  In this paper we develop a consistent, phenomenological methodology to
measure preferred-frame effects (PFEs) in binary pulsars that exhibit a high
rate of periastron advance. We show that in these systems the existence of a
preferred frame for gravity leads to an observable characteristic `signature'
in the timing data, which uniquely identifies this effect. We expand the
standard Damour-Deruelle timing formula to incorporate this `signature' and
show how this new PFE timing model can be used to either measure or constrain
the parameters related to a violation of the local Lorentz invariance of
gravity in the strong internal fields of neutron stars. In particular, we
demonstrate that in the presence of PFEs we expect a set of the new timing
parameters to have a unique relationship that can be measured and tested
incontrovertibly. This new methodology is applied to the Double Pulsar, which
turns out to be the ideal test system for this kind of experiments.The
currently available dataset allows us only to study the impact of PFEs on the
orbital precession rate, d omega/dt, providing limits that are, at the moment,
clearly less stringent than existing limits on PFE strong-field parameters.
However, simulations show that the constraints improve fast in the coming
years, allowing us to study all new PFE timing parameters and to check for the
unique relationship between them. Finally, we show how a combination of several
suitable systems in a "PFE antenna array", expected to be availabe for instance
with the Square-Kilometre-Array (SKA), provides full sensitivity to possible
violations of local Lorentz invariance in strong gravitational fields in all
directions of the sky. This PFE antenna array may eventually allow us to
determine the direction of a preferred frame should it exist.

Astrophysics