{"paper_id":"29a57aca-1296-4e4d-bc6b-5c8003f27950","body_text":"Emergence of the Hooke Tensor from Spacetime Permeability Kernels | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Emergence of the Hooke Tensor from Spacetime Permeability Kernels Andrea Giordano This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8616449/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Elastic wave propagation is conventionally described by local constitutive relations and differential operators, with the Hooke tensor and mass density introduced as fundamental material parameters. In this formulation, locality and instantaneous response are assumed a priori. In contrast, experimentally accessible information on elastic media is encoded in spacetime response functions relating applied forces to observable displacements. In this work we develop a response-based formulation of elasticity in which the primary dynamical object is a tensorial spacetime permeability kernel. No assumption of local constitutive behavior is imposed at the outset. We show that, under minimal and physically verifiable conditions, the inverse response kernel admits a controlled low-frequency and longwavelength expansion. The quadratic sector of this expansion defines an effective local elastic operator. Within this framework, the mass density and the Hooke tensor arise as second-order derivatives of the inverse spacetime permeability with respect to frequency and wavevector. For isotropic media, this structure reduces uniquely to the Lamé form, providing explicit expressions for the elastic moduli as emergent quantities. The domain of validity of the local elastic description is determined by the breakdown of the quadratic approximation, thereby quantifying the onset of non-local elastic behavior. This formulation places classical elasticity on the same conceptual footing as electromagnetic constitutive theory, where permeability is understood as a response function rather than a fundamental constant, and establishes a systematic basis for extending elastic descriptions to non-local media. Full Text Additional Declarations No competing interests reported. Supplementary Files SUPPHOOKE.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {\"props\":{\"pageProps\":{\"initialData\":{\"identity\":\"rs-8616449\",\"acceptedTermsAndConditions\":true,\"allowDirectSubmit\":false,\"archivedVersions\":[],\"articleType\":\"Research Article\",\"associatedPublications\":[],\"authors\":[{\"id\":576191208,\"identity\":\"6a906795-dc75-4b8c-912e-b58c44e742e5\",\"order_by\":0,\"name\":\"Andrea Giordano\",\"email\":\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFUlEQVRIiWNgGAWjYFCDwwyMB4CUAT+Ik1CAVy1jA1AhSAsDWItkA0iLATFaDkC1GIBIBjxa+Nvbnz/4ueOPHN9x3gMHfu6wMzY+vzrxwwMDBnl+sQNYtUicOWPY2HvGwFjyMF/Cwd4zyWZmN95ulgA6zHDm7ASsWgwkchgbeNsMEjcc5jE4wNt2wMbsxtkNIC0JBrdxaUl/2Pi3zaAepOXgX6AW4xlnN//AryXBsBloS4IBUMthoC1mBvy92/DaAvLLbNk2Y8OZQL8clm1LNpa4wbvNIsFAAqdfgCH24OPbNjl5vvNnDz5822Zn2N9/dvPNHxU28vzS2LUgAR6YxWCVEoSUI2vhP0CM6lEwCkbBKBhBAADI6mbhnoZPSAAAAABJRU5ErkJggg==\",\"orcid\":\"\",\"institution\":\"Independent Researcher\",\"correspondingAuthor\":true,\"prefix\":\"\",\"firstName\":\"Andrea\",\"middleName\":\"\",\"lastName\":\"Giordano\",\"suffix\":\"\"}],\"badges\":[],\"createdAt\":\"2026-01-16 08:36:51\",\"currentVersionCode\":1,\"declarations\":\"\",\"doi\":\"10.21203/rs.3.rs-8616449/v1\",\"doiUrl\":\"https://doi.org/10.21203/rs.3.rs-8616449/v1\",\"draftVersion\":[],\"editorialEvents\":[],\"editorialNote\":\"\",\"failedWorkflow\":false,\"files\":[{\"id\":104400811,\"identity\":\"f0b2a955-8a30-47fa-9550-bd41cfab5ecc\",\"added_by\":\"auto\",\"created_at\":\"2026-03-11 12:11:07\",\"extension\":\"pdf\",\"order_by\":1,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"manuscript-pdf\",\"size\":314934,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"hooke2.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-8616449/v1_covered_1bd20a18-ff74-4f34-884e-2cabc7224d0c.pdf\"},{\"id\":103792220,\"identity\":\"3f6f4f60-8ad2-46e0-8991-565019785e6f\",\"added_by\":\"auto\",\"created_at\":\"2026-03-03 02:46:45\",\"extension\":\"pdf\",\"order_by\":0,\"title\":\"\",\"display\":\"\",\"copyAsset\":false,\"role\":\"supplement\",\"size\":615166,\"visible\":true,\"origin\":\"\",\"legend\":\"\",\"description\":\"\",\"filename\":\"SUPPHOOKE.pdf\",\"url\":\"https://assets-eu.researchsquare.com/files/rs-8616449/v1/3920f90c440864ffcf3d662e.pdf\"}],\"financialInterests\":\"No competing interests reported.