The nontriviality of trivial general covariance: How electrons restrict ‘time’ coordinates, spinors (almost) fit into tensor calculus, and of a tetrad is surplus structure

Abstract

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus.On the other hand, the physics literature often claims that spinors as such cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is (nearly) Ockhamized, with most of the fluff removed. As developed * Forthcoming in Studies in History and Philosophy of Modern Physics 1 nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix diag(−1, 1, 1, 1), the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and values of the fields involved.Apart from coordinate order and the usual spinorial two-valuedness, (densitized) Ogievetsky-Polubarinov spinors form, with the (conformal part of the) met- The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited.

Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity

“…It is slightly tricky that some interesting nonlinear would-be geometric objects, such as the symmetric square root of an indefinite metric tensor, actually fail to be defined for some coordinate systems[67][68][69]. Such an entity is useful both for massive gravity theories and for spinors.…”

The dressing field method for diffeomorphisms: a relational framework

“…Such enforced Diff(M) symmetries do not have the same status as native ones, being grafted onto a theory mainly to hide its background non-dynamical structures. For that reason, these are called 'artificial symmetries' in the philosophy of physics literature [89][90][91][92] 32 . Theories obtained that way are only superficially and formally alike GR, but betray its key physical insight, i.e.…”

Against ‘functional gravitational energy’: a critical note on functionalism, selective realism, and geometric objects and gravitational energy

“…NB: Generically each pseudotensor has a different (viciously coordinate-dependent) transformation rule.9 It's worthwhile mentioning thatFriedman (1983), in contrast to the cited authors, restricts geometric objects to either tensors or connections. Thereby he-unduly-neglects, for instance, tensor densitities of arbitrary weight or projective connexions (cf Pitts 2006Pitts , 2012. Schouten 1954, p. 301).…”