  We calculate the thermal Euclidean correlators and the spectral functions of
the energy-momentum tensor for pure gauge theories, including at non-zero
spatial momentum, at leading order in perturbation theory. Our goal is to
improve the extraction of transport properties from Euclidean correlators that
are computable in lattice QCD. Based on our results and the predictions of
hydrodynamics for the structure of the spectral functions at low frequencies,
we show that the shear and bulk viscosities can advantageously be extracted
from the Euclidean correlators of the conserved charges, energy and momentum,
at small but non-vanishing spatial momentum. The spectral functions in these
channels are free of the ultraviolet $\omega^4$ term which represents a large
background to the thermal physics encoded in the correlators of the fluxes.
