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 GRs 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 GRs. 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.
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