Nonlinear Lagrangians and the significance of the metric

Recent investigations of nonlinear general relativistic gravitational Lagrangians have focused on the fact that these theories can be recast in standard Einstein form if the metric is redefined. It has been claimed that this result shows that these other theories are equivalent to the standard Einstein theory with additional fields. However, in this letter it is pointed out that the coupling to matter defines what the physically measured metric will be, so that the freedom of identification of an arbitrarily defined 2-tensor field as the metric is not justified. Simply, geodesics provide an operational definition of the metric.