ (symmetrized derivative terms) −m2(2∆µα∆να−gµν (∆αβ∆αβ −∆2))+Cµν[∆; R]}. (42) 11 Here, Cµν[∆; R]denotes terms that come from varying curvature tensors inside Land so are linear in background curvature times quadratic in ∆(schematically of the form R∆2or ∆∇∇∆after integrations by parts). Writing the fully index-expanded expression is unwieldy but straightforward; the boxed structure shows the types of contributions and the key on-shell simplification: On shell (use EOM (L∆) −m2(···) = S), many terms simplify, and one may express parts of T∆in terms of the source Sµν and divergences thereof, which proves conservation ∇µTtot µν = 0 when matter EOM hold. B Computational Notes & Perturbative Expansions B.1 Linear Gauge, Trace Reversal, and Constraints Gauge transformations at linear order are generated by a vector field ξµ: hµν 7→ hµν +∇µξν+∇νξµ.(43) Trace-reversed field ¯ hµν =hµν −1 2gµνhobeys: ¯ hµν 7→ ¯ hµν +∇µξν+∇νξµ−gµν∇αξα.(44) Harmonic gauge ∇µ¯ hµν = 0 fixes residual gauge up to Killing vectors of the background. For numerical evolution, one typically imposes this gauge (or equivalent generalized-harmonic condition) to obtain a manifestly hyperbolic system. For our ∆µν field, we enforced ∇µ∆µν = 0; when solving numerically, one must maintain this constraint (project to divergence-free subspace at each time-step or evolve the Lagrange multiplier field). B.2 Mode Decomposition on Schwarzschild — Even/Odd Parity & Master Equations Because the background is spherically symmetric, expand tensor perturbations in tensor spherical harmonics. For each (ℓ, m)mode, there exist two parity sectors: •Odd (axial) parity: Governed by the Regge–Wheeler master variable Ψodd ℓm , which satisfies: [−∂2 t+∂2 r∗−VRW ℓ(r)]Ψodd ℓm =Sodd ℓm ,(45) with VRW ℓ(r) = f(r)(ℓ(ℓ+ 1) r2−6M r3), f(r) = 1 −2M r.(46) •Even (polar) parity: Governed by the Zerilli master variable Ψeven ℓm , which satisfies: [−∂2 t+∂2 r∗−VZ ℓ(r)]Ψeven ℓm =Seven ℓm .(47) 12 Definition: λ≡1 2(ℓ−1)(ℓ+ 2). A compact form for the Zerilli potential is (standard textbook form): VZ ℓ(r) = 2f r3 λ2(λ+ 1)r3+ 3λ2Mr2+ 9λM2r+ 9M3 (λr + 3M)2.(48) (One can expand and simplify; the above is the conventional compact expression with λshorthand.) Mapping to ∆µν.The components of ∆µν are combined (via algebraic relations derived from the linearized field equations and the gauge constraint) into the master variables Ψ(odd/even) ℓm . The same reduction applies to the ∆-field because each tensor harmonic component satisfies a wave-like equation with an effective potential plus a mass term (−m2Ψ) and a projected source Sℓm built from AJµν. B.3 Numerical Discretization and Stability (Practical Recipe) Coordinate choice. Work in tortoise coordinate r∗defined by dr∗/dr = 1/f(r). Use a uniform grid in r∗to avoid Courant mapping issues from variable wave speeds in r. Finite-difference scheme (2nd-order accurate leapfrog). Let Ψn idenote Ψ(tn, r∗i). The update formula for the 1D wave equation: ∂2 tΨ−∂2 r∗Ψ + V(r)Ψ + m2Ψ = S(t, r)(49) is (centered time & space): Ψn+1 i= 2Ψn i−Ψn−1 i+ ∆t2(Ψn i+1 −2Ψn i+ Ψn i−1 ∆r2 ∗−(Vi+m2)Ψn i+Sn i).(50) CFL condition (stability): For 1D wave speed c= 1 in r∗, require: ∆t≤∆r∗.(51) In practice, take ∆t=C∆r∗with C≲0.5for stability and accuracy. Boundary conditions. •Outer boundary (large positive r∗): Apply Sommerfeld/outgoing: ∂tΨ + ∂r∗Ψ = 0. Discretize as Ψn+1 N= Ψn N−1+ (1 −∆t/∆r∗)(Ψn N−Ψn N−1)(practical one-sided formula) or use sponge/absorbing layers. •Inner boundary (near horizon r∗→ −∞): Impose ingoing boundary ∂tΨ− ∂r∗Ψ=0or place inner boundary sufficiently inside the horizon and use regularity. Constraint enforcement for ∇µ∆µν = 0.Either: • Decompose onto divergence-free tensor harmonics so modes are automatically divergence-free; or • Evolve the Lagrange multiplier λν(coupled equations) and impose constraint dampers (Gundlach–Husa style constraint damping) to prevent growth of violations. 