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On the non-existence of periodic, asymptotically flat spacetimes

On the non-existence of periodic, asymptotically flat spacetimes

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MotivationThe non-existence of asymptotically flat periodic spacetimesHelically symmetric solutionsHelically symmetric spacetimesBinary systems inspiralNewton’s theoryBinary system of two fluid bodies (Lichtenstein)Maxwell’s theoryTwo charged particlesorbiting about the same center (Schild)Motivation: models of elementary particlesOutline:find classical stable statescompute E, P, Lapply Bohr quantizationL = n Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesSchild’s solutionHelically symmetric spacetimesBinary systems inspiralFokker-Wheeler-Feynman time-symmetric actionS = −m ∫ ds − ¯m ∫ d¯s − eē ∫ ds d¯s ẋ µ ˙¯x µ δ ( (x − ¯x) 2)(x − ¯x) 2 = 0 → t ± = t ± |¯x(t ± ) − x(t)| retardation angle: v 2 + ¯v 2 + 2v ¯v cos θ = θ 2Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesSchild’s solutionHelically symmetric spacetimesBinary systems inspiralHelically symmetric solutionequation of motionmωv(1 − v 2 ) 1/2 = e ē(θ + v ¯v sin θ) 2 [(v + ¯v cos θ)(1 − v 2 )(1 − ¯v 2 )conserved quantitiesE = m(1 − v 2 ) 1/2 + ¯m(1 − ¯v 2 ) 1/2P = 0+(vθ + ¯v sin θ)(θ + v ¯v sin θ)]1+v ¯v cos θL z = eē θ+v ¯v sin θSpin can be included (Schlosser, Schild)Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesHelically symmetric solutionsHelically symmetric spacetimesBinary systems inspiralJ. Bičák, B. G. Schmidt: Helical symmetry in linear systems (2007)Helically symmetric solutions for scalar and electromagneticfield on Minkowski spaceAnalysis of asymptotic behaviour of the fieldsR. Beig, J. M. Heinzle, B. G. Schmidt: Helically symmetricN-particle solution (2007)Scalar gravityEquilibrium configurationof N point particlesorbiting about the centerTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesHelically symmetric spacetimesHelically symmetric spacetimesBinary systems inspiralNo helically symmetric solutions of Einstein’s equations are knownTwo orbiting bodies lose energy by radiation → inspiralIngoing radiation is needed to compensate the loss of energyTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesOutlineHelically symmetric spacetimesBinary systems inspiral1 MotivationHelically symmetric spacetimesBinary systems inspiral2 The non-existence of asymptotically flat periodic spacetimesTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesThe inspiral of binary systemHelically symmetric spacetimesBinary systems inspiralIsolated binary system of orbiting black holes or neutron starsSource of detectable gravitational wavesNumerical simulationsTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesHelically symmetric spacetimesBinary systems inspiralNo asymptotically flat and helically symmetric spacetimesBy physical reasons, isolated systems cannot move periodicallyProof given by Gibbons and Stewart(1984)AssumptionsAsymptotically flat spacetimeNo matter near infinityAnalytic initial data