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Gravitational Waves Theory, Sources and Detection

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Gravitational Waves Theory, Sources and Detection Gravitational Waves Theory, Sources and Detection Kostas Kokkotas September 28, 2016 Kostas Kokkotas Gravitational Waves Suggested Reading I Books I Gravitational Waves: Volume 1: Theory and Experiments by Michele Maggiore Oxford University Press (2007) I Gravitation and Spacetime by Ohanian, Hans C. and Ruffini, Remo Cambridge University Preess (Sep 20, 2013) I Gravitation by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Sep 15, 1973) W.H. Freeman I Review articles I Gravitational wave astronomy F.F. Schutz, Class. Quantum Grav. 16 (1999) A131– A156 I Gravitational wave astronomy: in anticipation of first sources to be detected L P Grishchuk, V M Lipunov, K A Postnov, M E Prokhorov, B S Sathyaprakash, Physics- Uspekhi 44 (1) 1 -51 (2001) I The basics of gravitational wave theory E.E. Flanagan and S.A. Hughes, New Journal of Physics 7 (2005) 204 I Gravitational Wave Astronomy K.D. Kokkotas Reviews in Modern Astrophysics, Vol 20, ”Cosmic Matter”, WILEY-VCH, Ed.S. Roeser (2008) arXiv:0809.1602 [astro-ph] Kostas Kokkotas Gravitational Waves Tensors I Tensor Transformations X ∂x ν X ∂x˜µ b˜ = b and˜ aµ = aν (1) µ ∂x˜µ ν ∂x ν ν ν ∂x˜α ∂x˜β ∂x˜α ∂x ν ∂x µ ∂x ν T˜ αβ = T µν , T˜ α = T µ & T˜ = T ∂x µ ∂x ν β ∂x µ ∂x˜β ν αβ ∂x˜α ∂x˜β µν Covariant Derivative φ;λ = φ,λ (2) ρ Aλ;µ = Aλ,µ − ΓµλAρ (3) λµ λµ λ αµ µ λα T ;ν = T ,ν + Γαν T + Γαν T (4) Parallel Transport λ ν δaµ = Γ µν aλdx for covariant vectors (5) µ µ λ ν δa = −Γ λν a dx for contravariant vectors (6) Kostas Kokkotas Gravitational Waves Tensors II The distance ds of two points P(x µ) and P0(x µ + dx µ) is given by 2 2 2 ds2 = dx 1 + dx 2 + dx 3 (7) in two different coordinate frames , 2 µ ν α β ds =g ˜µν dx˜ dx˜ = gαβ dx dx . (8) I Metric element for Minkowski spacetime ds2 = −c2dt2 + dx 2 + dy 2 + dz2 (9) ds2 = −c2dt2 + dr 2 + r 2dθ2 + r 2 sin2 θdφ2 (10) I For a sphere with radius R : ds2 = R2 dθ2 + sin2 θdφ2 (11) I The metric element of a torus with radii a and b ds2 = a2dφ2 + (b + a sin φ)2 dθ2 (12) I The Schwarzschild metric: 2 GM 2 GM −1 ds2 = 1 − c2dt2 − 1 − dr 2 − r 2 dθ2 + sin2 θdφ2 c2 r c2 r Kostas Kokkotas Gravitational Waves Tensors III Curvature Tensor or Riemann Tensor λ λ λ µ λ µ λ R βνσ = −Γ βν,σ + Γ βσ,ν − Γ βν Γ µσ + Γ βσΓ µν (13) Christoffel Symbols 1 Γα = g αν (g + g − g ) (14) µρ 2 µν,ρ νρ,µ ρµ,ν Geodesic Equations duρ d 2x ρ dx µ dx ν + Γρ uµuν = 0 or + Γρ = 0 ds µν ds2 µν ds ds Kostas Kokkotas Gravitational Waves The Ricci and Einstein Tensors The contraction of the Riemann tensor leads to Ricci Tensor λ λµ Rαβ = R αλβ = g Rλαµβ µ µ µ ν µ ν =Γ αβ,µ − Γαµ,β + Γαβ Γνµ − Γαν Γβµ (15) which is symmetric Rαβ = Rβα. Further contraction leads to the Ricci or Curvature Scalar α αβ αβ µν R = R α = g Rαβ = g g Rµανβ . (16) The following combination of Riemann and Ricci tensors is called Einstein Tensor 1 G = R − g R (17) µν µν 2 µν with the very important property: µ µ 1 µ G ν;µ = R ν − δ ν R = 0 . (18) 2 ;µ This results from the Bianchi Identity (how?) λ R µ[νρ;σ] = 0 (19) Kostas Kokkotas Gravitational Waves Flat & Empty Spacetimes I When Rαβµν = 0 the spacetime is flat I When Rµν = 0 the spacetime is empty Prove that : λ λ λ κ a ;µ;ν − a ;ν;µ = −R κµν a Kostas Kokkotas Gravitational Waves Einstein’s equations Thus the typical form of Einstein’s equations is: 1 Rµν − g µν R + Λg µν = κT µν . (20) 2 8πG whereΛ= c2 ρv is the so called cosmological constant. They can also be written as: 1 R = −κ T − g T (21) µν µν 2 µν Kostas Kokkotas Gravitational Waves GW: Linear Theory I Weak gravitational fields can be represented by a slightly deformed Minkowski spacetime : 2 gµν ' ηµν + hµν + O(hµν ) , |hµν | 1 (22) here hµν is a small metric perturbation. The indices will be raised and lowered by ηµν i.e. αβ αµ βν h = η η hµν (23) µν h = η hµν (24) g µν = ηµν − hµν (25) and we will define the traceless (φµν ) tensor: 1 φ = h − η h . (26) µν µν 2 µν