In hyperbolic reductions of the Einstein equations the evolution of gauge
conditions or constraint quantities is controlled by subsidiary systems. We
point out a class of non-linearities in these systems which may have the
potential of generating catastrophic growth of gauge resp. constraint
violations in numerical calculations.Comment: 7 page

Friedrich, Helmut

English

arXiv

In hyperbolic reductions of the Einstein equations the evolution of gauge conditions or constraint quantities is controlled by subsidiary systems. We point out a class of non-linearities in these systems which may have the potential of generating catastrophic growth of gauge resp. constraint violations in numerical calculations

Friedrich, H.

MPG.PuRe

arXiv:gr-qc/0504129 v1   26 Apr 2005On the non-linearity of the subsidiary systemsHelmut FriedrichAlbert-Einstein-InstitutMax-Planck-Institut für GravitationsphysikAm Mühlenberg 114476 GolmGermanyApril 26, 2005AbstractIn hyperbolic reductions of the Einstein equations the evolution of gaugeconditions or constraint quantities is controlled by subsidiary systems.We point out a class of non-linearities in these systems which may havethe potential of generating catastrophic growth of gauge resp. constraintviolations in numerical calculations.1 IntroductionMost numerical calculations of solutions to Einstein’s field equations are beingplagued by an undesirably fast growth of constraint violations. A huge varietyof reduced equations has been derived from Einstein’s equations with the hopeof finding versions with stable propagation properties and there have been sug-gested modifcations of the equations which were hoped to force back the solutionto the constraint manifold (we refer to [6] for further dicussion and references).More recently, there have been performed stability analyses of the subsidiarysystems which control the evolution of the gauge conditions or the constraintquantities (cf. [1], [8] and the references given there). These led to requirementson the coefficients of the equations which in the case of [1] resulted in geometricconditions on the foliation underlying the evolution by the main system. Nev-ertheless, the field appears to be wide open, most of the suggested remedies are1experimental, and the cause of the problems is not understood. In [1], [8] thesubsidiary systems have been considered as linear systems on given space-times.In contrast, we wish to emphasize in this note the non-linearity of the subsidiarysystem, which appears to have a potential of generating catastrophic constraintviolations. To understand the extent to which these non-linearities may affectthe numerical construction of space-times and to develop, if necessary, ways toavoid their effects, further investigations are needed.2 The hyperbolic reductionWe shall use a few facts about the hyperbolic reduction procedure by whichthe geometric initial value problem for Einstein’s field equations is reduced toa Cauchy problem for a hyperbolic system. There exist now many versionsof it, whose general underlying structure are, however, more or less the same.Because it leads to concise expressions, we will use the metric coeffficients asbasic unknowns and consider the representation of the field equations in whichtheir evolution is governed by a system of wave equations. To emphasize theindependence of our discussion of any particular coordinate system we shallemploy the notion of a gauge source function introduced in [2]. We refer to [3],[4] for more details on the reduction procedure.Let (M, g) denote a smooth 4-dimensional Lorentz space with smooth space-like Cauchy hypersurface S and U ∋ xλ → Fµ(xλ) ∈ V a smooth map of anopen subset U into another subset V of R4. With functions xµ and their (4-dimensional) differentials dxν prescribed on some open subset W of S one cansolve near W the Cauchy problem for the semi-linear system of wave equations∇ν ∇ν xµ = −Fµ(xλ).If the dxµ have been chosen linearly independent on W the solution will providea smooth coordinate system xλ on some neighbourhood of W in M . In termsof these coordinates the equations above take the form−Γµ(xλ) = −Fµ(xλ),where the Γµ denote the contracted Christoffel symbols Γµ = gνη Γνµη of gµν .This shows (ignoring subtleties arising in situations of low differentiability) thatthe contracted Christoffel symbols can locally be made to agree with any pre-scribed set of functions Fµ and that these function and the initial data determinethe coordinates uniquely. We refer to these functions as gauge source functions.Assume now that (M, g) is to be obtained by solving a Cauchy problem forEinstein’s vacuum field equations. We shall derive the properties which willhelp us formulate this problem as a Cauchy problem for hyperbolic equations.The contracted Christoffel symbols are of particular interest to us because theRicci tensor of g can be written in the formRµν = −12gλρ gµν,λρ + ∇(µΓν) + Γλ η µ gηδ gλρ Γρ δ ν + 2 Γδ λ η gδρ gλ(µ Γν) η ρ.(2.1)2Here the contracted Christoffel symbols (and the functions Fν = gνµ Fµ consid-ered in the following) are being formally treated as if they defined a vector field(which, of course, they do not). Thus Γν = gνµ Γµ and ∇µΓν = ∂µΓν−Γµ λ ν Γλ.The discussion above suggests replacing in (2.1) the functions Γν by freely cho-sen gauge source functions Fν (so that the resulting expression will depend ingeneral on the coordinates xλ not any longer only through the gµν). With thisreplacement the vacuum field equations take the form0 = RFµν ≡ −12gλρ gµν,λρ+∇(µ Fν)+Γλ η µ gηδ gλρ Γρ δ ν+2 Γδ λ η gδρ gλ(µ Γν) η ρ,(2.2)of a system of wave equations for the gµν . We refer to (2.2) as the main evolutionsystem or the reduced equations. For this system the Cauchy problem for gµνwith data satisfying the constraints on a space-like hypersurface S is well posed.Suppose that gµν is a solution of this problem near S. Since equation (2.2) isin fact of the formRµν = ∇(µ Qν), (2.3)where Qµ = Γµ − Fµ with the Γµ calculated from gµν , it is not clear a prioriwhether the solution g will indeed satisfy the gauge condition Γµ = Fµ and thusthe vacuum field equation Rµν = 0.The Bianchi identity, which holds for any metric, implies0 = 2∇µ(Rµν −12R gµν) = ∇µ ∇µ Qν + Rµ ν Qµ, (2.4)and thus a system of wave equations for the quantities Qν . We refer to thissystem as subsidiary system.The detailed analysis shows that if the Cauchy problem for the main evolutionsystem is arranged so that the initial data satisfy the constraints and the gaugecondition Qν = 0 on S, it follows for the solution of the main evolution systemthat also dQν and thus any ‘time derivative’ ∂t Qν transverse to S vanishes onS. The uniqueness property of the subsidiary system therefore implies that thesolution to the main evolution system does indeed satisfy Qν = 0 on the domainof dependence of S with respect to g. This reduces the local Cauchy problemfor Einstein’s field equations to the problem of solving equation (2.2), whichthus takes the central role in the analytic discussion. Of the subsidiary systemonly the homogeneity and the resulting uniqueness property are needed.3 The non-linearity of the subsidiary equationThe main evolution system is also central in numerical discussions. Becausethe initial data Qµ and ∂t Qν on the initial hypersurface S come with an error,the evolution properties of the subsidiary system will, however, also becomeimportant.There is no way to relate a numerical solution to the solution of the continuumproblem one wants to approximate. To get some idea how errors infiltrate into3the various systems, it is useful to consider an analogy accessible to analyticmethods. We assume the main evolution system to be satisfied by fields gµν ofclass C3 with an error term Eµν of class C1 so thatRµν = ∇(µ Qν) + Eµν . (3.1)Nothing will be assumed about the origin and structure of this error and theerrors