  The widely used Novikov-Thorne relativistic thin disc equations are only
valid down to the radius of the innermost stable circular orbit (ISCO). This
leads to an undetermined boundary condition at the ISCO, known as the inner
stress of the disc, which sets the luminosity of the disc at the ISCO and
introduces considerable ambiguity in accurately determining the mass, spin and
accretion rate of black holes from observed spectra. We resolve this ambiguity
by self-consistently extending the relativistic disc solution through the ISCO
to the black hole horizon by calculating the inspiral of an average disc
particle subject to turbulent disc forces, using a new particle-in-disc
technique. Traditionally it has been assumed that the stress at the ISCO is
zero, with material plunging approximately radially into the black hole at
close to the speed of light. We demonstrate that in fact the inspiral is less
severe, with several (~4-17) orbits completed before the horizon. This leads to
a small non-zero stress and luminosity at and inside the ISCO, with a local
surface temperature at the ISCO between ~0.15 and 0.3 times the maximum surface
temperature of the disc, in the case where no dynamically important net
magnetic field is present. For a range of disc parameters we calculate the
value of the inner stress/surface temperature, which is required when fitting
relativistic thin disc models to observations. We resolve a problem in
relativistic slim disc models in which turbulent heating becomes inaccurate and
falls to zero inside the plunging region.
