Since we do not have a third kind of device to specify the spatio-temporal location of the devices measuring the spacetime
geometry, we do not have any further operationally defined, maybe non-dynamical background, just in accordance with the
principle of equivalence. If there were some non-dynamical background metric
on
, then by requiring
we could reduce the almost arbitrary diffeomorphism
(essentially four arbitrary functions of four variables) to a
transformation depending on at most ten parameters.