Non-existence of time-periodic dynamics in general relativity

Transcription

1 Non-existence of time-periodic dynamics in general relativity Volker Schlue University of Toronto University of Miami, February 2, 2015

2 Outline 1 General relativity Newtonian mechanics Self-gravitating systems Gravitational Radiation 2 Final State Conjecture Stationarity of non-radiating systems Uniqueness of stationary spacetimes 3 Time-periodic vacuum spacetimes Statement of main result Idea of the proof 4 Uniqueness results for ill-posed hyperbolic p.d.e. s Unique continuation from infinity for linear waves Positive mass and the behavior of light rays

3 General Relativity General Relativity is a unified theory of space, time & gravitation. physics geometry GR analysis Some of the most fascinating developments: black holes (astronomy), expansion of the universe (cosmology); geometric analysis of hyperbolic equations (mathematics)

4 1 Gravity in Newton s and Einstein s theory

5 Newtonian Gravity Two-body problem in classical celestial mechanics: Kepler orbits. Space: R 3, Time: R, Newtonian potential: ψ(t, x). Sun Test body: Earth Newton s equation: ψ harmonic function. Note: In Newtonian theory the gravitational field can be periodic in time. (Action at a distance)

6 Figure: Albert Einstein

7 Einstein s Gravity Spacetime: (M, g) dimensional manifold endowed with Lorentzian metric (quadratic form of index 1). Sun Test body: Earth Σ R 3 Einstein s vacuum equations: Spacetime manifold is Ricci-flat.

8 Self-gravitating systems Slow motion / post-newtonian / weak field approximations: Einstein-Infeld-Hoffmann,..., Blanchet-Damour,..., Poisson & Will. Sun Earth Correction to Newtonian potential: (c speed of light) ψ post-newtonian = ψ + 1 c 2 ω

9 Gravitational Radiation The Einstein equations are of hyperbolic nature, and describe the non-linear dynamics of spacetime itself. Harmonic gauge: Choquet-Bruhat 52 C Σ t Σ 0 Global non-linear stability of flat space: Christodoulou-Klainerman 93, Lindblad-Rodnianski 10

10 Time-periodicity in general relativity Periodic motion should not exist in general relativity due to the emission of gravitational waves! Our main result is that any time-periodic vacuum spacetime is in fact time-independent.

11 Null infinity Future null infinity is the conformal boundary attached to an asymptotically flat spacetime, that can be thought of as the space of far away observers. (Penrose) I + R S 2 C t C 0

12 Monotone mass C u Σ I + I ι 0 Notions of mass: M[Σ] (Arnowitt-Deser-Misner) M[C u ] (Bondi): amount of energy in the system at time u. Positive mass theorem: M[Σ] 0, M[C u ] 0: Schoen-Yau, Witten The Bondi mass M(u) := M[C u ] is monotone, M(u) u 0 and for perturbations of flat space (Christodoulou-Klainerman) lim M(u) = M[Σ] lim u M(u) = 0. u

13 2 Final State Conjectures

14 Non-radiating spacetimes We say a spacetime is not radiating, if the Bondi mass M(u) = M 0 is constant. Conjecture. An asymptotically flat vacuum spacetime that is not radiating is necessarily stationary. Here stationary refers to the existence of a continuous isometry ψ t : M M of the spacetime manifold (M, g) generated by a vectorfield T which is time-like near infinity, L T g = lim t 0 t 1 (ψ t g g) = 0.

15 Aside: Gravitational wave experiments Figure: LIGO, Washington v v v u M

16 Stationary spacetimes It is expected that all stationary solutions to the Einstein vacuum equations can be completely classified, even if black holes have formed: M f = lim u M(u) > 0. Conjecture. Any smooth asymptotically flat black hole exterior solution to the Einstein vacuum equations that is stationary is isometric to the exterior of a Kerr solution (M, g M,a ). H + D I + I + : future null infinity H + : future event horizon D: black hole exterior Σ

17 Stationary black hole spacetimes The conjecture has been partially resolved in the perturbative regime. Black hole uniqueness theorems: Carter-Robinson (axi-symmetric case), Hawking (additional Killing vectorfield on horizon, and analytic case),..., Mars-Simon (local characterisation),..., Alexakis-Ionescu-Klainerman (rigidity results in the smooth case, for stationary vacuum spacetimes close to the Kerr solutions). H + I + H I

18 Final State Conjecture The final state conjecture gives a characterisation of all possible end states of the dynamical evolution in general relativity, as a result of the following scenario: (Penrose) Radiation: It is expected that due to the emission of gravitational waves any self-gravitating system settles down to a non-radiating state. Stationarity: It is conjectured that all self-gravitating systems that do not radiate to infinty are in fact stationary, i.e. time-independent. Uniqueness: It is believed that in vacuum the only stationary black hole exteriors are the Kerr solutions.

