Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 15Asthekar and Balachandran [70] noted that this ambiguity can be related, as in YM theory, to the possibility of adding tothe Einstein–Hilbert action the termS θ =iθ ∫d 4 x ∗ R R, (83)32π 2which is essentially the CS correction to the action when the scalar field ϑ = θ is constant and pulled out of the integral. Inthis sense a CS-like term arises naturally in LQG due to the requirement that wavefunctions, and thus physical observables,be invariant under large gauge transformations.But the θ-anomaly is not quite the same as CS modified gravity. After all, the above analysis is more reminiscent to thechiral anomaly in particle physics, discussed in Section 3.1. Recently, however, the connection between LQG and CS modifiedgravity has been completed, along a bit of an unexpected path. Taveras and Yunes [71] first investigated the possibility ofpromoting the Barbero–Immirzi (BI) parameter to a scalar field. This parameter is another quantization ambiguity parameterthat arises in LQG and determines the minimum eigenvalue of the discrete area and discrete volume operators [72]. At aclassical level, the BI parameter is a multiplicative constant that controls the strength of the dual curvature correction in theHolst action [69]. Taveras and Yunes realized that when this parameter is promoted to a field one essentially recovers GRgravity in the presence of an arbitrary scalar field at a classical level.Although the Holst action is attractive from a theoretical standpoint since it allows a construction of LQG in eitherAshtekar or Barbero form, this action has also been shown to lead to torsion and parity violation when one couples fermionsto the theory [73–75]. This issue can be corrected, while still allowing a mapping between GR and the Barbero–Ashtekarformalism, by adding to the Holst action a torsion squared term, essentially transforming the Holst term to the Nieh–Yaninvariant [76]. When one couples fermions to the Nieh–Yan modified theory, then the resulting effective theory remainstorsion free and parity preserving [77].Inspired by the work of Taveras and Yunes [71], Mercuri [78,79] and Mercuri and Taveras [80] considered the possibilityof promoting the BI parameter to a scalar field in the Nieh–Yan corrected theory. As in the Holst case, they found that the BIscalar naturally induces torsion, but this time when this torsion is used to construct an effective action they found that oneunavoidingly obtains CS modified gravity. In particular, one recovers Eq. (3) with ϑ = [3/(2κ)] 1/2 ˜β, with ˜β the BI scalarand α = 3/(32π 2 ) √ 3κ, while the scalar field action becomes Eq. (4) with β = 1 and vanishing potential.4. Exact vacuum solutionsOne of the most difficult tasks in any alternative theory of gravity is that of finding exact solutions, without the aid ofany approximation scheme. In the context of string theory, Campbell, et al. [23] showed that certain line elements, suchas Schwarzschild and FRW, lead to an exact CS three-form, which thus does not affect the modified field equations. In thecontext of CS modified gravity, Jackiw and Pi [22] showed explicitly that the Schwarzschild metric remains a solution ofthe non-dynamical modified theory for the canonical choice of CS scalar. Shortly after, Guarrera and Hariton [81] showedthat the FRW and Reissner–Nordstrom line elements also satisfy the non-dynamical modified field equations with the samechoice of scalar, verifying the results of Campbell, et al. [23]. Recently, Grumiller and Yunes [35] carried out an extensivestudy of exact solutions in the non-dynamical theory for arbitrary CS scalars, with the hope to find one that could representa spinning black hole. All of these investigations concern vacuum solutions in the non-dynamical framework (β = 0), withthe coupling constant choice α = κ. We shall also choose these conventions here.4.1. Classification of general solutionsLet us begin with a broad classification of general solutions in CS modified gravity. Grumiller and Yunes [35] classified thespace of solutions, a 2-dimensional representation of which is shown in Fig. 1, into an Einstein space, E, and a CS space, CS.Elements of the former satisfy the Einstein equations, while the elements of the later satisfy the CS modified field equations.The intersection of E with CS, P := E ∩ CS, defines the Pontryagin space, whose elements satisfy both the Einstein and theCS modified field equations independently. From the above definitions we can now classify solutions in CS modified gravity.Elements in P are GR solutions, because they satisfy the Einstein equations and possess a vanishing C-tensor and Pontryagindensity. Elements in CS \P are non-GR solutions, because they are not Ricci-flat but they do satisfy the Pontryagin constraintand the CS modified field equations.A full analytic study of exact solutions has been possible only regarding spacetimes with sufficient symmetries that allowfor the modify field equations to simplify dramatically. For such scenarios, however, the search for CS GR solutions have leadmostly to either Minkowski space or the Schwarzschild metric. This can be perhaps understood by considering the vacuumsector of P , where the C-tensor becomesC ab | Rab =0 = v cd∗ R d(ab)c = v cd∗ C d(ab)c = 0, (84)where C abcd and ∗ C are the Weyl tensor and its dual respectively [Eqs. (27) and (28)]. Such a condition implies the Weyltensor must be divergenceless via the contracted Bianchi identities, which leads to three distinct possibilities:(1) The (dual) Weyl tensor vanishes. In vacuum, elements of P are also Ricci flat, so this possibility leads uniquely toMinkowski space.