Nonlinear gravitational waves and their polarization Geometry, - PowerPoint PPT Presentation

Nonlinear gravitational waves and their polarization Geometry, Integrability and Quantization Varna, 8 - 13 June 2007 GEVilasi GEVilasi Dipartimento di Fisica Dipartimento di Fisica Università Università degli Studi di Salerno & INFN, Italy degli Studi di Salerno & INFN, Italy 1

Contents Geometric aspects Physical properties (Isotropic case) Christodoulou memory Sources � � Dust, γ -ray bursts (GRB) Reduction � � Asymptotic flatness History and notation � � Semi-adapted coordinates Cosmic strings � � Invariant metrics � Wave-like character � Killing leaves � Zelmanov’s criterion � The non isotropic case � Pirani criterion � Canonical form and normal form of metrics Petrov-Penrose classification � � Ricci-flat metrics with 3d Killing algebra & 2d leaves Energy � � The isotropic case Landau-Lifchitz pseudo-tensor � � Bel superenergy tensor Global aspects � � Polarization � J-complex structures � Pauli-Ljubanski vector Model solutions � � Spin The local geometry of leaves � � Detection Examples � � Jacobi equation Algebraic solutions � � Raychaudhuri equation Info-holes � � A star outside the universe � Gaetano Vilasi, Salerno University, Italy 2 The square root of Schwarzschild universe �

Collaboration � M. Baetchold*, F. Canfora, D. Catalano*, L.Parisi, G. Sparano*, G.Vilasi, A. M. Vinogradov*, L. Vitagliano* Università di Salerno & INFN, Italy � S. People, Diffiety Institute, Moscow � G. Marmo, P. Vitale, Università di Napoli & INFN � A. Ibort, Universidad Carlos III, Madrid Gaetano Vilasi, Salerno University, Italy 3

Some research lines Complete integrability in field theory � General relativity : � Reduction of Einstein field equations Nonlinear gravitational waves Integrable gravitational models and their quantization Gaetano Vilasi, Salerno University, Italy 4

Nonlinearity � The need of taking into full account the nonlinearity of Einstein's equations when studying gravitational waves from strong sources is generally recognized. � Despite the great distance of the sources from Earth (where most of detectors are located) there are situations where the nonlinear effects cannot be neglected. Gaetano Vilasi, Salerno University, Italy 5

Christodoulou memory � When the source is a coalescing binary a secondary wave is generated via the non linearity of Einstein's field equations . � The memory seems to be too weak to be detected from the present generation of interferometers (even if ω is in the optimal band for LIGO/VIRGO interferometers) Gaetano Vilasi, Salerno University, Italy 6

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Exact gravitational waves � However, the Christodoulou memory is of the same order as the linear effects related to the same source, thus stressing the relevance of the nonlinearity of the Einstein's equations also from an experimental (LIGO/VIRGO/ NAUTILUS) point of view. � For these reasons exact solutions of the Einstein equations deserve special attention when they are of propagative nature. Gaetano Vilasi, Salerno University, Italy 10

Role of exact solutions Explicit solutions enable to discriminate more easily among physical or pathological features. � Where are there singularities? � What is their character? � How do test particles and fields behavior in given background space-times? � What are their global structures? � Is a solution stable and generic? Gaetano Vilasi, Salerno University, Italy 11

Problem � Classification of gravitational fields (not only Ricci-flat metrics) invariant for a Lie algebra G of Killing vector fields, such that: The distribution D , generated by I. vector fields belonging to G , is 2-dimensional. Gaetano Vilasi, Salerno University, Italy 12

An integrable gravitational case � Ernst, Maison, Harrison, …,Belinsky, Zakharov: Einstein field equations for a metric of the form g = f ( z, t )( dt 2 -dz 2 ) + h 11 ( z, t ) dx 2 + h 22 ( z, t ) dy 2 + 2 h 12 ( z, t ) dxdy reduce essentially to ∂ ξ ( α H -1 ∂ η H ) + ∂ η ( α H -1 ∂ ξ H ) = 0 H = || h ab ||; ξ = ( t + z ) / √ 2 ; η = ( t - z ) / √ 2 ; α = √| det H|. � G-Inverse Scattering Transform yields solitary wave solutions . Geon Gaetano Vilasi, Salerno University, Italy 13

The choice of coordinates The choice of coordinates also � depends on the properties of the distribution, D ⊥ , 1. orthogonal to D, the rank of the metric restricted to the 2. leaves of D . Gaetano Vilasi, Salerno University, Italy 14

Several cases II. The distribution D ⊥ is: IIa integrable and transversal to D . IIb semintegrable and transversal to D IIc non integrable and transversal to D IId integrable and not transversal to D . IIe semintegrable and not transversal to D IIf non integrable and not transversal to D Gaetano Vilasi, Salerno University, Italy 15

