Stationarity of non-radiating spacetimes

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1 University of Warwick April 4th, 2016

2 Motivation Theorem Motivation Newtonian gravity: Periodic solutions for two-body system. Einstein gravity: Periodic solutions? At first Post-Newtonian order, Yes! Toy model, consider a wave equation on a Minkowski background: Non-radiating solutions are stationary (Friedlander).

3 Motivation Theorem A dynamical gravitational system loses energy through radiation towards infinity. There are no periodic solutions. THEOREM Suppose (M, g) is an asymptotically flat, non-radiating solution of the vacuum Einstein equations. Then there exists a time-like vector field T in a neighbourhood of infinity such that L T g = 0, that is, (M, g) is stationary in a neighbourhood of infinity.

4 Asymptotic Flatness Radiation Asymptotic flatness I We say that a function f has asymptotic expansion k=1 1 r k f (k) if for all N N there exists a constant C N such that for all r > r 0, f (r) N k=1 1 r k f (k) 1 < C N r N+1. We write O 2 (r 1 ) for the space of functions admitting an asymptotic expansion which is well-behaved with respect to derivatives up to order 2. Remark. The function f uniquely determines f (k) but not viceversa. For example, the zero function and e r.

5 Asymptotic Flatness Radiation Asymptotic flatness II DEFINITION A Lorentzian manifold (M, g) is said to be asymptotically flat and regular at spatial infinity if there are coordinates (t, r, θ, φ) such that for large r, the metric admits an expansion of the form, g = dt 2 + dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) + g, where the components of g belong to the class O 2 (r 1 ).

6 Asymptotic Flatness Radiation Mass Given a sphere S M there are two null directions orthogonal to T p S in T p M. Choose null vectors fields L and L along S generating those directions and normalised by g(l, L) = 2. The null second fundamental forms of S are the 2-covariant symmetric tensors on S χ := g( L, ) and χ := g( L, ). The Hawking mass enclosed by a 2-sphere S is defined to be m := r ( ) trχtrχ. 2 16π Here trχ and trχ are the null expansions of S and r is the area radius. S

7 Asymptotic Flatness Radiation Radiation Along the level sets of a null foliation given by u =constant, the Hawking mass converges to the Bondi mass as r : m(u, r) = M B (u) + O(r 1 ) Moreover, the limit of the Raychaudhuri equation implies the Bondi mass formula at I +, u M B = 1 Ξ(u, ) 2 32π S 2 where Ξ := lim u;r r ˆχ can be regarded as the radiated power per unit solid angle. DEFINITION An asymptotically flat spacetime is non-radiating towards future null infinity if M B is constant. Equivalently, if Ξ = 0 along I +.

8 Proof The proof is divided in three main steps: 1 Construction of the candidate Killing field: Use adapted out-going coordinates (u, r, θ, φ) such that T = u is Killing to leading order. Related to the freedom L al, L 1 a L. 2 Asymptotic quantities to all orders at infinity: Einstein equations + asymptotic expansions + Ξ = 0 = u (asymptotic coefficients)=0. 3 Unique continuation from infinity (Carleman estimates): g φ + a α α φ + V φ = 0 and r N ( φ 2 + φ 2 ) = 0 = φ 0 in a neighbourhood of infinity. Apply this idea to u Weyl.

9 REFERENCES. THANKS! Alexakis and Schlue. Non-existence of time-periodic vacuum spacetimes. arxiv preprint arxiv: (2015). Friedlander. An inverse problem for radiation fields. Proceedings of the London Mathematical Society 3.3 (1973):

10 Proof II: Asymptotic quantities Exploit structure equations: Riem = dω + ω ω =: Ω These split into trace (Einstein equations) and trace-less part (Weyl curvature), In general 0 = Ric = trω and Weyl = ˆΩ 0 = trω (k) = tr(dω (k) + ω (1) ω (k 1) +...) is a 1st order system of PDEs for the connection coefficients which allows us to compute them recursively in terms of χ (1) = δ + Ξ (and initial conditions). In particular, by taking the u-derivative we can conclude u ω (k) = 0 inductively, provided that Ξ = 0.

11 Proof III: Unique continuation techniques Consider Schwarzschild spacetime, ( g S = 2 1 2m ) dudv + r 2 (dθ 2 + sin 2 dφ 2 ). r And let f := 1, D := {0 < f < ω}, ω > 0 uv Consider an operator of the form L = + a α α + V with ( ) a u = O(v 1 r 1 2 ) a v = O ( u) 1 r 1 2 a I = O(f 1 2 r 3 2 ) V = O(f 1+η ), η > 0.

12 Proof III: Unique continuation techniques Theorem. Suppose φ is a C 2 function satisfying Lg = 0 and vanishing to all orders at infinity in the sense of r N ( φ 2 + φ 2 ) <, N N D Then there exists 0 < ω < ω such that φ 0 on D ω.