{"schemaVersion":1,"publication":{"id":"boundary-ball-readouts","title":"Boundary Ball Readouts and Linearized Einstein Gravity in a Source–Readout Architecture","subtitle":"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.","description":"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.","doi":"10.5281/zenodo.24029188","date":"2026-11-07","url":"/source-readout/boundary-ball-readouts","pdfUrl":"/publications/gravity/boundary-ball-readouts/paper.pdf","markdownUrl":"/publications/gravity/boundary-ball-readouts/complete.md","manifestUrl":"/source-readout/boundary-ball-readouts/manifest.json","paperSlug":"boundary-ball-readouts","headline":"What fixed-slice balls can see.","author":"Jeremy Rodgers","programme":"source-readout","series":"gravity-holography","supplementUrl":"/publications/gravity/boundary-ball-readouts/Faulkner_Verification_24029188.zip","previousDoi":"10.5281/zenodo.24019669","supplementDescription":"Full verification suite with errata checks"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"boundary-ball-readouts","slug":"01-preparation-and-ball-projections","title":"One perturbation, two ball readouts","description":"Define the common off-shell source, fix the ball boost normalization, and derive the boundary energy and extremal-area projections.","order":1,"url":"/source-readout/boundary-ball-readouts/01-preparation-and-ball-projections","sourcePath":"content/paper-integrations/boundary-ball-readouts/01-preparation-and-ball-projections.mdx","sections":[{"id":"introduction","title":"1. What the ball first law can recover"},{"id":"common-source","title":"2. Common source and the ball projections"},{"id":"ball-boost","title":"One tangent, many apertures"},{"id":"energy-projection","title":"The one-sided energy projection"},{"id":"area-projection","title":"The area projection and its cutoff endpoint"}],"anchors":["abstract","introduction","common-source","eq-1","ball-boost","energy-projection","area-projection","eq-2"],"words":885},{"paperId":"boundary-ball-readouts","slug":"02-two-proofs-of-the-charge-identity","title":"Two proofs of the off-shell charge identity","description":"Compute the same weighted Einstein residual from covariant phase space and directly from the scalar-curvature variation.","order":2,"url":"/source-readout/boundary-ball-readouts/02-two-proofs-of-the-charge-identity","sourcePath":"content/paper-integrations/boundary-ball-readouts/02-two-proofs-of-the-charge-identity.mdx","sections":[{"id":"off-shell-charge-identity","title":"3. The off-shell charge identity"},{"id":"theorem-3-1","title":"Theorem 3.1. Defect as a weighted residual"},{"id":"phase-space-proof","title":"Covariant phase-space proof"},{"id":"slice-proof","title":"Independent slice proof"}],"anchors":["off-shell-charge-identity","theorem-3-1","eq-3","phase-space-proof","slice-proof"],"words":502},{"paperId":"boundary-ball-readouts","slug":"03-recovering-the-local-equation","title":"Recovering the local equation","description":"Invert the half-ball integrals, polarize over a finite family of frames, and impose the radial Ward data.","order":3,"url":"/source-readout/boundary-ball-readouts/03-recovering-the-local-equation","sourcePath":"content/paper-integrations/boundary-ball-readouts/03-recovering-the-local-equation.mdx","sections":[{"id":"recovering-the-local-equation","title":"4. From readout data to the local equation"},{"id":"half-ball-inversion","title":"Recover the slice constraint"},{"id":"theorem-4-1","title":"Theorem 4.1. Recover the Einstein residual"},{"id":"determining-frames","title":"Proof: span the boundary contractions"},{"id":"radial-ward-data","title":"Proof: complete the radial equations"},{"id":"return-maps","title":"What the return map returns"}],"anchors":["recovering-the-local-equation","half-ball-inversion","theorem-4-1","determining-frames","radial-ward-data","return-maps"],"words":703},{"paperId":"boundary-ball-readouts","slug":"04-what-fixed-slice-balls-miss","title":"What fixed-slice balls miss","description":"Construct non-gauge lapse and cross-mode perturbations that evade all rest-frame balls, and isolate the effects of fixing a radius or center.","order":4,"url":"/source-readout/boundary-ball-readouts/04-what-fixed-slice-balls-miss","sourcePath":"content/paper-integrations/boundary-ball-readouts/04-what-fixed-slice-balls-miss.mdx","sections":[{"id":"fixed-frame-kernel","title":"5. Metric perturbations invisible to restricted families"},{"id":"compact-lapse","title":"A compactly supported lapse perturbation"},{"id":"ward-compatible-cross-mode","title":"A nonzero Ward-compatible boundary tensor"},{"id":"fixed-radius-and-center","title":"Fixing the radius or the center"}],"anchors":["fixed-frame-kernel","compact-lapse","ward-compatible-cross-mode","fixed-radius-and-center"],"words":645},{"paperId":"boundary-ball-readouts","slug":"05-verification-and-references","title":"Verification, scope and references","description":"Separate the reproduced checks from the analytic additions and state precisely what first-order ball data recover.","order":5,"url":"/source-readout/boundary-ball-readouts/05-verification-and-references","sourcePath":"content/paper-integrations/boundary-ball-readouts/05-verification-and-references.mdx","sections":[{"id":"verification-and-scope","title":"6. Verification and scope"},{"id":"funding","title":"Funding"},{"id":"verification-materials","title":"Verification materials"},{"id":"references","title":"References"}],"anchors":["verification-and-scope","funding","verification-materials","references","ref-1","ref-2","ref-3","ref-4","ref-5","ref-6","ref-7","ref-8"],"words":404}],"completeMarkdown":"/publications/gravity/boundary-ball-readouts/complete.md","originalPdf":"/publications/gravity/boundary-ball-readouts/paper.pdf"}