19 Figure: Artist s impression of binary star system J0806 rapidly approaching coalescence. (NASA)

20 3 Time-periodic vacuum spacetimes

21 Notion of time-periodicity Definition An asymptotically flat spacetime is called time-periodic if there exists a discrete isometry ϕ a with time-like orbits, that extends to a translation ϕ + a on future null infinity. ϕ a (p) I + R S 2 ϕ + a (u, ξ) = (u + a, ξ) ϕ a ι 0 p I

22 Stationarity of time-periodic spacetimes Theorem ( Alexakis-S.) Any asymptotically flat solution (M, g) to Einstein s vacuum equations Ric(g) = 0 which is smooth and time-periodic near infinity is stationary near infinity. I + ι 0 D T ι 0 I The theorem asserts that there exists a time-like vectorfield T on an arbitrarily small neighborhood D of infinity such that L T g = 0 on D.

23 Previous results Early work: Papapetrou (weak field approximation, non-singular solutions, strong time-periodicity assumption) Recent work: Gibbons-Stewart 84, Bicak-Scholz-Tod 10 (contains ideas how to exploit time-periodicity, stationarity inferred under much more restrictive analyticity assumption) Cosmological setting: Tipler 79, Galloway 84 (spatially closed case)

24 Strategy of Proof 1 Construction of time-like candidate vectorfield T, such that by time-periodicity lim r k L T R = 0 r k N 2 Use that by virtue of the vacuum Einstein equations, thus g R = R R g L T R = R L T R. Then apply our result that solutions to wave equations on asymptotically flat spacetimes are uniquely determined if all higher order radiation fields are known, to show that L T R = 0.

25 Unique continuation from infinity Theorem ( Alexakis-S.-Shao 14) Let (M, g) be an asymptotically flat spacetime with positive mass, and L g a linear wave operator L g = g + a + V with suitably decaying coefficients a, and V. If φ is a solution to L g φ = 0 which in addition satisfies r k φ 2 + r k φ 2 < D where D is an arbitrarily small neighborhood of infinity ι 0, then φ 0 : on D D.

26 Application of the theorem The application of the theorem is not immediate because g R = R R is not a scalar equation, but a covariant equation for the Riemann curvature tensor. Moreover, [ g, L T ] 0 and differentiating the equation produces additional terms which are not in the scope of the theorem: g L T R [ g, L T ]R = R L T R + L T g R 2 These obstacles can be overcome in the general framework of Ionescu-Klainerman 13 for the extension of Killing vectorfields in Ricci-flat manifolds.

27 Construction of candidate vectorfield T S u,s C + u L T L C C + 0 S 0 B d S 0 Σ 0 Define T = u : binormal to spheres S u. Then extend inwards by Lie transport along geodesics: [L, T ] = 0, L L = 0, g(l, L) = 0.

28 4 Uniqueness results for ill-posed hyperbolic p.d.e. s

29 Linear theory The conjecture that non-radiating spacetimes are stationary has a simple analogy in linear theory. Consider the linear wave equation on R 3+1 : φ := 2 φ t 2 + The radiation field is defined by 3 i=1 2 φ x i 2 = 0 Ψ(u, ξ) = lim r (rφ)(u + r, rξ). The assumption that a spacetime is not radiating corresponds in linear theory to the vanishing of the radiation field. One may thus ask: Ψ = 0 = (?) φ 0

30 Counterexamples in linear theory Question: Does the radiation field uniquely determine the solution, i.e. Ψ = 0 = φ 0? Answer: No, because 1 r is a solution, and thus also φ = 1 x i r 1, which is r 2 a non-trivial solution with Ψ = 0. Theorem ( Friedlander 61) For finite energy solutions to the classical wave equation φ = 0 the radiation field vanishes identically if and only if the initial data is trivial. Without a global finite energy condition, or a smoothness assumption at infinity, we are led to assume lim (r k φ)(u + r, rξ) = 0 k N. r

31 Obstructions to unique continuation from infinity There is an obstruction to localisation (near spacelike infinity ι 0 ) related to the behavior of light rays. Alinhac-Baouendi 83: Unique continuation fails across surfaces which are not pseudo-convex, in general.

32 Unique continuation from infinity for linear waves Theorem ( Alexakis-S.-Shao 13) Let (M, g) be a perturbation of Minkowski space, and L g a linear wave operator with decaying coefficients. If φ is a solution to L g φ = 0 which in addition satisfies an infinite order vanishing condition on at least half of future and past null infinity, then φ 0 in a neighborhood of infinity. D I + ɛ I ɛ

33 Pseudoconvexity The proof crucially relies on the construction of a family of pseudo-convex time-like hypersurfaces. Definition ( Calderon, Hörmander) A time-like hypersurface is pseudo-convex if it is convex with respect to tangential null geodesics. p We find a family of pseudo-convex hypersurfaces that foliate a neighborhood of infinity and derive a Carleman inequality to prove the uniqueness result.

34 Positive mass and the behaviour of light rays While the result cannot be localised on Minkowski space, we have seen that in the presence of a positive mass unique continuation does hold in an arbitrarily small neighborhood of spacelike infinity. M > 0 light ray ι 0 This is related to ideas of Penrose 90 to characterise the positivity of mass by the behavior of null geodesics near spacelike infinity. (Mason)

35 Thank you for your attention! Figure: Roger Penrose