Self trapped gravitational waves (geons) with anti-de Sitter asymptotics

Transcription

1 Self trapped gravitational waves (geons) with anti-de Sitter asymptotics Gyula Fodor Wigner Research Centre for Physics, Budapest ELTE, 20 March 2017 in collaboration with Péter Forgács (Wigner Research Centre for Physics, Budapest) Philippe Grandclément and Grégoire Martinon (Observatoire de Paris, Meudon) G. Fodor: AdS geons 1/41

2 Outline 1 Asymptotically flat geons 2 Anti-de Sitter spacetime (AdS) instability 3 Asymptotically AdS spacetimes conserved quantities 4 AdS geons nonlinear perturbative construction based on spherical harmonic decomposition helically symmetric rotating geons comparison to numerical results G. Fodor: AdS geons 2/41

3 Introduction: asymptotically flat geons John Archibald Wheeler, Phys. Rev. 97, 511 (1955) : gravitational-electromagnetic entity geon G. Fodor: AdS geons 3/41

4 Toroidal geons Wheeler 1955 regions of strong electric field Held together by the gravitational attraction of the mass associated with the electromagnetic field energy Wheeler studied very thin torus and very high wave numbers Small amplitude high frequency high angular momentum waves going around in a circle Brill and Hartle, Phys. Rev. 135, B271 (1964): similar geons formed by vacuum gravitational waves G. Fodor: AdS geons 4/41

5 Spherical geons: electromagnetic or gravitational waves Brill and Hartle 1964: Large number of identical-size thin toroidal geons with different orientations The metric becomes spherically symmetric and static on the large scale ds 2 = e ν dt 2 + e λ dr 2 + r 2 dω 2 There is a thin sphere active region, where the high frequency waves are concentrated inner region flat, outside Schwarzschild Spacetime averaging is used for small amplitude high frequency perturbations made precise by Anderson and Brill, Phys. Rev. D. 56, 4824 (1997) G. Fodor: AdS geons 5/41

6 Negative cosmological constant For Λ < 0 the spacetime of geons must tend asymptotically to the anti-de Sitter metric (AdS) Λ < 0 provides an effective attractive force formation of localized solutions is easier Localized linear perturbative modes of AdS are studied by Mukohyama, Phys. Rev. D 62, (2000) Kodama, Ishibashi and Seto, Phys. Rev. D 62, (2000) Ishibashi and Wald, Class. Quant. Grav. 21, 2981 (2004) There are one-parameter families of geon solutions emerging from the linear modes Dias, Horowitz and Santos, Class. Quant. Grav. 29, (2012) Horowitz and Santos, Surveys Diff. Geom. 20, 321 (2015) Dias and Santos, Class. Quant. Grav. 33, 23LT01 (2016) Rostworowski, arxiv: [gr-qc] (2017) Martinon, Fodor, Grandclément and Forgács, arxiv: [gr-qc] (2017) G. Fodor: AdS geons 6/41

7 Anti-de Sitter spacetime center timelike geodesic null geodesic t=π/2 infinity Global spatially compactified coordinates ds 2 = L2 ( dt 2 cos 2 + dx 2 + sin 2 x dω 2) x where L 2 = 3/Λ each point corresponds to a 2-sphere with radius L tan x metric is static in these coordinates t=-π/2 x=0 x=π/2 center is at x = 0, infinity at x = π 2 range of time coordinate: < t < radial outwards acceleration of constant x observers is sin x L timelike geodesics meet again at a point G. Fodor: AdS geons 7/41

8 Anti-de Sitter spacetime t=π/2 center t=-π/2 ρ τ infinity x=0 x=π/2 Cosmological-type coordinates ds 2 = L 2 [ dτ 2 + cos 2 τ ( dρ 2 + sinh 2 ρ dω 2)] Friedmann-Lemaître-Robertson-Walker universe with k = 1 constant τ slices are homogeneous hyperbolic spaces constant ρ lines are geodesics coordinate singularity at τ = ±π/2 Review of AdS coordinate systems in: Griffiths and Podolský, Exact Space-Times in Einsteins General Relativity, Cambridge 2009 We use static global coordinates t, x G. Fodor: AdS geons 8/41