\",\"formattedTitle\":\"Emergence of the Hooke Tensor from Spacetime Permeability Kernels\",\"fulltext\":[],\"fulltextSource\":\"\",\"fullText\":\"\",\"funders\":[],\"hasAdminPriorityOnWorkflow\":false,\"hasManuscriptDocX\":false,\"hasOptedInToPreprint\":true,\"hasPassedJournalQc\":\"\",\"hasAnyPriority\":true,\"hideJournal\":true,\"highlight\":\"\",\"institution\":\"\",\"isAcceptedByJournal\":false,\"isAuthorSuppliedPdf\":true,\"isDeskRejected\":\"\",\"isHiddenFromSearch\":false,\"isInQc\":false,\"isInWorkflow\":false,\"isPdf\":true,\"isPdfUpToDate\":true,\"isWithdrawnOrRetracted\":false,\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true},\"keywords\":\"\",\"lastPublishedDoi\":\"10.21203/rs.3.rs-8616449/v1\",\"lastPublishedDoiUrl\":\"https://doi.org/10.21203/rs.3.rs-8616449/v1\",\"license\":{\"name\":\"CC BY 4.0\",\"url\":\"https://creativecommons.org/licenses/by/4.0/\"},\"manuscriptAbstract\":\"\\u003cp\\u003eElastic wave propagation is conventionally described by local constitutive relations and differential operators, with the Hooke tensor and mass density introduced as fundamental material parameters. In this formulation, locality and instantaneous response are assumed a priori.\\u0026nbsp;\\u003c/p\\u003e\\n\\u003cp\\u003eIn contrast, experimentally accessible information on elastic media is encoded in spacetime response functions relating applied forces to observable displacements. In this work we develop a response-based formulation of elasticity in which the primary dynamical object is a tensorial spacetime permeability kernel. No assumption of local constitutive behavior is imposed at the outset. We show that, under minimal and physically verifiable conditions, the inverse response kernel admits a controlled low-frequency and longwavelength expansion. The quadratic sector of this expansion defines an effective local elastic operator.\\u003c/p\\u003e\\n\\u003cp\\u003eWithin this framework, the mass density and the Hooke tensor arise as second-order derivatives of the inverse spacetime permeability with respect to frequency and wavevector. For isotropic media, this structure reduces uniquely to the Lamé form, providing explicit expressions for the elastic moduli as emergent quantities. The domain of validity of the local elastic description is determined by the breakdown of the quadratic approximation, thereby quantifying the onset of non-local elastic behavior.\\u003c/p\\u003e\\n\\u003cp\\u003eThis formulation places classical elasticity on the same conceptual footing as electromagnetic constitutive theory, where permeability is understood as a response function rather than a fundamental constant, and establishes a systematic basis for extending elastic descriptions to non-local media.\\u003c/p\\u003e\",\"manuscriptTitle\":\"Emergence of the Hooke Tensor from Spacetime Permeability Kernels\",\"msid\":\"\",\"msnumber\":\"\",\"nonDraftVersions\":[{\"code\":1,\"date\":\"2026-03-03 02:46:39\",\"doi\":\"10.21203/rs.3.rs-8616449/v1\",\"editorialEvents\":[{\"type\":\"communityComments\",\"content\":0}],\"status\":\"published\",\"journal\":{\"display\":true,\"email\":\"info@researchsquare.com\",\"identity\":\"researchsquare\",\"isNatureJournal\":false,\"hasQc\":true,\"allowDirectSubmit\":true,\"externalIdentity\":\"\",\"sideBox\":\"\",\"snPcode\":\"\",\"submissionUrl\":\"/submission\",\"title\":\"Research Square\",\"twitterHandle\":\"researchsquare\",\"acdcEnabled\":true,\"dfaEnabled\":false,\"editorialSystem\":\"\",\"reportingPortfolio\":\"\",\"inReviewEnabled\":false,\"inReviewRevisionsEnabled\":true}}],\"origin\":\"\",\"ownerIdentity\":\"1a5cb772-6271-44e8-bf79-9d1b42d53680\",\"owner\":[],\"postedDate\":\"March 3rd, 2026\",\"published\":true,\"recentEditorialEvents\":[],\"rejectedJournal\":[],\"revision\":\"\",\"amendment\":\"\",\"status\":\"posted\",\"subjectAreas\":[],\"tags\":[],\"updatedAt\":\"2026-03-03T02:46:40+00:00\",\"versionOfRecord\":[],\"versionCreatedAt\":\"2026-03-03 02:46:39\",\"video\":\"\",\"vorDoi\":\"\",\"vorDoiUrl\":\"\",\"workflowStages\":[]},\"version\":\"v1\",\"identity\":\"rs-8616449\",\"journalConfig\":\"researchsquare\"},\"__N_SSP\":true},\"page\":\"/article/[identity]/[[...version]]\",\"query\":{\"redirect\":\"/article/rs-8616449\",\"identity\":\"rs-8616449\",\"version\":[\"v1\"]},\"buildId\":\"XKTyCvWXoU3ODBz1xrDgd\",\"isFallback\":false,\"isExperimentalCompile\":false,\"dynamicIds\":[84888],\"gssp\":true,\"scriptLoader\":[]}","source_license":"CC-BY-4.0","license_restricted":false}