Quasi-local energy for the Kerr space Liu Jian-Liang work with Sun - PowerPoint PPT Presentation

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Quasi-local energy for the Kerr space Liu Jian-Liang work with Sun Gang, Dr. Wu Ming-Fan, Prof. Chen Chiang-Mei and Prof. James M. Nester (supervisor) 2012.03.02, Fri. at YITP, Kyoto

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Outline • Hamiltonian and quasi-local quantities • Application to General Relativity • Preferred boundary term for GR • The choice of reference • Kerr space • Quasi-local energy, angular momentum for Kerr space • The extremal case for Kerr • Reference

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Hamiltonian and quasi-local quantities � • Action S = M L , ( M , g ) is a spacetime manifold with metric g . • First order Lagrangian 4-form for a k -form field ϕ is L = d ϕ ∧ p − Λ( ϕ, p ) • The variation of L δ L = d( δϕ ∧ p ) + δϕ ∧ δ L δϕ + δ L δ p ∧ δ p . (1) • Define the Euler-Lagrange equations by Hamilton’s principle ( E . L . p ) δ L δ p := d ϕ − ∂ p Λ = 0 , (2) ( E . L . ϕ ) δ L δϕ := − ς d p − ∂ ϕ Λ = 0 , ς := ( − 1) k . (3)

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Hamiltonian and quasi-local quantities • By the diffeomorphism invariant requirement (implies L N → δ ) d ι N L = L N L = d( L N ϕ ∧ p ) + L N ϕ ∧ δ L δϕ + δ L δ p ∧ L N p , L N ϕ ∧ δ L δϕ + δ L δ p ∧ L N p + d( L N ϕ ∧ p − ι N L � ) ≡ 0 . (4) � �� H (Apply Cartan formula: L N = d ι N + ι N d)

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • Hamiltonian is defined on the spatial hypersurface by � � ( N µ H µ + d B ) , H ( N ) = H = (5) Σ Σ where N µ H µ = ι N ϕ ∧ ( E . L . ϕ ) + ( E . L . p ) ∧ ι N p , B ( N ) = ι N ϕ ∧ p We obtain N µ H µ = ι N ϕ ∧ ( E . L . ϕ ) + ( E . L . p ) ∧ ι N p , which vanishes on shell. Consequently, � H ( N ) = B ( N ) . (6) ∂ Σ

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • Conserved quasilocal quantities and the corresponding symmetries 1. Quasilocal quantity H : the Hamiltonian boundary term B integrated over a closed space-like 2 − surface. 2. Conservation and symmetries conserved quantity H ↔ invariant under N energy time-like momentum space-like angular mumentum rotation center of mass boost

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • Boundary Variation principle From the variation of the Hamiltonian: � � � δ H = δ H = ( · · · ) + C . Σ Σ ∂ Σ δ H = − δϕ ∧ L N p + L N ϕ ∧ δ p − ι N [ δϕ ∧ ( E . L . ϕ ) + ( E . L . p ) ∧ δ p ] +d[ ι N ( δϕ ∧ p )] = − δϕ ∧ L N p + L N ϕ ∧ δ p + d[ ι N ( δϕ ∧ p )] “on shell” � � If ∂ Σ C = ∂ Σ ι N ( δϕ ∧ p ) vanishes, then the Hamiltonian is functional differentiable such that the Hamilton equations can be written L N ϕ = δ H L N p = − δ H δ p , δϕ

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • Boundary condition comes from � � C = ι N ( δϕ ∧ p ) = 0 , ∂ Σ ∂ Σ which means C = ι N ( δϕ ∧ p ) vanishes on the closed 2 − surface ∂ Σ. Note that if the 3 − region Σ is compact without � boundary, then ∂ Σ B of the Hamiltonian is automatically vanishing, which implies the Hamiltonian is certainly well-defined (i.e. functionally differentiabe). But we are usually interested in the region which is asymptotically flat ( R 3 is non compact), so we need the boundary conditions.

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • C.M. Chen’s improved boundary terms With ∆ α := α − ¯ α , replace the natural boundary term ι N ϕ ∧ p by � � � � ϕ p B ( N ) = ι N ∧ ∆ p − ς ∆ ϕ ∧ ι N (7) ϕ ¯ ¯ p the associated Hamiltonian variation boundary term has a symplectic form �� ι N δϕ ∧ ∆ p � � − ∆ ϕ ∧ ι N δ p �� δ H ( N ) ∼ d + ς . (8) − ι N ∆ ϕ ∧ δ p δϕ ∧ ι N ∆ p [Chen, Nester, Tung, PRD 72 , 104020, (21)-(24)]

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Regge-Teitelboim like asymptotic fall off and parity conditions: O + ( r − 1 ) + O − ( r − 2 ) , ∆ ϕ ∼ (9) O − ( r − 2 ) + O + ( r − 3 ) , ∆ p ∼ (10) with N µ = N µ 0 = λ [ µν ] 0 + λ µ 0 ν x ν , where N µ 0 , λ µν are 0 constant up to O + ( r − 1 ), being asymptotically Killing, the quasi-local quantities have finite values, and the boundary term in the Hamiltonian vanishes asymptotically.

