In order to couple Lorentz violating coefficients to fermions, one must work in the vierbein formalism (for a discussion see [272]). In Riemann-Cartan geometry the gravitational degrees of freedom are the vierbein and spin connection which give the Riemann and torsion tensors in spacetime. For the purposes of this review we will set the torsion to zero and work strictly in Riemannian geometry; for the complete Lorentz violating theory with torsion see [173] (for more general reviews of torsion in gravity see [143, 139]). The low energy action involving only second derivatives in the metric is given by
where The difficulty with this formulation is that it constitutes prior geometry and generically leads
to energy-momentum non-conservation, similar to the bimetric model in Section 2.4. Again
the matter stress tensor will not be conserved unless very restrictive conditions are placed on
and
(for example that they are covariantly constant). It is unclear whether or not
such restrictions can be consistently imposed in a complicated metric as would describe our
universe.
A more flexible approach is to presume that the Lorentz violating coefficients are dynamical, as has been
pursued in [122, 157
, 34, 185
, 220]. In this scenario, the matter stress tensor is automatically conserved if
all the fields are on-shell. The trade-off for this is that the coefficients
and
must be promoted
to the level of fields. In particularly they can have their own kinetic terms. Not surprisingly, this rapidly
leads to a very complicated theory, as not only must
and
have kinetic terms,
but they must also have potentials that force them to be non-zero at low energies. (If such
potentials were not present, then the vacuum state of the theory would be Lorentz invariant.)
For generic
and
, the complete theory is not known, but a simpler theory of a
dynamical “aether”, first looked at by [122] and expanded on by [157, 185, 104, 51] has been
explored.
The aether models assume that all the Lorentz violation is provided by a vector field
.12
With this assumption,
can be written as
, and
can always be reduced to an
term
due to the symmetries of the Riemann tensor. The most generic action in
dimensions that is quadratic
in fields is therefore
The aether models use a vector field to describe a preferred frame. Ghost condensate gives a more
specific model involving a scalar field. In this scenario the scalar field has a Lagrangian of the form
, where
.
is a polynomial in
with a minimum at some value
,
i.e.
acquires a constant velocity at its minimum. In a cosmological setting, Hubble friction drives the
field to this minimum, hence there is a global preferred frame determined by the velocity of
. This theory gives rise to the same Lorentz violating effects of aether theories, such as
Čerenkov radiation and spin dependent forces [33]. In general, systems that give constraints on the
coefficients of the aether theory are likely to also yield constraints on the size of the velocity
.
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