Physics Reports Chern–Simons modified general relativity

20 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55additional, non-vanishing, gravitomagnetic metric components. Up to the writing of this review, the only attempts to findsuch solutions have failed [35], due to the incredible complexity of the differential system.Consider next spacetimes with matter content. The Kerr BH is a ‘‘vacuum’’ solution in GR, but it does possess adistributional energy momentum tensor [97]. Moreover, in string theory and cosmology [28], CS modified gravity arisesfrom matter currents, so the inclusion of such degrees of freedom might in principle be important. In the dynamical scheme,for example, one could lift any GR solution to a solution of CS modified gravity by requiring thatR ab − 1 2 g abR = 8πT matab , (110)C ab = 8πT ϑ ab , (111)where T matabstands for the stress–energy of matter degrees of freedom (such as the distributional one associated with theKerr solution), while T ϑ abis the energy–momentum of the CS coupling. One would now have to solve the system of PDEsassociated with Eq. (111) for the background which satisfies Eq. (110). Such a task, however, would imply also solving theequation of motion for the CS scalar, thus reducing this analysis to the study of exact solutions in dynamical CS modifiedgravity, which has not yet been performed.5. Approximate vacuum solutionsApproximate schemes have been employed to solve the CS modified field equations in different limits. The first attemptalong these lines was that of Alexander and Yunes [32,7], who performed a far-field, PPN analysis of non-dynamical CSmodified gravity (α = κ, β = 0) with canonical ϑ. This study was closely followed by that of Smith, et al. [37] who carriedout a far-field investigation of non-dynamical solutions representing the gravitational field outside a homogeneous, rotatingsphere, taking careful account of the matching between interior and exterior solutions. Konno, et al. [98] investigated theslow-rotation limit of stationary and axisymmetric line elements in non-dynamical CS modified gravity with non-canonicalCS scalar fields. Finally, gravitational wave solutions of CS modified gravity have been studied by a number of authors, bothin Minkowski spacetime and in an FRW background [22,28,31,29,33,36]. The last two set of studies were carried out in thenon-dynamical formalism with the choices α = κ and β = 0. Little is known about GW propagation or generation indynamical CS modified gravity, although early efforts are being directed on that front [84,99].5.1. Formal post-Newtonian solutionThe PN approximation has seen tremendous success to model full general relativity in the slow-motion, weaklygravitating regime (for a recent review see [100]). This approximation is used heavily to study Solar System tests ofalternative theories of gravity in the PPN framework, as well as to describe gravitational waves from inspiraling compactbinaries, which could be observed in gravitational wave detectors in the near future. For these reasons, it is instructive tostudy the PN expansion of CS modified gravity, before submerging ourselves in other approximate solutions.The PN approximation is essentially a slow-motion and weak-gravity scheme in which the field equations of some theoryare expanded and solved perturbatively and iteratively. As such, this scheme makes use of multiple-scale perturbationtheory [101–104], where the perturbation parameters are the self-gravity of the objects (an expansion in powers of Newton’sconstant G) and their typical velocities v (an expansion in inverse powers of the speed of light). For example, matter densitiesρ are dominant over pressures p and specific energy densities Π, while spatial derivatives are dominant over temporal ones.The PN framework also requires the presence of external matter degrees of freedom, i.e. bodies that are self-gravitatingand slowly-moving. Such objects can be described in a point-particle approximation [100], or alternatively with a perfectfluid stress–energy tensor [105]:T ab = (ρ + ρΠ + p) u a u b + pg ab , (112)where u a is the object’s four-velocity. In GR, the internal structure of the gravitating objects can be neglected to rather largePN order [106], and thus one can effectively take the radius of the fluid balls to zero, which reproduces the point-particleresult. This statement is that of the effacing principle [106], which is the view we shall take in the next section when westudy CS modified gravity in the PPN framework. However, care must be taken, since the effacing principle need not holdin alternative theories of gravity. In fact, as we shall see later on, the effacing principle must be corrected in CS modifiedgravity due to modifications to the junction conditions [107–113,21,114].Perturbation theory, and thus the PN approximation, requires the use of a specific background and coordinate system.In the traditional PN scheme, one linearizes the field equation with the metric g ab = η ab + h ab , where η ab is the Minkowskibackground, since cosmological effects are usually subdominant. Moreover, a Lorentz gauge is usually chosen h ba,a = h ,b /2,which allows one to cast the field equations as a wave equation with non-trivial, non-linear source terms. One can showthat to first order in the metric perturbation, the linearized CS modified field equations in the non-dynamical formalism,with canonical CS scalar and in the Lorentz gauge, can be written as [32,7](□ η H ab = −16π T ab − 1 )2 g abT + O(h) 2 , (113)