The Geroch group in Einstein spaces Marios Petropoulos CPHT Ecole - PowerPoint PPT Presentation

The Geroch group in Einstein spaces Marios Petropoulos CPHT – Ecole Polytechnique – CNRS IHES – Bures-sur-Yvette March 2014

Highlights Motivations From four to three dimensions and back to four The sigma-model Conservation laws, integrability and solution generation Outlook

Framework Solution-generating algebraic methods for Einstein’s equations ◮ Always: give a deeper perspective on ◮ the structure of the space of solutions ◮ integrability properties ◮ Often: ◮ assume extra symmetry ◮ based on a mini-superspace analysis of the eoms ◮ Sometimes: provide new solutions

Here Explore Geroch’s approach for R ab = Λ g ab ◮ Originally: R ab = 0 [Ehlers ’59; Geroch ’71] ◮ ( M , g , ξ ) → ( S , h ) → ( S , h’ ) → ( M , g’ , ξ ′ ) ◮ h → h’: algebraic action of SL ( 2 , R ) ◮ no integrability discussion ◮ Before: Ernst method with 2 Killings [Ernst ’68] ◮ After: general integrability properties with 2 Killings → 2-dim sigma-models (Lax pairs, inverse scattering, . . . ) ◮ powerful and complementary wrt algebraic (Geroch) [Belinskii, Zakharov ’78; Maison ’79; Bernard, Regnault ’01] ◮ no mention of Λ : hard problem [Astorino ’12]

Results [Leigh, Petkou, Petropoulos, Tripathy ’14] Unified treatment for Λ = 0 or � = 0 thanks to the conformal mode κ ◮ Mapping to a 3-dim sigma-model: ( κ , ω , λ )-target space conformal to R × H 2 ◮ Geroch’s SL ( 2 , R ) ≡ isometry – partly broken by the potential ◮ reduced algebraic solution-generating action ◮ no effect on integrability ◮ Mini-superspace analysis: h on S ∝ R × S 2 ◮ particle motion on R × H 2 at zero energy ◮ integrability using Hamilton–Jacobi ◮ Λ : constant of motion as m and n

Highlights Motivations From four to three dimensions and back to four The sigma-model Conservation laws, integrability and solution generation Outlook

4-dim M With g = g ab d x a d x b ( − + ++ ) and a time-like Killing field ξ ◮ norm: λ = � ξ � 2 < 0 ◮ twist 1-form: Ω = − 2i ξ ⋆ d ξ Assuming Ric = Λ g d ⋆ d ξ = 2 Λ ⋆ ξ ⇓ d Ω = 0 Locally scalar twist Ω = d ω

3-dim S S : coset space obtained by modding out the group generated by ξ ◮ Natural pos. def. metric/projector: h ab = g ab − ξ a ξ b λ − 1 − λ η abcd ξ d ◮ Natural fully antisymmetric tensor: η abc = √ ◮ One-to-one correspondence between tensors on S and tensors T on M s.t. i ξ T = 0 and L ξ T = 0: T S n q T M b 1 ... b q a 1 . . . h m p n 1 . . . h b q n 1 ... n q = h m 1 a p h b 1 a 1 ... a p m 1 ... m p ◮ Induced connection on S – coinciding with Levi–Civita b 1 ... b q a 1 . . . h m p n 1 . . . h b q n 1 ... n q = h ℓ c h m 1 a p h b 1 n q ∇ ℓ T D c T a 1 ... a p m 1 ... m p with curvature � � R abcd = h p [ a h q b ] h r [ c h s R pqrs + 2 λ ( ∇ p ξ q ∇ r ξ s + ∇ p ξ r ∇ q ξ s ) d ]

Dynamics for g on M translates into dynamics for ( h , ω , λ ) on S ◮ Dynamics for g on M : R ab = Λ g ab ◮ Dynamics for ( h , ω , λ ) on S : 2 λ 2 ( D a ω D b ω − h ab D c ω D c ω ) + 1 1 = R ab 2 λ D a D b λ − 1 4 λ 2 D a λ D b λ + Λ h ab D 2 λ 2 λ ( D c λ D c λ − 2 D c ω D c ω ) − 2 Λ λ 1 = D 2 ω 2 λ D c λ D c ω 3 =

