Tails for the Einstein-Yang-Mills system
aa r X i v : . [ g r- q c ] J a n Tails for the Einstein-Yang-Mills system
Michael P¨urrer and Peter C Aichelburg
Institut f¨ur Theoretische Physik, Universit¨at Wien, 1090 Wien, AustriaE-mail:
[email protected],[email protected]
Abstract.
We study numerically the late-time behaviour of the coupled EinsteinYang-Mills system. We restrict ourselves to spherical symmetry and employ Bondi-like coordinates with radial compactification. Numerical results exhibit tails withexponents close to − i + and − I + .PACS numbers: 04.25.D-, 04.40.Nr, 03.65.Pm, 02.70.Bf
1. Introduction
Radiating systems relax to equilibrium by dissipating energy to infinity. The fall-offproperties of the field at late times are governed by so-called radiation tails. These tailsemerge from primary outgoing radiation that is backscattered. This far-field effect iseither due to a background or an effective potential produced by the nonlinearities ofthe radiation field itself, or in general, a mixture of both.The classical fall-off properties were based on liner perturbations about a givenbackground [1]. However, Bizon [2] has pointed out, based on previous results [3, 4, 5],that for certain nonlinear systems the nonlinear tails may dominate the long-timebehaviour, i.e. these tails fall off more slowly in time than the linear perturbations. As anexample, he studied the spherically symmetric Yang-Mills equations on Minkowski andSchwarzschild spacetimes [6]. While linear perturbation theory predicts a t − power lawdecay due the backscattering off the Schwarzschild curvature, the nonlinear part decaysonly as t − for observers near timelike infinity. This slower fall-off was also observednumerically. More recently, these results were confirmed and extended to the late-timebehaviour at future null infinity by Zengino˘glu [7], showing that tails die off as t − onSchwarzschild spacetime.These calculations are on a given background and the question arises what happensfor the coupled Einstein-Yang-Mills system. In the fully coupled case it is difficult todisentangle the different contributions. If one writes down a perturbation expansionstarting from flat spacetime, then the first order perturbation of the YM field evolvesin an effective potential produced by the back reaction of the YM-field to the metric.In addition, there will be the nonlinear effects from the YM equation already present in ails for the Einstein-Yang-Mills system
2. Model
We assume spherical symmetry with a regular center and choose Bondi-like coordinates { u, r, θ, φ } based upon outgoing null hypersurfaces u = const with the line-element [10] ds = − e β ( u,r ) V ( u, r ) r du − e β ( u,r ) dudr + r d Ω . (1)We consider the Yang-Mills theory with the gauge group SU(2) and assume themagnetic ansatz for the gauge connection [6, 11] A = wτ θ dθ + (cid:0) cot θτ r + wτ φ (cid:1) sin θdφ, (2)where w = w ( u, r ) is the Yang-Mills field and the τ a are the spherical generators of su (2), normalized such that [ τ a , τ b ] = iǫ abc τ c , where a, b, c ∈ { r, θ, φ } . They are relatedto the Pauli matrices σ a via τ a = σ a / F = dA + A ∧ A , then becomes F = ( ˙ wdu + w ′ dr ) ∧ (cid:0) τ θ dθ + τ φ sin θdφ (cid:1) − (cid:0) − w (cid:1) τ r dθ ∧ sin θdφ, (3)where ˙ w and w ′ denote partial derivatives of w ( u, r ) with respect to u and r , respectively.Using the above ansatz the trace of the Yang-Mills curvature becomestr ( F µν F µν ) = F aµν F aµν = − e − β r (cid:20) ww ′ − Vr ( w ′ ) (cid:21) + (1 − w ) r , (4)where Greek indices range over the four spacetime dimensions and Latin indices aregroup indices. The action for the Yang-Mills field coupled to Einstein’s equations is [11] S = Z d x √− g (cid:20) R πG − e F aµν F aµν (cid:21) , (5) ails for the Einstein-Yang-Mills system w − w ′ + (cid:18) Vr (cid:19) ′ w ′ + Vr w ′′ + e β r w (1 − w ) = 0 , (6)while variation with respect to the metric functions, β and V , yields two constraintequations V ′ = e β (cid:20) − πGe (1 − w ) r (cid:21) (7) β ′ = 8 πGe ( w ′ ) r . (8)The coupling constant [ G/e ] has dimension of length . Since it is notdimensionless, changing the coupling constant does not give rise to a one-parameterfamily of theories, but only changes the scale. To simplify the equations, we choose8 πGe = 1 . (9)The final form of the hypersurface equations then becomes V ′ = e β (cid:20) − (1 − w ) r (cid:21) (10) β ′ = ( w ′ ) r . (11)The regularity condition to be imposed on the Yang-Mills field at the origin is w = ± r ) (12)while the gauge ( u is chosen to be proper time at the regular center) and regularityconditions on the metric functions are β = O( r ) (13) V /r = 1 + O( r ) . (14)For the study of tail behaviour it is crucial to introduce a new field variable¯ w := w − w = ±
