Gravitational Waves Theory, Sources and Detection Kostas Kokkotas February 18, 2014 Kostas Kokkotas Gravitational Waves
Suggested Reading ◮ Books ◮ Gravitational Waves: Volume 1: Theory and Experiments by Michele Maggiore Oxford University Press (2007) ◮ Gravitation and Spacetime by Ohanian, Hans C. and Ruffini, Remo Cambridge University Preess (Sep 20, 2013) ◮ Gravitation by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Sep 15, 1973) W.H. Freeman ◮ Review articles ◮ Gravitational wave astronomy F.F. Schutz, Class. Quantum Grav. 16 (1999) A131ÐA156 ◮ 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 ś Uspekhi 44 (1) 1 ś 51 (2001) ◮ The basics of gravitational wave theory E.E. Flanagan and S.A. Hughes, New Journal of Physics 7 (2005) 204 ◮ 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
GR - Tensors Tensor Transformations ( x µ → ˜ x µ ) ∂ x ν x µ ∂ ˜ a µ = � � ˜ ∂ x ν a ν and (1) b µ = x µ b ν ˜ ∂ ˜ ν ν x α x β x α ∂ x ν T αβ = ∂ x µ ∂ x ν ∂ ˜ ∂ ˜ β = ∂ ˜ T αβ = ∂ x ν T µν , ˜ � ˜ T α x β T µ ν & ˜ x β T µν ∂ x µ ∂ x µ ∂ ˜ ∂ ˜ x α ∂ ˜ µν Covariant Derivative (2) φ ; λ = φ ,λ ∇ α A µ = ∂ α A µ + Γ µ A µ A µ ,α + Γ µ αλ A λ αλ A λ or (3) = ; α A λ,µ − Γ ρ (4) A λ ; µ = µλ A ρ αν T αµ + Γ µ T λµ T λµ ,ν + Γ λ αν T λα (5) = ; ν Parallel Transport Γ λ µν a λ dx ν for covariant vectors (6) δ a µ = δ a µ − Γ µ λν a λ dx ν for contravariant vectors (7) = Kostas Kokkotas Gravitational Waves
GR - Metric Tensor A space is called a metric space if a prescription is given attributing a scalar distance to each pair of neighbouring points The distance ds of two points P ( x µ ) and P ′ ( x µ + dx µ ) is given by dx 1 � 2 + � dx 2 � 2 + � ds 2 = � dx 3 � 2 (8) x µ , we will get In another coordinate system, ˜ ∂ x ν dx ν = � x α (9) x α d ˜ ∂ ˜ α which leads to: ds 2 = ˜ x ν = g αβ dx α dx β . x µ d ˜ (10) g µν d ˜ • Properties: we can now raise and lower indices of tensors: With g µν g µν A µ = A ν , g µν T µα = T α g µν g ασ T µα = T νσ g µν T µ ν , α = T αν , With g µν A µ = g µν A ν , T µν = g µρ T ν ρ = g µρ g νσ T ρσ (11) Kostas Kokkotas Gravitational Waves
GR - Christoffel Symbols & Riemann Tensor ◮ Christoffel Symbols µρ = 1 2 g αν ( g µν,ρ + g νρ,µ − g ρµ,ν ) Γ α (12) ◮ Riemann or Curvature Tensor R λ βνσ = − Γ λ βν,σ + Γ λ βσ,ν − Γ µ βν Γ λ µσ + Γ µ βσ Γ λ (13) µν Figure: Measuring the curvature for the space. Kostas Kokkotas Gravitational Waves
GR - Ricci & Einstein Tensors The contraction of the Riemann tensor leads to Ricci Tensor R λ αλβ = g λµ R λαµβ R αβ = Γ µ αβ,µ − Γ µ αµ,β + Γ µ αβ Γ ν νµ − Γ µ αν Γ ν (14) = βµ which is symmetric R αβ = R βα . Further contraction leads to the Ricci or Curvature Scalar R = R α α = g αβ R αβ = g αβ g µν R µανβ . (15) The following combination of Riemann and Ricci tensors is called Einstein Tensor G µν = R µν − 1 (16) 2 g µν R with the very important property: ν − 1 � � G µ R µ 2 δ µ = 0 . (17) ν ; µ = ν R ; µ Kostas Kokkotas Gravitational Waves
GR - Einstein’s Equations ◮ Einstein’s equations are : R µν − 1 2 g µν R + Λ g µν = κ T µν . (18) where κ = 8 π G is the coupling constant and c 4 Λ = 8 π G c 2 ρ v is the so called cosmological constant. ◮ Einstein’s equations can also be written as: T µν − 1 � � (19) R µν = − κ 2 g µν T ◮ Geodesic equation du ρ d 2 x ρ dx µ dx ν µν u µ u ν = 0 ds + Γ ρ ds 2 + Γ ρ or = 0 µν ds ds ◮ Flat & Empty Spacetimes ◮ When R αβµν = 0 the spacetime is flat ◮ When R µν = 0 the spacetime is empty Kostas Kokkotas Gravitational Waves
