A Strange Detail Concerning the Variational Principle of General Relativity Theory

TL;DR

This paper clarifies a mathematical complication in the variational principle of general relativity, resolving historical confusion and emphasizing the importance of tensor differentiation over component differentiation for accurate calculations.

Contribution

It identifies and clears up a formal assumption issue in the variational principle of general relativity, providing solutions and emphasizing the correct differentiation method.

Findings

Clarified the mathematical complication in Einstein-Hilbert variational principle.

Highlighted the importance of tensor differentiation over component differentiation.

Provided verified solutions and numerical modifications for calculating Einstein tensor density.

Abstract

A mathematical complication due to an unnecessary formal assumption concerning the variational principle of general relativity theory, which apparently bothered Einstein and Hilbert, is shown and cleared up. Some historical confusion seems caused by the impossibility to use the conventional Euler-Lagrange formalism directly there, which even otherwise is nothing but one of various possible procedures to apply the superior principle of least action. Correspondingly to the absence of any direct calculation in the literature so far, only a numerical modification in parts - explicitly taken into account now after once mentioned by Hilbert without implementation - would allow to compute the fundamental Einstein tensor density from these authors' initial formulae, which must not be taken literally. Nevertheless adhering to a merely symbolic Euler-Lagrange formalism, this needs a clear distinction between 'component differentiation' and 'tensor differentiation' defined here. Various corresponding solutions are shown including the probably most natural one. Two of them are additionally verified in the detailed supplementary material appended to the electronic edition of the note.