given on null infinityTheorem: Every asymptotically flat periodic solution ofvacuum Einstein’s equations is stationary near infinity.Possible generalizationsNon-analytic initial dataNon-vacuum spacetimes (EM field, scalar field)Drawbacks of the proofUsed conformal gauge is incompatible with periodicityProof does not apply to Minkowski spacetimeTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesGS-proofDouble-null coordinates x µ = (u, v, x 2 , x 3 )γ = γ(v) - null generators of I −x J , J = 2, 3 - coordinates on S, spacelike cut on I −γ ′ = γ ′ (u) - null generators orthogonal to Su, v - retarded/advanced time (u = t − r, v = t + r)NP null tetrad l = ∂ v , n = ∂ u + C J ∂ J , m = P J ∂ J Assumption: g ab periodic in v on I −Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesGS-proofGibbons and Stewart have shown:g ab is v−independentK = ∂ v is Killing vectorConclusion: spacetime is stationary (wrong)Problem‖K‖ 2 = 0, i.e. K is null, not timelikeExample: Minkowski spacetime in double-null coordinatesds 2 = du dv − 1 4 (u − v)2 (dθ 2 + sin 2 θdφ 2 )L ∂v g ≠ 0 !!!No null Killing vector tangent to I exists→ flat spacetime is not periodicReason: inappropriate definition of periodicityTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesSolutionPeriodicity on Iv ↦→ v + a(θ, φ)is isometryCoordinate system x µ = (v, r, θ, φ)l = ∂ v − H∂ r + C J ∂ J , n = ∂ r , m = P I ∂ J ‖∂ v ‖ 2 = 0 only on I − Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesConformal field equationsMaxwell’s equations∇ AA′ φ AB = 0Conformal Bianchi identities∇ D C ′ψ ABCD = 3 ¯φ A ′ B ′ φ (AB ∇ B′C) Ω + Ω ¯φ A ′ B ′ ∇B′ (C φ AB)Conformal factor∇ AA ′∇ BB ′Ω = Ω 3 φ AB ¯φ A ′ B ′ − Ω Φ ABA ′ B ′ + ɛ A ′ B ′ ɛ AB (F + Ω Λ)∇ AA ′F = Ω 2 φ B A¯φB′A ′ ∇ BB ′Ω − Φ ABA ′ B ′∇BB′ Ω + Λ ∇ AA ′ΩF ≡ 1 2 Ω−1 g ab (∇ a Ω)(∇ b Ω)Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesResultTheorem. A weakly-asymptotically simple, vacuum orelectrovacuum, time-periodic space-time which is analytic ina neighbourhood of I − necessarily has a Killing vector which istime-like in the interior and extends to a translation on I − .Corollary. In any weakly-asymptotically simple, stationaryelectrovacuum space-time which is analytic in a neighbourhood ofI − , the Maxwell field is also stationary.Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesScalar fieldsScalar field as a source of gravityTwo kinds of scalar fieldKlein-Gordon field(□ + m2 ) φ = 0Non-analytic behaviour for m > 0φ = O(e −mr )Argument does not apply (e.g. mini-boson stars)Only massless fields are consideredConformally invariant field(□ + 4Λ) φ = 0Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesMassless scalar fieldMassless KG equation in the physical spaceime˜□ ˜φ = 0Einstein’s equations˜Φ ABA ′ B ′ = 2 ˜ϕ (A(A ′ ˜¯ϕ B′ )B), 6 ˜Λ = − ˜ϕ c ˜¯ϕ cwhere ˜ϕ a ≡ ˜∇ a ˜φConformal Bianchi identities∇ D A ′ψ ABCD = 2φ ¯φs (C B′ Φ AB)A ′ B ′ + 4 (sϕ ¯ϕ) + 2φ (∇s ¯ϕ) + 2 ¯φ (∇sϕ)[ ]1+4Ω2 (∇ϕ ¯ϕ) − φ ¯φ 2 (sϕs) − ¯φφ 2 (s ¯ϕs) − 4Ω 2 φ ¯φ (sϕ ¯ϕ)where (XYZ) = X B′(C Y A(A ′Z B ′ )B)Rescaled wave equation∇ A′A ϕ BA(Λ ′ = 2 − Ω −2˜Λ)φ ɛ ABTod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesThe Bondi mass M BMeasures the mass of the sourceDecreases when the source radiatesPeriodicity requires M B = constantElectro-vacuum spacetimes (on I + )M B = − ∮ [ ψ2 0 + σ0 ˙¯σ 0] dSṀ B = − ∮ [ ˙σ 0 ˙¯σ 0 + φ 0 ¯φ]2 0 2M B =const. → ˙σ 0 = 0, φ 0 2 = 0Scalar field?Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesThe Bondi mass M BHeuristic derivationTime decrease of the Bondi mass ∝ energy carried by outgoing radiationT ab = T (n, n)l al b + T (l, l)n an b + 2T (n, l)l (a n b) + · · ·T (n, n) ∼ Φ ABA ′ B ′ιA ῑ A′ ι B ῑ B′ = Φ 22Scalar field: Φ 0 22 = ˙φ 0 ˙¯φ0Conjecture: Ṁ B ∝ − ∮ [ ˙σ 0 ˙¯σ 0 + k]˙φ 0 ˙¯φ0 dS Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesExact calculationAsymptotic twistor equation∇ A′(A τ B) = 0 on ICanonical symmetric [ 2−spinor and]antisymmetric tensorφ AB = 1 τ 2 (A ∇ C ′B) ¯τ C ′ − ¯τ C ′∇C ′(A τ B)F ab = φ AB ɛ A ′ ∮B ′ + ¯φ A ′ B ′ɛ ABM B (u) = lim F ab l a n b dSΩ→0S(Ω)ResultM B (u) = − ∮ [ ψ 0 2 + σ0 ˙¯σ 0 + 1 3 ∂v (φ0 ¯φ 0 ) ] dSṀ B (u) = − ∮ [ ˙σ 0 ˙¯σ 0 + 2 ˙φ 0 ˙¯φ 0 ]dSPeriodicity → ˙φ 0 = 0, i.e. ϕ 0 = 0 on I −Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesResultTheorem. A weakly-asymptotically simple, time-periodic solution ofEinstein-massless-Klein-Gordon equations, which is analytic ina neighbourhood of I − necessarily has a Killing vector which istime-like in the interior and extends to a translation on I − .Corollary. In any weakly-asymptotically simple, stationaryEinstein-massless-Klein-Gordon spacetime, which is analytic in aneighbourhood of I − , scalar field is also stationary.Remark. Theorem holds also for self-interacting field satisfying□φ + V (φ, ¯φ) = 0,if V = O(Ω 3 ), e.g. for φ 4 -theory, where V = λ|φ| 4Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesConformally invariant scalar fieldWave equation (□ + 4Λ)φ = 0Einstein’s equations(˜Φ ABA ′ B ′ = 1 − ˜φ 2) −1 [2 ˜ϕ (A(A ′ ˜ϕ B′ )B) − ˜φ]˜∇ (A(A ′ ˜ϕ B′ )B)˜Λ = 0Possible problem - singularity for ˜φ = 1Bondi massM B (u) = − 12 √ πṀ B (u) = − 12 √ π∮∮[dS Ψ (0)2 + σ (0) ˙¯σ (0)] ,[(dS ˙σ (0) ˙¯σ ) (0) 2+ 2 ˙φ(0) − φ(0) ¨φ(0)] .Tod, ScholtzPeriodic solutions


MotivationThe non-existence of asymptotically flat periodic spacetimesBibliographyJ. Bičák, B. G. Schmidt, Phys. Rev. D76, 104040 (2007)L. Lichtenstein, Mathematische Zeitschrift 12, 201(1922)J. K. Blackburn and S. Detweiler, Phys. Rev. D 46, 2318(1992).S. Detweiler, Phys. Rev. D 50,4929 (1994).A. Schild, Phys. Rev. 131(6), 2762 (1963)J. T. Whelan, W. Krivan, and R. H. Price, Class. QuantumGrav. 17, 4895 (2000)J. L. Friedman, K. Uryu, and M. Shibata, Phys. Rev.D 65, 064035 (2002); 70,129904(E) (2004);H. Friedrich, Proc. R. Soc. Lond. A 381, 361 (1982)G. W. Gibbons, J. M. Stewart, in Classical General Relativity, CambridgeUniversity Press (Cambridge U.K.)1984, p. 77-94J. Stewart, Advanced general relativity (Cambridge University Press, Cambridge,U.K., 1991).Tod, ScholtzPeriodic solutions

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