9 Instability of anti-de Sitter spacetime A light ray can travel to infinity and back in a finite time This is related to the (conjectured) instability of AdS Bizoń and Rostworowski, Phys. Rev. L. 107, (2011) a wave packet can bounce back many times to the center, and in the end always collapses to a black hole smaller amplitude more bounces needed demonstrated numerically for a spherically symmetric massless scalar field coupled to gravity AdS is linearly stable positive mass theorem cannot decay adding a small energy, a small black hole may form energy cannot disperse at infinity it is like a bounding box all natural boundary conditions are reflective G. Fodor: AdS geons 9/41

10 Asymptotically AdS spacetimes Definition using Penrose s conformal treatment of infinity Hawking, Phys. Lett. B. 126, 175 (1983) Ashtekar and Magnon, Class. Quant. Grav. 1, L39 (1984) Ashtekar and Das, Class. Quant. Grav. 17, L17 (2000) Hollands, Ishibashi and Marolf, Class. Quant. Grav. 22, 2881 (2005) Also gives definition for conserved quantities: total mass and 3 components of angular momentum G. Fodor: AdS geons 10/41

11 Weakly asymptotically AdS spacetimes (Λ < 0) Manifold M, metric g µν (M, g µν ) is weakly asymptotically AdS if, vacuum solution of Einstein s equations (i) there is a manifold ( M, g µν ) with boundary I, and a diffeomorphism ψ from M onto M \ I, such that g µν = Ω 2 g µν (ii) Ω = 0 on I, but its gradient is nonzero (iii) I is topologically S 2 R Infinity is part of the new manifold, it is at finite coordinates, and finite distance with respect to the unphysical metric Einstein s equations I timelike G. Fodor: AdS geons 11/41

12 AdS is asymptotically AdS Ω = cos x L, g µν = Ω 2 g µν, ψ is identity map ds 2 = ( L2 dt 2 + dx 2 + sin 2 x dω 2) d s 2 = dt 2 + dx 2 + sin 2 x dω 2 cos 2 x General metric: g µν = g AdS µν + h µν, we use t, x coordinates weekly asymptotically AdS if h µν diverges as ( π 2 x) 2 asymptotically AdS if h µν diverges as ( π 2 x) 1 G. Fodor: AdS geons 12/41

13 Conformal structure of infinity If a spacetime is asymptotically AdS with some Ω, then Ω = ωω is equally good, where ω any function for which ω 0 on I only the conformal geometry of I is meaningful For exact AdS, with our choice of coordinates, and choice of Ω d s 2 = dt 2 + dx 2 + sin 2 x dω 2 the induced metric on I, at x = π 2, is d s2 0 = dt2 + dω 2 it is the flat metric on S 2 R with general coordinates and general Ω d s 0 2 can be any conformally flat metric For general weakly asymptotically AdS spacetimes the induced metric on I can be anything this is a problem, since Asymptotic symmetries correspond to asymptotic Killing vector fields on M and they correspond to conformal Killing vector fields on I G. Fodor: AdS geons 13/41

14 Definition of asymptotically AdS spacetimes AdS has the symmetries corresponding to the AdS group O(3, 2) 10 conformal Killing vector fields on I For general weakly asymptotically AdS metric I may have no conformal Killing vector fields at all To each conformal Killing field a conserved quantity belongs should the number of conserved quantities depend on which metric we choose? (M, g µν ) is asymptotically AdS if it is weakly asymptotically AdS [ points (i), (ii), (iii) ] (iv) the metric on I induced by g µν is conformally flat Equivalent to the condition that the symmetry group is the AdS group G. Fodor: AdS geons 14/41

15 Asymptotic behavior of the Weyl tensor In order to have regular quantities at I we work with quantities belonging to the unphysical metric g µν Einstein s equations Weyl tensor C µνρσ = 0 on I we can define the leading order asymptotic Weyl at I by 1 K µνρσ = lim I Ω C µνρσ The one-form n µ = µ Ω is normal to I, and g µν n µ n ν = 1/L 2 on I, where L 2 = 3/Λ The electric part of the leading order asymptotic Weyl tensor is defined as E µν = L 2 K µρνσ n ρ n σ E µν n ν = 0 it is a tensor in I it is symmetric and trace-free G. Fodor: AdS geons 15/41