The Christoffel symbols & the Ricci tensor will become : 1 Γλ ≈ ηλρ (h + h − h ) (27) µν 2 ρν,µ µρ,ν µν,ρ 1 R ≈ Γα − Γα ≈ (hα + hα − h α − hα )(28) µν µν,α µα,ν 2 ν,µα µ,να µν,α α,µν µν αβ α,β R = η Rµν ≈ hαβ − hα,β (29) Kostas Kokkotas Gravitational Waves Finally, Einstein tensor gets the form: 1 G (1) = (hα + hα − h α − hα ) − η h ,αβ − hα,β (30) µν 2 ν,µα µ,να µν,α α,µν µν αβ α,β Einstein’s equations reduce to (how?): ,α ,αβ ,α ,α −φµν ,α − ηµν φαβ + φµα,ν + φνα,µ = κTµν (31) Then by using the so called Hilbert (or Harmonic or De Donder) gauge similar α ,α 1 to Lorenz gauge( A ,α = Aα = 0) in EM µα ,α φ ,α = φµα = 0 (32) we come to the following equation: 1 ∂2 φ ,α ≡ φ ≡ − − ∇2 φ = −κT (33) µν ,α µν c2 ∂t2 µν µν which is a simple wave equations describing ripples of spacetime propagating with the speed of light (why?). These ripples are called gravitational waves. 1The De Donder gauge is defined in a curved background by the condition µν √ ∂µ(g −g) = 0 Kostas Kokkotas Gravitational Waves GW: about Gauge conditions (*) By careful choice of coordinates the linearized Einstein equations can be simplified. We can fix ηµν = diag(−1, 1, 1, 1) and make small changes in the coordinates that leave ηµν unchanged but induce small changes in hµν . For example lets consider a change of the form: x 0µ = x µ + ξµ (34) where ξµ are 4 small arbitrary functions of the same order as hµν . Then ∂x 0µ ∂x µ = δµ + ∂ ξµ and = δµ − ∂ ξµ ∂x ν ν ν ∂x 0ν ν ν Thus, the metric transforms as: ∂x ρ ∂x σ g 0 = g = δρ − ∂ ξρ (δσ − ∂ ξσ)(η + h ) µν ∂x 0µ ∂x 0ν ρσ µ µ ν ν ρσ ρσ 0 ≈ ηµν + hµν − ∂µξν − ∂ν ξµ = ηµν + hµν (35) Then in the new coordinate system we get 0 hµν = hµν − ξµ,ν − ξν,µ (36) This transformation is called gauge transformation. Kostas Kokkotas Gravitational Waves GW: about Gauge conditions II (*) This is analogous to the gauge transformation in Electromagnetism. If Aµ is a solution of the EM field equations then another solution that describes precisely the same physical situation is given by (new) Aµ = Aµ − ψ,µ (37) where ψ is any scalar field. (new)µ (new) ,µ Then the gauge condition A ,µ = A µ = 0 means that ,µ ,µ µ ψ ,µ = ψ,µ =ψ = A ,µ. From (36) it is clear that if hµν is a solution to the linearised field equations then the same physical situation is also described by (new) φµν = φµν −ξµ,ν − ξν,µ = φµν −Ξµν (38) NOTE • This is a gauge transformation and not a coordinate one • We are still working on the same set of coordinates x µ and have defined a (new) new tensor φµν whose components in this basis are given by (38). Kostas Kokkotas Gravitational Waves GW: about Gauge conditions III (*) We can easily see that from (36) or (38) we can get (new) µρ µρ µ φ ,ρ = φ ,ρ − ξ (39) Therefore, if we choose the function ξµ so that to satisfy µ µρ ξ = φ ,ρ (40) we get the Hilbert gauge (new) µρ φ ,ρ = 0 (41) NOTE: This gauge condition is preserved by any further gauge transformation µ µ of the form (38) provided that the functions ξ satisfy ξ = 0 or equivalently µν Ξ = 0. Kostas Kokkotas Gravitational Waves µν I The choice of the Hilbert gauge φ ,ν = 0, gives 4 conditions that reduces the 10 independent components of the symmetric tensor hµν to 6! I Eqn (38) tells us that, from the 6 independent components of φµν which satisfy φµν = 0, we can subtract the functionsΞ µν , which depend on 4 independent arbitrary functions ξµ satisfying the same equation Ξµν = 0. I This means that we can choose the functions ξµ so that as to impose 4 conditions on φµν . 0 I We can choose ξ such that the trace φ = 0. Note that if φ = 0 then φµν = hµν . 0i I The 3 functions ξi can be chosen so that φ = 0. ,0 ,i I Then the Hilbert condition for µ = 0 will be written φ00 + φ0i = 0. 0i ,0 But since we fixed φ = 0 we get φ00 = 0, i.e. φ00 is a constant in time. I A time-independent part term φ00 corresponds to the static part of the grav. interactions i.e. to the Newtonian potential of the source. I The GW itself is the time-dependent part and therefore as far as ,0 the GW concerns h00 = 0 means h00 = 0.
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