Physics Reports ChernâSimons modified general relativity

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24 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55where the current J a := (−ρ,⃗ J) and ta = [1, 0, 0, 0] is a time-like unit vector. 6 Such a choice of stress–energy is equivalentot that of a pressureless perfect fluid in the limit as the fluid balls tend to particles. Moreover, Smith, et al. [37] focused onthe dynamical formulation of CS modified gravity, with the conventions α = −l/3 and β = −1. Such a study is forbiddenin the non-dynamical formalism, because the Pontryagin constraint does not allow for rotating solutions. In the dynamicalscheme, however, such solutions are allowed, provided the Pontryagin density is balanced by the dynamics of the scalarfield.In spite of working in the dynamical theory, Smith, et al. [37] chose a canonical ϑ, thus implicitly treating this field as nondynamical.Such a choice of scalar field is formally inconsistent with dynamical CS modified gravity, since the non-vanishingPontryagin constraint will force spatial variations on ϑ. Nonetheless, this choice could become a good approximation, if oneis only concerned with far-field solutions. This is because, as we have shown in Eq. (89) of Section 4.3, the exterior Pontryagindensity for a rotating source scales as ∗ R R ≈ aM 2 /r 7 , which then forcesβ□ϑ = −24α aM2r 7= 8l aM2 , (137)r 7with uncontrolled remainders of O(M/r) 4 . Thus, spatial variations will induce modifications to the canonical ϑ that are atmost of O(M/r) 3 , which is beyond the order considered in this section. A full dynamical study of this problem has yet to becarried out.Let us now return to the gravitoelectromagnetic formalism, in which certain components of the metric perturbation areidentified with a four-vector potential, namelyA a := − 1 ¯h ab t b , (138)4where ¯hab is the trace-reversed metric perturbation ¯hab := h ab − 1 2 η abh and h = η ab h ab is the trace of the latter. In analogywith electromagnetism, one can now define electric and magnetic fields viaE i = ∂ i A 0 − ∂ 0 A i , B i = ɛ 0ijk ∂ j A k , (139)where i, j stand for spatial indices only. The geodesic equations can then be written in terms of these fields asF i = −mE i − 4m (v × B) i , (140)where the cross product is that of flat Euclidean space, from which one can read off the Lorentz force acceleration a i = F i /m.We can now apply this formalism to the linearized CS modified field equations. In the Lorentz gauge, A a ,a= 0, the modifiedfield equations become [37]⃗∇ · ⃗ B = 0, (141)⃗∇ × E ⃗∂⃗ B = −∂t , (142)⃗∇ · E ⃗ = 4πG(ρ + Tϑ00 (143)⃗∇ × ⃗ B −∂⃗ E∂t − 1m cs□⃗ B = 4πG ⃗ J, (144)where m cs is a characteristic mass scale defined viam cs := 2κα ˙ϑ = − 6κl ˙ϑ . (145)In the above equations, the first two arise from the definitions of the four-vector potential, while the last two (Gauss’slaw in Eq. (143) and Ampère’s law in Eq. (144)) are modified in CS modified gravity, the former arising from the timetimecomponent of the modified field equation and the latter from the gravitomagnetic sector. Smith, et al. [37] neglect T ϑ 00because they associate it with the energy density of the scalar field, which they argue must be uniform throughout the SolarSystem, not larger than the mean cosmological energy density and negligible relative to ρ. Moreover, from Eq. (15), we seethat T ϑ 00 is at least quadratic in ˙ϑ and can thus be neglected, provided 3β/(2ρµ 2 ) ≪ 1.Let us now specialize the above treatment to that of homogeneous rotating source, with mass current⃗ J = ρ[⃗ω × ⃗r]Θ(R − r), (146)where R is the radius of the rotating body, ρ is its density, ⃗ω is its angular velocity, r is the distance from the origin, and Θis the Heaviside step function. One can now use this mass current in Ampère’s law to solve this equation for A i , imposing6 Recall that indices are raised and lowered with the Minkowski metric in the far-field approximation.
