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 nondynamic background, just in accordance with the
principle of equivalence. If there were some nondynamic 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.