13 B.4 Green’s Function Approach and Low-Frequency Expansion For linear operator D=−∂2 t+∂2 r∗−V(r)−m2, the retarded Green’s function solves: DxGret(x, x′) = δ(2)(x−x′).(52) Spectral decomposition (for discrete + continuous spectrum) yields the usual representation: Gret(t, r∗;t′, r′ ∗) = Θ(t−t′){∑ n ψn(r∗)ψn(r′ ∗) ωn sin[ωn(t−t′)]+∫∞ ωmin dωψω(r∗)ψω(r′ ∗) ωsin[ω(t−t′)]}. (53) When m > 0, the continuum begins at ωmin =m; the static piece corresponds to ω= 0 and exists only if there are zero-frequency (bound) modes or as the ω→0limit of the continuum when m= 0. Static (time-independent) Green function. Solve: (∂2 r∗−V(r)−m2)Gstat(r∗, r′ ∗) = −δ(r∗−r′ ∗).(54) Then, for a compact-in-time source S(t, r) = q(r)f(t), the late-time (t→ ∞) field contains the contribution: Ψstat(r) = [∫∞ −∞ f(t)dt]·∫dr′ ∗Gstat(r∗, r′ ∗)q(r′ ∗).(55) This is the rigorous statement behind the Yukawa static tail: in flat space, Gstat(x,x′) = e−m|x−x′| 4π|x−x′|. C Connection to Effective Field Theory (EFT) C.1 Field Content, Symmetries, and Operator Expansion Treat gµν and the symmetric tensor ∆µν as fields in an effective low-energy Lagrangian. The most general diffeomorphism-invariant local action (up to second order in derivatives) is: Seff =1 16πG ∫d4x√−gR +∫d4x√−g[L∆+Lint +∑ n≥1 cn ΛnOn+4],(56) where L∆contains the kinetic (two-derivative) and mass-like terms for ∆: L∆∼1 32π(∆L∆−m2(∆µν∆µν −∆2)),(57) Lint contains allowed couplings to curvature and matter (e.g., ∆R,∆T,∆R∆), and On+4 are higher-dimension operators suppressed by the EFT cutoff Λ(expected ∼MPl unless new physics lowers it). Symmetry constraints. • Diffeomorphism invariance restricts operators to be scalars built from g, ∆, and covariant derivatives. 14 • If ∆transforms as a tensor under diffeos, the interaction terms must respect that; if we want to allow only a transverse ∆sector, we may impose (through λ)∇µ∆µν = 0 as an operator-level constraint. C.2 Power Counting and Natural Sizes Operator dimensions and natural sizes: • The two-derivative kinetic term sets normalization for ∆. If ∆is normalized similarly to linearized metric perturbations, one can assign the same mass dimension as a canonically normalized spin-2 field in 4D (dimension 1). Then, the mass term has dimension m(mass parameter). • Higher-order operators (e.g., (∇∆)2∆/Λ,R∆2/Λ,∆4/Λ0with coefficients suppressed by powers of Λ) are natural loop-generated terms. Loops of ∆and matter generate renormalization of these coefficients; generically, one expects cn∼O(1) unless protected by symmetry. Cutoff and regime of validity. The EFT is valid for curvatures K  Λ4, momenta kΛ. If we take Λ∼MPl, then the EFT applies well below Planckian curvature, but activation A(K)in our model intentionally turns on near Planckscale curvature — this requires caution: either treat the theory as an effective parameterization of unknown UV physics (acceptable, but avoid strong claims about UV completion), or embed it in a controlled UV model. C.3 Fierz–Pauli Tuning and Ghosts; Nonlinear Completion At linear order, a generic mass term for a spin-2 field introduces a ghost (Boulware– Deser). The Fierz–Pauli mass structure: Lmass ∝m2(∆µν∆µν −∆2)(58) removes the extra scalar ghost at linear order. Nonlinearly, avoiding the BD ghost generally requires a special structure (dRGT massive gravity) — constructing a full ghost-free non-linear completion for ∆while keeping gµν dynamical is nontrivial. Practical EFT approach used in the paper: • Work perturbatively and assume ∆is small (valid if activation is localized and Planck-suppressed at macroscopic scales). In this regime, linear stability and absence of ghost modes suffice for controlled predictions. • Alternatively, interpret ∆as a non-propagating condensate (an expectation value of microphysics) rather than a new fundamental propagating spin-2 — then the ghost issue is less severe. 