  Motivated by the quantization of linearized gravity, we consider gauge-fixed
linearized Einstein equations and their Wick rotation near a Cauchy surface. We
show that Calder\'on projectors for the Wick-rotated equations induce Hadamard
bi-solutions on the Lorentzian level. On the other hand, we find smoothing
obstructions to gauge-invariance and positivity conditions needed in
quantization. These obstructions are primarily due to boundary terms arising in
the Wick-rotated theory and depend on the boundary conditions.
