TTHE φ KINK ON A WORMHOLE SPACETIME
ALICE WATERHOUSE
Abstract.
The soliton resolution conjecture states that solutions to solitonic equationswith generic initial data should, after some non–linear behaviour, eventually resolve intoa finite number of solitons plus a radiative term. This conjecture is intimately tied tosoliton stability, which has been investigated for a number of solitonic equations, includingthat of φ theory on R , . We study a modification of this theory on a 3 + 1 dimensionalwormhole spacetime which has a spherical throat of radius a , with a focus on the stabilityproperties of the modified kink. In particular, we prove that the modified kink is linearlystable, and compare its discrete spectrum to that of the φ kink on R , . We also study theresonant coupling between the discrete modes and the continuous spectrum for small butnon–linear perturbations. Some numerical and analytical evidence for asymptotic stabilityis presented for the range of a where the kink has exactly one discrete mode. Introduction: the φ kink on R , One dimensional φ theory is well–documented in the literature (see for example [9]).The aim of this section is to introduce some notation and some ideas about stability whichwill be useful when we come to consider the modified theory.The action takes the form S = (cid:90) R (cid:18) η ab ∂ a φ∂ b φ + 12 (1 − φ ) (cid:19) dx, where x a = ( t, x ) are coordinates on R , and η ab is the Minkowski metric with signature( − , +). Note that the potential has two vacua, given by φ = ±
1. Finiteness of the associatedconserved energy E = (cid:90) R (cid:18)
12 ( φ t ) + 12 ( φ x ) + 12 (1 − φ ) (cid:19) dx, requires that the field lies in one of these two vacua in the limits φ ± = lim x →±∞ [ φ ( x )]. Wecan thus classify finite energy solutions in terms of their topological charge N = ( φ + − φ − ) / {− , , } .The equations of motion are φ tt = φ xx + 2 φ (1 − φ ) (1.1)and we find a static solution φ = tanh( x − c ) which we call the flat kink. It interpolatesbetween the two vacua and thus has topological charge N = 1. The constant of integration c can be thought of as the position of the kink. We will henceforth use Φ to denote the statickink at the origin, that is, Φ ( x ) = tanh( x ). It is evident that no finite energy deformationcan affect N . For this reason, we say that the kink is topologically stable . Date : August 27, 2019. a r X i v : . [ g r- q c ] A ug ALICE WATERHOUSE
Linear Stability.
A second notion of stability which will be important to our dis-cussion is linear stability. On discarding non–linear terms, we find that small pertubations φ ( t, x ) = Φ ( x ) + e iωt v ( x ) satisfy the Schr¨odinger equation L v := − v (cid:48)(cid:48) − − ) v = ω v . (1.2)The potential V ( x ) = − − ( x ) ] exhibits a so–called “mass gap”, meaning that ittakes a finite positive value in the limits x → ±∞ . In this case, V ( ±∞ ) = 4. For ω > (cid:0) v ( x ) , ω (cid:1) = (cid:18) √
32 sech ( x ) , (cid:19) and (cid:0) v ( x ) , ω (cid:1) = (cid:18) √ √ x )tanh( x ) , √ (cid:19) , (1.3)where we have chosen the normalisation constant such that (cid:82) ∞−∞ v ( x ) dx = 1.The first of these is the zero mode of the kink. Its existence is guaranteed by the transla-tion invariance of (1.1), and up to a multiplicative constant it is equal to Φ (cid:48) ( x ). Excitationof this state corresponds to performing a Lorentz boost. In the non–relativistic limit, thisamounts to replacing Φ ( x ) with Φ ( x − vt ) for some v (cid:28) internal mode has non–zero frequency ω ,and is thus time periodic. In the full non–linear theory, it decays through resonant couplingto the continuous spectrum [8]. This phenomenon is of considerable interest in non–linearPDEs, and was studied in a more general setting in [12]. The corresponding process in themodified theory will be discussed in section 4.Linear stability of the kink is equivalent to the Schr¨odinger operator L in (1.2) havingno negative eigenvalues, so that linearised perturbations cannot grow exponentially withtime. One way to see that the kink is linearly stable is via the Sturm oscillation theorem: Theorem 1.1 (Sturm) . Let L be a differential operator of the form L = − d dx + V ( x ) on the smooth, square integrable functions u on the interval [0 , ∞ ) , with the boundary con-dition u (0) = 0 (corresponding to even parity) or u (cid:48) (0) = 0 (corresponding to odd parity).Let ω be an eigenvalue of L with associated eigenfunction u ( x ; ω ) . Then the number ofeigenvalues of L (subject to the appropriate boundary conditions) which are strictly below ω is the number of zeros of u ( x ; ω ) in (0 , ∞ ) . Note that the symmetry of (1.2) under x (cid:55)→ − x means that any solution on the interval[0 , ∞ ) has a corresponding solution on the interval ( −∞ , −∞ , ∞ ) as long as the boundary conditionsat x = 0 are chosen to ensure parity ±
1. Thus there is a one–to–one correspondencebetween solutions on [0 , ∞ ) and solutions on ( −∞ , ∞ ) which are smooth at x = 0. Sincethe eigenfunctions (1.3) have no zeros on the interval [0 , ∞ ), it follows that there can be noeigenfunctions with ω < ω = 0, and thus the kink is linearly stable. HE φ KINK ON A WORMHOLE SPACETIME 3
Asymptotic stability.
