{
  "abstract": "We present a self-contained coordinate-matrix formulation of General Relativity in the harmonic gauge, providing a compact, index-free representation of the gravitational field equations within the algebra of $4\\times 4$ real matrices $M_4$. By defining musical isomorphism operators ($\\sharp, \\flat$), metric adjoints ($\\dagger$), and matrix-valued derivative vectors ($\\bm{\\mathcal{B}}, \\bm{\\mathcal{C}}, \\bm{\\mathcal{D}}^\\sharp$), the Ricci tensor is recast into an algebraic matrix structure dominated by the Laplace--Beltrami operator $\\Box_{\\mathrm{LB}} \\coloneqq \\operatorname{tr}(\\mathcal{G}^\\sharp \\partial^2)$, where $\\partial^2 \\coloneqq \\bm{\\partial}\\bm{\\partial}^T$ represents the matrix Hessian operator of second partial derivatives. Within this setting, Einstein's field equations naturally take the form of a non-linear wave equation for the metric matrix $\\mathcal{G}$, giving rise to an index-free matrix representation of the gravitational energy-momentum pseudotensor $\\bar{\\mathcal{T}}$. We derive the explicit algebraic relation connecting $\\bar{\\mathcal{T}}$ with the classical Landau--Lifshitz energy-momentum complex $\\mathcal{T}_{\\mathrm{LL}}$, and express the associated local conservation law in a simple matrix divergence form, $\\bm{\\partial}^T \\left((-g)(\\mathcal{T}^{\\sharp} + \\mathcal{T}_{\\mathrm{LL}}^{\\sharp})\\right) = \\bm{0}^T$. Finally, we illustrate the scope and applicability of the matrix formalism through three physical applications: weak-field linearized gravity, the exact derivation of Friedmann cosmological dynamics in harmonic time, and the complete formulation of linear scalar cosmological perturbations in both the Newtonian and harmonic gauges.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07500",
  "author_total": 1,
  "authors": [
    {
      "name": "Jaume de Haro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0701jxcmk4q28.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07500",
  "fulltext_url": null,
  "id": "0701jxcmk4q28",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07500",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0701jxcmk4q28.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2611.07500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07500"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Algebraic representation of General Relativity in harmonic gauge",
  "title_text": "Algebraic representation of General Relativity in harmonic gauge",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:41Z",
  "url": "https://research.draconitic.com/paper/0701jxcmk4q28",
  "venue": "arXiv",
  "versions": []
}