Physics Reports Chern–Simons modified general relativity

8 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55decide what ϑ is, one loses the predictive power of the Einstein equations and replaces it by a family of possible solutions.Moreover, all choices of ϑ so far explored are rather unnatural or unphysical, in that they lead to a field with infinite energy,since the field’s kinetic energy is constant. Such fields are completely incompatible with the dynamical framework, whichimplies that results arrived at in the non-dynamical framework cannot be directly extended into the dynamical scheme.Second, the Pontryagin constraint can be thought of as a selection rule, that eliminates certain metrics from the space ofallowed solutions. Such a selection rule has been found to overconstrain the modified field equations, to the point that onlythe trivial zero-solution is allowed in certain cases [34].Having said this, the non-dynamical framework has been useful to qualitatively understand the effect of the CS correctionon gravitational parity violation. Only recently has there been a serious, albeit limited, effort to study the much more difficultdynamical formulation, and preliminary results seem to indicate that solutions found in this framework do share manysimilarities with solutions found in the non-dynamical scheme. The non-dynamical formulation should thus be viewed as atoy-model that might help us gain some insight into the more realistic dynamical framework.2.3. Dynamical CS gravityThe dynamical framework is defined by allowing β and α to be arbitrary, but non-zero constants. In fact, β cannot beassumed to be close to zero (or much smaller than α), because then the evolution equation for ϑ becomes singular. Thisformulation was initially introduced by Smith, et al. [37], with the particular choice α = −l/3 and β = −1, which implies[ϑ] = L −1 . In this model, the CS scalar field is thus not externally prescribed, but it instead evolves driven by the spacetimecurvature. The Pontryagin constraint is then superseded by Eq. (21), which does not impose a direct and hard constraint onthe solution space of the modified theory. Instead, it couples the evolution of the CS field to the modified field equations.The dynamical formulation, however, is not completely devoid of arbitrariness. Most of this is captured in the potentialV(ϑ) that appears in Eq. (4), since this is a priori unknown. In the context of string theory, the CS scalar field is a moduli field,which before stabilization has zero potential (i.e. it represents a flat direction in the Calabi–Yau manifold). Stabilization of themoduli field occurs via supersymmetry breaking at some large energy scale, thus inducing an (almost incalculable) potentialthat is relevant only at such a scale. Therefore, in the string theory context, it is reasonable to neglect such a potential whenconsidering classical and semi-classical scenarios.The arbitrariness aforementioned, however, still persists through the definition of the kinetic energy contained in thescalar field. For example, there is no reason to disallow scalar field actions of the form∫S new ϑ = − 1 2Vd 4 x √ −g [ β 1 (∂ϑ) 2 + β 2 (∂ϑ) 4] , (32)which leads to the following stress–energy tensorT new ϑab[= β 1 1 + 2β2 (∂ϑ) 2] (∇ a ϑ) (∇ b ϑ) − 1 [2 g ab β1 (∂ϑ) 2 + β 2 (∂ϑ) 4] , (33)where we have used the shorthand (∂ϑ) 2 := g ab (∇ a ϑ) (∇ b ϑ). The Pontryagin constraint would then be replaced byβ 1 □ϑ + 2β 2 ∇ a[(∇ a ϑ ) (∂ϑ) 2] = −α κ 4∗ R R. (34)Of course, the choice of Eq. (4) is natural in the sense that it corresponds to the Klein–Gordon action, but it should actuallybe the more fundamental theory, from which CS modified gravity is derived, that prescribes the scalar field Hamiltonian. Inthe string theory context, however, the moduli field possess a canonical kinetic Hamiltonian, suggesting that Eq. (32) is thecorrect prescription [27].Another natural choice for the potential of the CS coupling is the C-tensor itself. In other words, consider the possibilityof placing the C-tensor on the right-hand side of the modified field equations and treating it as simply a non-standardstress–energy contribution. Such a possibility was studied by Grümiller and Yunes [35] for certain background metrics,which are solutions in GR but not in CS modified gravity. The Kerr metric is an example of such a background, for which theyfound that the induced C-tensor stress–energy violates all energy conditions. No classical matter in the observable universeis so far known to violate all energy conditions, thus rendering this possibility rather unrealistic.2.4. Parity violation in CS modified gravityPrecisely what type of parity violation is induced by the CS correction? Let us first define parity violation as the purelyspatial reflection of the triad that defines the coordinate system. The operation ˆP [A] = λp A is then said to be even,parity-preserving or symmetric when λ p = +1, while it is said to be odd, parity-violating or antisymmetric if λ p = −1.[ ] [By definition, we then have that ˆP eIi = −eIi , where eI iis a spatial triad, and thus ˆP ] eaijk= −e aijk . Note that paritytransformations are slicing-dependent, discrete operations, where one must specify some spacelike hypersurface on whichto operate. On the other hand the combined parity and time-reversal operations is a spacetime operation that is slicingindependent.