Non-existence of time-periodic dynamicsin general relativity

Volker Schlue

University of Toronto

University of Miami, February 2, 2015

Outline

1 General relativityNewtonian mechanicsSelf-gravitating systemsGravitational Radiation

2 Final State ConjectureStationarity of non-radiating systemsUniqueness of stationary spacetimes

3 Time-periodic vacuum spacetimesStatement of main resultIdea of the proof

4 Uniqueness results for ill-posed hyperbolic p.d.e.’sUnique continuation from infinity for linear wavesPositive mass and the behavior of light rays

General Relativity

General Relativity is a unified theory of space, time & gravitation.

geometry

GR

analysis

physics

Some of the most fascinating developments:black holes (astronomy), expansion of the universe (cosmology);geometric analysis of hyperbolic equations (mathematics)

1 Gravity in Newton’s and Einstein’s theory

Newtonian Gravity

Two-body problem in classical celestial mechanics: Kepler orbits.Space: R3, Time: R, Newtonian potential: ψ(t, x).

Sun

Earth

Test body:

Newton’s equation: ψ harmonic function.

Note: In Newtonian theory the gravitational field can be periodicin time. (Action at a distance)

Figure: Albert Einstein

Einstein’s Gravity

Spacetime: (M, g) 3 + 1-dimensional manifold endowed withLorentzian metric (quadratic form of index 1).

Sun

Earth

Test body:

Σ ' R3

Einstein’s vacuum equations: Spacetime manifold is “Ricci-flat”.

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)

ψpost-Newtonian = ψ +1

c2ω

Gravitational RadiationThe Einstein equations are of hyperbolic nature, and describe thenon-linear dynamics of spacetime itself.

Harmonic gauge:

Choquet-Bruhat ’52

Σ0

ΣtC

Global non-linear stability of flat space:

Christodoulou-Klainerman ’93 , Lindblad-Rodnianski ’10

Time-periodicity in general relativity

Periodic motion should not exist in general relativity due to theemission of gravitational waves!

Our main result is that any time-periodic vacuum spacetimeis in fact time-independent.

Null infinity

Future null infinity is the conformal boundary attached to anasymptotically flat spacetime, that can be thought of as the spaceof “far away observers”. (Penrose)

C0

Ct

I+ ' R× S2

Monotone mass

I+

I−

ι0Σ

Cu

Notions of mass:

M[Σ] (Arnowitt-Deser-Misner)M[Cu] (Bondi): amount of energy inthe system at time u.

Positive mass theorem:M[Σ] ≥ 0, M[Cu] ≥ 0:Schoen-Yau, Witten

The Bondi mass M(u) := M[Cu] is monotone,

∂M(u)

∂u≤ 0

and for perturbations of flat space (Christodoulou-Klainerman)

limu→−∞

M(u) = M[Σ] limu→∞

M(u) = 0 .

2 Final State Conjectures

Non-radiating spacetimes

We say a spacetime is not radiating,if the Bondi mass M(u) = M0 is constant.

Conjecture.

An asymptotically flat vacuum spacetime that is not radiating isnecessarily stationary.

Here stationary refers to the existence of a continuous isometryψt :M→M of the spacetime manifold (M, g) generated by avectorfield T which is time-like near infinity,

LTg = limt→0

t−1(ψ∗t g − g) = 0 .

Aside: Gravitational wave experiments

Figure: LIGO, Washington

v

v

v ∼ ∂uM

Stationary spacetimes

It is expected that all stationary solutions to the Einstein vacuumequations can be completely classified, even if black holes haveformed: Mf = limu→∞M(u) > 0.

Conjecture.

Any smooth asymptotically flat black hole exterior solution to theEinstein vacuum equations that is stationary is isometric to theexterior of a Kerr solution (M, gM,a).

I+H+

Σ

D

I+: future null infinity

H+: future event horizon

D: black hole exterior

Stationary black hole spacetimes

The conjecture has been partially resolvedin the perturbative regime.

Black hole uniqueness theorems:

Carter-Robinson (axi-symmetriccase), Hawking (additional Killingvectorfield on horizon, and analyticcase), . . . , Mars-Simon (localcharacterisation), . . . ,Alexakis-Ionescu-Klainerman

(rigidity results in the smooth case, forstationary vacuum spacetimes close tothe Kerr solutions).

H+ I+

I−H−

Final State Conjecture

The final state conjecture gives a characterisation of all possibleend states of the dynamical evolution in general relativity, as aresult of the following scenario: (Penrose)

Radiation: It is expected that due to the emission ofgravitational waves any self-gravitating system“settles down” to a non-radiating state.

Stationarity: It is conjectured that all self-gravitating systems thatdo not radiate to infinty are in fact stationary, i.e.time-independent.

Uniqueness: It is believed that in vacuum the only stationaryblack hole exteriors are the Kerr solutions.

Figure: Artist’s impression of binary star system J0806 rapidlyapproaching coalescence. (NASA)

3 Time-periodic vacuum spacetimes

Notion of time-periodicity

Definition

An asymptotically flat spacetime is called time-periodic if thereexists a discrete isometry ϕa with time-like orbits, that extends toa translation ϕ+

a on future null infinity.

p

ϕa(p)

I−

ι0

I+ ' R× S2

ϕ+a (u, ξ) = (u+ a, ξ)

ϕa

Stationarity of time-periodic spacetimes

Theorem (Alexakis-S.)

