Algebraic representation of General Relativity in harmonic gauge
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× 4 real matrices M4. By defining musical isomorphism operators (♯, ♭), metric adjoints (†), and matrix-valued derivative vectors (ℬ, 𝒞, 𝒟♯), the Ricci tensor is recast into an algebraic matrix structure dominated by the Laplace--Beltrami operator BoxLB coloneqq tr(𝒢♯ ∂2), where ∂2 coloneqq ∂∂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 𝒢, giving rise to an index-free matrix representation of the gravitational energy-momentum pseudotensor 𝒯̅. We derive the explicit algebraic relation connecting 𝒯̅ with the classical Landau--Lifshitz energy-momentum complex 𝒯LL, and express the associated local conservation law in a simple matrix divergence form, ∂T ((−g)(𝒯♯ + 𝒯LL♯)) = 0T. 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.
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BibTeX
@misc{haro2026algebraic,
title = {{Algebraic representation of General Relativity in harmonic gauge}},
author = {Jaume de Haro},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07500},
eprint = {2611.07500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07500},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic representation of General Relativity in harmonic gauge
AU - Jaume de Haro
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07500
UR - https://arxiv.org/abs/2611.07500
AB - 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× 4 real matrices M_4. By defining musical isomorphism operators (♯, ♭), metric adjoints (†), and matrix-valued derivative vectors (ℬ, 𝒞, 𝒟^♯), the Ricci tensor is recast into an algebraic matrix structure dominated by the Laplace--Beltrami operator Box_LB coloneqq tr(𝒢^♯ ∂^2), where ∂^2 coloneqq ∂∂^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 𝒢, giving rise to an index-free matrix representation of the gravitational energy-momentum pseudotensor 𝒯̅. We derive the explicit algebraic relation connecting 𝒯̅ with the classical Landau--Lifshitz energy-momentum complex 𝒯_LL, and express the associated local conservation law in a simple matrix divergence form, ∂^T ((−g)(𝒯^♯ + 𝒯_LL^♯)) = 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.
ER -