16 Conserved quantities If there are no matter fields, then D µ E µν = 0, where D µ belongs to the metric γ µν induced by g µν on I I is S 2 R, and has 10 conformal Killing fields ξ µ For any choice of ξ µ and for any 2-sphere cross section C of I, a conserved quantity can be defined by Q ξ = L E µν ξ µ u ν d S 8π C where d S is the volume element on C, and u ν is the unit normal to C, both with respect to the metric γ µν G. Fodor: AdS geons 16/41

17 Mass and angular momentum The conserved quantity ξ = L E µν ξ µ u ν d S 8π C is independent of the choice of the hypersurface C is independent of the choice of the conformal factor Ω The simplest choice for C is to take a constant t section Coordinates on I are x µ = (t, θ, φ) Choices for the conformal Killing field ξ µ : ξ µ = ( 1 L, 0, 0) Q ξ gives the total mass ξ µ = (0, 0, 1) ξ µ = (0, sin φ, cot θ cos φ) ξ µ = (0, cos φ, cot θ sin φ) they are time independent 3 components of angular momentum G. Fodor: AdS geons 17/41

18 AdS geons Localized time-periodic vacuum solutions formed by gravitational waves, with regular center and no horizon for Λ < 0 typical size given by the length-scale L = 3 Λ Λ < 0 makes the formation of geons easier there are solutions without high-frequency waves even in the small amplitude linear limit In this sense, they are similar to the sine-gordon breathers Also similar to spherically symmetric scalar AdS breathers formed by a self-gravitating massless real Klein-Gordon field Fodor, Forgács and Grandclément, Phys. Rev. D 92, (2015) There are no spherically symmetric geon solutions for vacuum G. Fodor: AdS geons 18/41

19 One-parameter families of asymptotically AdS solutions Consider a one-parameter family of solutions depending on a parameter ε, and expand the metric as g µν = ε k g µν (k) k=0 where g µν (0) is the AdS metric ds(0) 2 = L2 [ dt 2 cos 2 + dx 2 + sin 2 x(dθ 2 + sin 2 θdφ 2 ) ] x g (0) µν has components that diverge as ( π 2 x) 2 at infinity We require that for k 1 all g (k) µν diverge at most as ( π 2 x) 1 metric induced at I remains the same as for AdS g µν is asymptotically AdS G. Fodor: AdS geons 19/41

20 Time-periodic solutions The physical frequency ω of geon families generally depend on ε in the small ε limit ω approaches integer values For technical simplicity we use a time coordinate for which the coordinate frequency ω remains an ε independent integer for this we have to rescale the time coordinate as t α(ε)t Since g (0) tt = L2, it follows that the asymptotic behavior of the cos 2 x total metric component g tt becomes ε dependent ( π2 ) ] 2 lim [g tt x = ν x π 2 where ν depends on ε, but independent of the coordinates The relation between the two frequencies becomes ω = ω ν G. Fodor: AdS geons 20/41

21 Decomposition along spheres We decompose along the symmetry spheres of the background AdS a, b, c... = 1, 2, i, j, k... = 3, 4 coordinates along the time-radius plane: x a = (x 1, x 2 ) = (t, x) coordinates along the symmetry spheres: x i = (x 3, x 4 ) = (θ, φ) We decompose AdS as where g ab = L2 ( 1 0 cos 2 x 0 1 ds 2 (0) = g abdx a dx b + r 2 γ ij dx i dx j ), r = L tan x, γ ij = ( sin 2 θ ) G. Fodor: AdS geons 21/41

22 Spherical harmonic decomposition We use real spherical harmonics S lm defined for l 0 and l m l integers φ dependence is cos(mφ) for m 0, and sin( m φ) for m < 0 Tensors can be decomposed into scalar part vector part tensor part only for higher than 4 dimensional spacetimes Vector spherical harmonics has components V (lm)θ = 1 1 S lm l(l + 1) sin θ φ, V (lm)φ = 1 l(l + 1) sin θ S lm θ Perturbations for each l, m can be considered separately they are only coupled by lower order terms in the ε expansion G. Fodor: AdS geons 22/41

23 Vector-type perturbations of the metric We consider vector-type first, because it is technically simpler Even if we start with scalar-type perturbations at linear order in ε, vector-type components appear at ε 2 order V (lm)i components at ε k order in the expansion (a, b = 1, 2; i, j = 3, 4) g (k) ab = 0, g (k) ai = Z a V i, g (k) ij = 0 Only two unknown functions: Z a = (Z t, Z x ) they depend on the coordinates x a = (t, x) The l = 1 spherical harmonics have to be considered separately for the linear order only the trivial solution Z a = 0 for higher order determines the angular momentum G. Fodor: AdS geons 23/41

