We perform numerical simulations of the critical gravitational collapse of a
spherically symmetric scalar field in 6 dimensions. The critical solution has
discrete self-similarity. We find the critical exponent \gamma and the
self-similarity period \Delta.Comment: 8 pages, 3 figures RevTe

C. Gundlach

Curt Cutler

D. Christodoulou

D. Garfinkle

D. S. Goldwirth

David Garfinkle

G. Comer Duncan

M. Choptuik

R. Myers

Y. Peleg

English

arXiv

Crossref

Choptuik scaling in six dimensions

We perform numerical simulations of the critical gravitational collapse of a spherically symmetric scalar field in 6 dimensions. The critical solution has discrete self-similarity. We find the critical exponent γ and the self-similarity period Δ

Garfinkle, D.

Cutler, C.

Comer, D.

MPG.PuRe

PHYSICAL REVIEW D, VOLUME 60, 104007Choptuik scaling in six dimensionsDavid Garfinkle*Department of Physics, Oakland University, Rochester, Michigan 48309Curt Cutler†Max Planck Institute for Gravitational Physics, D-14476 Golm, GermanyG. Comer Duncan‡Department of Physics and Astronomy, Bowling Green State University, Bowling Green, Ohio 43403~Received 13 August 1999; published 14 October 1999!We perform numerical simulations of the critical gravitational collapse of a spherically symmetric scalarfield in 6 dimensions. The critical solution has discrete self-similarity. We find the critical exponentg and theself-similarity periodD. @S0556-2821~99!00424-5#PACS number~s!: 04.25.Dm, 04.40.2b, 04.50.1h, 04.70.Bwdckalepetaleck--oinesinneceis-nthriesit-e-mtricI. INTRODUCTIONCritical behavior in gravitational collapse, as first founby Choptuik @1#, occurs at and near the threshold of blahole formation@2#. For a one-parameter family of initial datslightly above the threshold, the mass of the black hoMBH , scales like (p2p*)g. Herep is the parameter,p* isits critical value andg is a constant that depends on the tyof matter, but not on the family of data. For initial daslightly below the threshold, the maximum curvature scalike (p* 2p)22g whereg is the same constant as in the blahole mass scaling law@3#. @The definition of the scaling exponentg must be slightly generalized inspacetime dimen-sions, whereMBH has dimension (length)n23; see Sec. II.#The critical solution (p5p*) has either continuous self similarity or discrete self similarity, depending on the typematter.While critical gravitational collapse has been studiedmany types of matter, the work has, in general, been don4 spacetime dimensions.~The exception is work on analogof gravitational collapse in 3 spacetime dimensions@4#.! Onemight therefore wonder whether critical behavior occursgravitational collapse inn spacetime dimensions forn.4,and if so, how the properties of the critical behavior depeon n.In this work, we perform numerical simulations of thcollapse of a spherically symmetric scalar field in 6 spatime dimensions. We find that the critical solution has dcrete self-similarity. We find the scaling exponentg and theself-similarity periodD. Section II reviews the Schwarzschild solution inn dimensions and shows how the definitioof theg exponent should be generalized to then-dimensionalcase. Section III gives the equations for the evolution ofscalar field and the metric in a form suitable for our numecal simulations. The numerical method is presented in SIV and our results in Sec. V.*Email address: garfinkl@oakland.edu†Email address: cutler@aei-potsdam.mpg.de‡Email address: gcd@chandra.bgsu.edu0556-2821/99/60~10!/104007~5!/$15.00 60 1040,sfind--e-c.II. SCHWARZSCHILD METRIC IN N DIMENSIONSThe Schwarzschild metric in spacetime dimensions i@5#ds252S 12 16p~n22!An22 Mr n23D dt21S 12 16p~n22!An22 Mr n23D21dr21r 2dVn222 ~1!whereAn2252p(n21)/2/G@(n21)/2# is the area of the unit(n22)-sphere. The quantityM in the metric~1! is the space-time mass. M is normalized so that when Eq.