5.3 Cavity experiments
From the Michelson-Morley experiments onward, interferometry has been an excellent method of testing
relativity. Modern cavity experiments extend on the ideas of interferometry and provide very precise tests
on the bounds of certain photon parameters. The main technique of a cavity experiment is to detect the
variation of the resonance frequency of the cavity as its orientation changes with respect to a stationary
frequency standard. In this sense, it is similar to a clock comparison experiment. However, since one of the
clocks involves photons, cavity experiments constrain the electromagnetic sector of the mSME as
well.
The analysis of cavity experiments is much easier if we make a field redefinition of the electromagnetic
sector of the mSME [179
]. In analogy to the theory of dielectrics, we define two new fields
and
by
The
coefficients are related to the mSME coefficients by
With this choice of fields, the modified Maxwell equations from the mSME take the suggestive form
This redefinition shows that the Lorentz violating background tensor
can be thought of as a
dielectric medium with no charge or current density. Hence we can apply much of our intuition about the
behavior of fields inside a dielectric to construct tests of Lorentz violation. Note that since
and
depend on the components of
, the properties of the dielectric are orientation
dependent.
Constraints from cavity experiments are not on the
parameters themselves, but rather on the linear
combinations
,
, and
are all parity even, while
and
are parity odd. The usefulness
of this parameterization can be seen if we rewrite the Lagrangian in these parameters [179
],
From this expression it is easy to see that
corresponds to a rotationally invariant shift in the speed of
light. It can be shown that
and
also yield a shift in the speed of light, although in a
direction dependent manner. The coefficients
and
control birefringence. Cavity
experiments yield limits on
and
, while birefringence (see Section 6.3) bounds
and
.
The most straightforward way to constrain Lorentz violation with cavity resonators is to study the
resonant frequency of a cavity. Since we have a cavity filled with an orientation dependent dielectric,
the resonant frequency will also vary with orientation. The resonant frequency of a cavity is
where
is the mode number,
is the speed of light,
is the index of refraction (including Lorentz
violation) of any medium in the cavity, and
is the length of the cavity.
can be sensitive to Lorentz
violating effects through
,
, and
. Depending on the construction of the cavity some effects can
dominate over others. For example, in sapphire cavities the change in
due to Lorentz violation
is negligible compared to the change in
. This allows one to isolate the electromagnetic
sector.
In general, all cavities are sensitive to the photon
parameters. In contrast to sapphire, for
certain materials the strain induced on the cavity by Lorentz violation is large. This allows
sensitivity to the electron parameters
at a level equivalent to the photon parameters.
Furthermore, by using a cavity with a medium, the dependence of
on
gives additional electron
sensitivity [226
].
The complete bounds on the mSME coefficients for cavity experiments are given
in [23
, 67, 227, 226
, 280, 207, 32
, 261
]. The strongest bounds are displayed in Table 1. Roughly, the
components of
and
are bounded at
while
is bounded at
. The
difference arises as
enters constraints suppressed by the boost factor of the earth relative to the
solar “rest” frame where the coefficients are taken to be constant.