Type II critical behavior of gravitating magnetic monopoles
TType II critical behavior of gravitating magnetic monopoles
Ben Kain
Department of Physics, College of the Holy Cross, Worcester, Massachusetts 01610, USA
I study type II critical collapse in the spherically symmetric gravitating magnetic monopole system.This is an Einstein-Yang-Mills-Higgs system with two matter fields: a field parametrizing the scalarfield gauged under SU (2) and a field parametrizing the gauge field. This system offers interestingdifferences compared to what is commonly found for type II collapse in other systems. For example,instead of the critical solution sitting between collapse and complete dispersal of the matter fields,on the non-black hole side of the critical solution, the matter fields settle down to a static andstable configuration. More interesting, however, is that I find strong evidence for the existence oftwo critical solutions, each with their own set of scaling and echoing exponents, which I determinenumerically. I. INTRODUCTION
Critical gravitational phenomena was first discov-ered by Choptuik [1] in the form known as type II.In type II, a spacetime evolves such that it even-tually leads to the formation of a black hole ordoes not. For example, consider regular initial data,parametrized by a single parameter, p , such that for p > p ∗ the spacetime dynamically evolves from theinitial data to a spacetime containing a black hole,while for p < p ∗ a black hole does not form andthe matter fields disperse to infinity. The spacetimewith p = p ∗ is called the critical solution and theremarkable behavior at or near p ∗ is what is meantby type II critical phenomena.What is remarkable is that near-critical space-times, i.e. spacetimes for which p is near p ∗ , exhibitself-similarity. In the case of discrete self-similarity,a scale invariant function, Z , obeys Z ( τ + ∆ , ln r + ∆) = Z ( τ, ln r ) , (1)where the echoing exponent, ∆, is universal, in thatit is independent of initial data. In the above equa-tion τ = ln( T ∗ − T ), where T is the central propertime (i.e. the proper time at the origin) and T ∗ is aconstant called the accumulation time. Also remark-able is that the mass of the black hole at collapseobeys the scaling relation m BH ∼ | p − p ∗ | γ , (2)where the scaling exponent, γ , like the echoing ex-ponent, is universal. This scaling relation indicatesthat in type II collapse a black hole can form witharbitrarily small mass. Gundlach [2] and Hod andPiran [3] showed that the scaling relation (2) is nota strict proportionality, but on top of the linear re-lationship is a periodic wiggle with period ∆ / (2 γ ).The scaling relation (2) applies only to supercrit-ical evolutions, i.e. evolutions during which a blackhole forms. Scaling relations for subcritical evolu-tions, i.e. evolutions during which a black hole does not form, are known and offer additional means fordetermining γ . Garfinkle and Duncan [4] showedthat the maximum value over the total evolution ofthe central value of the Ricci scalar can obey R ≡ max t R µµ ( t, ∼ | p − p ∗ | − γ , (3)where R µν is the Ricci tensor and R µµ is the Ricciscalar. In some systems the above formula is not use-ful. For example, in the Einstein-Yang-Mills systemwith SU (2) the Ricci scalar vanishes. Garfinkle andDuncan suggested other possibilities [4], the simplestof which is R ≡ max t | R µν ( t, R µν ( t, | / ∼ | p − p ∗ | − γ . (4)They further argued that the scaling relations (3)and (4), like the black hole mass scaling (2), shouldhave a periodic wiggle on top of the linear relation-ship, again with period ∆ / (2 γ ).In addition to type II, there is type I and type IIIcritical phenomena. In type I, originally discoveredby Choptuik, Chmaj, and Bizo´n in their study ofgravitational SU (2) [5], one again considers single-parameter initial data that evolves either to a space-time containing a black hole or to one that does not.In this case, however, the black hole must form withfinite mass. Further, the critical solution is a staticgravitational solution with a single decay mode. Forexample, for the SU (2) system studied in [5], thecritical solution is the n = 1 Bartnik-McKinnon so-lution [6]. The closer the evolution is to the criticalsolution, i.e. the closer p is to p ∗ , the longer the evo-lution spends near the static critical solution beforeevolving away to one of its two possible end states.Type III critical phenomena was discovered byChoptuik, Hirschmann, and Marsa [7], again in astudy of gravitational SU (2). In this case, bothend states of the evolution contain a black hole, butthe final system is distinctly different depending onwhether p > p ∗ or p < p ∗ . Type III shares similari-ties with type I in that the critical solution is a static a r X i v : . [ g r- q c ] M a y gravitational solution with a single decay mode (butin this case the static solution contains a black hole)and the closer the evolution is to the