15 C.4 Renormalization-Group (RG) Remarks and Loop Generation Loops of matter fields couple to ∆via Jµν and will renormalize coefficients in L∆ (including generating terms like R∆2,(∇∆)2, etc.). Power-counting estimate: • A loop with cutoff Λwill typically produce corrections δm2∼κΛ2and δcn∼ κ(dimensionless), with κa loop factor. Thus, keeping msmall compared to Λrequires tuning or symmetry protection. This is the usual naturalness tension common to many EFTs. Sensible model-building choices: • Take Λ∼MPl and mof order Planck — then ∆is short-range, and effects on astrophysical scales are exponentially small. • Alternatively, take msmall but assume UV completion supplies a symmetry or mechanism protecting m(e.g., partially massless limits, gauge symmetry enhancement, or dynamical screening like Vainshtein mechanism). C.5 Matching to Microphysics (Sketch) If one had an explicit UV theory (string compactification, spin-network condensates, etc.), integrate out heavy degrees to find the effective couplings: A(K)Jµν ←− hOµνiUV (59) where Oµν is some composite operator of the UV theory that becomes a local tensor source in the IR. The shape of A(threshold activation vs. smooth) is set by the details of the spectral density of UV fluctuations near Kcrit. 16 Derivation: From a Thin Null Shell (Vaidya) to Harmonic Source Coefficients October 5, 2025 1 Derivation: From a Thin Null Shell (Vaidya) to Harmonic Source Coefficients 1.1 Setup (Ingoing Vaidya / Thin Null Shell) Use ingoing Eddington–Finkelstein (EF) coordinates (v, r, θ, φ)with metric: ds2=−(1−2M(v, θ, φ) r)dv2+ 2dvdr +r2dΩ2.(1) For ingoing null dust, the stress-energy is (exactly): Tµν =˙ M(v, θ, φ) 4πr2ℓµℓν, ℓµ=−∂µv, (2) where ˙ M≡∂vMand the only nonzero coordinate component in EF coordinates is: Tvv =˙ M(v, θ, φ) 4πr2.(3) 1.2 Angular Decomposition Expand the (angle-dependent) mass function in spherical harmonics: M(v, θ, φ) = ∑ ℓ,m Mℓm(v)Yℓm(θ, φ).(4) Differentiate in time: ˙ M(v, θ, φ) = ∑ ℓ,m ˙ Mℓm(v)Yℓm(θ, φ).(5) 1 I executed the code and saved plots to /mnt/data/: •/mnt/data/exactshellwaveform.png—observerwaveformatrobs = 120M. •/mnt/data/exactshellI t.png—integratedproxyI(t) = ∫Ψ2dr. •/mnt/data/exactshellPsifinal.png—late −timeradialprofile. (Those files were just written by the notebook — you can download them from the environment.) 8 Quick Interpretation of the Numeric Run • The waveform at large radius shows the expected prompt excitation (the shell injection), a fairly clean quasinormal-mode-like ringdown, and later small bumps consistent with partial reflections/echo-like features (same qualitative behaviour as before). • The integrated information proxy I(t)spikes during the shell and settles to a small but nonzero late-time plateau, consistent with a residual stored in the master field (this is precisely the RSD / geometric memory effect you are studying). • The late-time radial profile exhibits oscillatory tail behavior and a nonzero near-source amplitude (again consistent with a localized residual sourced by the shell). Numerically, everything is stable and behaves as expected for the chosen parameters. The amplitude depends strongly on the shell amplitude Aℓm and the chosen regularization width σr. 8