Gravitational-Wave Observations as a Tool for Testing Relativistic Gravity

The Polarizations of Gravitational Waves

“…The relation between {Ψ 2 , Ψ 3 , Ψ 4 , Φ 22 } and the polarizations of the GW can be found by examining the linearized geodesic deviation equation [17],…”

“…where ϑ ∈ [0, 2π) and ρ is complex [17]. Using these, six classes are defined below [17], Class II 6 Ψ 2 = 0; for any observer, there is the same Ψ 2 = 0 mode, but all other modes are observer-dependent; Class III 5 Ψ 2 = 0, Ψ 3 = 0; for any observer, there are the Ψ 2 = 0 mode and the same Ψ 3 = 0 mode, but the remaining modes Ψ 4 and Φ 22 are observer-dependent;…”

“…E(2) classification is a framework [17,18] to categorize the null GWs in a generic, local Lorentz invariant metric theory of gravity using the Newman-Penrose variables [16]. For the GW traveling in the +z direction, the suitable null tetrad basis,…”

“…This classification can be applied to any metric theory of gravity which respects the local Lorentz invariance and predicts null GWs, such as Brans-Dicke theory, the simplest scalar-tensor theory [19]. In this theory, there are plus and cross modes Ψ 4 due to the massless graviton, and there also exists the transverse breathing mode Φ 22 induced by the massless scalar field [17].…”

“…According to their transformation properties under the little group E(2) of the Lorentz group, the six polarizations of the null plane GWs can be classified in terms of the Newman-Penrose (NP) variables: Ψ 2 , Ψ 3 , Ψ 4 and Φ 22 [16][17][18]. Among them, Ψ 4 represents the plus and the cross polarizations, Φ 22 donates the transverse breathing polarization, Ψ 3 corresponds to the vector-x and vector-y polarizations, and Ψ 2 is for the longitudinal polarization.…”