An Ambiguous Statement Called 'Tetrad Postulate' and the Correct Field Equations Satisfied by the Tetrad Fields

TL;DR

This paper clarifies the definitions of tetrad terms, addresses misconceptions caused by the 'tetrad postulate', and derives intrinsic wave equations for cotetrad fields using modern mathematical tools, correcting previous errors.

Contribution

It introduces precise definitions for tetrad terminology, clarifies the misconceptions surrounding the tetrad postulate, and derives correct intrinsic wave equations for cotetrad fields using Clifford bundle formalism.

Findings

Clarified the meanings of tetrad, tetrads, cotetrads.

Identified and corrected errors in existing cotetrad wave equations.

Derived intrinsic wave equations for cotetrad fields using modern mathematical tools.

Abstract

The names tetrad, tetrads, cotetrads, have been used with many different meanings in the physical literature, not all of them, equivalent from the mathematical point of view. In this paper we introduce unambiguous definitions for each one of those terms, and show how the old miscellanea made many authors to introduce in their formalism an ambiguous statement called `tetrad postulate', which has been source of many misunderstandings, as we show explicitly examining examples found in the literature. Since formulating Einstein's field equations intrinsically in terms of cotetrad fields theta^{a}, a = 0,1,2,3 is an worth enterprise, we derive the equation of motion of each theta^{a} using modern mathematical tools (the Clifford bundle formalism and the theory of the square of the Dirac operator). Indeed, we identify (giving all details and theorems) from the square of the Dirac operator some noticeable mathematical objects, namely, the Ricci, Einstein, covariant D'Alembertian and the Hodge Laplacian operators, which permit to show that each theta^{a} satisfies a well defined wave equation. Also, we present for completeness a detailed derivation of the cotetrad wave equations from a variational principal. We compare the cotetrad wave equation satisfied by each theta^{a} with some others appearing in the literature, and which are unfortunately in error.