Foundations from Minimal Change

Research article · Jinku Guo

A Spectral Criterion for Emergent Gravity

arXiv SSRN Zenodo Read manuscript PDF Files and citation

Author: Jinku Guo

Work type: Research article

arXiv: 2607.21621

SSRN: Abstract and download page

Zenodo: 10.5281/zenodo.22093341

Repository snapshot: 23 October 2026

Research overview

The question behind the construction

An emergent description of gravity must explain how microscopic degrees of freedom produce a long-range interaction between material bodies. An internal zero mode is only part of that explanation. The mode must couple to matter, survive elimination of the unobserved variables, propagate in physical space, and give the same leading coupling to different kinds of localized sources. This paper brings these requirements into a single spectral criterion.

The central object is the response from a specified matter source to a specified observation. Its singularities encode the interactions that can actually be measured. Keeping the source and observation maps in the construction makes it possible to distinguish an internal spectral degeneracy from a physical exchange channel. It also supplies a precise way to decide whether an effective description preserves the gravitational response of the underlying system.

From internal response to long-range exchange

The analysis begins with a stationary response problem and its linearization. Exact channel elimination is performed by a Schur or Feshbach reduction. The propagator, source insertion, observation map and direct response are transported together. A simple critical direction then gives a pole whose residue depends on both the spectral crossing and the overlaps with the physical sources.

This dependence matters when an approximation moves the critical point. The paper compares responses at the displaced critical root and controls the corresponding residues. Comparing two kernels at the old root would miss part of the observable error. The resulting criterion therefore concerns a complete source-to-observation response and remains meaningful under reduction and approximation.

Positive spectral representations provide the next step. A pole at zero invariant mass corresponds to an atom of the Kallen-Lehmann measure, including the case where a continuum begins at the same endpoint. The paper also establishes a structural obstruction for a complete positive Lorentz-covariant measure of a conserved microscopic stress tensor. Under those hypotheses its null-shell contribution is longitudinal and has no transverse-traceless atomic weight. This identifies exactly which realization must be examined when constructing a physical two-helicity exchange channel.

An explicit compact realization supplies such a reduced channel. Its internal tensor response has a simple zero. Scalar and one-form matter sectors provide concrete source vertices, and the inverse response and its slope determine the matching to external propagation. The construction consequently connects the internal spectrum, the observable residue and the external helicity-two covariance.

Universal matter coupling and the force law

A particularly important result concerns binding energy. The matter vertex is built from the complete energy-momentum response. For a localized stationary source, the integrated stress identity reduces the leading long-wavelength coupling to its total energy. Internal stresses and binding contributions enter the same accounting. When two bodies couple through a common critical direction, their mixed residue factors into their total energies and a common propagation coefficient.

Spatial resolvent estimates then turn this response into a potential and a force. In three spatial dimensions the leading behavior is the inverse-distance potential and inverse-square force. The estimates also control finite-size corrections and their gradients. This closes the passage from an internal spectral calculation to an interaction between extended material bodies.

The effective metric description follows through the usual helicity-two consistency requirements. Positivity, universal soft coupling and local nonlinear consistency organize an Einstein-matter completion. A dimensional gravitational normalization fixes the physical unit, while the dimensionless exchange strength is determined by the spectral residue.

What the paper establishes

The main achievement is a usable criterion for emergent gravitational exchange, accompanied by a compact realization and quantitative reduction estimates. It combines several obligations that are often treated separately:

The generating perspective is concrete here. Collective geometry is obtained by following a response through its construction, its reductions and its physical readout. The spectrum supplies the candidate mode, matter supplies its observable meaning, and the residue fixes the strength of the interaction. Their compatibility is what makes the emergent gravitational description possible.

Research subjects and results

Paper keywords

Emergent gravity; Massless spin-2 fields; Universal gravitational coupling; Newtonian limit; Spectral positivity; Källén–Lehmann representation.

Primary subjects

Emergent gravity; Massless spin-2 fields; Universal gravitational coupling; Newtonian limit; Spectral positivity; Källén–Lehmann representation; Gravitational interaction; Spectral representation; Stress-energy tensor; Källén-Lehmann representation; Effective field theory of gravity; Long-range interactions; Graviton exchange; Positive spectral measures.

Research subjects

Conserved sources; Energy-momentum conservation; Tensor correlators; Transverse-traceless modes; Helicity-2 states; Massless excitations; Spectral atoms; Vacuum correlations; Static gravitational potential; Inverse-square law; Finite-size corrections; Planck scale; Lorentz covariance; Einstein-matter systems; Source coupling.

Methods

Spectral theory; Operator theory; Analytic Fredholm theory; Schur complements; Resolvent estimates; Pole residues; Spectral projections; Contour integration; Finite-rank approximation; Operator convergence; Channel elimination; Positive measures; Linear response theory; Effective field theory; Spectral approximation.

Results and research questions

Spectral criterion for emergent gravity; Universal coupling from conserved matter sources; Newtonian potential from tensor exchange; Massless spectral atoms at continuum thresholds; Two-helicity long-range gravitational response; Finite-size bounds for gravitational exchange; Stability of critical pole residues under approximation; Vanishing transverse-traceless atomic weight for a positive conserved stress tensor.

Chinese subject terms

涌现引力; 无质量自旋二场; 引力子交换; 普适引力耦合; 牛顿极限; 引力有效场论; 能动张量; 正谱测度; 无质量谱原子; 长程张量相互作用; 普朗克尺度; 谱论.

Search aliases

Kallen-Lehmann representation; Kallen Lehmann spectral representation.

Public file

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Citation and archival records

Jinku Guo. A Spectral Criterion for Emergent Gravity. 2026. doi:10.48550/arXiv.2607.21621.

The repository PDF is identified by its snapshot tag and checksum. The public article records identify the work or their respective deposited versions.

Research context

Related research and results · Programme overview · Proofs and calculations