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One null pencil, three readouts.

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.

Jeremy Rodgers · Independent Researcher · 2026-11-07
doi:10.5281/zenodo.24029166 · Corrected edition · Earlier version

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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.

Area entropy, local equilibrium, Unruh normalization and stress conservation give the shrinking-pencil equation. A weighted prescription on a finite generator segment does not establish the same result for a pencil of finite transverse width.

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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 prepared horizon, three readouts

    Put heat, temperature and entropy on the same null pencil, and normalize the boost before comparing them.

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  2. The Clausius coefficient and the directions that determine it

    Derive the leading field equation in both directions, reconstruct its trace-free part from nine null directions, and exhibit the kernel of a circle.

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  3. The first finite-length correction

    Track the cubic residual along one generator, the width estimate needed to transfer it, and the separate weighted expansion prescription.

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  4. Preparation, assumptions and verification

    Separate pointwise equilibrium from stronger screen conditions, state the physical inputs, and check what the construction establishes.

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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.

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.

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: Local Horizon Thermodynamics and Einstein Gravity in a Source–Readout Architecture. Read the original PDF alongside the web treatment.