Is exponential metric a natural space-time metric of Newtonian gravity ?

We show how Newtonian gravity with effective (actually observed) masses, obeying the mass-energy relation of special relativity, can explain all observations used to test General relativity. Dynamics of a gravitationally coupled binary system is considered in detail, and the effective masses of constituents are determined. Interpreting our results in terms of motion in a curved space-time background, we are led, using the Lagrangian formalism, to consider the exponential metric as a natural space-time metric of Newtonian gravity.