The final notion of stability that we will consider is thatof asymptotic stability. Stated simply, asymptotic stability of the kink means that forsufficiently small initial pertubations, solutions of (1.1) will converge locally to Φ ( r ) or itsLorentz boosted counterpart. This was proved in [7] for odd perturbations, but has notbeen proved in the general case.Generalisation of the finite energy φ kink to higher dimensional Minkowski spacetimesis prohibited by a scaling argument due to Derrick [4]. In order to construct a higherdimensional φ kink, we must add curvature. In the next section we introduce a curvedbackground, and show that a modified φ kink exists on this background. We will alsoexamine a limit in which the modified kink reduces to the flat kink. In section 3 weconsider linearised perturbations around the modified kink, proving that it is linearly stableand comparing its discrete spectrum to that of the flat kink. In section 4 we examine themode of decay to the modified kink in the full non–linear theory, in particular the resonantcoupling of its internal modes to the continuous spectrum.2. The static kink on a wormhole
We now replace the flat R , background with a wormhole spacetime ( M, g ), where g = − dt + dr + ( r + a )( dϑ + sin ϑdϕ )for some constant a >
0, and −∞ < r < ∞ . This spacetime was first studied by Ellis [5]and Bronnikov [3], and has featured in a number of recent studies about kinks and theirstability [1, 2]. Note the presence of asymptotically flat ends as r → ±∞ , connected by aspherical throat of radius a at r = 0.Our action is the modified by the presence of a non-flat metric: S = (cid:90) (cid:18) g ab ∂ a φ∂ b φ + 12 (1 − φ ) (cid:19) √− gdx, where x a are now local coordinates on M . Variation with respect to φ gives (cid:3) g φ + 2 φ (1 − φ ) = 0 (2.1)where (cid:3) g φ = √− g ∂ a ( g ab √− g∂ b φ ). We assume φ is independent of the angular coordinates( ϑ, ϕ ), so (2.1) can be written explicitly as φ tt = φ rr + 2 rr + a φ r + 2 φ (1 − φ ) . (2.2)The conserved energy in the theory is given by E = (cid:90) + ∞−∞ (cid:18)
12 ( φ t ) + 12 ( φ r ) + 12 (1 − φ ) (cid:19) ( r + a ) dr, which we require to be finite. This imposes the condition φ → r → ±∞ , so that thefield lies at one of the two vacua at both asymptotically flat ends.Static solutions φ ( r ) satisfy φ (cid:48)(cid:48) + 2 rr + a φ (cid:48) = − ddφ (cid:18) −
12 (1 − φ ) (cid:19) , (2.3) ALICE WATERHOUSE which, can be thought of as a Newtonian equation of motion for a particle at position φ moving in a potential U ( φ ) = − (1 − φ ) /
2, with a time dependent friction term.In addition to the two vacuum solutions, we have a single soliton solution which interpo-lates between the saddle points at ( − ,
0) and (1 ,
0) in the in the ( φ, φ (cid:48) ) plane. Its existenceand uniqueness among odd parity solutions follow from a shooting argument: suppose theparticle lies at φ = 0 when r = 0. If its velocity φ (cid:48) (0) is too small, it will never reachthe local maximum of the potential at φ = 1, but if φ (cid:48) (0) is too large it will overshoot themaximum so that U ( φ ) → −∞ as r → ∞ , thus having infinite energy. Continuity ensuresthat there is some critical velocity φ (cid:48) (0) such that the particle reaches φ = 1 in infinitetime and has zero velocity upon arrival. This corresponds to the non–trivial kink solution,which we call Φ( r ). Time reversal implies that φ → − r → −∞ , and that the anti–kink φ ( r ) = − Φ( r ) is also a solution.We can find Φ( r ) numerically using a shooting method for the gradient at r = 0. Figure1 shows such numerically generated kinks for several values of a . Note that the absolutevalue of Φ( r ) is always greater than or equal to that of the flat kink Φ ( r ), and that atfixed non–zero r , the absolute value of Φ decreases as a increases. The reason for this willbecome clear in section 3. In section 2.1 we examine Φ( r ) in the limit where a is large,finding that it reduces to the flat kink Φ ( r ), and examining its departure from the flat kinkat first order in 1 /a .We again label the values at the boundary as φ ± := lim r →±∞ Φ( r ) ∈ {± } . Since no finite energy deformation can change the value of the topological charge N =( φ + − φ − ) / ∈ {− , , } , we again conclude that Φ( r ) is topologically stable.2.1. Large a limit. As a → ∞ , equation (2.2) becomes the standard equation (1.1) forthe flat kink. It is thus helpful to expand the modified kink in (cid:15) := 1 /a for small (cid:15) , sincewe can then solve both (2.2) and (3.2) analytically up to O ( (cid:15) ). We shall denote the statickink by Φ (cid:15) ( r ) in this limit. It satisfiesΦ (cid:48)(cid:48) (cid:15) + 2 r(cid:15) (cid:15) r + 1 Φ (cid:48) (cid:15) = − (cid:15) (1 − Φ (cid:15) ) . (2.4)Setting Φ (cid:15) ( r ) = Φ ( r ) + (cid:15) Φ ( r ) + O ( (cid:15) ) we obtain at order zero the equation (1.1) of astatic kink on R , . This has solution Φ ( r ), where we have chosen the kink at the originto restrict to solutions with odd parity.At order (cid:15) we find that Φ ( r ) must satisfyΦ (cid:48)(cid:48) + 2 r sech r = 2Φ (2 − r ) . The unique solution which is odd and decays as r → ±∞ is given byΦ ( r ) = 124 sech r ( f ( r ) + f ( r ) + f ( r )) , HE φ KINK ON A WORMHOLE SPACETIME 5 − . − . − . − . . . . . r − . . . . Φ( r ) Φ ( r ) a = a = a = Figure 1.
