We consider a rapidly spinning black hole surrounded by an equatorial,
geometrically thin, slowly accreting disk that is stationary and axisymmetric.
We analytically compute the broadening of electromagnetic line emissions from
the innermost part of the disk, which resides in the near-horizon region. The
result is independent of the details of the disk model and therefore universal.
This is an example of critical behavior in astronomy that is potentially
observable by current or future telescopes.