GR - Metric Tensor ◮ Metric element for Minkowski spacetime − dt 2 + dx 2 + dy 2 + dz 2 ds 2 (20) = − dt 2 + dr 2 + r 2 d θ 2 + r 2 sin 2 θ d φ 2 ds 2 (21) = ◮ A typical solution of Einstein’s equations describing spherically symmetric spacetimes has the form: ds 2 = e ν ( t , r ) dt 2 − e λ ( t , r ) dr 2 − r 2 � d θ 2 + sin 2 θ d φ 2 � (22) ◮ The Kerr solution is probably the most import an solution of Einstein’s equations relevant to astrophysics. dt − a sin 2 θ d φ 2 � 2 + sin 2 θ ( r 2 + a 2 ) d φ − adt � 2 + ρ 2 ds 2 = − ∆ ∆ dr 2 + ρ 2 d θ 2 � � ρ 2 ρ 2 where 1 ρ 2 = r 2 + a 2 cos 2 θ ∆ = r 2 − 2 Mr + a 2 and (23) here a = J / M = GJ / Mc 3 is the angular momentum per unit mass. For our Sun J = 1 . 6 × 10 48 g cm 2 / s which corresponds to a = 0 . 185. 1 If ∆ = r 2 − 2 Mr + a 2 + Q 2 the we get the so called Kerr-Newman solution which describes a stationary, axially symmetric and charged spacetime. Kostas Kokkotas Gravitational Waves
GR - Schwarzschild Solution 1 − 2 GM 1 − 2 GM � − 1 ds 2 = � � c 2 dt 2 − � dr 2 − r 2 � d θ 2 + sin 2 θ d φ 2 � rc 2 rc 2 ◮ Sun: M ⊙ ≈ 2 × 10 33 gr and R ⊙ = 696 . 000 km 2 GM ≈ 4 × 10 − 6 rc 2 ◮ Neutron star : M ≈ 1 . 4 M ⊙ and R ⊙ ≈ 10 − 15 km 2 GM ≈ 0 . 3 − 0 . 5 rc 2 ◮ Neutonian limit: g 00 ≈ η 00 + h 00 = 1 + 2 U U = − GM ⇒ c 2 r Kostas Kokkotas Gravitational Waves
GW: Linear Theory I Weak gravitational fields can be represented by a slightly deformed Minkowski spacetime : g µν ≃ η µν + h µν + O ( h µν ) 2 , | h µν | ≪ 1 (24) here h µν is a small metric perturbation. The indices will be raised and lowered by η µν i.e. h αβ η αµ η βν h µν (25) = η µν h µν (26) h = η µν − h µν g µν (27) = and we will define the traceless ( φ µν ) tensor: φ µν = h µν − 1 (28) 2 η µν h . The Christoffel symbols & the Ricci tensor will become : 1 2 η λρ ( h ρν,µ + h µρ,ν − h µν,ρ ) Γ λ (29) = µν µα,ν = 1 α − h α Γ α µν,α − Γ α 2 ( h α ν,µα + h α α,µν ) (30) R µν = µ,να − h µν,α αβ − h α η µν R µν = h αβ ,β (31) R = α ,β Kostas Kokkotas Gravitational Waves
Finally, Einstein tensor gets the form: µν = 1 α − h α ,αβ − h α G ( 1 ) 2 ( h α ν,µα + h α ,β � � (32) µ,να − h µν,α α,µν ) − η µν h αβ α ,β Einstein’s equations reduce to (how?): ,αβ + φ µα ,α ,α ,α (33) − φ µν ,α − η µν φ αβ ,ν + φ να ,µ = κ T µν Then by using the so called Hilbert (or Harmonic or De Donder) gauge similar ,α = 0) in EM 2 to Lorenz gauge ( A α ,α = A α ,α = 0 φ µα (34) ,α = φ µα we come to the following equation: � � 1 ∂ 2 ,α ∂ t 2 − ∇ 2 (35) φ µν ,α ≡ � φ µν ≡ − φ µν = − κ T µν c 2 which is a simple wave equations describing ripples of spacetime propagating with the speed of light (why?). These ripples are called gravitational waves. 2 The De Donder gauge is defined in a curved background by the condition ∂ µ ( g µν √− 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 η µν = diag ( − 1 , 1 , 1 , 1 ) and make small changes in the coordinates that leave η µν unchanged but induce small changes in h µν . For example lets consider a change of the form: x ′ µ = x µ + ξ µ (36) where ξ µ are 4 small arbitrary functions of the same order as h µν . Then ∂ x ′ µ ∂ x µ ∂ x ν = δ µ ν + ∂ ν ξ µ ∂ x ′ ν = δ µ ν − ∂ ν ξ µ and Thus, the metric transforms as: µν = ∂ x ρ ∂ x σ g ′ δ ρ µ − ∂ µ ξ ρ � ( δ σ ν − ∂ ν ξ σ ) ( η ρσ + h ρσ ) � ∂ x ′ ν g ρσ = ∂ x ′ µ η µν + h µν − ∂ µ ξ ν − ∂ ν ξ µ = η µν + h ′ (37) ≈ µν Then in the new coordinate system we get h ′ (38) µν = h µν − ξ µ,ν − ξ ν,µ 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 A µ is a solution of the EM field equations then another solution that describes precisely the same physical situation is given by A ( new ) (39) = A µ + ψ ,µ µ ,µ = 0 means where ψ is any scalar field. Then the gauge condition A µ ,µ = A µ ,µ = � ψ = 0. that ψ ,µ ,µ = ψ ,µ From (38) it is clear that if h µν is a solution to the linearised field equations then the same physical situation is also described by φ ( new ) (40) = φ µν − ξ µ,ν − ξ ν,µ = φ µν − Ξ µν µν 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 tensor φ ( new ) whose components in this basis are given by (40). µν Kostas Kokkotas Gravitational Waves
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