Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 29essentially demands that the CS correction be a small deformation of the Kerr line element, i.e. the CS coupling is assumedsmall relative to the GR one. In this way, one finds the solution 7 [84]:α 2ds 2 = ds 2 + 5 aKerr(1 + 128 βκ r 4 7ϑ = 5 α a cos(θ)(1 + 2M8 β M r 2 rMr + 27 )M 2sin 2 θ dφdt,10 r 2+ 18M25r 2 ), (174)where ds 2 Kerris the slow-rotation limit of the Kerr metric, M is the BH mass and J = a · M is the BH angular momentumto leading order. This solution is valid to second order in the slowly-rotation expansion parameter a/M, as well as in thestrength of the coupling α/(βκM 4 ). Notice that this solution, derived under the dynamical formulation, is perfectly wellbehavedat spatial infinity, remaining asymptotically flat.Eq. (174) is the first 8 rotating BH solution in dynamical CS modified gravity, and can be thought of as a small deformationof a Kerr black hole with additional CS scalar ‘‘hair’’ of finite energy. Although this is a ‘‘hairy’’ solution, Sopuerta andYunes [99] have shown that the solution is still entirely described by the mass and the angular momentum of the source. Theno-hair theorem is, however, violated in that the relation between higher-multipoles and the mass quadrupole and currentdipole is CS modified at l = 4 multipole due to the CS correction in Eq. (174) of the gravitomagnetic sector.The solution found in the dynamical theory presents interesting parity properties. Since the dynamics of the CS scalarfield are determined by the Pontryagin density, this field is parity-violating (i.e. it is a pseudo-scalar). Both the Kerr metricand the CS correction to it are also parity-violating, but the latter is induced by a curvature-scalar field interaction, instead ofdue to the Kerr distributional stress–energy. In fact, the parity violation introduced by the CS correction becomes dominantin regions of high curvature.Moreover, the dynamical solution presented above shows remarkable similarities with some of the far-field results foundin the non-dynamical framework. In particular, it is only the gravitomagnetic sector of the metric that is CS modified,thus implying that the frame-dragging effect will be primarily corrected. Notice, however, that the CS correction is highlysuppressed by large inverse powers of radius, which suggest that weak-field tests will not be able to constrain the dynamicalframework. We shall investigate this possibilities further in Section 7.5.6. Gravitational wave propagationGW solutions have been studied by a large number of authors [22,28,29,33,36,81,99], but mostly in the non-dynamicaltheory, which we shall concentrate on here. The first GW investigation in non-dynamical CS modified gravity was carried outby Lue, Wang and Kamionkowski [134], who studied the effect of GWs in the cosmic microwave background (see Section 8).Jackiw and Pi [22] also studied GWs in CS modified gravity, concentrating on the generation of such waves and the powercarried by them in the modified theory. Shortly after, such waves were used to explain baryogenesis during inflation [28] andto calculate the super-Hubble power spectrum [29]. The generation of GWs was also studied in the dynamical formalism [81]through the construction of an effective stress–energy tensor and the Isaacson scheme [99]. Recently, GW tests have beenproposed to constrain CS gravity with space-borne [33] gravitational wave interferometers.We shall here discuss GW solutions in non-dynamical CS modified gravity, and postpone any discussion of GW generationto the next section. Moreover, we shall not discuss, in this section, cosmological power spectra, since these will besummarized in Section 8. GW propagation in CS gravity has only been studied in the non-dynamical formalism with β = 0and α = κ. Let us begin with a discussion of GW propagation in a FRW background. Consider, then, the backgroundds 2 = a 2 (η) [ −dη 2 + ( δ ij + h ij)dχ i dχ j] , (175)where a(η) is the conformal factor, η is conformal time and χ i are comoving coordinates. The quantity h ij stands for thegravitational wave perturbation, which we take to be transverse and traceless (TT), h := h i i = δ ij h ij = 0 and ∂ i h ij = 0. Onecan show that a coordinate system exists, such that the gravitational wave perturbation can be put in such a TT form. Forthe remainder of this section, i, j, k stand for spatial indices only.With such a metric decomposition, one can linearize the action to find the perturbed field equations. In doing so, one mustchoose a functional form for the CS scalar and we shall here follow Alexander and Martin [29], who chose ϑ = ϑ(η). Onecan show that the linearized action (the Einstein–Hilbert piece plus the CS piece) to second order in the metric perturbationyieldsS EH + S CS = κ ∫[ d 4 x a 2 (η) ( h i j,η h j i,η − h i j,k h j k) i, − ϑ ,η ˜ɛ ( )] ijk h q i,ηh kq,jη − h q i, r h kq,rj + O(h)3(176)4V7 Notice that the solution for ϑ is identical to that found by Campbell [85] and Reuter [87] and discussed in Eq. (90), except that there the backreactionof this field on the metric was ignored.8 Shortly after publication of this result, Konno, et al. employed slightly different methods to verify that the solution found by Yunes and Pretorius indeedsatisfies the modified field equations [133].