  The anticipated discovery of a pulsar in orbit with a black hole is expected
to provide a unique laboratory for black hole physics and gravity. In this
context, the next generation of radio telescopes, like the Five-hundred-metre
Aperture Spherical radio Telescope (FAST) and the Square Kilometre Array (SKA),
with their unprecedented sensitivity, will play a key role. In this paper, we
investigate the capability of future radio telescopes to probe the spacetime of
a black hole and test gravity theories, by timing a pulsar orbiting a
stellar-mass-black-hole (SBH). Based on mock data simulations, we show that a
few years of timing observations of a sufficiently compact pulsar-SBH (PSR-SBH)
system with future radio telescopes would allow precise measurements of the
black hole mass and spin. A measurement precision of one per cent can be
expected for the spin. Measuring the quadrupole moment of the black hole,
needed to test GR's no-hair theorem, requires extreme system configurations
with compact orbits and a large SBH mass. Additionally, we show that a PSR-SBH
system can lead to greatly improved constraints on alternative gravity theories
even if they predict black holes (practically) identical to GR's. This is
demonstrated for a specific class of scalar-tensor theories. Finally, we
investigate the requirements for searching for PSR-SBH systems. It is shown
that the high sensitivity of the next generation of radio telescopes is key for
discovering compact PSR-SBH systems, as it will allow for sufficiently short
survey integration times.
