Late-time tails of a Yang-Mills field on Minkowski and Schwarzschild backgrounds

TL;DR

This paper analyzes the late-time decay of spherically symmetric Yang-Mills fields on Minkowski and Schwarzschild backgrounds, showing solutions decay as t^{-4} and deriving a formula for the tail amplitude.

Contribution

It provides the first detailed analysis of late-time tails for Yang-Mills fields on these backgrounds, including a third-order formula for the tail amplitude on Minkowski space.

Findings

Solutions decay as t^{-4} at timelike infinity.

Derived a third-order formula for tail amplitude on Minkowski background.

Numerical confirmation of the tail amplitude formula.

Abstract

We study the late-time behavior of spherically symmetric solutions of the Yang-Mills equations on Minkowski and Schwarzschild backgrounds. Using nonlinear perturbation theory we show in both cases that solutions having smooth compactly supported initial data posses tails which decay as $t^{-4}$ at timelike infinity. Moreover, for small initial data on Minkowski background we derive the third-order formula for the amplitude of the tail and confirm numerically its accuracy.

Figures

Equations

$$A=w{\tau }_{1}d\theta +(\cot \theta {\tau }_3+w{\tau }_2)\sin \theta d\varphi ,$$

$$\ddot{w}-w^{\prime \prime }-\frac{1}{r^2}w(1-w^2)=0,$$

$$\mathcal{L}f:=\ddot{f}-f^{\prime \prime }-\frac{2}{r}{f}^{\prime }+\frac{2}{r^2}f=-f^3-\frac{3}{r}{f}^2.$$

$$f(0,r)=\epsilon \alpha (r),\dot{f}(0,r)=\epsilon \beta (r).$$

$$f=\epsilon f_1+{\epsilon }^2{f}_2+{\epsilon }^3{f}_3+...,$$

$$\mathcal{L}{f}_1$$

$$\mathcal{L}{f}_2$$

$$\mathcal{L}{f}_3$$

$$f_1(t,r)=\frac{a^{\prime }(t-r)+a^{\prime }(t+r)}{r}+\frac{a(t-r)-a(t+r)}{r^2},$$

$$a(\xi )=-\frac{1}{2}\xi \underset{\xi }{\overset{\infty }{\int }}\alpha (s)ds+\frac{1}{4}\underset{\xi }{\overset{\infty }{\int }}(s^2-{\xi }^2)\beta (s)ds.$$

$$G(t-t^{\prime },r,r^{\prime })=[|r-r^{\prime }|\le t-t^{\prime }\le r+r^{\prime }]\frac{r^2+{r^{\prime }}^2-(t-t^{\prime }{)}^2}{4r^2}.$$

$$f(t,r)=\frac{1}{8r^2}\underset{|t-r|}{\overset{t+r}{\int }}dv\underset{-v}{\overset{t-r}{\int }}K(t,r;u,v)N(u,v)du,$$

$$f_2(t,r)=-\frac{3}{4r^2}\underset{|t-r|}{\overset{t+r}{\int }}dv\underset{-v}{\overset{t-r}{\int }}K(t,r;u,v)\frac{f_1^2(u,v)}{v-u}du.$$

$$f_2(t,r)=-\frac{3}{r^2}\underset{-R}{\overset{R}{\int }}du\underset{t-r}{\overset{t+r}{\int }}\frac{(v-t)(t-u)+r^2}{(v-u{)}^3}{\left(a^{\prime }(u)+\frac{2a(u)}{v-u}\right)}^{2}dv.$$

$$f_2(t,r)=8r\underset{-R}{\overset{R}{\int }}\frac{a(u)}{(t-u{)}^2-r^2}\frac{d}{du}\left(\frac{a(u)}{(t-u{)}^2-r^2}\right)du,$$

$$f_3^{(1)}(t,r)$$

$$f_3^{(2)}(t,r)$$

$$f_3^{(1)}(t,r)=4r\underset{-R}{\overset{R}{\int }}\left(\frac{a(u)a^{\prime }(u{)}^2}{z^2}+\frac{4(t-u)a^{\prime }(u)a^2(u)}{z^3}+\frac{4((t-u{)}^2+\frac{1}{5}{r}^2)a^3(u)}{z^4}\right)du,$$

