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

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 θ a , a = 0, 1, 2, 3 is an worth enterprise, we derive the equation of motion of each θ 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 θ 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 θ a with some others appearing in the literature, and which are unfortunately in error. 1 IntroductionIn what follows we identify an ambiguous statement called 'tetrad postulate' (a better name, as we shall see would be 'naive tetrad postulate') that appears often in the Physics literature (see e.g., [5,12,24,47,49,50], to quote only a few examples here). We identify the genesis of the wording 'tetrad postulate' as a result of a deficient identification of some mathematical objects of differential geometry. Note that we used the word ambiguous, not the word wrong. This is because, as we shall show, the equation dubbed 'tetrad postulate' can be rigorously interpreted as meaning that the components of a covariant derivative in the direction of a vector field ∂ µ of a certain tensor field Q (Eq.(34)) are null (see Eq. (80)). This equation is not a postulate. Indeed, it is nothing more than the intrinsic expression of an obvious identity of differential geometry that we dubbed the freshman identity (Eq. (63)). However, if the freshman identity is used naively as if meaning a 'tetrad postulate' misunderstandings may arise, and in what follows we present some of them, by examining some examples that we found in the literature. We comment also on a result called 'Evans Lemma' of differential geometry and claimed in [12] to be as important as the Poincaré lemma. We show that 'Evans Lemma' as presented in [12] is a false statement, the proof offered by that author being invalid because in trying to use the naive tetrad postulate he did incorrect use of some fundamental concepts of differential geometry, as, e.g., 1 his (wrong) Eq.(41E). We explain all that in details in what follows. We observe also that in [12,13,14,15,16,17] it is claimed that 'Evans Lemma' is the basic pil...

An Assessment of Evans’ Unified Field Theory I

“…Such a statement, which overlaps with the last subsection, seems necessary since there are numerous inconsistencies and mistakes in Evans' work, see Bruhn [3][4][5][6][7][8][9][10][11][12] and Rodrigues et al [19,61], such that it is necessary to distiguish between the relevant and the irrelevant parts of Evans' articles.…”

“…Evans arrives at a RC-geometry by means of what he calls the "tetrad postulate", see [25], Eqs. (32) and (33), and Rodrigues et al [61]. In Sec.2 we will display the geometric properties of a RC-geometry in the 4-dimensional spacetime in quite some detail.…”

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“…6 that any Lorentzian spacetime admitting spinor fields must have a global tetrad. 33 Sometimes in the written of some formulas in the next chapters it is convenient to use the notation 35 -defined by the following parallel transport rule: a vector is parallel transported along a curve, if at any x 2 S 2 the angle between the vector and the vector tangent to the latitude line passing through that point is constant during the transport (see Fig. 4.5).…”

“…(4.313) becomes sometimes misunderstood as meaning r h a or r C h a , thus generating a big confusion. For a discussion of this issue see [33]. 39 Recall that other authors prefer the notations .r @ V/˛WD VW and .r @ C/˛WD C˛W .…”

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