Tensor gauge condition and tensor field decomposition

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.