Any asymptotically flat solution (M, g) to Einstein’s vacuumequations Ric(g) = 0 which is smooth and time-periodic nearinfinity is stationary near infinity.

I+

I−

ι0 D T ι0

The theorem asserts that there exists a time-like vectorfield T on an

arbitrarily small neighborhood D of infinity such that LTg = 0 on D.

Previous results

Early work:

Papapetrou ’57-’58 (weak field approximation, “non-singular”solutions, strong time-periodicity assumption)

Recent work:Gibbons-Stewart ’84, Bicak-Scholz-Tod ’10 (containsideas how to exploit time-periodicity, stationarity inferred undermuch more restrictive analyticity assumption)

Cosmological setting:

Tipler ’79, Galloway ’84 (spatially closed case)

Strategy of Proof

1 Construction of time-like candidate vectorfield T , such thatby time-periodicity

limr→∞

rkLTR = 0 ∀k ∈ N

2 Use that by virtue of the vacuum Einstein equations,

�gR = R ∗ R

thus“ �gLTR = R ∗ LTR ” .

Then apply our result that solutions to wave equations onasymptotically flat spacetimes are uniquely determined if allhigher order radiation fields are known, to show that

LTR = 0 .

Unique continuation from infinity

Theorem (Alexakis-S.-Shao ’14)

Let (M, g) be an asymptotically flat spacetime with positivemass, and Lg a linear wave operator

Lg = �g + a · ∇+ V

with suitably decaying coefficients a, and V . If φ is a solution toLgφ = 0 which in addition satisfies∫

Drkφ2 + rk |∂φ|2 <∞

where D is an arbitrarily small neighborhood of infinity ι0, then

φ ≡ 0 : on D′ ⊂ D .

Application of the theorem

The application of the theorem is not immediate because

�gR = R ∗ R

is not a scalar equation, but a covariant equation for theRiemann curvature tensor. Moreover, [�g ,LT ] 6= 0 anddifferentiating the equation produces additional terms which arenot in the scope of the theorem:

�gLTR − [�g ,LT ]R = R ∗ LTR + LTg ∗ R2

These obstacles can be overcome in the general framework ofIonescu-Klainerman ’13 for the extension of Killingvectorfields in Ricci-flat manifolds.

Construction of candidate vectorfield T

Σ0

Bd∗ S0

C−∗

C+0

S∗0

C+u

LL

T

Su,sSu,s

Define T = ∂∂u : binormal to spheres S∗u . Then extend inwards by

Lie transport along geodesics: [L,T ] = 0, ∇LL = 0, g(L, L) = 0.

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

Linear theory

The conjecture that non-radiating spacetimes are stationary has asimple analogy in linear theory.Consider the linear wave equation on R3+1:

�φ := −∂2φ

∂t2+

3∑i=1

∂2φ

∂x i 2= 0

The radiation field is defined by

Ψ(u, ξ) = limr→∞

(rφ)(u + r , rξ) .

The assumption that a spacetime is not radiating corresponds inlinear theory to the vanishing of the radiation field. One maythus ask:

Ψ = 0(?)

=⇒ φ ≡ 0

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 = ∂x i1r ∼ 1

r2, which is

a non-trivial solution with Ψ = 0.

Theorem (Friedlander ’61)

For finite energy solutions to the classical wave equation �φ = 0the radiation field vanishes identically if and only if the initial datais trivial.

Without a global finite energy condition, or a smoothnessassumption at infinity, we are led to assume

limr→∞

(rkφ)(u + r , rξ) = 0 ∀k ∈ N .

Obstructions to unique continuationfrom infinity

There is an obstruction to localisation (near spacelike infinity ι0)related to the behavior of light rays.

Alinhac-Baouendi ’83: Unique continuation fails acrosssurfaces which are not pseudo-convex, in general.

Unique continuation from infinity forlinear waves

Theorem (Alexakis-S.-Shao ’13)

Let (M, g) be a perturbation ofMinkowski space, and Lg a linear waveoperator with decaying coefficients. If φis a solution to Lgφ = 0 which inaddition satisfies an infinite ordervanishing condition on “at least half” offuture and past null infinity, then

φ ≡ 0

in a neighborhood of infinity.

D

I−ε

I+ε

Pseudoconvexity

The proof crucially relies on the construction of a family ofpseudo-convex time-like hypersurfaces.

Definition (Calderon, Hormander)

A time-like hypersurface is pseudo-convex ifit is convex with respect to tangential nullgeodesics.

p

We find a family of pseudo-convex hypersurfaces that foliate aneighborhood of infinity and derive a Carleman inequality toprove the uniqueness result.

Positive massand the behaviour of light rays

While the result cannot be localised on Minkowski space, we haveseen that in the presence of a positive mass unique continuationdoes hold in an arbitrarily small neighborhood of spacelike infinity.

ι0M > 0

light ray

This is related to ideas of Penrose ’90 to characterise thepositivity of mass by the behavior of null geodesics near spacelikeinfinity. (Mason)

Thank you for your attention!

Figure: Roger Penrose