in the initial data Qµ and ∂t Qν on S.Using the Bianchi identity with (3.1) gives the analogue∇µ ∇µ Qν + Rµ ν Qµ = −2∇µ(Eµν −12gµν Eρρ), (3.2)of the subsidiary equation (2.4). This equation will in general not be homo-geneous any longer. There may be a way to avoid this problem. Assume asplitting of the formEµν = ∇(µ eν) + fµν ,with some C2 vector field eµ and some symmetric C1 tensor field fµν . With(3.1) this givesRµν = ∇(µ Q′ν) + fµν with Q′ν = Qν + eν . (3.3)If fµν would vanish this would just amount to a redefinitionFµ → F ′µ = Fµ − eµ,of the gauge source function which may be quite harmless. Moreover, we wouldget a homogenous wave equation for Q′µ. If Eµν is known but not of the form∇(µ eν), a homogenous system can be obtained by choosing the splitting suitably.If we require eµ to solve the system of wave equations∇µ ∇µ eν + Rµ ν eµ = 2∇µ(Eµν −12gµν Eρρ),it follows from the splitting above that∇µ(fµν −12gµν fρρ)= 0,and the Bianchi identity implies for Q′µ the homogeneous equation∇µ ∇µ Q′ν + Rµ ν Q′µ = 0. (3.4)One could now try to analyse how errors in Qµ and ∂t Qν on S are propagated,in dependence of Eµν , by (3.2) or how errors in Q′µ and ∂t Q′ν on S will bepropagated by (3.4). Assuming in this analysis Eµν and gµν whence Rµν [g]as suitably bounded but given with no further information, standard energyestimates admit but cannot exclude exponential growth of Qµ resp. Q′µ. Onemight be able to work out conditions on Rµν [g] which will allow one to keep the4errors under control. However, by itself this will be of limited use. Inserting(3.1) into (3.2) and, observing the splitting, into (3.4), gives∇µ ∇µ Qν + Qλ(∇(λ Qν) + Eλν) = −2∇µ(Eµν −12gµν Eρρ), (3.5)and∇µ ∇µ Q′ν + Q′λ(∇(λ Q′ν) + fλν) = 0, (3.6)respectively. The relative size of the errors Eλν and fλν is not clear. Moreimportant is that in this form the subsidiary system shows its non-linearity.If Q′µ and ∂t Q′ν vanish on S, equation (3.6) will still imply that Q′µ vanishes inthe domain of dependence. If Q′µ and ∂t Q′ν do not vanish on S, however, thefollowing observations indicate that (3.6) may imply a growth of the solution Q′µwhich is worth than exponential. We assume g = dt2−δab dxa dxb (so that (3.6)decouples from (3.3)), fλν = 0, and initial data which satisfy for a = 1, 2, 3,Q′a = 0, ∂tQ′a = 0, ∂a Q′0 = 0, ∂a ∂tQ′0 = 0 on {t = 0},so that the error resides only in the constant functions a = Q′0, b = ∂tQ′0 on{t = 0}.Equation (3.6) then implies Q′a ≡ 0 and reduces in fact to ∂t Q′0 = 12 (c −Q′20 ) with c = 2 b + a2. The integration gives Q′0 = a = const. if b = 0,Q′0 =√ca+√c tanh√c t2√c+a tanh√c t2if 0 6= 2 b > −a2, Q′0 = 2 aa t+2 if a2 = −2 b, andQ′0 =√|c| tan{−√|c| t2 + arctana√|c|}if 2 b < −a2. The solutions thus re-main bounded for t ≥ 0 if b ≥ 0 or if a ≥ 0 and −a2 < 2 b < 0, while theydevelop poles at some t∗ > 0 if b < 0 and a ≤ 0 or if 2 b < −a2 and a > 0.(If the corresponding initial data would be modified outside the intersection ofthe hypersurface {t = 0} with the backward light cone of the point (t∗, xa), thesolution would still become singular at (t∗, xa).)We note that in general any Killing vector field K of a vacuum solution gµνsatisfies equation (2.4) with Qµ = Kµ in the coordinates xµ. In the presentcase this gives solutions which grow linearly.For us the following observation is important:With our assumption on gµν it follows that each neighbourhood of the initialdata Qa = 0 and ∂tQa = 0 contains initial data for (2.4) for which the solutionQa(xµ) become unbounded at some finite x0 = t∗ > 0.54 Concluding remarksThe only purpose of the discussion above was to indicate the behaviour of solu-tions to the subsidiary system. We expect to find a similar singular behaviourof the solutions to equation (2.4) if the latter is given with a general smoothmetric gµν (extending sufficiently far into the future). It would be interestingto know whether the singular data set, i.e. the subset of data which