1, (i.e. where the fieldstrength vanishes), in order to avoid the tails being swamped by accumulated numericalerrors. Since the matter field equation and the constraint equations are invariant underreflection symmetry, w ( u, r ) → ˜ w ( u, r ) := − w ( u, r ), we may specialize to one of the twovacua without loss of generality.There are a number of different ways to go about solving the Yang-Mills waveequation (6) in Bondi coordinates, e.g. the diamond integral approach due to Gomezand Winicour [12, 13] which we used in [10] or Goldwirth and Piran [14] and Garfinkle’s[15] way of rewriting the equation with a total time derivative along the ingoing nullgeodesics. The latter allows for employing standard method of lines (MOL) techniques,i.e. one first discretizes the spatial derivatives (albeit non-equispaced), which in turn, ails for the Einstein-Yang-Mills system h := ¯ w ,r (16)so as to eliminate the mixed ur derivative in equation (6). In addition, we also keep thelocations of gridpoints (in time) at fixed values of r . Here, MOL discretizations, usingstandard stencils for equidistant grids, are applicable and the treatment of the origin istrivial modulo boundary conditions. Moreover, we are here not interested in strong fieldregions, where the focussing of ingoing null geodesics would provide a natural increaseof resolution near the center. Rather, we want to study late time tails for subcriticalevolutions, which entails tracking the field at locations of constant r through time.The evolution equation then becomes˙ h = 12 (cid:18) Vr (cid:19) ′ h + 12 Vr h ′ − e β F (¯ h ) r , (17)where F (¯ h ) = 2¯ h + 3¯ h + ¯ h , (18)and ¯ h ≡ ¯ w .We now have an added constraint to solve (in addition to the two geometryequations (10), (11))¯ h = Z r h ( u, ˜ r ) d ˜ r. (19)Note that equation (17) is of advection type. For flat space and without the YMself-interaction term, it reduces to˙ h = 12 h ′ , (20)which is equivalent to the flat space wave equation for φ = 1 r Z r h ( u, ˜ r ) d ˜ r. (21)Given initial data h ( u = 0 , r ) = f ( r ) with r ∈ R the solution of the advection equation(20) is simply h ( u, r ) = f ( r + 12 u ) , r ∈ R , u > . (22)The characteristic curves r + u/ const are purely ingoing and thus, there is noboundary condition at the origin of spherical symmetry. The outgoing characteristic ofthe wave equation comes into play via the constraint equation (21).The outer boundary needs a different treatment. While it is possible to specifyan outgoing wave boundary condition we prefer to use compactification, which has the ails for the Einstein-Yang-Mills system x := r r , (23)which maps r ∈ [0 , ∞ ] x ∈ [0 , m = r − g rr ] = r (cid:20) − Vr e − β (cid:21) (24)as an evolution variable, thereby eliminating V . This is necessary, since the compactifiedconstraint equation for V is singular at future null infinity I + , similar to the scalarfield case treated in [10]. Instead of h we introduce a new field variable˜ h := ¯ w ,x = ¯ h ,x , (25)which finally leads to a manifestly non-singular evolution system.The evolution equation then becomes˜ h u = 12 (cid:20) e β (cid:18) − m − xx (cid:19) (1 − x ) ˜ h (cid:21) ,x − e β F (¯ h ) x , (26)and the constraints are¯ h = Z x ˜ hd ˜ x (27) β ,x = (1 − x ) x (˜ h ) (28) m ,x = (cid:20) − m − xx (cid:21) (1 − x ) (˜ h ) + ¯ h (4 + 4¯ h + ¯ h )2 x (29)The regularity conditions at the origin for the compactified scheme become, using r = O( x ), ˜ h = O( x ) ¯ h = O( x ) (30) β = O( x ) m = O( x ) (31)There is no boundary condition at future null infinity, since it is a characteristic.