The kink solution for several values of a , along with the flat kink Φ ( r ).where f ( r ) = r (cid:2) − r ) − cosh(4 r ) (cid:3) ,f ( r ) = sinh(2 r ) (cid:2) r )) − (cid:3) + sinh(4 r )log(2cosh( r )) ,f ( r ) = π r + 6Li ( − e − r ) , and Li ( z ) is the dilogarithm function, i.e. the special case s = 2 of the polylogarithmLi s ( z ) = ∞ (cid:88) k =1 z k k s . To show that Φ ( r ) is odd, note that sech r is an even function, and that f and f areconstructed from products of even and odd functions, and hence are odd. To see that f isalso odd, we use Landen identity for the dilogarithm:Li ( − e − r ) + Li ( − e r ) = − π − (cid:2) log(e − r ) (cid:3) = − π − r , thus verifying f ( r ) + f ( − r ) = 0. ALICE WATERHOUSE − − r − . − . − . . . . Φ ( r ) Figure 2.
The order (cid:15) perturbation to the static kink on R , .We now turn to the behaviour of Φ ( r ) as r → ∞ . Since sech r ∼ − r for large r , weneed only consider terms in the { f i } of order e r or higher. We first note thatlog(2cosh r ) = log(e r (1 + e − r )) = r + log(1 + e − r )= r + e − r + O (e − r ) . Then f ( r ) = − r e r − r r + O (e r ) f ( r ) = 12 e r (8 r + 8e − r −
1) + 12 e r ( r + e − r ) + O (e r )= 4 r e r + r r + O (e r ) , so f ( r ) + f ( r ) = O (e r ). Since f ( r ) = O ( r ) for large r , we see that Φ ( r ) vanishes as r → ∞ , as we expect. Note that its vanishing as r → −∞ then follows using parity. A plotof Φ ( r ) is shown in figure 2.3. Linearised perturbations around the kink
To study the linear stability of the kink, we first plug φ ( t, r ) = Φ( r ) + w ( t, r ) (3.1) HE φ KINK ON A WORMHOLE SPACETIME 7 r V ( r ) a = a = 1 / p a = a = a = Figure 3.
The potential of the 1–dimensional quantum mechanics problemarising from the study of stability of the soliton for values of a between a = 10 and a = 0 .
3. In particular, note that those with a < / √ w . Imposing the fact that Φ( r ) satisfies(2.3), we find w tt = w rr + 2 rr + a w r + 2 w (1 − ) . For w ( t, r ) = e iωt ( r + a ) − / v ( r ), this becomes a one–dimensional Schr¨odinger equation Lv := ( − ∂ r ∂ r + V ( r )) v = ω v, (3.2)where the potential is given by V ( r ) = a ( r + a ) − − ) . (3.3)Figure 3 shows the potential V ( r ) for several values of a . Note that for large a it hasa single well with a minimum at r = 0, close to the potential V corresponding to the flatkink. As a decreases, the critical point at r = 0 becomes a maximum with minima on eitherside, creating a double well. We find numerically that this happens at about a = 0 . Proposition 3.1.
The kink solution Φ( r ) is linearly stable. ALICE WATERHOUSE
Proof.
We first decompose the potential V ( r ) in (3.2) as V = V + V + V a , where V = − − ( r ) ] , V = 6[Φ( r ) − Φ ( r ) ] , V a = a ( r + a ) . As discussed above, we know that the operator L = − ∂ r ∂ r + V has no negative eigenvalues.It then follows that L itself has no negative eigenvalues as long as the functions V ( r ) and V a ( r ) are everywhere non–negative.The latter is obvious; to prove the former we recall that we can think of Φ( r ) and Φ ( r )as the trajectories of particles moving in a potential U ( φ ), where r is imagined as thetime coordinate. The particle corresponding to Φ( r ) suffers an increased frictional forcecompared to Φ ( r ), i.e.Φ (cid:48)(cid:48) = − ∂ U ∂φ (cid:12)(cid:12)(cid:12)(cid:12) φ =Φ , Φ (cid:48)(cid:48) + 2 rr + a Φ (cid:48) = − ∂ U ∂φ (cid:12)(cid:12)(cid:12)(cid:12) φ =Φ . (3.4)Both Φ and Φ interpolate between the maxima of U at φ = ±
1; reaching the minimum( φ = 0) when r = 0.Multiplying the equations (3.4) by Φ (cid:48) and Φ (cid:48) respectively, then integrating from r to ∞ ,we have that at every instant of time12 (Φ (cid:48) ) + U (Φ ) = 0 ,
12 (Φ (cid:48) ) + U (Φ) = (cid:90) ∞ r rr + a (Φ (cid:48) ) dr. (3.5)These equations are equivalent to conservation of energy for each of the particles. Notethat the integral on the RHS is non–negative for r ≥
0, and vanishes only at r = ∞ . Inparticular, when r = 0 we have U (Φ) = U (Φ ) = − /
2, so Φ (cid:48) (0) > Φ (cid:48) (0). This means V ( r )is initially increasing from zero.For V ( r ) to return to zero at some finite r = r , we would need that Φ( r ) = Φ ( r ) at apoint where Φ (cid:48) ( r ) ≤ Φ (cid:48) ( r ). However, this is made impossible by equations (3.5), since atsuch a point U (Φ) = U (Φ ) and the integral on the RHS is positive. Hence V ( r ) remainsnon–negative for all r >
0, and thus for all r since it is even in r . (cid:3) Finding internal modes numerically.
