  Gaia BH3 is the most massive known stellar-origin black hole in the Milky
Way, with a mass $M_\bullet \approx 33 \, \rm M_\odot$. Detected from Gaia's
astrometry, this black hole is in the mass range of those observed via
gravitational waves, whose nature is still highly debated. Hosted in a binary
system with a companion giant star that is too far away for Roche-lobe mass
transfer, this black hole could nonetheless accrete at low levels due to
wind-driven mass-loss from its companion star, thus accreting in
advection-dominated accretion flow, or ADAF, mode. Using stellar wind models,
we constrain its Eddington ratio in the range $10^{-9} < f_{\rm Edd} <
10^{-7}$, corresponding to radiative efficiencies $5\times10^{-5} < \epsilon <
10^{-3}$, compatible with radiatively inefficient accretion modes. Chandra
ACIS-S observed this object and obtained the most sensitive upper bound of its
[2-10] keV flux: $F_X < 3.25 \times 10^{-15} \, \rm erg \, s^{-1} \, cm^{-2}$
at $90\%$ confidence level corresponding to $L_{[2-10]}< 2.10 \times 10^{29} \,
\rm erg \, s^{-1}$. Using ADAF emission models, we constrained its accretion
rate to $f_{\rm Edd}< 4.91 \times 10^{-7}$ at the apastron, in agreement with
our theoretical estimate. At the periastron, we expect fluxes $\sim 50$ times
larger. Because of the inferred low rates, accretion did not significantly
contribute to black hole growth over the system's lifetime. Detecting the
electromagnetic emission from Gaia BH3 will be fundamental to informing stellar
wind and accretion disk models.
