  This paper studies the linearized gravitational field in the presence of
boundaries. For this purpose, $\zeta$-function regularization is used to
perform the mode-by-mode evaluation of BRST-invariant Faddeev-Popov amplitudes
in the case of flat Euclidean four-space bounded by a three-sphere. On choosing
the de Donder gauge-averaging term, the resulting $\zeta(0)$ value is found to
agree with the space-time covariant calculation of the same amplitudes, which
relies on the recently corrected geometric formulas for the asymptotic heat
kernel in the case of mixed boundary conditions. Two sets of mixed boundary
conditions for Euclidean quantum gravity are then compared in detail. The
analysis proves that one cannot restrict the path-integral measure to
transverse-traceless perturbations. By contrast, gauge-invariant amplitudes are
only obtained on considering from the beginning all perturbative modes of the
gravitational field, jointly with ghost modes.