Bound states of the potential (3.3) corre-spond to internal modes of the kink like the odd solution of (1.2) in (1.3). In contrast,for frequencies greater than ω = 2, solutions to (3.2) are interpreted as radiation. It ispossible to search for bound states of (3.3) numerically by setting v (0) = 1 , v (cid:48) (0) = 0 or v (0) = 0 , v (cid:48) (0) = 1 depending on the required parity and then using a bisection method tosearch for the value of ω for which v ( r ) goes to zero at the boundaries. This procedurewill only be effective within the range of r for which Φ( r ) is calculated.For large a , the potential has both an even and an odd bound state which look quali-tatively similar to the internal modes (1.3) of the φ kink on R , . The bound states forseveral values of a can be found in figures 4 and 5. As a decreases, the eigenvalues ω of thebound states increase, until they disappear into the continuous spectrum ( ω > a in figure 6. HE φ KINK ON A WORMHOLE SPACETIME 9 − − − − r − . . . . Φ( r ) √ v ( r ) / a = a = Figure 4.
Even bound states of the potential V ( r ) and of the potential V ( r ) for two different values of a .3.2. Large a limit. We can also perturbatively expand the eigenvalues of the eigenvalueproblem (3.2). Consider solutions to (2.2) of the form φ (cid:15) ( r ) = Φ (cid:15) ( r ) + e iωt v (cid:15) ( r ), where v (cid:15) is small. These satisfy v (cid:48)(cid:48) (cid:15) + 2 r(cid:15) (cid:15) r + 1 v (cid:48) (cid:15) + 2(1 − (cid:15) ) v (cid:15) = − ω (cid:15) v (cid:15) . (3.6)Let ( v (cid:15) , ω (cid:15) ) be a solution to (3.6) with ω (cid:15) = ω + (cid:15) ξ + O ( (cid:15) ) and v (cid:15) ( r ) = v ( r ) + (cid:15) v ( r ) + O ( (cid:15) ) . Our aim will be to find ξ . Substituting into (3.6), at zero order we obtain the equation(1.2) which controls the linear stability analysis of the φ kink on R , .The terms of order (cid:15) in (3.6) give us v (cid:48)(cid:48) + 2 rv (cid:48) + 2(1 − ) v − Φ v = − ω v − ξv . (3.7) Note that in section 3 we considered perturbations v ( r ) which differ from v (cid:15) ( r ) by a factor of ( r + a ) − / ,since such perturbations are described by a Schr¨odinger problem. Here it will be simpler to remove thisfactor; however there is a one–to–one correspondence between v ( r ) and v (cid:15) ( r ). − − − − r − . − . − . . . . . . Φ( r ) √ v ( r ) / a = a = Figure 5.
Odd bound states of the potential V ( r ) and of the potential V ( r ) for two different values of a .We multiply equation (3.7) by v , and subtract from this v multiplied by equation (1.2).Integrating the result from r = −∞ to r = ∞ , we find (cid:90) ∞−∞ (cid:0) v (cid:48)(cid:48) v − v (cid:48)(cid:48) v (cid:1) dr + (cid:90) ∞−∞ rv (cid:48) v dr − (cid:90) ∞−∞ Φ Φ v dr = − ξ. In the first term the integrand is a total derivative, and the second term is easily found tobe − ξ = 1 + 12 (cid:90) ∞−∞ Φ Φ v dr, (3.8)which we can evaluate for each of the solutions (1.3) using the symbolic computation facilityin Mathematica. We find ξ = 2 in the case of the zero mode and ξ = π − v, ω ) numericallyfor a range of small values of (cid:15) and comparing ω to the ω + ξ(cid:15) predicted here. Thecorresponding plots are shown in figures 7 and 8.3.3. Critical values of a . It is interesting to investigate the values of a at which theinternal modes disappear into the continuous spectrum. The larger of these, at which theodd internal mode disappears, we shall call a . The smaller one, at which the even internalmode disappears, we shall call a . HE φ KINK ON A WORMHOLE SPACETIME 11 a √ ω Figure 6.