$$f_3^{(1)}(t,r)\sim c_{1}r{t}^{-4},c_1=4\underset{-\infty }{\overset{+\infty }{\int }}a(u){a^{\prime }(u)}^{2}du.$$

$$f_2(u,v)=\frac{h^{\prime }(u)}{v-u}+\frac{2h(u)}{(v-u{)}^2}+\frac{2g(u)}{(v-u{)}^2}+\mathcal{O}(v^{-3}),$$

$$f_3^{(2)}(t,r)\sim c_{2}r{t}^{-4},c_2=4\underset{-\infty }{\overset{+\infty }{\int }}\left[\frac{d}{du}(h(u)a(u))+g(u)a^{\prime }(u)\right]du=-12\underset{-\infty }{\overset{+\infty }{\int }}a(u)a^{\prime }(u{)}^{2}du,$$

$$f_3(t,r)\sim cr{t}^{-4},c=-8\underset{-\infty }{\overset{+\infty }{\int }}a(u)a^{\prime }(u{)}^{2}du.$$

$$\underset{t-r}{\overset{t+r}{\int }}\frac{(v-t)(t-u)+r^2}{(v-u{)}^3}dv=0$$

$$d{s}^2=-\left(1-\frac{2m}{r}\right)d{t}^2+{\left(1-\frac{2m}{r}\right)}^{-1}d{r}^2+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),r>2m,$$

$${\left(1-\frac{2m}{r}\right)}^{-1}\ddot{w}-{\left(\left(1-\frac{2m}{r}\right)w^{\prime }\right)}^{\prime }-\frac{1}{r^2}w(1-w^2)=0.$$

$$x=r+2m\ln \left(\frac{r}{2m}-1\right),h(t,x)=w(t,r(x))-1,$$

$$\ddot{h}-\frac{d^{2}h}{d{x}^2}+\left(1-\frac{2m}{r}\right)\frac{2}{r^2}h=-\left(1-\frac{2m}{r}\right)\frac{1}{r^2}(3h^2+h^3),$$

$$f_2(u,r)=-\frac{3}{4r}\underset{|u|}{\overset{u+2r}{\int }}d{v}^{\prime }\underset{-v^{\prime }}{\overset{u}{\int }}\frac{u^{\prime }+v^{\prime }-2u+(u{u}^{\prime }+u{v}^{\prime }-u^{\prime }{v}^{\prime }-u^2)/r}{v^{\prime }-u^{\prime }}{f}_1^2(u^{\prime },v^{\prime })d{u}^{\prime }.$$

$$I(u,ϵ):=-\frac{3}{2}\underset{|u|}{\overset{u+2/ϵ}{\int }}d{v}^{\prime }\underset{-v^{\prime }}{\overset{u}{\int }}\frac{u^{\prime }+v^{\prime }-2u+ϵ(u{u}^{\prime }+u{v}^{\prime }-u^{\prime }{v}^{\prime }-u^2)}{v^{\prime }-u^{\prime }}{f}_1^2(u^{\prime },v^{\prime })d{u}^{\prime }.$$

$$A(u)$$

$$B(u)$$

$$h(u)$$

Full text

I Introduction

In a classical paper em Eardley and Moncrief proved that solutions of the Yang-Mills equations on the $3+1$ Minkowski spacetime starting from smooth initial data remain smooth for all future times. A different proof allowing for initial data with only finite energy was given later by Klainerman and Machedon km . Once global regularity was established, the problem of asymptotic behavior of solutions for $t\rightarrow\infty$ was studied by many authors gs ; ch1 ; cpc ; shu ; sch who obtained various decay estimates using different techniques and assumptions about initial data. In this paper we are concerned with the simplest possible situation, namely spherically symmetric initial data with compact support. In this case it follows from the conformal method of Christodoulou that the Yang-Mills curvature decays as $t^{-4}$ at timelike infinity ch1 . The purpose of this paper is threefold. First, we rederive Christodoulou’s result using the nonlinear perturbation theory. The advantage of our approach lies in its wide applicability; in contrast to the conformal method which is very powerful (in the sense of giving sharp decay rates) only for conformally invariant equations.

Second, we go beyond qualitative decay estimates and give the third-order formula for the amplitude of solution which provides a precise quantitative information about the tail. We wish to point out that although our result depends crucially on spherical symmetry, the assumption of compact support for initial data is made for simplicity and can be relaxed by imposing a suitable fall-off condition at spatial infinity (which can be implemented via appropriately weighted norms). However, some kind of localization condition is necessary in order to avoid a situation where the tail in time is induced entirely by the tail of initial data at spatial infinity (due, for instance, to nonzero charge).