~1! is the ex-terior metric of a nearly Newtonian static fluid ball, thenM is*T00dV. With this normalization, the force on a static unmass particle at distancer approaches@8p(n23)/(n22)An22#M /rn24 as r→`.It is important to note thatM has dimension (length)n23.This implies the following scaling behaviors for critical phnomena inn dimensions. If below threshold the maximucurvature @which has dimension (length)22] scales like(p* 2p)22g, then above thresholdMBH scales as (p2p*) (n23)g.III. EVOLUTION EQUATIONSOur matter model is the one used in Ref.@1#: a massless,minimally coupled, self gravitating scalar fieldf. The stress-energy of the field isTab5¹af¹bf212¹cf¹cfgab . ~2!In n spacetime dimensions, a spherically symmetric mehas the formds252e2ndu222en1ldudr1r 2dS2 ~3!wheredS2 is the metric of the unit (n22)-sphere andn andl are functions ofu andr. The coordinater is a generaliza-tion of the usual area radius. The coordinateu is constant on©1999 The American Physical Society07-1t orualdqule-ne-rtheor-er:iesinon.ricalduy-ofin-ince.eon-itccu--DAVID GARFINKLE, CURT CUTLER, AND G. COMER DUNCAN PHYSICAL REVIEW D60 104007outgoing radial null geodesics. Here, we also require thathe world line of the central observer,u is equal to the propetime of that observer. This gives rise to the conditionn50 atr 50. DefineEab[Gab28pTab whereGab is the Einsteintensor, so the Einstein-scalar equations are justEab50. Notethat as a consequence of the Bianchi identities,f must sat-isfy the wave equation.In 4 spacetime dimensions, Christodoulou@8# showedhow to write the spherically symmetric Einstein-scalar eqtions as an integro-differential equation for the scalar fieWe now generalize the method of Ref.@8# to then-dimensional case. Define the null vectorsl a andna byl a5e2lS ]]r Da~4!na5e2nS ]]uDa212e2lS ]]r Da. ~5!Then some straightforward but tedious calculations giveGablal b5n22re22l]]r~l1n!, ~6!Gablanb5n222 Fn23r 2 ~12e22l!2 e22lr]]r~n2l!G .~7!One can show that the equationsEablal b50, Eablanb50, ¹a¹af50 ~8!plus the condition of regularity at the origin implyEab50everywhere. Thus we need to impose only those three etions.Using Eqs.~6!, ~7! and ~2!, we find that the first two ofEqs.~8! yieldn22r]]r~l1n!28pS ]f]r D250 ~9!22n2r Fe22l ]]r ~n2l!2n23r ~12e22l!G50. ~10!Defineg[en1l and ḡ[en2l. Then the solution of Eqs.~9!and ~10! isg5expF 8pn22E0r r̂ S ]f] r̂ D2dr̂G ~11!ḡ5n23r n23E0rr̂ n24g~ r̂ !dr̂. ~12!The wave equation forf in this metric is2]2f]u]r1n22r]f]u2r 22n]]r S r n22ḡ]f]r D50. ~13!10400n-.a-In principle, Eqs.~11!–~13! could be used for a numericatreatment of the Einstein-scalar equations. Givenf on aninitial light cone, Eqs.~11! and~12! could be used to find themetric on that light cone. Equation~13! could then be used toevolve f to a nearby light cone. However, in 4 spacetimdimensions, Christodoulou@8# finds that a change of variables gives rise to a nicer form of the wave equation. Defithe operatorD by D[enna¹a . ThenD is a derivative alongingoing light rays and we haveD5]]u212ḡ]]r. ~14!Now in 4 spacetime dimensions defineh[(]/]r )(rf). Thenthe wave Eq.~13! becomesDh512r~g2ḡ!~h2h̄! ~15!whereh̄[r 21*0r hdr. For the purposes of numerical simulations, the nice property of Eq.~15! is that it involves a de-rivative only in the direction of ingoing light rays. Anothenice feature of Eq.~15! is that in the absence of gravity (g5ḡ51), the right hand side vanishes. This is related tofact that in general, the right hand side vanishes liker 2 asr→0. ~The metric is smooth, and therefore in a neighbhood of the origin it behaves like a flat metric to some ordin r.! This property is important for the following reasonspherically symmetric metrics have coordinate singularitat the origin. These coordinate singularities are reflectedthe appearance of inverse powers ofr in Eq. ~13!, which canlead to instabilities or inaccuracies in a numerical simulatiThese considerations suggest that instead of a numesimulation of Eq.