  If a class of stars orbits the central black hole in our galaxy in short
period (~ 0.1 year), high eccentricity (~ 0.9) orbits, they will experience
precessions of their orbital planes induced by both relativistic frame-dragging
and the quadrupolar gravity of the hole, at levels that could be as large as 10
microarcseconds per year, if the black hole is rotating faster than 1/2 of its
maximum rotation rate. Astrometric observations of the orbits of at least two
such stars can in principle lead to a determination of the angular momentum
vector J of the black hole and its quadrupole moment Q_2. This could lead to a
test of the general relativistic no-hair theorems, which demand that Q_2 = -
J^2/M. Future high-precision adaptive infrared optics instruments make make
such a fundamental test of the black-hole paradigm possible.
