The twistor theory of the Ernst equations

Transcription

1 The twistor theory of the Ernst equations Lionel Mason The Mathematical Institute, Oxford November 21, 2005 Credits: R.S.Ward, N.M.J.Woodhouse, J.Fletcher, K.Y.Tee, based on classic work on Ernst equations (Ehlers, Ernst, Geroch, Belinsky, Zakharov, Breitenlohner, Maison, Kramer, Neugebauer, Cosgrove, Alekseev, Hauser, Hoenselaar, Griffiths,...). References: Mason & Woodhouse (1988), Fletcher & Woodhouse (1991), Integrability, self-duality and twistor theory, Mason & Woodhouse, OUP, (1996).

2 Twistor theory Basic aim: find 1 1 correspondences { } Solutions to physical equations: Yang-Mills, Einstein... { Deformed complex structures on twistor space Hope: twistor space is correct geometric arena for physics. Twistor correspondences for self-dual Yang-Mills & Einstein equations suggests the larger programme. }

3 Integrable systems Spin off: Symmetry reduction of self-duality equations many/most integrable systems. Twistor correspondences reduce to prove complete integrability. Programme: (1) Classify integrable systems that arise as reductions of the self-duality equations. (2) Derive theory of reduced equations from twistor correspondences.

4 Stationary axisymmetric self-dual Yang-Mills Yang s form of Self-dual Yang-Mills on R 4 with coords (u, v) = (z + it, ρ e iθ ), metric ds 2 = dudū + dvd v is: ( ) 1 J J + ( ) 1 J J ū u v v where J = J(u, v, ū, v) is a Hermitian matrix function. t and θ-independence ρ (ρj 1 ρ J) + z (ρj 1 z J) = 0.

5 The reduced vacuum equations Ward s reduction (1983) Metric: ds 2 = ±e 2k (dρ 2 + dz 2 ) ± J ij dx i dx j k = k(ρ, z), J ij = J ij (ρ, z). / x i, i = 1,... n 2 are 2-surface orthogonal Killing vectors. The Vacuum field equations ρ 2 = det J 4i k ζ = ρ tr ((J 1 ζ J) 2 ) 1 ρ, ρ (ρj 1 ρ J) + z (ρj 1 z J) = 0 We focus on last equation and n = 4. ζ = z + iρ.

6 The Ernst equations L. Witten s reduction (1979) Pick one Killing vector X, set X = g(x, ) and f = g(x, X ), dψ = (X dx ) Then define Vacuum equations are K = 1 f ( f 2 + ψ 2 ) ψ ψ 1 ρ (ρk 1 ρ K) + z (ρk 1 z K) = 0. ( 3 3 reductions of both types for Electrovacuum solutions.)

7 The Lax pair Let U H = {(z, ρ) R 2 ρ 0} and set ζ = z + iρ. For (ζ, λ) U CP 1 define Then L := iλ(λ i) ζ + 2ρ(λ + i) λ + i λ + i J 1 ζ J, L := iλ(λ + i) ζ 2ρ(λ i) λ i λ i J 1 ζ J [L, L] = 0 ρ (ρj 1 ρ J) + z (ρj 1 z J) = 0

8 Twistor space For a given U H define the reduced twistor space to be T(U) = U CP 1 /{l, l} where {l, l} is the distribution spanned by l = ζ + iλ(λ i) 2ρ(λ + i) iλ(λ + i) λ, l = ζ 2ρ(λ i) λ. Points of T(U) are the leaves of {l, l} given by constant γ = z + ρ 2 ( 1 λ λ) T(U) is a non-hausdorff Riemann surface with holomorphic coordinate γ (or 1/γ) so γ : T(U) CP 1.

9 Reduced Ward correspondence Thus we have p : U CP 1 T(U) and T(U) is a covering of CP 1. Theorem Solutions to the stationary axisymmetric SDYM equations on U are in 1:1 correspondence with holomorphic vector bundles E T(U) such that p E is trivial over each CP 1 in U CP 1. (Condition is generically satisfied for small enough U.) E γ = solutions Ψ to LΨ = LΨ = 0 on leaf of constant γ. Theorem Suppose Ē = pull-back of E by complex conjugation γ γ interchange of sheets of T(U) CP 1 sends E E, then J is real and symmetric and is a solution of the reduced vacuum equations.