determinesingular solutions of equation (2.4), is open or of lower dimension in the set ofall data if the metric g is given. While the singular data set itself will depend onthe metric g, such a characterization may be independent of the chosen metricand may help avoid entering the singular sector.In a consistent discussion of the growth of Qµ one would have to consider themain system and the subsidiary systemRµν = ∇(µ Qν), ∇µ ∇µ Qν + Qλ∇(λ Qν) = 0,as a coupled system. The main equation can be studied independently. Thesubsidiary system, implicit in the main system, depends on the metric definedby the latter. If Qµ tends to grow, the main system will react to it, whetherfor the better or the worth is not clear. Whether with changing gµν the non-linearity of the subsidiary system can still imply a blow up of Qµ at a finitetime needs to be analysed.For this purpose it might be interesting to study under simplifying assumptionssuch as spherical symmetry whether the model systemRµν = ∇(µ qν), ∇µ ∇µ qν + qλ∇(λ qν) = 0,considered as Einstein equations coupled to a source field given by a vector fieldqµ, will develop a blow up for suitable data. The reduction of these equationsis obtained by a slight modification of the one described above. The systemcan be simplified further by assuming qµ to be a differential qµ = ∇µ f of somefunction f . The second equation will then be implied if f satisfies∇µ ∇µf + ∇µf ∇µf = const.,which is a wave map equation in the case where the constant on the righthand side vanishes. In any case the results might lead to an identifcation ofa mechanism responsible for the growth of constraint violations and to thedevelopment of methods to avoid them.Some authors add terms built from Qν to the main system, which appear toreduce the growth of the constraint violations in certain calculation [5], [7]. Thismay be related to the different effects of the non-linearities which are obtainedin the appropriately modified subsidiary systems.There are available now many different types of reductions. Depending onseveral choices, the subsidiary system may control the preservation of the gaugeor the preservation of constraints or a mixture thereof. In spite of the differentappearance of the resulting main and subsidiary systems we expect that similar6non-linearities as the one discussed above will occur in any subsidiary system,though, depending on the system, they may have different effects.In a numerical scheme for the second order wave equations the subsidiary sys-tem, which is of third order in the metric, can, of course, hardly be identifiedany longer as a kind of identity and the relations between the two systems isobscured. But if the non-linearity of the subsidiary system can have for non-vanishing initial data Qµ and ∂t Qµ drastic effects in the continuum model, theyare likely to be reflected in numerical calculations.References[1] J. Frauendiener, T. Vogel. On the stability of constraint propagation.http://de.arXiv.org/abs/gr-qc/0410100[2] H. Friedrich. On the hyperbolicity of Einstein’s and other gauge field equa-tions. Comm. Math. Phys. 100 (1985) 525–543.[3] H. Friedrich. Hyperbolic Reductions for Einstein’s Equations. Class. Quan-tum Grav. 13 (1996) 1451–1469.[4] H. Friedrich, A. Rendall. The Cauchy Problem for the Einstein Equations.In: B. Schmidt (ed.): Einstein’s Field Equations and Their Physical Impli-cations. Springer, Lecture Notes in Physics, Berlin 2000.[5] C. Gundlach. Private communication.[6] L. Lehner, O. Reula. Status quo and open problems in the numerical con-struction of space-times. In: P. T. Chruściel, H. Friedrich (eds.): TheEinstein equations and the large scale behaviour of gravitational fields.Birkhäuser, Basel, 2004.[7] F. Pretorius. Private communication.[8] G. Yoneda, H. Shinkai. Diagonalizability of constraint propagation matri-ces. Class. Quantum Grav. 20 (2003) L31 - L36.7

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