Huygens’ principle and anomalously small radiation tails Piotr Bizon´ M. Smoluchowski Institute of Physics, Jagiellonian University, Krak´ow, Poland (Dated: February 2, 2008) 8 Abstract 0 0 2 This is a short account of recent joint work with T. Chmaj and A. Rostworowski on asymptotic n behavior oflinear andnonlinear waves, as presentedat theconference devoted to MyronMathisson a J held at the Banach Center, Warsaw 2007. 4 1 ] h p - h t a m [ 1 v 3 1 0 2 . 1 0 8 0 : v i X r a 1 I. INTRODUCTION The work presented in this talk is part of a long-time project aimed at the detailed quan- titative description of the process of relaxation to equilibrium for nonlinear wave equations defined on spatially unbounded manifolds. By equilibrium we mean here a stable stationary solution, like a soliton, a black hole, or just a flat space. The convergence to these solutions occurs through a mechanism of radiating an excess energy to infinity. For a large class of physically interesting systems the late stages of this process are universal: for intermediate times the convergence has the form of exponentially damped oscillations (called quasinormal modes) and asymptotically it has the form of polynomial decay (called a tail). This very last stage of the relaxation process, the tail, is the subject of my talk. The presentation of this talk at the conference devoted to Myron Mathisson is justified by the fact our results shed new light on Huygens’ principle, one of the main subjects of Mathisson’s mathematical interests. Recall that a wave equation is said to satisfy Huygens’ principle if: (i) the solution at a point P depends only on the initial data at the intersection of the past light cone of P with the Cauchy hypersurface or, equivalently, (ii) the solution vanishes at all points which cannot be reached from the initial data by a null geodesic (i.e., there is no tail). A prototype equation satisfying Huygens’ principle is the ordinary wave equation in d + 1 dimensional Minkowski spacetime for odd d 3. Actually, according ≥ to Hadamard’s conjecture [1] this is the only (modulo trivial transformations) huygensian linear second-order hyperbolic equation of the form gµν(x) φ+Aµ(x) φ+B(x)φ = 0. (1) µ ν µ ∇ ∇ ∇ Mathisson proved this conjecture in the case of four dimensional Minkowski spacetime [2]. Counterexamples to Hadamard’s conjecture, which have been found later (see [3] and Roy McLenaghan’s talk at this conference), do not change the fact that Huygens’ property is a very rare and unstable phenomenon. Thus, it is natural to ask if there are perturbations of the free wave equation which preserve Huygens’ property approximately, in the sense that the tail which is left behind the wave front is very small. The existence of such special perturbations in higher even dimensions is a byproduct of our studies of tails. 2 II. MODEL AND ASSUMPTIONS We consider equations of the form (cid:3)φ+V(x)φ+N(φ, φ,x) = 0, (cid:3) = ∂2 ∆, (t,x) Rd+1, (2) ∇ t − ∈ for spherically symmetric smooth initial data with compact support. Since we want the free part to satisfy Huygens’ property, we restrict ourselves to odd spatial dimension d 3. ≥ Apart from obvious mathematical motivations, there are at least two physical reasons for studying higher dimensions d > 3. First, for linear wave equations higher dimensions are equivalent to higher spherical harmonics. This follows from the identity d 1 1 2 l(l+1) ∂2 ∂2 − ∂ φ = ∂2 ∂2 ∂ + (rlφ), l = (d 3)/2, (3) (cid:18) t − r − r r(cid:19) rl (cid:18) t − r − r r r2 (cid:19) − which relates the l = 0 radial wave operator in d space dimensions with the radial wave operator in three space dimensions for the lth spherical harmonic with l = (d 3)/2. − Second, some geometric wave equations in 3+1 dimensions are equivalent