  For radio pulsars in orbit with a compact companion, pulsar timing
observations have proved to be a powerful tool for identifying the physical
nature of the companion. Unfortunately, perhaps the most intriguing system
where such a tool could be used, a pulsar in orbit with a black hole, has yet
to be discovered. In this paper we give a detailed investigation of what one
can learn about the black hole companion via timing observations of the pulsar.
We present an analytic calculation for the propagation delay caused by the
frame-dragging effect and show that it has the same functional behavior as the
modulation of the observed rotational phase of the pulsar caused by the
deflection of the radio signals in the gravitational field of the companion
(bending delay). Thus, contrary to statements of other authors, the
frame-dragging delay is unlikely to be separately measurable in pulsar binaries
where the companion is a stellar mass black hole. We demonstrate, however, that
the precession of the binary orbit caused by the relativistic spin-orbit
coupling can lead to observable effects which can be used to set a lower limit
to the black-hole spin, or possibly allow the determination of its magnitude
and orientation. We give parameter estimates for two possible systems, a 10
solar mass black hole in orbit either with a young (~0.1 s) pulsar or with a
millisecond pulsar. Finally, we discuss the measurability of the quadrupole
moment of the rotating black hole companion which would test the presence of a
Kerr black hole. As an interesting side result of our calculations, we can give
a further argument why the companion of PSR J0045-7319 cannot be a Kerr black
hole.