Third, we argue that the same tail is present in the scattering of spherically symmetric Yang-Mills fields off the Schwarzschild black hole. In this case the global existence of solutions follows from the work of Chruściel and Shatah cs who generalized the proof of Eardley and Moncrief to arbitrary globally hyperbolic Lorentzian 4-manifolds. The late-time tail of the Yang-Mills field on the Schwarzschild background was studied in cw , however the fall-off $t^{-5}$ derived there on the basis of the linear perturbation analysis is not correct. As we shall see, the error in cw is due to the fact that the late-time tail is not governed by the linearized evolution. At first sight that might seem odd but upon reflection it is easy to understand. A rough intuitive explanation is that the tail is a far-field effect hence the flat space tail $t^{-4}$ is expected to persist in any asymptotically flat spacetime as long as the backscattering on the curvature does not produce a more slowly decaying tail. A similar example of the failure of linear perturbation analysis was recently observed in the scattering of skyrmions bcr .

II Minkowski background.

We consider the Yang-Mills theory with the gauge group $SU(2)$ and assume the spherically symmetric ansatz for the connection fm

[TABLE]

where $w=w(t,r)$ and $\tau_{i}$ ($i=1,2,3$) are the usual generators of $su(2)$. The Yang-Mills equations $d\ast F=0$, where $F=dA+A\wedge A$ is the Yang-Mills curvature, reduce then to the semilinear radial wave equation

[TABLE]

where primes and dots denote derivatives with respect to $r$ and $t$, respectively. For our purposes it is convenient to define the function $f(t,r)=(w(t,r)-1)/r$ and rewrite equation (2) in the following form

[TABLE]

Note that $\mathcal{L}$ is the radial wave operator for the $l=1$ spherical harmonic.

We consider the late-time evolution of solutions of equation (3) for smooth compactly supported initial data

[TABLE]

The prefactor $\varepsilon$ is added for convenience and to emphasize that our initial data are assumed to be small. Regularity at the origin is ensured by the boundary condition $f(t,r)\sim b(t)r$ for $r\rightarrow 0$. As follows from gs such solutions decay to zero on any compact region of space as $t\rightarrow\infty$. To determine the asymptotic behavior of solutions we define the perturbative expansion

[TABLE]

where $\varepsilon f_{1}$ satisfies initial data (4) and all $f_{n}$ with $n>1$ have zero initial data. Substituting the expansion (5) into equation (3) up to the third order we get

[TABLE]

We solve these equations recursively. The first order solution is given by the general regular solution of the free radial wave equation for the $l=1$ spherical harmonic

[TABLE]

where the function $a(\xi)$ is determined by the initial data

[TABLE]

For compactly supported initial data the function $a(\xi)$ has compact support as well (note that the functions $\alpha(s)$ and $\beta(s)$ in (10) are odd extensions of initial data to the whole line), hence $f_{1}$ has no tail in agreement with Huygens’ principle.

To solve equations for the higher order perturbations we use the retarded Green’s function of the operator $\mathcal{L}$

[TABLE]

It follows from (11) that the solution of the inhomogeneous equation $\mathcal{L}f=N(t,r)$ with zero initial data has the form (using null coordinates $u=t^{\prime}-r^{\prime},v=t^{\prime}+r^{\prime}$) gs

[TABLE]

where the kernel $K(t,r;u,v)=(v-t)(t-u)+r^{2}$. In the second order, i.e. for equation (7), the representation (12) yields

[TABLE]

Somewhat surprisingly, Huygens’ property is preserved in the second order. To see this, let us assume that $a(\xi)=0$ for $|\xi|\geq R$. Then, for $t>r+R$ we may change the order of integration in (13) and rewrite it as (see Fig. 1)

[TABLE]

Performing the inner integral we get

[TABLE]

which after integration gives zero. Thus, $f_{2}(t,r)$ vanishes identically for $t>r+R$ and consequently there is no tail up to the second order.