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for • Each distinct choice of Hamiltonian boundary quasi-local expression is associated with a physically distinct boundary condition. • In order to accommodate suitable boundary conditions one must, in general, introduce certain reference values ¯ p , ¯ ϕ , which represent the ground state of the field—the “vacuum” (or background field) values.

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Application to General Relativity • Lagrangian density is L [ g , ∂ g ] = R η, where η is the 4-D volume element √− g d x 0 ∧ d x 1 ∧ d x 2 ∧ d x 3 . In the differential form language and using the orthonormal frame basis rather then the coordinate basis L [ ϑ µ , Γ µ ν ] = R α β , β ∧ η α (11) where R αβ = dΓ αβ + Γ αλ ∧ Γ λβ is the curvature two-form, and η αβ = 1 2 h βλ ǫ αλµν ϑ µ ∧ ϑ ν is the dual basis two-form, h µν is the flat metric diag( − 1 , +1 , +1 , +1).

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Field variables are ϑ α ↔ g and Γ µν ↔ ∂ g . Recall the first order Lagrangian L = d ϕ ∧ p − Λ implies ν − 1 ν + D ϑ α ∧ τ α − V µ L = D Γ µ ν ) , ν ∧ ρ µ ν ∧ ( ρ µ 2 κη µ where τ α , ρ µν are conjugate momenta w.r.t ϑ α and Γ µν ; V µν is the role multiplier.(Note that τ α = 0 as the construction go back to the original Lagrangian.) ν : D Γ µ ν = R α β = V µ δρ µ ν ; δτ α : D ϑ α = 0 (torsion free); ν = 1 δ V µ ν , ν : ρ µ 2 κη µ δϑ α : D τ α = R α β β ∧ η α µ = G µ (Einstein three form); ν = D η µ ν = 0 (followed by torsion free) . δ Γ µ ν : D ρ µ

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Preferred Boundary Term for GR Chen, Nester, Tung, Phys Lett A 203 , 5 (1995) [also found by Katz, Biˇ c´ ak & Lynden-Bel] B ( N ) = 1 β + ¯ 2 κ (∆Γ α D β N α ∆ η α β ) β ∧ ι n η α It corresponds to holding the metric fixed on the boundary: δ H ( N ) ∼ di N (∆Γ α β ) β ∧ δη α (12)

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for The choice of reference Given a spacetime manifold ( M , g ), and pick a local coordinate system { x µ } . The corresponding physical variables are the metric g µν and the connection (Christoffel symbol) Γ µνλ . • Take a closed space-like two surface S • Define the reference variables 1. The reference metric ¯ g , and 2. the reference connection ¯ Γ (note that it is not unique) • Then the quasi-local expression is covariant.

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for The strategy of choosing reference 1. Directly defined from the physical variables: e.g. for Kerr case here, let m = a = 0 ¯ ¯ g := g ( m = a = 0); Γ := Γ ( m = a = 0) . (13) 2. Determined by the local transformation only on S . (For the spherical symmetric case, see [Phys.Lett.A 374 3599 (arXiv:0909.2754), and PRD 84 084047 (arXiv:1109.4738)]) ¯ g ab d y a d y b ; g := ¯ ¯ Γ = 0 , (14) � ∂ y a � ∂ x µ ∂ y a ∂ y b ¯ Γ µ g µν = ¯ ¯ ∂ x ν ; ν = − d ∂ x ν . g ab ∂ x µ ∂ y a

Hamiltonian and quasi-local quantities Application to General Relativity The choice of reference Kerr space Quasi-local energy for Kerr space d s 2 = − ∆ − a 2 sin 2 θ d t 2 + 4 mar sin 2 θ d t d φ ρ 2 ρ 2 +sin 2 θ �� � r 2 + a 2 � 2 − a 2 ∆ sin 2 θ d φ 2 ρ 2 + ρ 2 ∆d r 2 + ρ 2 d θ 2 , (15) where ∆ = r 2 + a 2 − 2 mr , ρ 2 = r 2 + a 2 cos 2 θ .