24 Scalar function determining vector-type perturbations For l 2, from Einstein s equations follows that there exists a scalar function φ such that Z t = φ x + Z t, Z x = φ t where Z t is an already known function fixed by lower order perturbations zero at linear order Defining a rescaled scalar function by φ = rφ, r = L tan x from Einstein s equations follows that 2 Φ t Φ l(l + 1) x 2 sin 2 x Φ = Φ sin 2 x where Φ is a known function of t, x determined at lower order in ε G. Fodor: AdS geons 24/41

25 Scalar-type perturbations of the metric For each l 2 and m, for scalar-type perturbations also exists a scalar function φ = rφ, which satisfies the same equation 2 Φ t Φ l(l + 1) x 2 sin 2 x Φ = Φ sin 2 x The metric perturbations can be obtained from Φ as follows: First, calculate the quantities Z ab and Z Z tt = t 2 φ tan x x φ + φ cos 2 x + Z tt Z tx = t x φ tan x t φ + Z tx Z xx = x 2 φ tan x x φ φ cos 2 x + Z xx Z = cos2 x L 2 (Z xx Z tt ) + Z where Z ab and Z are determined at lower order in ε G. Fodor: AdS geons 25/41

26 Second step: from Z ab and Z calculate H L = r 2 2 Z, H ab = Z ab 1 2 Zg ab The general S lm scalar-type metric perturbations at ε k order are (a, b = 1, 2; i, j = 3, 4) g (k) ab = H abs, g (k) ai = 0, g (k) ij = H L γ ij S The l = 0, 1 scalar-type perturbations have to be treated separately no generating scalar function only gives gauge modes at linear order in ε l = 0 mode determines the mass at higher orders We have made the most natural gauge choice in the higher order generalization of the gauge invariant formalism of Mukohyama-Kodama-Ishibashi-Seto-Wald G. Fodor: AdS geons 26/41

27 Scalar equation For l 2 all scalar or vector perturbations are determined by 2 Φ t Φ l(l + 1) x 2 sin 2 x Φ = where Φ is given from lower order in ε results Φ sin 2 x The boundary conditions at infinity are different in the two cases: The generated metric perturbation will be asymptotically AdS if for vector-type perturbations lim Φ = 0 x π 2 for scalar-type perturbations lim x π 2 dφ dx = 0 G. Fodor: AdS geons 27/41

28 Periodic solutions at linear order There is no inhomogeneous source term: Φ = 0 Search solutions in the form Φ = p(x) cos(ωt δ) Centrally regular and asymptotically AdS solutions only exist: scalar-type perturbations: ω = l n, n 0 integer p(x) = α n! L sinl+1 x (l ) P (l+ 1 2, 1 2 ) n (cos(2x)) n vector-type perturbations: ω = l n, n 0 integer p(x) = α n! L sinl+1 x cos x (l ) P (l+ 1 2, 1 2 ) n (cos(2x)) n where the Pochhammers Symbol is (c) n = Γ(c + n)/γ(c) and Pn α,β (z) are Jacobi polynomials n gives the number of radial nodes (zero crossings) G. Fodor: AdS geons 28/41

29 Inhomogeneous equation at higher orders in ε 2 Φ t Φ l(l + 1) x 2 sin 2 x Φ = Φ sin 2 x homogeneous solutions with frequency ω scalar-type: ω = l n vector-type: ω = l n source terms of the type Φ = p 0 (x) sin(ω s t) or Φ = p 0 (x) cos(ω s t) If ω ω s for all n 0 integers, there are always time-periodic solutions which are asymptotically AdS and have a regular center If ω = ω s for some n, it is the resonant case generally, all regular asymptotically AdS solutions are blow-up solutions of the type t cos(ωt) time-periodic solutions only exist if a consistency condition holds G. Fodor: AdS geons 29/41

30 Consistency conditions In the resonant case, time-periodic centrally regular asymptotically AdS solutions only exist, if a consistency condition holds on the source term Φ = p 0 (x) sin(ω s t) π 2 0 f l,n (x)p 0 (x)dx = 0 f l,n (x) is a function determined by the homogeneous solutions The consistency condition determines the change of physical frequency ω as a function of ε ratio of the modes included at linear order If the consistency conditions cannot be satisfied = terms with linearly increasing amplitude t cos(ωt) shift of energy to higher frequency modes turbulent instability black hole formation G. Fodor: AdS geons 30/41

