Tensor gauge condition and tensor field decomposition

TL;DR

This paper explores various methods of decomposing tensor fields into gauge-invariant and pure-gauge parts, relating these to gauge-fixing approaches, and analyzing their mathematical and physical properties in the context of general relativity.

Contribution

It systematically investigates multiple tensor gauge conditions and their decompositions, highlighting the lack of a universally optimal choice and clarifying their relation to gauge fixing.

Findings

Multiple tensor gauge conditions are mathematically explored.

No single gauge condition is superior in all aspects.

Tensor decomposition relates closely to gauge-fixing methods.

Abstract

We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant components. Such tensor field decomposition is intimately related to the effort of identifying the real gravitational degrees of freedom out of the metric tensor in Einstein's general relativity. We show that, as for a vector field, the tensor field decomposition has exact correspondence to, and can be derived from, the gauge-fixing approach. The complication for the tensor field, however, is that there are infinitely many complete gauge conditions, in contrast to the uniqueness of Coulomb gauge for a vector field. We make an extensive exploration of these tensor gauge conditions and their corresponding tensor field decompositions, regarding mathematical structures, equations of motion, nonlinear properties; and show that apparently no single choice is superior in all aspects.