The frequencies of the internal modes of the kink plotted againstthe wormhole radius a . The choice of axis ticks will be motivated in section4.The most convenient method of estimating a and a is based on the Sturm OscillationTheorem 1.1. The points at which the even and odd internal modes disappear into thecontinuous spectrum are the points at which the zeros of the even and odd eigenfunctions of L with ω = 4 disappear. We can thus examine the number of zeros of the odd eigenfunctionwith ω = 4 to determine the number of odd bound states with ω <
4. The critical value a which we are searching for can then be found using a bisection method. An equivalentmethod using even bound states will yield an estimate of a .One problem with this method is that we need the number of zeros in the interval (0 , ∞ ),and the shooting method we use to generate Φ( r ) and V ( r ) is only accurate up to a finitevalue of r . Since zeros of the eigenfunction with ω = 4 disappear at r = ∞ , this limits theaccuracy with which we can determine a and a .For the finite integration range which is accessible based on the shooting method, theodd state disappears at a ≈ . a ≈ . V ( r ) which decays sufficiently quickly at the bound-aries, the condition I := (cid:90) ∞−∞ V ( r ) dr < − ∂ r ∂ r + V ( r ) has at least one bound state. In fact,the condition I ≤ .
000 0 .
002 0 .
004 0 .
006 0 .
008 0 . (cid:15) . . . . . ω numericalpredicted: ω = 2 (cid:15) Figure 7.
A comparison of the predicted and numerical calculations for theenergy of the zero mode as a function of (cid:15) for small (cid:15) . The numerical cal-culations were executed by finding the even bound states and their energiesas described in section 3.1.potentials which have at least one bound state where (3.9) is not satisfied. It is interestingto investigate the disappearance of our ground state in this context.Note that V ( r ) must go to zero as r → ±∞ to ensure that the integral converges, meaningthat the relevant choice for us is V ( r ) = V ( r ) −
4. We then examine the value of this integralfor the critical value a = a when the ground state disappears. We find that I ≈ a ≈ . a atwhich I = 0; this also occurs at around a ≈ .
3. Thus our results would be consistent withthe conjecture that (3.2) has no bound states for
I >
Resonant Coupling of the Internal Modes to the Continuous Spectrum
We now move on to consider time dependent perturbations of the form φ ( t, r ) = Φ( r ) + ( r + a ) − / w ( t, r ) , where we consider non–linear terms in w ( t, r ). Substituting into (2.2) we find w tt = − Lw + f ( w ) , (4.1)where we have defined f ( w ) = − w Φ √ a + r − w a + r , (4.2) HE φ KINK ON A WORMHOLE SPACETIME 13 .
000 0 .
002 0 .
004 0 .
006 0 .
008 0 . (cid:15) . . . . . ω numericalpredicted: ω = 3 + ( π − (cid:15) Figure 8.
A comparison of the predicted and numerical calculations forthe energy of the odd vibrational mode as a function of (cid:15) for small (cid:15) . Thenumerical calculations were executed by finding the odd bound states andtheir energies as described in section 3.1.suppressing the dependence of f on r to simplify the notation. We will not have much needfor the expression for f other than to note that it contains terms which are quadratic andcubic in w .If a is large enough to allow internal modes, then these can only decay through resonantcoupling to the continuous spectrum of L . The analogous process of decay to the φ kinkon R , was discussed in [8], and the general theory was developed in [12]. In the followingsections we investigate this decay in the case of a single internal mode, before comparingour result with numerical data.4.1. Conjectured decay rate in the presence of a single internal mode.
In thissection we follow the analysis in [2]. Looking at figure 6, we note that for a ∈ (0 . , .
8) wehave spec L = { ω } ∩ [ m , ∞ ) , ω < m < ω (4.3)where m = 4. As above, we denote the unique normalised eigenfunction of L by v , so that Lv = ω v . We will use (cid:104)· , ·(cid:105) to denote the usual inner product on R .We decompose the perturbation as w ( t, r ) = α ( t ) v ( r ) + η ( t, r ) , (4.4) where v ( r ) refers to the single even internal mode of the kink and η is a superposition ofstates from the continuous spectrum of L . Where there is only one internal mode present,its frequency ω always lies in the upper half of the mass gap: 1 < ω <