to scalar wave equations in d + 1 dimensions for d > 3. For example, equivariant wave maps from R3+1 into S3 satisfy the following equation 2 sin(2ψ) ∂2 ∂2 ∂ ψ + = 0, (4) (cid:18) t − r − r r(cid:19) r2 which, after substitution ψ = rφ, becomes the nonlinear scalar wave equation in 5 + 1 dimensions 4 4 ∂2 ∂2 ∂ φ+ φ3 +higher order terms = 0. (5) (cid:18) t − r − r r(cid:19) 3 III. TOOLS In this section we recall two elementary tools from the theory of linear wave equations. The first tool is a formula for the spherically symmetric solution of the free wave equation d 3 (cid:3)φ = 0 in d+1 dimensions (hereafter we shall use l = − instead of d): 2 1 l 2k−l(2l k)! φ(t,r) = − rk a(k)(t r) ( 1)ka(k)(t+r) . (6) r2l+1 k!(l k)! − − − Xk=0 − (cid:0) (cid:1) This solution, which is a superposition of ingoing and outgoing waves, is parameterized by a single function a(r) uniquely determined by initial data (the superscript in round brackets denotes the k-th derivative). 3 The second tool is the Duhamel formula for the solution of the inhomogeneous free wave equation (cid:3)φ = F(t,r) with zero data t t+r−τ 1 r2 +ρ2 (t τ)2 φ(t,r) = dτ ρl+1P (µ) F(τ,ρ)dρ, µ = − − , (7) 2rl+1 Z Z l 2rρ 0 |t−r−τ| where P (µ) is the Legendre polynomial of degree l This expression can be easily obtained l from the standard Green’s function formula by integrating out the angular variables [4]. In terms of null coordinates u = τ ρ and v = τ +ρ the Duhamel formula (7) takes a more − convenient form t+r t−r 1 r2 +(v t)(t u) φ(t,r) = dv F(u,v)(v u)l+1P (µ)du, µ = − − . (8) 2l+3rl+1 Z Z − l r(v u) − |t−r| −v The formulae (6) and (8) will be used repeatedly below. IV. LINEAR TAILS For the clarity of presentation we first consider the linear equation with a potential (cid:3)φ+λVφ = 0, (φ(0,r),∂ φ(0,r)) = (f(r),g(r)). (9) t The prefactor λ, introduced for convenience, will be assumed small and used as a pertur- bation parameter. In order to determine the long-time behavior of φ(t,r) we define the perturbation series φ = φ +λφ +λ2φ +..., (10) 0 1 2 where φ satisfies initial data (9) and all higher φ have zero data. Substituting this series 0 n into equation (9) we get the iterative scheme (cid:3)φ = 0, (cid:3)φ = Vφ , (cid:3)φ = Vφ , etc. (11) 0 1 0 2 1 − − which can be solved recursively using the formulae (6) and (8). Assuming that V(r) r−α ∼ (α > 2) for r , we showed in [5] that the leading order asymptotic behavior at timelike → ∞ infinity (fixed r and t ) is given by → ∞ C(l,α) 1 φ (t,r) = A+ , (12) 1 tα+2l (cid:20) O(cid:18)t(cid:19)(cid:21) 4 where1 +∞ 2α+2l−1 α 3 l α l C(l,α) = − and A = a(u)du. (13) −(2l+1)!! (cid:18) 2 (cid:19) (cid:16)2(cid:17) Z −∞ The constant A is the only trace of initial data. The expression (12) was first derived by Ching et al. [6] who used Fourier transform methods. We claim that the first iterate provides a good approximation of the entire tail if λ is sufficiently small, that is φ(t,r) λφ (t,r) (λ2)t−(α+2l). (14) 1 − ∼ O This basically follows from the fact that all higher-order iterates φ (t,r) decay in the same n manner (or faster) as φ (t,r). Of course, the main issue is whether the perturbation series is 1 convergent; for l = 0 (i.e., in three space dimensions) this was proved in [7]) but for higher l the problem is open. Note however that for practical purposes it is sufficient that the series is asymptotic to the solution. The numerical verification of (14) shows perfect agreement with analytic predictions [5]. We remark that numerical simulations of tails are not quite trivial even in the radial case because discretization errors generate artificial tails which might mask the true behavior. To eliminate such artifacts one has to use high-order finite difference schemes. In addition, quadruple precision is needed to