Any new solution ( h ′ , ω ′ , λ ′ ) on S translates into a new solution g ′ on M with Killing ξ ′ – a new Einstein space with symmetry ◮ Define a 2-form on S : F ′ = h ′ d ω ′ ( − λ ′ ) 3 / 2 ⋆ 3 1 ◮ Check it is closed ◮ Locally: F ′ = d η ′ ◮ Promote η ′ on M by adding a longit. comp. s.t. i ξ η ′ = 1 ◮ New Killing on M : ξ ′ = η ′ λ ′ ab + ξ ′ a ξ ′ ◮ New Einstein metric on M : g ′ ab = h ′ b λ ′

Highlights Motivations From four to three dimensions and back to four The sigma-model Conservation laws, integrability and solution generation Outlook

Introduce a reference metric ˆ h: h ab = κ ˆ h ab λ (in Geroch ˜ h ab = λ h ab = κ ˆ h ab ) Eqs. for ˆ h , κ , τ = ω + i λ follow from � � S d 3 x ˆ S = h L � ˆ � √ D a κ ˆ D a τ ˆ ˆ D a κ D a ¯ τ κ τ ) 2 + ˆ L = − − κ + 2 R − 4 i Λ 2 κ 2 ( τ − ¯ τ − ¯ τ ◮ ˆ h ab : gravity in 3 dim with dilaton-Einstein–Hilbert action ◮ κ , τ : matter with sigma-model kinetic term plus potential

Symmetries Kinetic term for the matter fields κ , ω , λ : target space � κ 2 + d ω 2 + d λ 2 � √ − d κ 2 d s 2 target = − κ λ 2 ◮ Conformal to R × H 2 ◮ Conformal isometry group: R generated by ζ = 1 2 κ∂ κ ◮ Isometry group: SL ( 2 , R ) generated by � λ 2 − ω 2 � ξ + = ∂ ω ξ − = ∂ ω − 2 ωλ∂ λ ξ 2 = ω∂ ω + λ∂ λ [ ξ + , ξ − ] = − 2 ξ 2 [ ξ + , ξ 2 ] = ξ + [ ξ 2 , ξ − ] = ξ −

Potential for the matter fields κ , ω , λ : √ � � R − 2 Λ κ ˆ V = − κ λ Λ breaks ξ − and ξ 2

Next ◮ Integrability properties and solution generation ◮ Assume a further Killing for g: 2-dim Ernst-like sigma model (Lax pairs, inverse scattering, . . . ) ◮ Freeze ˆ h: 1-dim sigma model – particle motion (Hamilton–Jacobi) ◮ Role of the dilaton-like field κ

Mini-superspace analysis Freeze ˆ h to R × S 2 – motivation: Taub–NUT, Schwarzschild s 2 = d σ 2 + d Ω 2 d ˆ R ab d x a d x b = ˆ ◮ d Ω 2 : 2-dim, σ -independent → ˆ R 2 d Ω 2 ◮ Matter: κ ( σ ) , ω ( σ ) and λ ( σ )

Impose in equations and check consistency ◮ In ˆ h ab equations ◮ Trace part: κ -equation (as in the generic case) ◮ Transverse part: consistency condition 2 κ 1 κ 2 ˆ τ ˙ R = τ + 4 i Λ τ + τ ) 2 ˙ ¯ 2 κ 2 ˙ ( τ − ¯ τ − ¯ ◮ extended symmetry: ˆ R = 2 ℓ , ℓ = 1 , 0 , − 1 ◮ constraint (first-order equation) ◮ Dynamics: particle motion on d s 2 target with V subject to H = 0 √− κ � ˙ � �� � 2 � τ ˙ κ ˙ τ ¯ κ L = − − 4 τ ) 2 − 4 ℓ − 2 i Λ 2 ( τ − ¯ τ − ¯ κ τ

In summary 4-dim Einstein space with symmetry ( M , g , ξ ) ↓ ξ 3-dim sigma-model ( S , ˆ h , κ , τ )   R × S 2     extra ˆ  R 3 h isometries     R × H 2 � 1-dim “time”- σ particle dynamics Case under investigation: 1 extra Killing field for h ⇒ 3-dim sigma-model → 2-dim sigma-model (Ernst-like with dilaton)

Highlights Motivations From four to three dimensions and back to four The sigma-model Conservation laws, integrability and solution generation Outlook