In the third order, i.e. for equation (8), the representation (12) gives $f_{3}=f^{(1)}_{3}+f^{(2)}_{3}$, where

[TABLE]

To calculate $f^{(1)}_{3}(t,r)$ for $t>r+R$, as above we change the order of integration and perform the integral over $v$ with the result (using the abbreviation $z=(t-u)^{2}-r^{2}$)

[TABLE]

which has the following asymptotic behavior near timelike infinity ($r=const$ and $t\rightarrow\infty$)

[TABLE]

In the formula above we replaced $R$ by $\infty$ in the limits of integration to emphasize that the result holds not only for strictly compactly supported initial data but also for initial data which fall off sufficiently fast at spatial infinity.

To calculate the contribution to the tail coming from $f^{(2)}_{3}(t,r)$ we need to know both the leading and the subleading terms in the asymptotic expansion of $f_{2}(u,v)$ near null infinity ($u=const$ and $v\rightarrow\infty$). This calculation is deferred to the appendix where we show that near null infinity

[TABLE]

where $h(u)$ and $g(u)$ are defined by (32) and (33), respectively. Note that the first two terms in (20) represent the ”free” part of the iterate $f_{2}(t,r)$; as we shall see in a moment this part does not affect the behavior of $f^{(2)}_{3}(t,r)$ at timelike infinity. Substituting (20) into (17) and proceeding along the same lines as in the derivation of the expression (19) we obtain the following asymptotic behavior near timelike infinity

[TABLE]

where the last expression follows from (33) and integration by parts. Putting equations (19) and (21) together we finally get the leading asymptotic behavior near timelike infinity

[TABLE]

This is our main result. We claim that the expression (22) provides a very good approximation of the tail for solutions having sufficiently small initial data. More precisely, we conjecture that for any given smooth compactly supported functions $\alpha(r)$ and $\beta(r)$ one can choose $\varepsilon$ such that for each fixed $r>0$ and $t\rightarrow\infty$ the remainder $|t^{4}f(t,r)-\varepsilon^{3}c\,r|$ is as small as one pleases. The numerical evidence supporting this conjecture is shown in Fig. 2. The obvious issue remains as to whether the perturbation expansion corresponding to given initial data is convergent for sufficiently small values of $\varepsilon$. Without a proof of convergence our analysis is not mathematically rigorous, nevertheless we believe that it gives a rather convincing and, most importantly, quantitative description of the late-time tail.

A few remarks are in order: Remark 1: It should be clear from the above derivation that the simplicity of the final result (22) is due to some amazing cancelations (notably those occurring in equations (15) and (21)) which in turn are attributed to the particular form of the nonlinearity of the Yang-Mills equations. In this respect the Yang-Mills equations are exceptional and particularly interesting mathematically; for most other nonlinear perturbations of the wave equation the tail is a second-order phenomenon which is much easier to analyze (e.g., see bcr ).

Remark 2: Note that all iterates $f_{k}$ behave as $\mathcal{O}[(v-u)^{-1}]$ near null infinity and therefore at each order of the perturbation expansion the sources behave as $\mathcal{O}[(v-u)^{-3}]$. For such sources one might expect by dimensional arguments that there would be a $t^{-3}$ tail. Fortunately, due to the identity

[TABLE]

all coefficients of hypothetical $t^{-3}$ tails vanish identically and thus all higher-order terms in the perturbation expansion decay as $t^{-4}$. This fact is crucial since otherwise the third-order approximation would break down for late times; for instance a nonzero fourth-order term $\sim\varepsilon^{4}/t^{3}$ would make the formula (22) useless for times $t\gtrsim 1/\varepsilon$. Remark 3: Note that equation (3) has the scaling symmetry: if $f(t,r)$ is a solution, so is $f_{\lambda}(t,r):=\lambda f(\lambda t,\lambda r)$. Under this scaling the energy scales as $E(f_{\lambda})=\lambda E(f)$, hence given any finite energy initial datum one can scale it down to an arbitrarily small amplitude and energy. Note, however, that for compactly supported initial data such a rescaling spreads the support by a factor $1/\lambda$ and for this reason it cannot make large data smaller in the sense of our perturbation expansion. This follows immediately from the fact that all iterates $f_{k}(t,r)$ in (5) scale in the same way. In other words, the rescaling does not change the convergence properties of the perturbation expansion.

