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

Wick rotation of linearized gravity in Gaussian time and Calder\'on projectors