( G 2 , r ) -type metrics The case, in which the metric g restricted to � any integral (2-dimensional) submanifold ( Killing leaf ) D of the distribution is degenerate, splits naturally into two sub- cases according to whether the rank r of g restricted to Killing leaves is 1 or 0. In order to distinguish various cases occurring � in the sequel, the notation ( G 2 , r ) will be used: metrics satisfying the conditions I and IIa will be called of ( G 2 ,2) -type; metrics satisfying conditions I and II d,e or f ( D and D ⊥ , are not transversal ) will be called of ( G 2 ,1)- type or of ( G 2 ,0)- type according to the rank of their restriction to Killing leaves. Gaetano Vilasi, Salerno University, Italy 16

2-dimensional Lie algebra of isometries D ⊥ , r=2 D ⊥ , r=1 D ⊥ , r=0 integrable integrable G 2 integrable integrable G 2 semi-integrable semi-integrable semi-integrable non-integrable non-integrable G 2 non-integrable non-integrable A 2 integrable integrable integrable A 2 semi-integrable semi-integrable semi-integrable non-integrable non-integrable non-integrable non-integrable A 2 non-integrable Gaetano Vilasi, Salerno University, Italy 17

The integrable case. Local aspects � Complete classification of gravitational fields (not only Ricci-flat metrics) invariant for a Lie algebra G of Killing vector fields, such that: The distribution D , generated by I. vector fields belonging to G , is 2-dimensional. II. The distribution D ⊥ , orthogonal to D , is integrable and transversal* to D . Gaetano Vilasi, Salerno University, Italy 18

The integrable case. Global aspects � Global solutions of the Einstein field equations can also be constructed. Two cases: dim G =2 or dim G =3. They are qualitatively different but all manifolds satisfying the assumptions I and II are in a sense fibered over ζ -complex curves. | → global solutions of vacuum Einstein equations. � If dim G =3, condition II follows from I. Gaetano Vilasi, Salerno University, Italy 19

History � Only two 2-d Lie algebras: A 2 and G 2 . � A gravitational field g satisfying I and II, with G = A 2 or G 2 , is said to be G -integrable. � 1916 A 2 -integrable gravitational fields by Weyl. � 1937 A 2 -integrable grav. waves by Einstein-Rosen. � 1958 A 2 -integrable grav. fields by Kompaneyets and Landau. � 1979 A 2 -integrable grav. solitary waves by Belinsky-Zakharov. � 2000 G 2 -integrable gravitational fields by G.S, G.V, A.V. Gaetano Vilasi, Salerno University, Italy 20

Notation ∞ , Manifolds M are connected and C Metric : a non-degenerate symmetric (0,2) tensor field, Kil(g) : the Lie algebra of all Killing fields of a metric g , Killing algebra : a sub-algebra of Kil(g) Killing leaves of g : integral sub-manifolds of the distribution generated by vector fields of Kil(g) A 2 : a 2d Abelian Lie algebra, G 2 : a 2d non-Abelian Lie algebra, G -integrable metric : metric satisfying I & II, with G = A 2 or G 2 Gaetano Vilasi, Salerno University, Italy 21

Semi-adapted coordinates � Let g be a metric on the space-time M (a connected smooth manifold), G 2 =Span(X,Y) one of its Killing algebra [ X,Y ]= sY , s =0,1 � The Frobenius distribution D (possibly with singularities) generated by X and Y is 2d . � In a neighbourhood of a non-singular point of D a chart ( x μ ) ( semi-adapted ) exists such that X= ∂ 3 , Y=e sx3 ∂ 4 Gaetano Vilasi, Salerno University, Italy 22

Invariant gravitational fields � A gravitational field g admits X and Y as Killing fields iff in a s-a chart has the form g| S =g ij dx i dx i +2(l i +sm i ) dx i dx 3 -2m i dx i dx 4 + [ s 2 λ ( x 4 ) 2 -2s μ x 4 + ν ] dx 3 dx 3 +2 [ μ - s λ x 4 ] dx 3 dx 4 + λ dx 4 dx 4 � with g ij , m i , l i , λ, μ,ν functions of ( x 1 , x 2 ) � (Note: det H = λν −μ 2 ) Gaetano Vilasi, Salerno University, Italy 23

Invariant anholonomic basis e i = ∂ i , e 3 = ∂ 3 + s ∂ 4 , e 4 = - ∂ 4 ; [ e μ , e ν ] = c αμν e α � θ i = dx i , θ 3 = dx 3 , θ 4 = sx 4 dx 3 - dx 4 ; 2d θ α =- c αμν θ μ Λ θ ν (g ij ) (l i ) (m i ) (l i ) ν - μ (m i ) + - μ λ Gaetano Vilasi, Salerno University, Italy 24