  Type I X-ray bursts are very common in neutron star X-ray binaries, but no
Type I burst has been seen in the dozen or so binaries in which the accreting
compact star is too massive to be a neutron star and therefore is identified as
a black hole candidate. We have carried out a global linear stability analysis
of the accumulating fuel on the surface of a compact star to identify the
conditions under which thermonuclear bursts are triggered. Our analysis, which
improves on previous calculations, reproduces the gross observational trends of
bursts in neutron star systems. It further shows that, if black hole candidates
have surfaces, they would very likely exhibit instabilities similar to those
that lead to Type I bursts on neutron stars. The lack of bursts in black hole
candidates is thus significant, and indicates that these objects have event
horizons. We discuss possible caveats to this conclusion.
