  Two possible explanations for the low luminosity of the supermassive black
hole at the center of our galaxy are (1) an accretion rate of order the
canonical Bondi value (roughly 10^{-5} solar masses per year), but a very low
radiative efficiency for the accreting gas or (2) an accretion rate much less
than the Bondi rate. Both models can explain the broad-band spectrum of the
Galactic Center. We show that they can be distinguished using the linear
polarization of synchrotron radiation. Accretion at the Bondi rate predicts no
linear polarization at any frequency due to Faraday depolarization. Low
accretion rate models, on the other hand, have much lower gas densities and
magnetic field strengths close to the black hole; polarization may therefore be
observable at high frequencies. If confirmed, a recent detection of linear
polarization from Sgr A$^*$ above 150 GHz argues for an accretion rate of order
10^{-8} solar masses per year, much less than the Bondi rate. This test can be
applied to other low-luminosity galactic nuclei.