suppress the accumulation of round-off errors during long- time simulations. For these reasons the simulations of tails are computationally expensive. V. NONLINEAR TAILS Inthissection we consider equation(2) without apotential. Forsimplicity, wetake apure power nonlinearity with aninteger exponent p 3 (the generalization to other nonlinearities ≥ is straightforward) (cid:3)φ φp = 0, (φ(0,r),∂ φ(0,r)) = (εf(r),εg(r)). (15) t − 1 Here we use the notation: x0 :=1, xk :=x (x 1) (x (k 1)), k >0, · − ····· − − x0 :=1, xk :=x (x+1) (x+(k 1)), k >0. · ····· − 5 This time the amplitude of initial data ε plays the role of a small parameter in the pertur- bation series: φ = εφ +ε2φ +ε3φ +... (16) 0 1 2 Substituting (16) into equation (15) we get the iteration scheme (cid:3)φ = 0, (cid:3)φ = φp, etc. (17) 0 p 0 As above, we get φ using the formula (6) and then evaluate φ using the Duhamel formula 0 p (8). In the limit of timelike infinity we obtain [8] C˜(l,p) 1 φ (t,r) = A˜+ , (18) p t(l+1)p−1 (cid:20) O(cid:18)t(cid:19)(cid:21) where +∞ 2(l+1)(p+1)−1 C˜(l,p) = ( 1)l [(l+1)(p 1) 2]l and A˜ = [a(l)(u)]pdu. (19) − (2l+1)!! − − Z −∞ The remarks given above in the linear case apply verbatim to the nonlinear case as well; in particular the perturbation series (16) is known to converge for l = 0 [7] and is at least asymptotic to the full solution for l > 0. VI. COMPETITION BETWEEN LINEAR AND NONLINEAR TAILS The most interesting situation occurs when both the potential and the nonlinearity are present in equation (2). Then, each of these terms produces its own tail: linear tail t−(α+2l) vs. nonlinear tail t1−(l+1)p. (20) ∼ ∼ Clearly, the tail with slower decay rate is dominant asymptotically, that is φ(t,r) t−γ, γ = min α+2l,(l+1)p 1 . (21) ∼ { − } For l = 0 this result (without a coefficient) was first proved by Strauss and Tsutaya [9]. To give an example of the competition of linear and nonlinear tails, let us consider the Skyrme model. Under a spherical symmetry reduction (corotational ansatz), this model reduces to the single nonlinear wave equation for the function F(t,r) sin2F ∂ (w∂ F) ∂ (w∂ F)+sin(2F)+sin(2F) +(∂ F)2 (∂ F)2 = 0, (22) t t r r r t − (cid:18) r2 − (cid:19) 6 where w = r2 +2sin2F. Regular finite energy solutions of (22) must satisfy the boundary conditions F(t,0) = 0 and F(t, ) = mπ, where an integer m has the interpretation of the ∞ topological degree of the solution. For m = 1, equation (22) has a regular static solution S(r) called the skyrmion. This solution is linearly stable and plays the role of a global attractor, that is, every solution starting from smooth finite energy initial data of degree one remainsgloballyregularforalltimesandasymptoticallyconvergestoS(r). Theperturbation φ(t,r) = √w(F(t,r) S(r))/r2 satisfies the equation − 4 4 ∂2 ∂2 ∂ φ+V(r)φ+ φ3 +higher order terms = 0, (23) (cid:18) t − r − r r(cid:19) 3 where the potential V(r) has no bound states and falls off as r−6 for r . For this → ∞ equation we have d = 5(l = 1), α = 6 and p = 3, hence from (20) linear tail t−8 and nonlinear tail t−5. (24) ∼ ∼ Thus, the nonlinear tail is dominant [10]. This example shows that one has to be cautious in drawing conclusions about the asymptotic behavior of solutions of nonlinear wave equations onthebasisoflinear perturbationanalysis –evenforsmallamplitudesolutionsthenonlinear effects can be dominant. VII. ANOMALOUS TAILS An advantage of our approach, in contrast to decay estimates in the form of inequalities, is that we control the coefficient of the leading order term of the tail. This allows us to identify those exceptional cases in which this coefficient vanishes and the decay is faster. We shall refer to such tails as anomalous. Let us first consider the linear case with the pure inverse power potential near