S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 25continuity and finiteness at the origin. The actual expressions for the four-vector potential can be found in [37], but itsassociated gravitomagnetic field is given by B ⃗ = BGR ⃗ + BCS ⃗ , where the GR piece is simply⎧⎪⎨(5 − 3 r )2⃗ω + 3 r 2ˆr × (ˆr × ⃗ω), r ≤ R,⃗B GR = 4πGρR2 R 2 R2 (147)15 ⎪⎩ R 3 [ ]2⃗ω + 3ˆr × (ˆr × ⃗ω) , r ≥ R,r 3while the CS correction is given by⃗B CS = 4πGρR 2 { D 1 (r) ⃗ω + D 2 (r) ˆr × ⃗ω + D 3 (r) ˆr × (ˆr × ⃗ω) } . (148)The function D 1,2,3 (r) are actually functionals of spherical Bessel functions of the first j l (x) and second y l (x) kind, givenexplicitly in [37]. Remarkably, the gravitoelectromagnetic analogy allows for a solution to the linearized modified fieldequations for an extended rotating source, which possesses both a poloidal and toroidal gravitomagnetic field. In otherwords, the metric does not only contain gravitomagnetic components co-aligned with the spin axis of the rotating extendedbody, but also along other axis, perpendicular to the plane defined by ⃗ω and ⃗r. Such terms cannot indeed be removed via acoordinate transformation since B i φ is generically non-vanishing.The solution for the metric perturbation found in the gravitoelectromagnetic analogy for extended sources differs slightlyfrom that found in the PN scheme for point particles. Some of these differences arise because in the latter the point particleswere allowed to possess non-vanishing angular and linear momentum. The main difference, however, is not due to thestress–energy tensor studied, but to the fact that extended and point particle treatments are not equivalent in CS modifiedgravity. In the former, an additional oscillatory behavior is needed, encoded in the appearance of Bessel functions, in order toguarantee continuity across the surface of the sphere. Such terms are necessary because the C-tensor modifies the junctionconditions of GR [107–113,21,114].The gravitoelectromagnetic analogy also sheds some light on the study of exact solutions to CS modified gravity, reviewedin Section 4. The CS correction to Ampère’s law, Eq. (144) changes the character of the differential system, from a first-orderone to a second-order one. Such a result is in fact expected from the structure of the modified field equations [ Eq. (18)], sincethe C-tensor depends on third-order derivatives of the metric. One then could also expect that approximate solution to CSmodified theory need not be easily obtained as deformations of GR solutions. This is because, in principle, the CS correctioncould produce dramatic changes to the dynamical behavior of the solution (i.e. the solution, for example, could be an elementof CS \ P in the nomenclature of Section 4, which would hold little resemblance to GR solutions). CS modified theory andGR, however, have both now been sufficiently tested to allow for a perturbative treatment of the modified theory, at leastin the Solar System. Whether such treatment is allowed in more non-linear, strong field scenarios remains to be studiedfurther.5.4. Perturbations of the Schwarzschild spacetimeBlack hole perturbation theory has been incredibly important in GR, leading for example to a better understanding ofsubtle issues related to BH physics and the extraction of gravitational waves in dynamic spacetimes. Moreover, through thediscovery of quasinormal ringing, new tests of GR have been proposed (see eg. [124] for a review). These accomplishmentsalone suggest a study of BH perturbation theory in CS modified gravity could lead to interesting results. Such a study wascarried out by Yunes and Sopuerta [34] in the non-dynamical formalism with α = κ and β = 0.Let us begin by introducing the basics of BH perturbation theory, as developed in GR. Consider then the followingperturbed metricg ab = ḡ ab + h ab , (149)where ḡ ab is the background Schwarzschild metric and h ab is a generic metric perturbation. Henceforth, objects associatedwith the background will be denoted with an overhead bar. The spherical symmetry of the background allows one to expandthe metric perturbation in tensor spherical harmonics, thus separating the angular i = {θ, φ} and temporal-radial µ = {t, r}dependence in the perturbed field equations and yielding a 1 + 1 PDE system. This system can be further simplified bydistinguishing between polar and axial parity harmonics: polar (axial) modes acquire a (−1) l [(−1) l+1 ] prefactor upon paritytransformations. The aforementioned simplification comes about because in GR the perturbative field equations decoupleinto two subsystem that can be classified by their parity.With these considerations in hand, we split the metric perturbation into h ab = h a + ab hp ab, where each of these pieces isdecomposed in tensor spherical harmonics: the temporal-radial sector becomesh a µν = 0,hp µν = ∑ l,mh p,lmµν = ∑ h lmµν Y lm , (150)l,mthe temporal-angular sector is given byh a µi = ∑ l,mh a,lmµi= ∑ h lmµ Slm i, h p = ∑ µih p,lmµi= ∑ p lmµ Y lmi, (151)l,ml,ml,m
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