  We study reflected X-ray emission that returns to the accretion disc in the
strong gravitational fields around black holes using General Relativistic ray
tracing and radiative transfer calculations. Reflected X-rays that are produced
when the inner regions of the disc are illuminated by the corona are subject to
strong gravitational light bending, causing up to 47 per cent of the reflected
emission to be returned to the disc around a rapidly spinning black hole,
depending upon the scale height of the corona. The iron K line is enhanced
relative to the continuum by 25 per cent, and the Compton hump by up to a
factor of three. Additional light travel time between primary and secondary
reflections increases the reverberation time lag measured in the iron K band by
49 per cent, while the soft X-ray lag is increased by 25 per cent and the
Compton hump response time is increased by 60 per cent. Measured samples of
X-ray reverberation lags are shown to be consistent with X-rays returning to
the accretion disc in strong gravity. Understanding the effects of returning
radiation is important in interpreting reverberation observations to probe
black holes. Reflected X-rays returning to the disc can be uniquely identified
by blueshifted returning iron K line photons that are Compton scattered from
the inner disc, producing excess, delayed emission in the 3.5-4.5keV energy
range that will be detectable with forthcoming X-ray observatories,
representing a unique test of General Relativity in the strong field limit.