~13! in n dimensions, we should insteasearch for a new variableh that satisfiesDh50 in the ab-sence of gravity. Note that a quantity that satisfiesDh50 isconstant along ingoing light rays and therefore satisfies Hgens’ principle. However, it is well known that solutionsthe wave equation in flat spacetime satisfy Huygens’ prciple only in even spacetime dimensions@9#. Therefore, weshould only expect to find an appropriate new variableeven spacetime dimensions. Letn be even and definem[(n22)/2. For anyf defineh[~m21!!~2m21!!r 12mS ]]r Dm~r 2m21f!. ~16!~Here, the numerical factor is chosen for later convenien!If f is a solution of the flat space wave equation@Eq. ~13!with ḡ51], thenh satisfiesDh50. That is, in the absencof gravity, a solution of the wave equationf gives rise to anh that is constant along ingoing light rays. One can demstrate this property ofh by using Eq.~13! ~with ḡ51) andmathematical induction onm.The advantage of usingh as the basic variable is thattends to make computer simulations more stable and arate, especially near the origin. The disadvantages are~1! themethod only works in even dimensions,~2! a separate com7-2CHOPTUIK SCALING IN SIX DIMENSIONS PHYSICAL REVIEW D60 104007FIG. 1. h plotted vs the rescaled coordinatesRandT.et,avethinare-eanfnedputer code must be written for each value ofn and ~3! asngets larger, the equations become more complicated. Wepect that some numerical technique can be used to evolveEinstein-scalar equations inn dimensions with a single codewith n as a free parameter, stably and with enough accurto treat critical gravitational collapse. However, we habeen unable to devise such a technique. Therefore, inwork we useh as our basic variable. Due to the increasecomplication with increasingn, we treat only the case of 6spacetime dimensions.We now specialize to the case of 6 dimensions. The vableh is given byh[16r]2]r 2~r 3f!. ~17!We must now use Eq.~11! to expressg in terms ofh andintegrals involvingh. Define the quantities and s̄ bys[2r 2E0rr̂ h~ r̂ !dr̂, ~18!s̄[3r 3E0rr̂ 2s~ r̂ !dr̂. ~19!Then, using Eqs.~17!–~19! and ~11! we findg5expF18pE0r dr̂r̂~s2 s̄!2G . ~20!Equation ~12! specialized to 6 spacetime dimensions bcomesḡ53r 3E0rr̂ 2g~ r̂ !dr̂. ~21!10400x-hecyisi--Using Eqs.~17!–~21! in Eq. ~13! we find that the wave equation for f in 6 spacetime dimensions becomesDh53r~g2ḡ!~h1 s̄22s!127p2gr~s2 s̄!3. ~22!Equations~17!–~22! are the full set of equations that arevolved in 6 spacetime dimensions.IV. NUMERICAL METHODThe numerical method used is that of Garfinkle@6#. Thismethod is based on numerical work of Goldwirth and Pir@7#, which is in turn based on the analytical work oChristodoulou@8#. The spatial grid is a set of points on aoutgoing null cone, and each spatial grid point is evolvalong an ingoing light ray. Givenh on a light cone, the codeFIG. 2. h plotted vsR at two times when the minimum ofh is atr 50. Note that the two curves agree.7-3aeDAVID GARFINKLE, CURT CUTLER, AND G. COMER DUNCAN PHYSICAL REVIEW D60 104007FIG. 3. lnRmax plotted vs ln(p*2p) alongwith the best straight line fit. The curve isstraight line with a periodic wiggle and the slopof the line is22g.rietiofugrtsteotehesdeonfoty-oflftheeva-i-eangaheateol-e totu-se-ofpu-performs several integrals in turn: Eqs.~18!–~21!. Near theorigin, these integrals are approximated by a Taylor seand evaluated using the slope ofh at r 50. At all otherpoints, the integrals are evaluated using Simpson’s rule.For each time step the quantityh is evolved using Eq.