In order to couple Lorentz violating coefficients to fermions, one must work in the vierbein formalism (for a discussion see ). 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  (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
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  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 that the theory suffers stability problems unless is of the form , where is a Lagrange multiplier. The theory is also ghost free with this potential and the further assumption that . Assuming these conditions, aether theories possess a set of coupled aether-metric modes which act as new gravitational degrees of freedom that can be searched for with gravitational wave interferometers or by determining energy loss rates from inspiral systems like the binary pulsar. The same scenario generically happens for any tensor field that acquires a VEV dynamically (see Section 7.1), which implies that Lorentz violation can be constrained by the gravitational sector as well as by direct matter couplings.
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 . 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 .
© Max Planck Society and the author(s)