  Einstein-aether theory provides a model to test the validity of local Lorentz
invariance in gravitational interactions. The speed of gravitational waves as
measured from the binary neutron star event GW170817 sets stringent limits on
Einstein-aether theory, but only on a combination of the theory's free
parameters. For this reason, a significant part of the theory's parameter space
remains unconstrained by observations. Motivated by this, we explore the
propagation of gravitational waves in Einstein-aether theory over an
inhomogeneous background (i.e., gravitational wave lensing) as a potential
mechanism to break the degeneracies between the theory's free parameters, and
hence enable new constraints on the theory to be obtained. By bringing the
field equations into the form of the so-called kinetic matrix and applying a
formalism known as the propagation eigenstate framework, we find that the speed
of gravitational waves is modified by inhomogeneities in the aether field.
However, the modification is common to both gravitational polarizations and
vanishes in the limit in which gravitational waves propagate with luminal
speed. This lens-dependent gravitational wave speed contrasts with the
lens-induced birefringence observed in other theories beyond general
relativity, like Horndeski's theory. While the potential to improve tests based
on gravitational-wave speed is limited, our formalism sets the basis to fully
describe signal propagation over inhomogeneous spacetimes in Einstein-aether
theory and other extensions of general relativity.