III Schwarzschild background

On the exterior Schwarzschild spacetime

[TABLE]

the spherically symmetric Yang-Mills equation corresponding to the ansatz (1) takes the form

[TABLE]

When $m=0$ this equation reduces of course to (2). In terms of the new variables

[TABLE]

equation (25) becomes

[TABLE]

where $r=r(x)$. Dropping the nonlinear terms on the right side of (27) one gets the linear $1+1$ dimensional wave equation on the real axis $-\infty<x<\infty$ with the effective potential $V(x)=2/r^{2}-4m/r^{3}$. This equation describes the propagation of the dipole ($l=1$) electromagnetic perturbation of the Schwarzschild black hole. For intermediate times the linearized approximation is very good; this stage of evolution has the form of exponentially damped oscillations dominated by the fundamental (i.e., least damped) quasinormal mode. We recall that quasinormal modes are solutions of the linearized equation satisfying the outgoing wave conditions $h(t,x)\sim e^{-ik(t\mp x)}$ for $x\rightarrow\pm\infty$. In the case at hand the fundamental quasinormal mode has the eigenvalue $k=0.49653-0.18498i$ (in units where $2m=1$) l .

The quasinormal mode decays exponentially so for late times it becomes negligible and eventually a polynomial tail is uncovered. Since the pioneering work of Price p it has been known that the tail of the $l$-th multipole decays as $t^{-2l-3}$, thus for the dipole the linearized theory predicts the tail $t^{-5}$ and this is exactly the result derived in cw . We claim that this prediction is incorrect and the actual tail behaves in the same manner as in Minkowski spacetime, namely it decays as $t^{-4}$. Regarding equation (25) as the perturbation (for $r\gg 2m$) of equation (2), one can see from dimensional considerations that the failure of linearization is due to the fact that the linear terms in (25) corresponding to nonzero curvature (proportional to $m$) are of shorter range (using PDE jargon) than the nonlinear terms. Thus, the presence of the black hole should not alter the flat space tail $t^{-4}$. The numerical substantiation of this handwaving argument is shown in Fig. 3.

Unfortunately, for the Schwarzschild background we were not able to derive a quantitative formula, analogous to (22), relating the amplitude of the tail to initial data. An attempt to repeat the perturbation analysis from section II encounters serious difficulties on Schwarzschild background which are caused by the violation of Huygens’ principle in $1+1$ dimensions and the presence of the potential. It would be interesting to pursue this problem further, perhaps borrowing ideas from an approach proposed some time ago by Barack b . Although Barack considered only the linear wave equation, we wish to emphasize that there are many similarities between his work and our analysis in section II.

IV Conclusions

Using third order nonlinear perturbation theory we determined the late-time tail of spherically symmetric Yang-Mills equations on Minkowski background. We also gave heuristic arguments that the same tail is present on Schwarzschild background. In both cases we provided numerical evidence supporting our results. We hope that our approach will trigger more rigorous mathematical analyses of these physically important phenomena.

We remark that the ideas presented here can be applied to other nonlinear wave equations. For example, one can show by similar methods that for the semilinear wave equation $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+|\phi|^{p}=0$ on Minkowski background the tail decays as $t^{1-p}$ for $p>1+\sqrt{2}$, while on the Schwarzschild background the tail changes its character at $p=4$ from linear (Price’s law $\phi\sim t^{-3}$ for $p\geq 4$ dr ) to nonlinear ($\phi\sim t^{1-p}$ for $1+\sqrt{2}<p<4$). A systematic analysis of the competition between linear and nonlinear effects in scattering for semilinear wave equations in Minkowski spacetime will be given elsewhere bcrs . Acknowledgments: P.B. thanks Bernd Schmidt, Helmut Friedrich, Alan Rendall, and Nikodem Szpak for helpful discussions. This research was supported in part by the Polish Research Committee grant 1PO3B01229.

Appendix

We derive here the equation (20). Our starting point is the equation (13) in which we relabel coordinates $(u,v)\rightarrow(u^{\prime},v^{\prime})$ and use the retarded time $u=t-r$:

[TABLE]

We let $\epsilon=1/r$ and define the quantity

[TABLE]

Expanding this in Taylor’s series $I(u,\epsilon)=A(u)+B(u)\epsilon+\mathcal{O}({\epsilon}^{2})$ we obtain

[TABLE]

where

[TABLE]

An elementary calculus exercise shows that

[TABLE]

Putting all the above equations together and noting that $r=(v-u)/2$ we get equation (20).