~22!and the quantityr is evolved usingDr 52ḡ2. ~23!These are essentially a set of uncoupled ordinary differenequations~ODEs!, one for each grid point. In the scheme@6# as the evolution proceeds grid points that pass thror 50 are lost. When half of the grid points are lost, they aput back, interpolated between the remaining grid poinThe critical solution is found by a search of a 1-paramespace of evolved data to find the boundary between thdata that form black holes and those that do not. The oumost gridpoint is chosen to be the light ray that hits tsingularity of the critical solution. This choice maintainresolution throughout the evolution.V. RESULTSAll runs were done with 300 spatial grid points. The cowas run in quadruple precision on Dec alpha workstatiand in double precision on a Cray YMP8. The initial datathe scalar field were chosen to be of the formf~0,r !5pr2exp@2~r 2r 0!2/s2#. ~24!Here, p is our parameter, andr 0 and s are constants. Thecritical solution was found to have discrete self-similariLet u* be the value ofu at the origin at the singularity of thecritical solution. Define the coordinatesT[2 ln(u*2u) andR[reT. Then discrete self similarity means thath, consid-ered as a function ofT and R, is periodic in T. Figure 1showsh plotted as a function ofT and R. Note that aftersome initial transient behavior,h becomes periodic. To ex10400salhe.rser-sr.amine the periodicity more closely, in Fig. 2 we ploth vs Rat two different times where the minimum ofh occurs atr50. Note that the curves agree, demonstrating periodicityh. The period ofh is D53.03. Thus, in each period a typicalength of the system shrinks by a factor ofeD'21. For com-parison, recall that forn54, D53.445.To compute the scaling exponentg we use the method oRef. @3#. We evolve data for a range of parameters belowthreshold of black hole formation. For each evolution wfind the maximum of the absolute value of the scalar curture on the world line of the central observerRmax. Plottingln Rmax vs ln(p*2p) the result is a straight line with a perodic wiggle, where the slope of the line is22g. We use 50values ofp equally spaced in ln(p*2p). Figure 3 is a plot ofln Rmax vs ln(p*2p) along with the best straight line fit to thpoints. The slope allows us to findg. The result isg50.424, soMBH}(p2p*)1.27. Recall that forn54, g50.374. Note thatg can also be found by evolving databove the threshold of black hole formation and plottiln MBH vs ln(p2p*). We have done this and the result isstraight line with a periodic wiggle where the slope of tline is '1.27. However, our method gives a more accurtreatment of subcritical collapse than of supercritical clapse. For that reason, we have used subcritical collapscalculateg.Comparison of our results for 6 dimensions with Chopik’s results for 4 dimensions@1# seems to indicate thatD is adecreasing function of the dimensionalityn of spacetime.The quantityg seems to be an increasing function ofn. Itwould be interesting to obtain more information about thefunctions by studying critical collapse for other values ofn.ACKNOWLEDGMENTSThis work was partially supported by NSF grant PHY9722039 to Oakland University. We acknowledge the usethe Ohio Supercomputer Center, where some of the comtations were performed.7-4-CHOPTUIK SCALING IN SIX DIMENSIONS PHYSICAL REVIEW D60 104007@1# M. Choptuik, Phys. Rev. Lett.70, 9 ~1993!.@2# For a review see C. Gundlach, Adv. Theor. Math. Phys.2, 1~1998! and references therein.@3# D. Garfinkle and G. C. Duncan, Phys. Rev. D58, 064024~1998!.@4# Y. Peleg and A. Steif, Phys. Rev. D51, 3992~1995!.@5# R. Myers and M. Perry, Ann. Phys.~N.Y.! 172, 304 ~1986!.10400@6# D. Garfinkle, Phys. Rev. D51, 5558~1995!.@7# D. S. Goldwirth and T. Piran, Phys. Rev. D36, 3575~1987!.@8# D. Christodoulou, Commun. Math. Phys.105, 337 ~1986!.@9# Y. Choquet-Bruhat, C. De Witt-Morette, and M. DillardBleick, Analysis, Manifolds and Physics~North Holland, Am-sterdam, 1977!, p. 432.7-5

Garfinkle, David

Cutler, Curt

Comer, Duncan

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Choptuik scaling in six dimensions

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