2. This is importantbecause it means that 2 ω lies within the continuous spectrum.We substitute this into (4.1) and project onto and away from the internal mode direction,obtaining the following equations for α and η :¨ α + ω α = (cid:104) v, f ( αv + η ) (cid:105) (4.5)¨ η + Lη = P ⊥ f ( αv + η ) , (4.6)where P ⊥ is the projection onto the space of eigenstates of L which are orthogonal to v ,given by P ⊥ ψ = ψ − (cid:104) v, ψ (cid:105) v. (4.7)These equations have initial conditions α (0) and η (0 , r ) such that φ (0 , r ) = Φ( r ) + ( r + a ) − / ( α (0) v ( r ) + η (0 , r )) , and˙ φ (0 , r ) = ( r + a ) − / ( ˙ α (0) v ( r ) + ˙ η (0 , r )) . In the following analysis we investigate decay of α ( t ). Equation (4.5) has a homogeneoussolution consisting of oscillations with frequency ω . Since 2 ω lies within the continuousspectrum of L , there will be a resonant interaction between the these oscillations and theradiation modes in η with frequencies ± ω , arising from the term of order α in the RHSof (4.6). Thus, to leading order, (4.6) is a driven wave equation with driving frequency 2 ω .This resonant part of η will have a back–reaction on α through (4.5), which will result indecay of the internal mode oscillations.We now define α = α, ωα = ˙ α so that (4.5) becomes (cid:40) ˙ α = ωα , ˙ α = − ωα + ω (cid:104) v, f ( α v + η ) (cid:105) , or equivalently ˙ A = − iωA + iω (cid:28) v, f (cid:18)
12 ( A + ¯ A ) v + η (cid:19)(cid:29) , (4.8)where A = α + iα . Next we write η = η, η = ˙ η , converting (4.6) to (cid:40) ˙ η = η , ˙ η = − Lη + P ⊥ f (cid:0) ( A + ¯ A ) v + η (cid:1) . (4.9)We will regard the right hand sides of (4.8) and (4.9) as power series in A and η . Termswhich we expect to be higher order will not be treated rigorously; for this reason, our analysiswill produce only a conjecture about the decay rate. Numerical evidence concerning theconjecture will be discussed in section 4.2.It will be helpful to introduce the notation O p ( A, η ) to mean terms of at least order p in A, ¯ A, η , η , so that A , η and ¯ Aη are all examples of terms which are O ( A, η ). Currently, HE φ KINK ON A WORMHOLE SPACETIME 15 the coupling between (4.8) and (4.9) is O ( A, η ). We will write f (cid:18)
12 ( A + ¯ A ) v + η (cid:19) = (cid:88) k + l ≥ f kl A k ¯ A l + (cid:88) k + l ≥ n ≥ f kln η A k ¯ A l where k, l, n are non–negative, to elucidate the lowest order terms in (4.9). Note that f kl and f kln are decaying functions of r defined by (4.2). We can then write P ⊥ (cid:20) f (cid:18)
12 ( A + ¯ A ) v + η (cid:19)(cid:21) = (cid:88) k + l =2 P ⊥ [ f kl ] A k ¯ A l + (cid:88) k + l =1 P ⊥ [ f kl η ] A k ¯ A l + O ( A, η ) . Terms in (4.8) with imaginary coefficients correspond to rotation in the complex plane,and thus to oscillatory behaviour in α . At first order, A oscillates with frequency ω . Thisis exactly the behaviour expected in the linearised theory discussed in section 3. In fact, apriori, all the terms in the power series for ˙ A have coefficients which are purely imaginary.The next step in our analysis will be to attempt a change of variable η i (cid:55)→ ˜ η i in (4.9) sothat its right hand side is O ( A, ˜ η ), meaning ˜ η is O ( A ). It will turn out that the requiredchange of variables is complex. The result will be a term in (4.8) which is O ( A ) and has areal coefficient. This will be the lowest order term with a real coefficient, and thus the keyresonant damping term.We write the change of variables as η = ˜ η + (cid:88) k + l =2 b kl A k ¯ A l , η = ˜ η + (cid:88) k + l =2 c kl A k ¯ A l , (4.10)where b kl and c kl are functions of r which are so far undetermined. Differentiating withrespect to time and using (4.8), we find˙ η = ˙˜ η − iω (cid:88) k + l =2 b kl ( k − l ) A k ¯ A l + O ( A, ˜ η ) , ˙ η = ˙˜ η − iω (cid:88) k + l =2 c kl ( k − l ) A k ¯ A l + O ( A, ˜ η ) . We equate these to the right hand sides of (4.9), substituting from (4.10) and requiring that˙˜ η = ˜ η + O ( A, ˜ η ) , ˙˜ η = − L ˜ η + O ( A, ˜ η ) . (4.11)This yields − iωb kl ( k − l ) = c kl and − iωc kl ( k − l ) = − Lb kl + P ⊥ [ f kl ]for k + l = 2, where we have discarded (cid:88) k + l =1 P ⊥ [ f kl η ] A k ¯ A l = (cid:88) k + l =1 P ⊥ [ f kl ˜ η ] A k ¯ A l + (cid:88) k + l =1 p + q =2 P ⊥ [ f kl b pq ] A k + p ¯ A l + q = O ( A )because ˜ η is at least third order in A .The change of variables (4.10) is now given by the solution to (cid:0) L − ω ( k − l ) (cid:1) b kl = P ⊥ [ f kl ] . (4.12) Because of the spectrum of L given in (4.3), for ( k, l ) ∈ (2 , ∪ (0 ,
