### 7.1 Gravitational waves

In the presence of dynamical Lorentz violation, where the entire action
is diffeomorphism invariant, one generically expects new gravitational wave
polarizations.
The reason is simple. Any dynamical Lorentz violating tensor field must have kinetic terms involving
derivatives of the form , where is the Lorentz violating tensor. Furthermore, must
take a non-zero vacuum expectation value if it violates Lorentz invariance. At linear order in the
perturbations (where ), the connection terms in the
covariant derivative are also first order, for example . Upon varying the linearized metric,
these terms contribute to the graviton equations of motion. The extra terms in the graviton
equations give rise to new solutions. Since the potential that forces to take a non-zero vacuum
expectation value must involve the metric, variations in are usually coupled to metric variations,
implying that the new graviton modes mix with excitations of the Lorentz violating tensor
fields.
There is a large literature on gravitational wave polarizations in theories of gravity other
than general relativity. For a thorough discussion, see [277] and references therein. Many of
the models with preferred frame effects are similar to the types of theories that give rise to
dynamical Lorentz violation. For example, the vector-tensor theories of Will, Hellings, and
Nordtvedt [278, 237, 145] have many similarities to the aether theory of Section 4.4. The aether model’s
wave spectrum has been calculated in [158, 136] and limits from the absence of Čerenkov
emission of these modes by cosmic rays has been studied in [105] (see Section 6.5.7). Other
consequences of dynamical Lorentz violation in Riemann-Cartan spacetimes have been examined
in [57].
Unfortunately, few constraints currently exist on dynamical Lorentz violation from gravitational wave
observations as the spectrum is only part of the story. Currently, the expected rate of production of these modes
from astrophysical sources as a function of the coefficients in the Lagrangian is unknown. However, both the
energy loss from inspiral systems due to gravitational radiation and gravitational wave observatories such as
LIGO and LISA should produce strict bounds on the possibility of dynamical Lorentz violating
fields.
We note that aether type theories seem to be free of certain obvious problems such as a van
Dam-Veltman-Zakharov type discontinuity [136]. The theories can therefore be made arbitrarily close to
GR by tuning the coefficients to be near zero.