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Shadow Theory

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What fixed-slice balls can see.

First-order ball data on a fixed boundary slice recover a projected Einstein constraint. The fixed-frame kernel shows why recovering the full field equation needs more frame data and boundary assumptions, and why that recovery is distinct from reconstructing the metric.

Jeremy Rodgers · Independent Researcher · 2026-11-07
doi:10.5281/zenodo.24029188 · Corrected edition · Earlier version

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Boundary Ball Readouts and Linearized Einstein Gravity in a Source–Readout Architecture

First-order ball data on a fixed boundary slice recover a projected Einstein constraint. The fixed-frame kernel shows why recovering the full field equation needs more frame data and boundary assumptions, and why that recovery is distinct from reconstructing the metric.

A fixed slice recovers the Hamiltonian constraint and leaves a non-gauge kernel. Full equation recovery requires a determining set of Lorentz frames, the boundary constraints and Bianchi propagation. That result does not reconstruct the metric from the ball data.

Continue with the same balls at second order

Second-Order Detection by Boundary Balls → follows an explicit transverse perturbation with zero first-order area and energy response. Its positive all-radii bulk canonical energy makes the second-order detection exact.

Explore the related Atlas structure →

The argument in full

Follow the chapters in order or enter at a particular question. Definitions, mathematics, proofs, constructions and substantive appendices belong to this reading edition.

  1. One perturbation, two ball readouts

    Define the common off-shell source, fix the ball boost normalization, and derive the boundary energy and extremal-area projections.

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  2. Two proofs of the off-shell charge identity

    Compute the same weighted Einstein residual from covariant phase space and directly from the scalar-curvature variation.

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  3. Recovering the local equation

    Invert the half-ball integrals, polarize over a finite family of frames, and impose the radial Ward data.

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  4. What fixed-slice balls miss

    Construct non-gauge lapse and cross-mode perturbations that evade all rest-frame balls, and isolate the effects of fixing a radius or center.

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  5. Verification, scope and references

    Separate the reproduced checks from the analytic additions and state precisely what first-order ball data recover.

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Where the arguments meet

Transport and Projective Reduction in a Source–Readout Derivation of Einstein Gravity →
Reconstruct the affine connection from smooth transport, remove the projective ambiguity with a volume-restored observable, and follow the complete off-shell spin-two bootstrap to Einstein–matter dynamics within the admitted realization.

Modular Transport and Spherical Holographic Entanglement Entropy in a Source–Readout Architecture →
One vacuum preparation supplies a restricted diamond state and a thermal state on the hyperbolic cylinder. Modular transport fixes the temperature; the regulated entropy calculation adds the holographic dictionary.

Local Horizon Thermodynamics and Einstein Gravity in a Source–Readout Architecture →
Heat, temperature and area readouts of a shrinking null pencil recover the Einstein equation under the stated thermodynamic and conservation assumptions. A finite generator segment and finite transverse width pose different questions.

Second-Order Detection by Boundary Balls in a Source–Readout Architecture →
An explicit transverse AdS perturbation has zero first-order area and energy variations on the fixed-slice balls. The same balls register positive bulk canonical energy at second order. A boundary relative-entropy interpretation adds a holographic dictionary and an admissible state family.

Published source: Boundary Ball Readouts and Linearized Einstein Gravity in a Source–Readout Architecture. Read the original PDF alongside the web treatment.