31 Helically symmetric rotating solution linear order The smallest frequency is for the l = 2, n = 0 nodeless scalar-type solution: ω = l n = 3 then the scalar function is Φ = α L sin3 x cos(ωt δ) We take a combination of the m = 2 and m = 2 modes S 2,2 = π sin2 θ cos(2φ), S 2, 2 = π sin2 θ sin(2φ) with time shifted phases δ = 0 and δ = π 2 and same amplitude Even at higher orders, all time dependence will be through terms cos[k(3t 2φ)] and sin[k(3t 2φ)] helical symmetry with Killing vector d dt This is the only ω = 3 solution with nonzero angular momentum and reflection symmetry with respect to the equatorial plane d dφ G. Fodor: AdS geons 31/41

32 Helically symmetric rotating solution higher orders We constructed the (l, m, n) = (2, ±2, 0) helically symmetric solution up to fourth order in ε had to solve consistency conditions at orders ε 3 and ε 5 to fix the unspecified constants we have a one-parameter family of solutions The reparametrization freedom ε f (ε) is fixed by setting the angular momentum J = 27πL2 128 ε2 This solution was studied first analytically by Dias, Horowitz and Santos, Class. Quant. Grav. 29, (2012) and then numerically by Horowitz and Santos, Surveys Diff. Geom. 20, 321 (2015) G. Fodor: AdS geons 32/41

33 Helically symmetric rotating solution higher orders The expansion of the physical frequency is ( J ωl = ω 1 L 2 + ω J 2 ) 2 L where ω 1 = π ω 2 = π π The expansion of the total mass is M L = 3 ( J 2 L 2 + ω 1 2 J 2 L 4 + ω 2 3 J 3 ) L They satisfy the identity dm dj = ω 2 first law of geon dynamics G. Fodor: AdS geons 33/41

34 Frequency angular momentum ωl/m rd order 5 th order numerical J AMD /L 2 G. Fodor: AdS geons 34/41

35 Mass angular momentum M AMD /L rd order th order numerical J AMD /L 2 G. Fodor: AdS geons 35/41

36 Numerical method KADATH library multi-domain spectral method developed by Philippe Grandclément at Paris Observatory Ads geons PhD topic of Grégoire Martinon, 2017 Maximal slicing (K = 0) and harmonic coordinates in space De-Turck method Start from a linearized solution increase the amplitude in steps typical resolution: radial 37, angular 9 9 typical running time: several days on hundreds of processors G. Fodor: AdS geons 36/41

37 Geons without radial nodes (l, m, n) (2, 2, 0) (4, 4, 0) (6, 6, 0) G. Fodor: AdS geons 37/41

38 Geons with one radial node Trying to construct a helically symmetric geon from the (l, m, n) = (2, ±2, 1) linear mode contradiction at ε 3 order no one-parameter family of solutions is generated the frequency of this mode is ω = l n = 5 the angular frequency is ω a = ω m = 5 2 There are two other linear modes with the same frequencies: (l, m, n) = (4, ±2, 0) scalar mode: ω = l n = 5, ω a = 5 2 (l, m, n) = (3, ±2, 0) vector mode: ω = l n = 5, ω a = 5 2 none of them generates a one-parameter family of solutions G. Fodor: AdS geons 38/41

39 Combining three modes at linear order (2, ±2, 1) with amplitude α (4, ±2, 0) with amplitude β (3, ±2, 0) with amplitude γ We get three conditions at ε 3 order can be transformed to a 13-th degree polynomial, it has 3 real and 10 complex roots we have 3 one-parameter families of solutions The ratios of the amplitudes are β α γ α family I family II family III G. Fodor: AdS geons 39/41

40 Angular frequency angular momentum ωl/m I 2.25 II III J AMD /L 2 G. Fodor: AdS geons 40/41

41 Outlook things to do Investigate non-rotating AdS geons with or without axial symmetry analytically and numerically Improve numerical method to reach maximal mass geons Study the stability of geons dimensional time-evolution code Construction of asymptotically flat geons G. Fodor: AdS geons 41/41