At Λ = 0 : Geroch The full Lagrangian is SL ( 2 , R ) -invariant ◮ Algebraic scan of the space of solutions τ → τ ′ = a τ + b κ frozen c τ + d ◮ Integrable with space of solutions: m , n � SO ( 2 ) ⊂ SL ( 2 , R ) : rotation in ( m , n ) N ⊂ SL ( 2 , R ) : homothetic transformation in ( m , n )

At Λ � = 0 : generalization Summary ◮ Only ξ + leaves L invariant ◮ Integrability unaltered ( SL ( 2 , R ) not crucial) ◮ ξ + and ξ 2 generate constants of motion ◮ Constants of motion: Λ , m , n ◮ Under N ⊂ SL ( 2 , R ) : ( Λ , m , n ) → ( a 2 Λ , m / a , n / a ) ◮ κ ( σ ) depends on Λ , m , n : freezing κ → missing solutions

In some detail r = ( − κ ) 3 / 2 Change time d ˆ d σ and go to the Hamiltonian − λ κ − λ 3 H = λ λ ) + 2 ℓ λ ˆ 2 p 2 2 κ 2 ( p 2 ω + p 2 κ − 2 Λ contraint to ˆ H = 0 ◮ Λ no longer any role in the symmetry: reduced to ξ + ∀ Λ ◮ SL ( 2 , R ) algebra on the phase space: ˆ ˆ F + = p ω F 2 = ω p ω + λ p λ + 2 Λ ˆ r F − = − 2 ωλ p λ − ( ω 2 − λ 2 ) p ω − 4 Λ ω ˆ ˆ r

◮ Action on ˆ H : � = 0 � = − ˆ � ˆ � ˆ H , ˆ H , ˆ F + F 2 H − 2 Λ � = 2 ω ˆ � ˆ � � r λ 3 p ω H , ˆ ω + ˆ F − H + 4 Λ κ 2 ◮ Conserved quantities: d ˆ d ˆ F + F 2 r = − ˆ r = 0 H d ˆ d ˆ d ˆ r λ 3 p ω F − H + 4 Λ ˆ r = 2 ω ˆ κ 2 d ˆ Under the constraint ˆ H = 0 : ˆ F + and ˆ F 2 conserved

Hamilton–Jacobi integration Hamilton–Jacobi: � ∂ S � + ∂ S ˆ ∂ q i , q i H r = 0 ∂ ˆ not fully separable but integrable – irrespective of Λ ◮ With q i = ( κ , ω , λ ) � � q i , ˆ ◮ find principal solution S r ; α i ◮ use β i = ∂ S ∂α i to get q i = q i ( ˆ r ; α j , β k ) ◮ use p i = ∂ S r ; α j , β k ) ∂ q i to get p i = p i ( ˆ

◮ Partial separation: 2 commuting first integrals ˆ F + and ˆ H with values 2 ν and ˆ E S = W + 2 νω − ˆ E ˆ r with W ( κ , λ ; α i ) solving a pde wrt κ , λ and α 1 = ˆ E + 2 Λ α 2 = ν α 3 = α ˆ E set to zero at the end Relevant constants ( α 1 , α 2 , β 3 ) ⇔ ( Λ , n , m ) the others can be reabsorbed in various redefinitions – Λ : effective constant of motion relaxing the Hamiltonian constraint

General solution κ , ω , λ with the reference ˆ h 4-dim metric g: general (A)dS Schwarzschild Taub–NUT h � � d r 2 �� � � √ � � χ � � 2 ∆ λ κ m 2 + ℓ 2 n 2 f ℓ + d Ω 2 − d T + 4 n d ψ + ( m 2 + ℓ 2 n 2 ) κ 2 λ ∆ ���� d σ 2 � �� � ˆ h r traded for r and f ℓ ( χ ) = sin 2 χ , χ 2 , sinh 2 χ for ℓ = 1 , 0 , − 1 ˆ ◮ ∆ = ℓ ( r 2 − n 2 ) − 2 mr − Λ / 3 � r 4 + 6 r 2 n 2 − 3 n 4 � ◮ κ = − ∆ / m 2 + ℓ 2 n 2 � � Λ r + 3 ℓ r − 3 m − 4 Λ n 2 r ◮ ω = − 2 n / 3 ( m 2 + ℓ 2 n 2 ) r 2 + n 2 ◮ λ = − ∆ / ( m 2 + ℓ 2 n 2 )( r 2 + n 2 )