  A coherently oscillating ultra-light axion can behave as dark matter. In
particular, its coherently oscillating pressure perturbations can source an
oscillating scalar metric perturbation, with a characteristic oscillation
frequency which is twice the axion Compton frequency. A candidate in the mass
range $10^{(-24,-21)}{\rm eV}$ can provide a signal in the frequency range
tested by current and future Pulsar Timing Array (PTA) programs. Involving the
pressure perturbations in a highly nonlinear environment, such an analysis
demands a relativistic and nonlinear treatment. Here, we provide a rigorous
derivation of the effect assuming weak gravity and slow-motion limit of
Einstein's gravity in zero-shear gauge and show that dark matter's velocity
potential determines the oscillation phase and frequency change. A
monochromatic PTA signal correlated with the velocity field would confirm the
prediction, for example, by cross-correlating the PTA results with the future
local velocity flow measurements.
