  A resistor at finite temperature produces white noise fluctuations of the
current called Johnson-Nyquist noise. Measuring the amplitude of this noise
provides a powerful primary thermometry technique to access the electron
temperature. In practical situations, however, one needs to generalize the
Johnson-Nyquist theorem to handle spatially inhomogeneous temperature profiles.
Recent work provided such a generalization for ohmic devices obeying the
Wiedemann-Franz law, but there is a need to provide a similar generalization
for hydrodynamic electron systems, since hydrodynamic electrons provide unusual
sensitivity for Johnson noise thermometry but they do not admit a local
conductivity nor obey the Wiedemann-Franz law. Here we address this need by
considering low-frequency Johnson noise in the hydrodynamic setting for a
rectangular geometry. Unlike in the ohmic setting, we find that the Johnson
noise is geometry-dependent due to non-local viscous gradients. Nonetheless,
ignoring the geometric correction only leads to an error of at most 40% as
compared to naively using the ohmic result.