2) the solution b kl isin general a complex function of r , whilst for k = l = 1 the solution is real and decay-ing. The reason for this can be understood using the variation of parameters method forinhomogeneous ODEs.Let g ( r ) be such that (cid:104) g, g (cid:105) is finite, and λ ≥ L − λ ) b ( r ) = g ( r )is given by b ( r ) = Z ( r ) (cid:90) r −∞ W ( r (cid:48) ) Z ( r (cid:48) ) g ( r (cid:48) ) dr (cid:48) + Z ( r ) (cid:90) ∞ r W ( r (cid:48) ) Z ( r (cid:48) ) g ( r (cid:48) ) dr (cid:48) , where { Z , Z } is a basis for solutions to the homogeneous equation with Wronskian W ( r ) = Z Z (cid:48) − Z Z (cid:48) . The basis must be chosen so that the above integrals converge.For k = l = 1, so that λ = 0 and hence λ < m , we can choose a basis such that W = 1 and Z , Z are both real, and they decay to zero in the limits r → −∞ and r → ∞ respectively. Then b ( r ) = Z ( r ) (cid:90) r −∞ Z ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) + Z ( r ) (cid:90) ∞ r Z ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) . (4.13)For λ ≥ m , we cannot choose a real solution in general. In the case ( k, l ) ∈ (2 , ∪ (0 , { j ± } , defined by j ± ( r ) ∼ e ± iξr as r → ∞ , where ξ = √ ω − m . Their Wronskian is then W ( j + , j − ) = − iξ , and we write thesolution as b ( r ) = b ( r ) = ij − ( r )2 ξ (cid:90) r −∞ j + ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) + ij + ( r )2 ξ (cid:90) ∞ r j − ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) . (4.14)Finally, we use (4.13) and (4.14) to change variable η i (cid:55)→ ˜ η i in (4.8), obtaining˙ A = − iωA + iω (cid:32) (cid:88) ≤ k + l ≤ (cid:104) v, f kl (cid:105) A k ¯ A l + (cid:88) k + l =1 p + q =2 (cid:104) v, f kl b pq (cid:105) A k + p ¯ A l + q + O ( A ) (cid:33) , (4.15)where we have ignored terms containing ˜ η since these are at least fourth order in A . Wecan now see that, of the terms which we have written explicitly, the only ones that can givea real contribution to ˙ A are those containing b and b . We thus find ddt | A | = ˙ A ¯ A + A ˙¯ A = 2 R e[ ˙ A ¯ A ]= 2 ω R e (cid:34) i (cid:88) k + l =1 (cid:18) (cid:104) v, f kl b (cid:105) A k +2 ¯ A l +1 + (cid:104) v, f kl b (cid:105) A k ¯ A l +3 (cid:19)(cid:35) + O ( A )= − ω I m (cid:20) (cid:104) v, f b (cid:105) ( A ¯ A + A ¯ A + A ¯ A + ¯ A ) (cid:21) + O ( A ) . HE φ KINK ON A WORMHOLE SPACETIME 17
In particular, the term A ¯ A = | A | is real and non–oscillating, giving a contribution ddt | A | ∼ − ω I m (cid:2) (cid:104) v, f b (cid:105) (cid:3) | A | . The terms A ¯ A , A ¯ A and ¯ A , on the other hand, would be expected to oscillate at frequen-cies 2 ω and 4 ω at first order, and thus time average to zero.Hence we conclude | A | ∼ (cid:18) Γ t + 1 | A (0) | (cid:19) − / , Γ := 2 ω I m (cid:2) (cid:104) v, f b (cid:105) (cid:3) . (4.16)The constant Γ is a function of a which can be calculated explicitly. Using (4.2) and(4.14) gives (cid:104) v, f b (cid:105) = (cid:90) ∞−∞ dr v ( r ) f ( r ) (cid:18) ij − ( r )2 ξ (cid:90) r −∞ j + ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) + ij + ( r )2 ξ (cid:90) ∞ r j − ( r (cid:48) ) P ⊥ [ f ]( r (cid:48) ) dr (cid:48) (cid:19) . We now use the facts that f = vf /
4, and P ⊥ [ f ] = f − (cid:104) v, f (cid:105) v . Note that f is anodd function of r , so in fact (cid:104) v, f (cid:105) = 0 and so P ⊥ [ f ] = f . We thus obtain (cid:104) v, f b (cid:105) = 2 iξ (cid:18) (cid:90) ∞−∞ dr (cid:90) r −∞ dr (cid:48) f ( r ) j − ( r ) f ( r (cid:48) ) j + ( r (cid:48) )+ (cid:90) ∞−∞ dr (cid:90) ∞ r dr (cid:48) f ( r ) j + ( r ) f ( r (cid:48) ) j − ( r (cid:48) ) (cid:19) . The two double integrals are integrals over complementary halves of the ( r, r (cid:48) ) plane, andthus sum to a single integral over the full plane. Hence (cid:104) v, f b (cid:105) = 2 iξ (cid:90) ∞−∞ f ( r ) j + ( r ) dr (cid:90) ∞−∞ f ( r (cid:48) ) j − ( r (cid:48) ) dr (cid:48) = 2 iξ |(cid:104) f , j + (cid:105)| , since j ± are complex conjugates.Combining this with (4.16) gives Γ = 4 ωξ |(cid:104) f , j + (cid:105)| . The so–called Fermi Golden Rule then reads |(cid:104) f , j + (cid:105)| (cid:54) = 0 . Numerical investigation of the conjectured decay rate.
In order to integrate thePDE (2.2) to large times t , we employ the method of hyperboloidal foliations and scri–fixing[17]. Following [1, 2], we define s = ta − (cid:114) r a + 1 , y = arctan (cid:16) ra (cid:17) , resulting in a hyperbolic equation ∂ s ∂ s F + 2sin( y ) ∂ y ∂ s F + 1 + sin ( y )cos( y ) ∂ s F = cos ( y ) ∂ y ∂ y F + 2 a F (1 − F )cos ( y ) . (4.17) s -1 φ ( , s ) a t e x t r e m a e − tan y e − tan y (1 + tanh y ) e − tan y . s − / Figure 9.
The decay of internal mode oscillations for various initial con-ditions when a = 0 .
