The Nontriviality of Trivial General Covariance: How Electrons Restrict 'Time' Coordinates, Spinors (Almost) Fit into Tensor Calculus, and 7/16 of a Tetrad Is Surplus Structure

TL;DR

This paper revisits the construction of spinors in curved spacetime, showing they can be represented in coordinates without the tetrad formalism, revealing new insights into general covariance and gravitational energy localization.

Contribution

It demonstrates that Ogievetsky and Polubarinov's coordinate-based spinors are self-sufficient, avoiding surplus structures of tetrads, and clarifies their coordinate restrictions and geometric properties.

Findings

OP spinors admit any coordinates with time listed first

They form nonlinear geometric objects with the metric

OP spinors enable gauge-invariant localization of gravitational energy

Abstract

It is a commonplace that any theory can be written in any coordinates via tensor calculus. But it is claimed that spinors as such cannot be represented in coordinates in a curved space-time. What general covariance means for theories with fermions is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov (OP) constructed spinors in coordinates in 1965, helping to spawn nonlinear group representations. Locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient. The tetrad formalism is de-Ockhamized, with 6 extra fields and 6 compensating gauge symmetries. OP spinors, as developed nonperturbatively by Bilyalov, admit any coordinates at a point, but "time" must be listed first: the product of the metric components and the matrix diag(-1,1,1,1) must have no negative eigenvalues to yield a real symmetric square root function of the metric. Thus the admissible coordinates depend on the types and values of the fields. Apart from coordinate order and spinorial two-valuedness, OP spinors form, with the metric, a nonlinear geometric object, with Lie and covariant derivatives. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in GR. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of a matrix with 16 components, one can use weighted OP spinors coupled to the 9-component symmetric unimodular square root of the conformal metric density. The surprising mildness of the restrictions on coordinates for the Schwarzschild solution is exhibited. (edited)