Physics Reports Chern–Simons modified general relativity

18 S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55The set of equations imposed by Eq. (86) can be solved exactly, yielding either flat space, the Schwarzschild solution orthe following two solutions:ds 2 = − 1 z dt2 + zdz 2 + z 2 (dρ 2 + ρ 2 dφ 2 ) (95)and)( 2mds 2 = −z− 1( 2mdt 2 +z− 1) −1dz 2 + z 2 (dρ 2 + sinh 2 ρ dφ 2 ). (96)These solutions, however, contain undesirable, non-physical features, such as the existence of a naked singularity at z = 0[Eq. (95)], or the existence of a spacelike Killing vector k a = (∂ t ) a in the ‘‘outside’’ region z > 2m [Eq. (96)].The above results can also be obtained by noticing that the spatial sector of C ab reduces identically to the 3-dimensionalCotton–York tensor for static and axisymmetric line elements, which by the decoupling of the field equations must vanishexactly [35]. This implies the metric must be spatially conformally flat. Lukács and Perjés [91] have shown that such solutionsto the Einstein equations reduce to Minkowski, Schwarzschild or the line elements of Eqs. (95) and (96), as described above.The above results hold for any CS scalar in the familyϑ = ϑ 1 (t, ρ, z) + ϑ 2 (ρ, z, φ). (97)For even more general CS scalars, as would arise, for instance, in the dynamical formulation, the modified field equationsdo not necessarily decouple and, thus, new and interesting solutions could arise. Unfortunately, when this is the case, thesystem of partial differential equations becomes too difficult to study analytically and has not really been analyzed in detail.4.5. Stationary and axisymmetric spacetimesNow consider stationary and axisymmetric spacetimes [42]:ds 2 = −V (dt − wdφ) 2 + V −1 ρ 2 dφ 2 + Ω 2 ( dρ 2 + Λdz 2) , (98)where V , Ω, Λ and w depend on ρ and z, with the latter identified with angular velocity of rotation about the Killing axis.Let us first consider the canonical choice of CS scalar, for which the modified field equations decouple once more. TheEinstein equations can be used to set Λ to unity and put the line element in Lewis–Papapetrou–Weyl form [42]ds 2 = −e 2U (dt − wdφ) 2 + e −2U [ e 2k (dρ 2 + dz 2 ) + ρ 2 dφ 2] , (99)where U and k replace V and Ω.The vanishing of the Ricci tensor reduces to a set of partial differential equations similar to Eq. (93), with a non-trivialsource that depends on w and an additional equation for this function. The vanishing of the C-tensor, however, does notlead to a vanishing Cotton–York tensor, and thus, it does not necessarily imply spatial conformal flatness. Grumiller andYunes [35] have argued that it seems unlikely that other non-trivial and physically interesting solutions besides the staticones could arise, because R ab = 0 and C ab = 0 is a strongly over-constrained differential system. This is an example ofhow the non-dynamical theory can lead to an overconstrained system of equations when searching for physically relevantsolutions, but a strict proof remains elusive.The argument presented above holds for any ϑ in the familyϑ = t µ + φ ν , (100)with constants µ and ν. When the CS coupling is not of the form of Eq. (100), then the field equations do not decouple, thearguments presented above do not hold and the system become more difficult to analyze analytically.In the non-dynamical framework, another route to exact solutions for generic CS scalars is through the Pontryaginconstraint. For axisymmetric spacetimes, ∗ R R ≠ 0, and this leads to a complicated set of partial differential equationsfor w, U and k. Solutions of ∗ R R = 0 have been found to be of Petrov type II [35], which correspond to the Van Stockumclass [42]ds 2 = ρΩdt 2 − 2ρdtdφ + 1 √ ρ(dρ 2 + dz 2) , (101)where Ω = Ω(ρ, z) is arbitrary and there is no dφ 2 component.The reduced Van-Stockum class of metrics leads to a complete decoupling of the modified field equations, except forthe dt 2 component. The vanishing of the C-tensor can be achieved if ϑ = ϑ(ρ, z), while the dt 2 component of the fieldequations can be solved to findΩ = c, ϑ = ϑ(ρ, z), (102)Ω = c +d √ ρ, ϑ = 2 3√ ρ z + ˜ϑ(ρ), (103)