5. Note that φ (0 , s ) is used as a proxy for the internalmode amplitude, and we use a log–log scale to elucidate the dependence on s − / in the large s limit. The lines are labelled in the legend by the initialconditions which produced them, with the exception of the gradient line4 . s − / .for F ( s, y ) = φ ( t, r ).We solve the corresponding initial value problem at space–like hypersurfaces of constant s , specifying φ ( s = 0 , y ) and ∂ s φ ( s = 0 , y ). No boundary conditions are required, since theprincipal symbol of (4.17) degenerates to ∂ s ( ∂ s ± ∂ y ) as y → ± π/
2, so there are no ingoingcharacteristics. This reflects the fact that no information comes in from future null infinity.Following [1, 18] we define the auxiliary variablesΨ = ∂ y F, Π = ∂ s F + sin y∂ y F to obtain the first order symmetric hyperbolic system ∂ s F = Π − Ψsin y (4.18) ∂ s Ψ = ∂ y (Π − Ψsin y ) (4.19) ∂ s Π = ∂ y (Ψ − Πsin y ) + 2tan y (Ψ − Πsin y ) + 2 a F (1 − F )cos y , (4.20) HE φ KINK ON A WORMHOLE SPACETIME 19 which we solve numerically using the method of lines. Kreiss–Oliger dissipation is requiredto reduce unphysical high–frequency noise. We also add the term − . − ∂ y F ) to theright hand side of equation (4.19) to suppress violation of the constraint Ψ = ∂ y F .We are interested in the range of values a < a < a for which the kink has exactlyone internal mode. We find that, for fixed but arbitrary y , F ( s, y ) oscillates in s with afrequency close to the internal mode frequency, and that these oscillations tend towards adecay rate of s − / , as we expect from section 4.1. Plots demonstrating this decay at y = 0for a = 0 . . Expected decay rates in other regimes.
From figure 6, we see that the secondinternal mode appears before the frequency of the first internal mode moves out of therange ( m/ , m ). In the presence of more than one internal mode, we expect complicatedcoupling between their amplitudes, making the behaviour at large times very difficult topredict. However, if we restrict to odd initial data, the solution to (4.1) remains odd. Thismeans that the even internal mode can never be excited, so the system effectively has onlyone internal mode. In this case, the analysis in section 4.1 still applies, since the frequencyof the odd internal mode always lies in the range ( m/ , m ), so we expect its amplitude todecay like s − / .For general initial data and wormhole radii a > a , we cannot produce a concrete con-jecture about the decay rate. However, we expect the behaviour of the system to dependon the locations of the two internal mode frequencies within the mass gap. The analysisin section 4.1 for a single internal mode suggests that for frequencies less than m/
2, a realcontribution to ˙ A does not appear until at least O ( A ). In this case, we require anotherchange of variable in the radiation equation to rule out the contribution of the radiativeterm at higher order. We then expect to solve an equivalent of (4.12) where k + l = 3 to findthe required change of variable. If 9 ω < m so that the solution is still real, we can proceedby induction, changing the variable until the solution is complex. A real contribution to(4.15) will only be obtained for k + l = N such that N ω > m . This would mean a realcontribution to ˙ A at O ( A N +1 ), and hence result in a decay rate of s − /N . Further detailcan be found in [2].Although the presence of a second internal mode complicates the dynamics, we stillexpect the smallest N such that the even internal mode frequency ω satisfies N ω > m to be an important factor in the behaviour at large s . The axis ticks in figure 6 show thevalue of N for a range of a . 5. Summary and Discussion
We have found that the modified kink is topologically and linearly stable, and investigatedits asymptotic stability for the range of a where exactly one discrete mode is present. Itwould be interesting to expand the investigation in section 4 to the case when both discretemodes are present. This problem is much more complicated because of the extra terms in(4.6) and (4.5) coming from the amplitude of the second internal mode. Similar problemshave been discussed in [15], although no such analysis has been done for non–linear Klein–Gordon equation of this type with two discrete modes. The φ theory on the wormhole presents a useful setting to undertake such analysis because the kink has exactly two discretemodes for any a > a , and because their frequencies can be controlled by the parameter a .This model shares an interesting property with its sine–Gordon counterpart in that weexpect a discontinuous change in decay behaviour when a moves out of the range a
The author would like to thank Maciej Dunajski, Piotr Bizo´n andMichal Kahl for helpful discussions. References [1] Bizo´n, P. and Kahl, M. (2015) Wave maps on a wormhole. Phys. Rev. D b4 , 1252 3[5] Ellis, H. G. (1973) Ether flow through a drainhole: A particle model in general relativity. J. Math. Phys. , 104–118 3[6] Hall, B. C. (2013) Quantum Theory for Mathematicians. Graduate Texts in Mathematics , Springer[7] Kowalczyk, M., Martel, Y. and Mu˜noz, C. Kink Dynamics in the φ Model: Asymptotic Stability forOdd Perturbations in the Energy Space. 3[8] Manton, N. S. and Merabet, H. (1997) φ Kinks - Gradient Flow and Dynamics. Nonlinearity ϕ and sine-Gordon theory. Journal of Mathematical Physics ,1439–1444[11] Simon, B. (1975) The Bound State of Weakly Coupled Schr¨odinger Operators in One and Two Dimen-sions. Annals of Physics H scattering for nonlinear Klein–Gordonequations with metastable states[17] Zenginoglu, A. (2008) Hyperboloidal foliations and scri-fixing. Classical and Quantum Gravity25