Lecture Notes on Gravitational Waves and Linear Perturbations, Study notes of Physics

Download Lecture Notes on Gravitational Waves and Linear Perturbations and more Study notes Physics in PDF only on Docsity! Lecture Notes for 12/2/08 Benjamin Hall and Hidris Pal December 4, 2008 0.1 Continuing linear perturbation As stated in the last lecture, we decompose the metric perturbation hµν as: hµν = (Φ,Ψ, wi, Sij) (1) where Sij is traceless. The vector part wi is decomposed further: wi = wi⊥ + w i ‖ where ∂iw i ⊥ = 0 (2) ijk∂iw ‖ k = 0. (3) These conditions imply that wi⊥ has 2 degrees of freedom and w i ‖ = ∂iξ has only one degree of freedom. Sij can be similarly decomposed: Sij = Sij⊥ + S ij S + S ij ‖ where ∂iS ij ⊥ = 0 (4) ∂i∂jS ij S = 0 (5) ijk∂j∂lS l ‖k = 0. (6) Again, Sij‖ has only 1 d.o.f, S ij S has 2, and S ij ⊥ has the remaining 2. This is because we can write Sij‖ = (∂i∂j − 1 2 ∇2δij)Θ (7) SSij = ∂(iΞj). (8) 1 As was stated in previous classes, gravitational waves are spin 2 (under spatial rotations) traceless tensors; the only part of the perturbation match- ing this criterion is Sij, specifically Sij⊥ , as it is the only spin-2 part of the metric perturbation. For the remainder of these notes we will use the traditional notation hijTT ≡ Sij⊥ , where the “TT” stands for “transverse traceless.” If expanded in spherical harmonics (for example) hijTT has only terms with L ≥ 2, so it doesn’t include the gravitational monopole (∆m, S = 0, L = 0), the gravitational dipole (relating to motion against the background, S = 0, L = 1) or the (static) angular momentum (S = 1, L = 1) terms that may be present from the source. Various choices of gauge can push the gravitational waves to other parts of the perturbation by reducing spin, but cannot change L. 0.2 Gravitational Waves For this section we will consider Sij⊥ = h ijTT = Aexp[i(ωt− ~k · ~x)]; kiSij⊥ = 0 and also set ~k = (0, 0, kz). Under these assumptions we can write: hµν = h+ hx 0 hx −h+ 0 0 0 0 (9) The action of a pure h+ or hx polarized gravitational wave on a circular array of masses is shown in Figure 1. Note, this formalism only describes the weak-field limit (small sources or large separation between source and detector). It is totally insufficient to describe the generation of gravitational waves close to strong sources such as black holes. We can thus write Gµν(η + h) = 8πTµν for weak sources and slow motion. We know from conservation that time and spatial derivatives of T µν can be related: ∂µT µ0 = 0 −→ ∂0T 00 = −∂iT i0 (10) ∂µT µi = 0 −→ ∂0T 0i = −∂jT ji. (11) After much work, we can then write the metric perturbation as: hij = 2G r d2 dt2 [Iij(t− r)] (12) 2