3. Numerics
From a numerical point of view, we do not even need to know that we are solving acharacteristic initial value problem. Rather, we may simply take the evolution systemand solve the nonlinear advection equation (26) using a standard MOL approach inconjunction with computing the constraint ODEs (27), (28), and (29).We may discretize the advection term in (26) by centered, fully upwind or upwind-biased stencils. Combined with an ODE integrator for time, such as, the classical 4thorder Runge-Kutta method (RK4), the resulting schemes will then exhibit differentnumerical errors. One has to make a choice between added dispersion, in the case ofcentered approximations, and added dissipation for upwind schemes. These errors affect ails for the Einstein-Yang-Mills system ∂H∂x ( x i ) = 2 H i − − H i − − H i + 80 H i +1 − H i +2 + 8 H i +3 − H i +4 x + O(∆ x ) , (32)where H i = H ( x i ). The above upwind stencil is the one closest to the centered stenciland it also has the lowest error term among the upwind stencils [16]. The two alternateupwind stencils do not lead to stable evolutions in our case. The weights for suchfinite-difference stencils can conveniently be computed in a computer algebra package,such as Mathematica, using Fornberg’s compact algorithm [17]. In contrast to centeredschemes, where it is often necessary to add some artificial dissipation to have numericalstability, the dissipation is already “built-in” in our chosen scheme.The term F (¯ h ) /x in (26) forces a regularity boundary condition at the origin, sothat ˜ h ( u, x = 0) = 0 . (33)We enforce it by choosing initial data that satisfy this condition (to machine precision)and then make sure that its time derivative is zero, i.e. ˜ h u ( u, x = 0) = 0 for all timesteps.We choose Gaussian initial data¯ h (0 , x ) = A exp (cid:2) − x − / (cid:3) . (34)The constraint equations for ¯ h and β are integrated using cumulative Newton-Cotes quadrature rules of order 6, which are given in the appendix. Since the righthand side of the constraint equation for the Misner-Sharp mass-function (29) dependson the unknown m , we integrate it with RK4 and use 4th order polynomial interpolationfor computing ˜ h and ¯ h at points x i +1 / in between actual gridpoints, as required by theRunge-Kutta method.In comparison to the established methods for solving characteristic initial valueproblems [12, 13, 14, 15], the method used here has a number of advantages for theproblem at hand. It relies on a standard MOL discretization using equidistant stencilsand thus allows for the use of high order schemes. Moreover, the boundary treatmentis simple and does not require Taylor series near the origin. It also allows us to trackthe field at lines of constant r without further interpolation.In general, we expect the code to be 4th order convergent, see figure 1. For smalldata, however, max 2 m/r is also small. Since m only appears in the wave equation inthis combination, errors in the computation of m may be allowed to be larger than thosein other fields, and we may therefore have close to 6th order spatial accuracy, in thiscase. If the Courant number C := 12 ∆ u ∆ x (35)is small enough, so that the errors from the RK4 integrator are of order O(∆ x ), then,we may in fact achieve 6th order convergence. It is, however, impractical to have small ails for the Einstein-Yang-Mills system h u = 12 (1 − x ) ˜ h x . (36)After freezing the nonconstant coefficient (1 − x ) at its maximum 1, Fourier analysisleads to the stability limit (similar to the 6th order centered case discussed in [19]) C = 12 ∆ u ∆ x ≤ .
29 (37)for a MOL discretization with the 6th order upwind-biased stencil (32) and RK4 astime-integrator. For the coupled system, we have found that Courant numbers of up to C ≤ .
25 yield stable evolutions. C f PSfrag replacements uC f Figure 1.
This figure shows the convergence factor C f = log k ¯ h − ¯ h kk ¯ h − ¯ h k in the l C = 1 .
25. Initially the code is 4thorder convergent. Since the initial amplitude A = 0 .
28 is quite large and 2 m/r ≈ . m/r becomes very small and the spatial discretization becomes roughly 6th orderaccurate. At even later times the convergence order decreases slowly - probably dueto the standard degrading of convergence in long evolutions. We have used numerical Python to conveniently automate the determination of tailexponents via fitting, while the core of the code was written in C++. ails for the Einstein-Yang-Mills system
4. Results
Perturbation theory predicts the late-time behaviour of the solution to be¯ h ( u, x ) ∼ Cu p , (38)where p is the tail exponent. A first test case for our code is to correctly reproducethe know tail behaviour on Minkowski background [6]. In figure 2 we find that thetail exponent tends to p = − i + and the exponent p = − I + . This also corresponds to the decay found onSchwarzschild backgrounds [7]. In terms of the compactified radial coordinate x , theobservers are located at x = (1 , . , . , . , . , . , . , . , . , . C = 1 .
25 and an initial data amplitude of A = 0 .
