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The Einstein-Vlasov system in spherical symmetry II: spherical perturbations of staticsolutions
Carsten Gundlach(Dated: 24 August 2017)
We reduce the equations governing the spherically symmetric perturbations of static sphericallysymmetric solutions of the Einstein-Vlasov system (with either massive or massless particles) to asingle stratified wave equation ,tt = H, with H containing second derivatives in radius, andintegrals over energy and angular momentum. We identify an inner product with respect to whichH is symmetric, and use the Ritz method to approximate the lowest eigenvalues of H numerically.For two representative background solutions with massless particles we find a single unstable modewith a growth rate consistent with the universal one found by Akbarian and Choptuik in nonlinearnumerical time evolutions.
I. Introduction 1A. The Einstein-Vlasov system 1B. Motivation for this paper 1C. Plan of the paper 2
II. Background equations 2A. Field equations in spherical symmetry 2B. Static solutions 3C. Reduction of the field equations for m = 0 4
III. Spherical perturbation equations 4A. Perturbation ansatz 4B. Change of variable from z to Q 5C. Static perturbations 5D. Reduction to a single stratified wave
equation 5E. Preliminary classification of non-static
perturbations 6F. Massless case 7
IV. Perturbation spectrum 7A. Inner product 7B. Quadratic form of the Hamiltonian 8C. Ritz method 9D. The space of test functions 9E. Numerical examples of the Ritz method for
massless particles 10
V. Conclusions 12
A. The Einstein-Vlasov system
The Vlasov-Einstein system describes an ensemble ofparticles of identical rest mass, each of which follows ageodesic. The particles interact with each other only
through the spacetime curvature generated by their col-lective stress-energy tensor, whereas particle collisionsare neglected.
For massive particles, this is a good physical modelof a stellar cluster. For either massive or massless par-ticles, the Einstein-Vlasov system also serves as a well-behaved toy model of matter in general relativity. In par-ticular, spherically symmetric solutions of the Einstein-Vlasov system with small data are known to exist glob-ally in time for massive  and massless particles .Self-similar spherically symmetric solutions with mass-less particles have been analyzed in  and . Theexistence of spherically symmetric static solutions withmassive particles was proved in , and there is numeri-cal evidence that at least some are stable within spheri-cal symmetry . Spherically symmetric static solutionswith massless particles were analysed and constructednumerically in , and we investigate their linear per-turbations here. See also  for a review and additionalreferences.
B. Motivation for this paper
This is the second paper in a series motivated byAkbarian and Choptuiks  (from now on, AC) re-cent study of numerical time evolutions of the masslessEinstein-Vlasov system in spherical symmetry. AC foundtwo apparently contradictory results:
I) Taking several 1-parameter families of genericsmooth initial data and fine-tuning the parameter tothe threshold of black hole formation, AC found what isknown as type-I critical collapse: in the fine-tuning limitthe time evolution goes through an intermediate staticsolution. The lifetime of this static solution increaseswith fine-tuning to the collapse threshold as
' ln |p p|+ const, (1)
where p is the parameter of the family, p its value atthe black-hole threshold, and is the proper time atthe centre, in units of the total mass of the critical so-lution. (1) implies the existence of a single unstablemode growing as exp(/). AC found that was ap-proximately universal (independent of the family), with
value ' 1.4 0.1, and that the metric of the inter-mediate static solution was also approximately universal(up to an overall length and mass scale). In particu-lar, its compactness := max(2M/r) was in the range ' 0.800.01 and its central redshift Zc 0 was in therange Zc ' 2.45 0.05.
II) Conversely, constructing static solutions by ansatz,AC found that these covered much larger ranges of and Zc, but that each one was at the threshold of col-lapse. That is, adding a small generic perturbation to thestatic initial data and evolving in time with their nonlin-ear code, they found that for one sign of the perturbationthe perturbed static solution collapsed while for the othersign it dispersed. They found that was in the narrowrange ' 1.43 0.07, compatible with the value above.
Result II suggests that in spherical symmetry withmassless particles, the black hole-threshold coincideswith the space of static solutions. If so, then each staticsolution would have precisely one unstable mode (withits sign deciding between collapse and dispersion), withall other modes either zero modes (moving to a neigh-bouring static solution) or purely oscillating.
