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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 (1View Equation) 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 (1View Equation). We will include gravity in the usual way by adding the Einstein-Hilbert Lagrangian for the metric. The resultant action is
integral [ ( ) ] S = d4x V~ -g --1---R + 1gab \~/ af \~/ bf + 1-gab + t ab \~/ ay \~/ by , (4) 16pG 2 2
and the corresponding field equations are
a 0 = \~/ \~/ af, (5) 0 = \~/ a\ ~/ ay + \~/ a(t ab \~/ by), (6) 1 1 1 Gab = \~/ af \~/ bf - -- \~/ sf \~/ sfgab + \~/ ay \~/ by - - \~/ sy \~/ sygab - -t rs \~/ ry \~/ sygab. (7) 2 2 2
Taking the divergence of Equation (7View Equation) and using the f,y equations of motion yields
0 = - \~/ by\ ~/ r (trs \~/ sy) - 1 \~/ b (trs \~/ ry\ ~/ sy) , (8) 2
since \~/ aGab vanishes by virtue of the Bianchi identities.

The right hand side of Equation (8View Equation) does not in general vanish for solutions to the field equations and therefore Equation (8View Equation) is not in general satisfied unless one restricts to very specific solutions for y. This is not a useful situation, as we would like to have the full space of solutions for y yet maintain energy conservation. The solution is to make all Lorentz violating tensors dynamical [173Jump To The Next Citation Point157Jump To The Next Citation Point], 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.


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