infinity, l α 3 that is V(r) = λr−α for r > R. Then, it follows from (13) that C(l,α) − = 0 if ∝ (cid:18) 2 (cid:19) α is an odd integer 2l +1, hence there is no tail in the first order. This means that the ≤ system (cid:3)φ = 0, (cid:3)φ = Vφ (25) 0 1 0 − is huygensian. In order to find the tail in this exceptional case we need to solve the second iteration equation (cid:3)φ = Vφ via the Duhamel formula. After a long calculation (which 2 1 − 7 requires the asymptotic expansion of φ at null infinity) we get (see [5] for the details) 1 D(l,α) 1 φ(t,r) λ2φ (t,r) = λ2 A+ , (26) ≈ 2 t2(α+l−1) (cid:20) O(cid:18)t(cid:19)(cid:21) where the coefficient D(l,α) is given by a complicated but explicit expression (see Eq.(23) in [5]). Next, consider the pure power nonlinearity (cid:3)φ = φp. It follows from (19) that C˜(l,p) ∝ [(l+1)(p 1) 2]l = 0 if p = 2 and l 1. Thus, in higher even dimensions the first order − − ≥ tail vanishes for the quadratic nonlinearity. This implies that the system (cid:3)φ = 0, (cid:3)φ = φ2 (27) 0 1 0 is huygensian. As before, the leading order behavior of the tail can be obtained by solving the second order equation (cid:3)φ = 2φ φ via the Duhamel formula. The result (see [8] for 2 0 1 the details) is c(l) 23l ∞ φ(t,r) ε3φ (t,r) ε3 , c = ( 1)l a(l−1)(η)[a(l)(η)]2dη. (28) ≈ 2 ∼ t3l+1 − 2l(2l+1) Z −∞ Note that quadratic nonlinearities occur frequently in nonlinear perturbation theory so anomalous tails are in fact quite common. As an example, consider the Yang-Mills field in four dimensions with the SO(3) gauge group, so that the potential A (x) is the skew- α symmetric 3 3 matrix Aij(x). For the spherically-symmetric ansatz × α Aij(x) = δjxi δixj φ(t,r), (29) µ µ − µ (cid:0) (cid:1) the Yang-Mills equation ∂ Fαβ + [A ,Fαβ] = 0, where F = ∂ A ∂ A + [A ,A ], α α αβ α β β α α β − reduces to the scalar semilinear wave equation in 5+1 dimensions 4 (∂2 ∂2 ∂ )φ+3φ2 +r2φ3 = 0. (30) t − r − r r The quadratic term in (30) produces an anomalous tail (28) which is of the same order, (ε3), as the standard tail (18) produced by the cubic term. Combined together they give O (see [11] for the derivation) +∞ φ(t,r) ε3φ (t,r) ε3ct−4, c = 8 a(u)a′(u)2. (31) 2 ≈ ∼ − Z −∞ 8 Finally, we remark that although our analysis of tails was restricted to the flat back- ground, many conclusions carry over to more general asymptotically flat spacetimes, in par- ticular black hole spacetimes. For example, applying the ideas of section VII one can show that the massless scalar field propagating outside a higher even-dimensional Schwarzschild black hole decays anomalously fast as φ t−(3d−5) [5]. This suggests that the problem of ∼ asymptotic stability of the Schwarzschild black hole is easier in higher dimensions. Acknowledgment: I would like to thank Professor Andrzej Trautman for inviting me to give this talk at the Banach Center. This research was supported in part by the MNII grants 1PO3B01229 and SPB/189/6 PR EU/2007/7. [1] R. Courant and D. Hilbert, Methods of mathematical physics, vol.2, Interscience, New York, 1962. [2] M. Mathisson, Acta Math. 71, 249 (1939). [3] P. Gu¨nther, Huygens’ principle and hyperbolic differential equations, Academic Press, San Diego, 1988. [4] H. Lindblad and C. D. Sogge, Amer. J. Math. 118, 1047 (1996). [5] P. Bizon´, T. Chmaj, and A. Rostworowski, Phys. Rev. D76, 124033 (2007). [6] E. S. C. Ching et al., Phys. Rev. D52, 2118 (1995). [7] N. Szpak, P. Bizon´, T. Chmaj, A. Rostworowski, arXiv:math-ph/0712.0493 [8] P. Bizon´, T. Chmaj, and A. Rostworowski, in preparation [9] W. Strauss and K. Tsutaya, Discrete Cont. Dyn. Sys. 3, 175 (1997). [10] P. Bizon´, T. Chmaj, A. Rostworowski, Phys. Rev. D75, 121702(R) (2007). [11] P. Bizon´, T. Chmaj, and A. Rostworowski, Class. Quantum. Grav. 24, F55 (2007). 9