In a framework describing manifestly covariant relativistic evolution using a scalar time #tau#. consistency demands that #tau#-dependent fields be used. In recent work by the authors, general features of a classical parametrized theory of gravitation, paralleling eneral relativity where possible, were outlined. The existence of a preferred "time" coordinate #tau# changes the theory significantly. In particular, the Hamiltonian constraint for #tau# is removed From the Euler-Lagrange equations. Instead of the 5-dimensional stress-energy tensor, a tensor comprised of 4-momentum density and flux density only serves as the source. Building on that foundation, in this paper we develop a linear approximate theory of parametrized gravitation in the spirit of the flat spacetime approach to general relativity. Using a modified form of Kraichnan's flat spacetime derivation of general relativity, we extend the linear theory to a family of nonlinear theories in which the flat metric and the gravitational field coalesce into a single effective curved metric.
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