01. For bigger, but still subcritical(in the coupled case), amplitudes and/or smaller Courant numbers, the tail decay isessentially the same.Figure 3 encodes our main results showing the late-time behavior for the coupledEinstein-Yang-Mills system. We find essentially the same fall-off as for the Yang-Mills field on Minkowski background. As mentioned in the introduction, in terms ofa perturbative approach, tails are generated on the one hand by the the nonlinearityof the YM field itself, and, on the other by the contribution of the field to the metric.What we see numerically is a superposition of both effects which we can not separate.Hod[20] has studied linear wave tails in time dependent potentials. He finds that for acertain class of potentials that go to zero asymptotically in time, the fall-off behaviourof the tails is a power law depending on the time dependence of the potential.
5. Conclusion
Using Bondi-like coordinates and radial compactification we have written the Einstein-Yang-Mills system as an advection equation plus three constraints. In this form, MOLdiscretizations are straightforward to apply, the origin treatment is easy and the outerboundary, I + , being a characteristic does not require boundary data. Compared tothe diamond integral scheme [12, 13] and Goldwirth, Piran and Garfinkle’s [14, 15] wayof solving characteristic initial value problems, the approach used here is very clean,simple to implement and allows the use of high order schemes.We have found that the spherically symmetric coupled Einstein-Yang-Mills systemshows the same fall-off behaviour at late times as Yang-Mills on Minkowski orSchwarzschild backgrounds. Although such a result could have been expected it isby no means evident, because so far it is not known how tails arising from the backreaction of the the Yang-Mills field to the metric decay. Our results indicate that theydecay as fast or faster than the nonlinear tails on Minkowski background.Our compactified code has allowed us to also study the fall-off behavior at futurenull infinity. As for the coupled Einstein massless scalar field [10], the fall-off on I + isslower than for observers approaching timelike infinity. Since realistic observers are ails for the Einstein-Yang-Mills system u | ¯ h ( u , x = c o n s t ) | PSfrag replacements p ( u , x ) u Figure 2.
The upper plot shows the decay of the field ¯ h at x = const versus retardedtime u , while the lower plot depicts the respective tail exponents for the same evolutionon Minkowski background. On I + the tail exponent is close to p = −
2. Observerslocated at finite x (or r ) approach timelike infinity i + with the exponent p = − I + . located only at finite distances from the center, what then is the practical relevance toknow the decay conditions on I + ? It has been pointed out in [10] and also in [7], thatfor astrophysical observers, the relevant decay rate is the one along null infinity. Thishas to do with the observation that the tail exponents for observers far out start closeto the exponent on I + and only slowly decrease to the value for timelike observers. ails for the Einstein-Yang-Mills system | ¯ h ( u , x = c o n s t ) | u PSfrag replacements p ( u , x ) u Figure 3.
Similar to figure 2 these plots depict the late-time tails and the respectiveexponents for the coupled Einstein-Yang-Mills system. The exponents coincide withthe behaviour on Minkowski space, being p = − I + and p = − i + , respectively.The zero-crossing in ¯ h at u ≈
50 depends on the initial data amplitude, i.e. the fieldgoes through zero earlier for smaller amplitudes.
Acknowledgments
We thank Piotr Bizon for helpful discussions and comments on the manuscript. Thiswork has been supported by the Austrian Fonds zur F¨orderung der wissenschaftlichenForschung (FWF) (project P19126-PHY). Partial support by the Fundaci´on Federicoand the hospitality at the Mittag-Leffler Institute (Sweden) is also acknowledged. ails for the Einstein-Yang-Mills system Appendix
The quadrature formulas below have been obtained by simply integrating the (quartic)interpolating polynomial P ( f | x i − , x i , x i +1 , x i +2 , x i +3 ) on an equidistant grid withspacing h over the intervals [ x i − , x i ] until [ x i +2 , x i +3 ], respectively. Z x i x i − f dx = h
720 (251 f i − + 646 f i − f i +1 + 106 f i +2 − f i +3 ) + O( h ) (A.1) Z x i +1 x i f dx = h
720 ( − f i − + 346 f i + 456 f i +1 − f i +2 + 11 f i +3 ) + O( h ) (A.2) Z x i +2 x i +1 f dx = h
720 (11 f i − − f i + 456 f i +1 + 346 f i +2 − f i +3 ) + O( h ) (A.3) Z x i +3 x i +2 f dx = h
720 ( − f i − + 106 f i − f i +1 + 646 f i +2 + 251 f i +3 ) + O( h ) (A.4)Summing these formulas together, yields the classical Boole’s or Milne’s rule. Z x i +3 x i − f dx = 2 h
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