  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.
