### 2.4 Diffeomorphism invariance and prior geometry

If Lorentz violating effects are to be embedded in an effective field theory, then new tensors must be
introduced that break the Lorentz symmetry (cf. the bimetric theory (1) of Section 2.1). If we are
considering only special relativity, then keeping these tensors as constant is viable. However, any complete
theory must include gravity, of course, and one should preserve as many fundamental principles of
general relativity as possible while still introducing local Lorentz violation. There are three
general principles in general relativity relevant to Lorentz violation: general covariance (which
implies both passive and active diffeomorphism invariance [247]), the equivalence principle, and
lack of prior geometry. As we saw in Section 2, general covariance is automatically a property
of an appropriately formulated Lorentz violating theory, even in flat space. The fate of the
equivalence principle we deal with below in Section 2.5. The last principle, lack of prior geometry, is
simply a statement that the metric is a dynamical object on the same level as any other field.
Coupled with diffeomorphism invariance this leads to conservation of matter stress tensors (for a
discussion see [73]). However, a fixed Lorentz violating tensor constitutes prior geometry in the
same way that a fixed metric would. If we keep our Lorentz violating tensors as fixed objects,
we immediately have non-conservation of stress tensors and inconsistent Einstein equations.
As a specific example, consider again the bimetric theory (1). We will include gravity in the
usual way by adding the Einstein-Hilbert Lagrangian for the metric. The resultant action is
and the corresponding field equations are
Taking the divergence of Equation (7) and using the equations of motion yields
since vanishes by virtue of the Bianchi identities.
The right hand side of Equation (8) does not in general vanish for solutions to the field equations and
therefore Equation (8) is not in general satisfied unless one restricts to very specific solutions for . This
is not a useful situation, as we would like to have the full space of solutions for yet maintain energy
conservation. The solution is to make all Lorentz violating tensors dynamical [173, 157], thereby removing
prior geometry. If the Lorentz violating tensors are dynamical then conservation of the stress tensor is
automatically enforced by the diffeomorphism invariance of the action. While dynamical Lorentz violating
tensors have a number of effects that are testable in the gravitational sector, most researchers
have concentrated on flat space tests of Lorentz invariance where gravitational effects can be
ignored. Hence for most of this review we will treat the Lorentz violating coefficients as fixed
and neglect dynamics. The theoretical consequences of dynamical Lorentz violation will be
analyzed only in Section 4.4, where we discuss a model of a diffeomorphism invariant “aether”
which has received some attention. The observational constraints on this theory are discussed in
Section 7.