The Heuristic Approach to General Relativity in the Laplace-Beltrami Formalism

The Laplace-Beltrami formalism, in which the Ricci tensor in the Einstein field equations (EFEs) is formulated at leading-order in terms of the partial-differential Laplace-Beltrami operator, was previously applied to coalescing compact binaries (CCBs) generating gravitational waves (GWs). Supposing that the CCB is an effective singular body -- a hollow mass-shell -- that follows a Kerr metric Ansatz, the EFEs were approached variationally such that the Ansatz geometric signature dictates the energetic output via $G_{\mu\nu}=8\pi GT_{\mu\nu}$. For the CCB mass-shell representation, the generated GW energy is treated as radiated surface energy via $E:=T_{00}V$. This surface energy yielded a close approximation to the cataloged GW coalescence energy, as previously shown in past comparisons. Given this success, it is logical to ask whether the Laplace-Beltrami formalism can be applied to other general relativistic systems, whether ``simple" or ``perturbative", beyond CCBs. This heuristic work focuses broadly on the EFEs themselves under the Laplace-Beltrami formalism, considering all differential orders up to second-order. This namely includes a deeper analysis on the variational methodology employed on the EFEs in the second-order sector, utilized in previous works, and the benchmark analysis of the lower first- and zeroth-order terms. This all-order report utilizes representative examples and select metric Ans\"atze to explore the formalism's practicality and its limitations; this is shown that the first-order decomposition showcases heuristically the mechanics of vector and scalar fields upon a curved spacetime.