non-existence of time-periodic dynamics in general relativityschlue/miami-slides.pdf · general...
Post on 09-Jun-2020
7 Views
Preview:
TRANSCRIPT
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
top related