10 The twistor data The general complexified case Assume U connected, simply connected & disjoint from ρ = 0. Let V CP 1 be the glued down region in T(U). Definition of γ γ (z + iρ) γ (z iρ) = ( ) λ + i 2. λ i V = {γ = z ± iρ (z, ρ) U} U Ũ. Twistor data can be normalized w.r.t. choice of (z 0, ρ 0 ) U. Twistor data is a pair of SL(2, C) valued functions P( γ (z 0 + iρ 0 )) on U, and P( γ (z 0 iρ 0 ) on Ũ, subject to certain involutions. Metric constructed via Riemann-Hilbert problem in λ-plane.

11 Gowdy/colliding plane waves K.Y.Tee, D.Phil. thesis For Gowdy/colliding plane waves, τ := iρ R is timelike, z, x i spacelike. τ = 0 is generically singular, in past/future for Gowdy, future for colliding plane waves. Choice of (z 0, τ 0 ) U is natural for characteristic IVP for colliding plane waves. P and P are naturally real functions of the real variables ζ = z + τ, ζ = z τ respectively. P and P are equivalent to the characteristic data on ζ 0 =const., ζ 0 =const. respectively.

12 Axis simple case The Ward ansatze Assume glued down region V is connected & simply connected. T(U) = CP 1 0 glued to CP 1 over V. There is a canonical normalization of twistor data: E CP 1 0 = E CP 1 = O(p) O(q), (O(p) = C-line bundle, c1 = p) Symmetric P(γ) : V SL(2, C) patches E CP 1 0 to E CP 1. Metric is obtained from Riemann-Hilbert problem in λ-plane G 0 (z, ρ, λ) = ( ρ p λ 0 p ρ 0 q λ q ) ( ) ( ρλ) p 0 P(γ) 0 ( ρλ) q G (z, ρ, λ), γ = z + ρ 2 ( 1 λ λ), G 0 holomorphic on λ 1, G on λ 1. J(z, ρ) = G 0 (z, ρ, 0)G (z, ρ, ) 1. P is obtained from finite order ρ-expansion of J at ρ = 0.

13 Cyindrical symmetry Woodhouse (1989) Here, ρ is space-like, ρ = 0 is axis, and t = iz R is time. Ward ansatze applies with V being (nhd of) real axis. Ward s metric reduction requires p = 1, q = 0, L.Witten s Ernst reduction requires p = q = 0. ρ = 0 is not a natural Cauchy surface. However, the P(γ) of the Ward ansatze has a canonical representation as a path-ordered exponential of cauchy data at t = 0.

14 Stationary axisymmetric case James Fletcher s thesis (1990) Here (x 1, x 2 ) = (t, φ), φ defined mod 2π, space-like, t time. Ward ansatze applies, V = U Ū where U is region on which solution exists. Ward s metric reduction requires p = 1, q = 0, L.Witten s Ernst reduction requires p = q = 0. Fletcher characterizes the intersections of the axis and horizons in terms of the twistor data (simple pole in P). Proves the stationary-axisymmetric black-hole uniqueness theorem using extension of Liouville s theorem on twistor data.

15 Geroch group Hidden symmetries, M. & Woodhouse (1988) Geroch group is group of hidden symmetries. Generated by interplay of L. Witten s Ernst reduction, K, and Ward s metric reduction, J. Map J K corresponds to twisting (conjugating) twistor data by ( ) 0 1 γ 1. 0 Generates action of loop group g(γ) : S 1 SL(2, C), S 1 = {γ, γ = 1}. Acts by twisting bundle. Can classify orbits in general case & prove transitivity in axis regular case.

16 Summary Twistor theory provides a geometric unifying framework for integrable systems approaches to the Ernst equations. Can be adapted to all the standard applications. Further projects: Project with A. Gray & M. Singer on signature case aims at classification of toric Ricci-flat 4-manifolds. Work with D. Calderbank. Deformations of T(U) solutions to spinor-vortex equation: Riemann surface Σ, metric g, Dirac op. D and spinor field Ψ, DΨ = 3 Ψ, R = 4 2 Ψ 2 toric anti-self-dual conformal structures.

17 The end