One aspect of Result I, namely that the spacetime ofthe critical solution is universal, would imply that thisuniversal solution has one unstable mode (as before), butthat all its other modes (including those tangential to theblack hole threshold) are decaying ones, so that the at-tracting manifold of the critical solution is precisely theblack hole threshold. Indeed, this is the familiar pic-ture of type-I critical collapse in other matter-Einsteinsystems. However, this is in apparent contradiction toResult II.
In the first paper in this series  (from now PaperI), we used a symmetry of the massless spherically sym-metric Einstein-Vlasov system to reduce its number ofindependent variables from four to three. We then nu-merically constructed static solutions with compactnessin the range 0.7 ' 8/9. Based on this, we conjec-tured that the apparent contradiction above is resolvedby the critical solution seen in fine-tuning generic initialdata being universal only to leading order, and that thisleading order is selected by the way in which it is ap-proached during the evolution of generic smooth initialdata.
To make further progress, it seems essential to analysethe spectrum of linear perturbations directly. This is theprogramme of the current paper. In contrast to the staticsolutions investigated in Paper I, their perturbations donot simplify significantly for m = 0, and hence all of ouranalysis, except for the numerical examples in Sec. IV E,will be for m 0.
C. Plan of the paper
In order to make the presentation self-contained andto establish notation, we review some material from Pa-per I in Sec. II. We begin in Sec. II A by presenting the
equations of the time-dependent spherically symmetricEinstein-Vlasov system. We do this in a form in whichthe massless particle limit is regular and leads to a reduc-tion of the number of independent variables. We discussstatic solutions in Sec. II B, and the massless limit, forboth the time-dependent and static case, in Sec. II C.
In Sec. III we then derive the spherical perturbationequations. In Sec. III A we perturb the Vlasov and Ein-stein equations about a static background, splitting theperturbation of the Vlasov distribution function f intoparts and that are even and odd, respectively un-der reversing time. In Sec. III B we change independentvariables from momentum to energy, as we did for thebackground solutions. We quickly dispense with staticperturbations in Sec. III C, and in Sec. III D we reducethe perturbed Vlasov and Einstein equations to a singleintegral-differential equation ,tt = H. In Sec. III Ewe dispense with the relatively trivial perturbations onregions of phase space where the background solution isvacuum. We state the massless limit in Sec. III F.
In Sec. IV we attempt to find the spectrum of eigen-values. In Sec. IV A we identify a positive definite in-ner product with respect to which H is symmetric. InSec. IV B we rewrite the Hamiltonian as H = AADDwhere D is bounded, giving us at least a lower bound onH. We then switch to an approximation method, theRitz method, which we review in Sec. IV C. In Sec. IV Dwe specify some properties of the function space V inwhich to look for perturbation modes, that is eigenfunc-tions of H. In Sec. IV E we pick two specific backgroundsolutions that we obtained numerically in Paper I anduse the Ritz method numerically. We find values of inagreement with AC.
Sec. V contains a summary and outlook. Throughoutthe paper, a := b defines a, and we use units such thatc = G = 1.
II. BACKGROUND EQUATIONS
A. Field equations in spherical symmetry
We consider the Einstein-Vlasov system in sphericalsymmetry, with particles of mass m 0. We write themetric as
ds2 = 2(t, r)dt2+a2(t, r)dr2+r2(d2+sin2 d2). (2)
To fix the remaining gauge freedom we set (t,) = 1.The Einstein equations give the following equations forthe first derivatives of the metric coefficients:
2r+ 4ra2p, (3)
= a2 12r
+ 4ra2, (4)
= 4ra2j, (5)
where p, and j are the radial pressure, energy densityand radial momentum density, measured by observers atconstant r. The fourth Einstein equation, involving thetangential pressure pT , is a combination of derivativesof these three, and is redundant modulo stress-energyconservation.
The Vlasov density describing collisionless matter ingeneral relativity is defined on the mass shell pp =m2 of the cotangent bundle of spacetime, or f(t, xi, pi)in coordinates. We define the square of particle angularmomentum
F := p2 + sin2 p2. (6)
In spherical symmetry this is conse