  We propose a new strategy for determining the equation of state of a
relativistic thermal quantum field theory by considering it in a moving
reference system. In this frame an observer can measure the entropy density of
the system directly from its average total momentum. In the Euclidean path
integral formalism, this amounts to compute the expectation value of the
off-diagonal components T_{0k} of the energy-momentum tensor in presence of
shifted boundary conditions. The entropy is thus easily measured from the
expectation value of a local observable computed at the target temperature T
only. At large T, the temperature itself is the only scale which drives the
systematic errors, and the lattice spacing can be tuned to perform a reliable
continuum limit extrapolation while keeping finite-size effects under control.
We test this strategy for the four-dimensional SU(3) Yang-Mills theory. We
present precise results for the entropy density and its step-scaling function
in the temperature range 0.9 T_c - 20 T_c. At each temperature, we consider
four lattice spacings in order to extrapolate the results to the continuum
limit. As a byproduct we also determine the ultraviolet finite renormalization
constant of T_{0k} by imposing suitable Ward identities. These findings
establish this strategy as a solid, simple and efficient method for an accurate
determination of the equation of state of a relativistic thermal field theory
over several orders of magnitude in T.
