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 $4\times 4$ 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.