critical solu-tion, the longer the evolution spends near the staticsolution before evolving away to one of its two pos-sible end states. For SU (2) [7, 8], the critical solu-tions are the colored black hole static solutions [9–11]. For reviews of gravitational critical phenomenasee those by Gundlach et al. [12, 13] and for studiesof the critical behavior of gravitational SU (2) see[5, 7, 8, 14–19].In this paper I study the gravitating ’t Hooft-Polyakov magnetic monopole system: sphericallysymmetric SU (2) with a scalar field in the adjointrepresentation coupled to gravity [20–22]. I pre-viously studied this Einstein-Yang-Mills-Higgs sys-tem in [18] with respect to type III critical behav-ior. The well-known solutions for static gravitatingmonopoles [23–26] include both stable and unsta-ble black hole monopole solutions and the Reissner-Nordstr¨om solution. I showed in [18] that the un-stable static black hole monopole solutions are typeIII critical solutions with the stable static blackhole monopole solutions and the static Reissner-Nordstr¨om solution as the two possible end states.There exist regular solutions for excited staticgravitating monopoles [25] which are expected to beunstable. Further, if the vacuum value of the scalarfield is sufficiently large, a branch of unstable fun-damental regular static solutions appear [27]. Bothof these are good candidates for a type I critical so-lution and it would be interesting to study type Icritical phenomena in this system.My focus in this work is on type II critical be-havior of gravitating monopoles. This system offersinteresting differences compared to type II collapsefound in other systems. For example, instead ofthe critical solution sitting between a black hole andcomplete dispersal of the matter fields, the matterfields on the non-black hole side do not completelydisperse, but instead settle down to a stable andstatic gravitating monopole [18, 27].More interesting is that the monopole system ap-pears to contain two type II critical solutions, eachwith their own set of scaling and echoing exponents.I note, however, that one solution is more exact thanthe other. For the solution I present first, near-critical evolutions exhibit precise self-similarity and,within the scope of initial data that leads to thecritical solution, universal scaling and echoing ex-ponents. The second solution has all the standardsigns for a type II critical solution, but the scalingand echoing exponents have a small spread in val-ues over different initial data and the self-similarityof near-critical evolutions is not as precise. For thissecond solution, then, it might be that exact self- similarity and universality are lost, a possibility thathas also been seen recently in pure SU (2) by Mali-borski and Rinne [17].In the next section I present equations, boundaryconditions, and aspects of the code I use to studytype II collapse. In Sec. III I present the first criticalsolution and in Sec. IV I present the second criticalsolution. I conclude in Sec. V. II. EQUATIONS, BOUNDARYCONDITIONS, AND NUMERICS
I gave the full set of equations for the sphericallysymmetric gravitating monopole in [18]. I quicklylist the equations here and I refer the reader to [18]for additional information. All results will be pre-sented in radial-polar gauge. This gauge has beenused in many studies of type II critical phenomena,including the original study [1] and the first study ofpure SU (2) [5]. In this gauge, the spherically sym-metric metric takes a particularly simple form: ds = − α dt + a dr + r (cid:0) dθ + sin θdφ (cid:1) (5)(here and throughout I set c = 1), where the metricis parametrized in terms of the lapse α ( t, r ) and themetric function a ( t, r ).The matter sector contains two fields: a real scalarfield, ϕ , which parametrizes the real triplet scalarfield gauged under SU (2), and what is effectively areal scalar field, w , which parametrizes the gaugefield. That there is only one field parametrizing thegauge field is because the monopole system is withinwhat is called the magnetic ansatz (see, for example,[7, 18] for details). For simplicity I shall refer to ϕ as the scalar field and w as the gauge field. Fromthese follow the auxiliary fields:Φ( t, r ) = ∂ r ϕ Π( t, r ) = aα ∂ t ϕQ ( t, r ) = ∂ r w P ( t, r ) = aα ∂ t w. (6)From the Einstein field equations, the metric func-tions obey the constraint equations ∂ r aa = 4 πGra ρ − a − r∂ r αα = 4 πGra S rr + a − r , (7)where G is the gravitational constant and ρ = Φ + Π a + w ϕ r + V + (1 − w ) g r + Q + P g a r S rr = Φ + Π a − w ϕ r − V − (1 − w ) g r + Q + P g a r (8)follow from the energy-momentum tensor. V is thescalar potential, which I give below, and g is thegauge coupling constant. The equations of motionfor the matter fields are ∂ t ϕ = αa Π ∂ t Φ = ∂ r (cid:16) αa Π (cid:17) ∂ t Π = 1 r ∂ r (cid:18) αr a Φ (cid:19) − αa ∂V∂ϕ − αar w ϕ∂ t w = αa P∂ t Q = ∂ r (cid:16) αa P (cid:17) ∂ t P = ∂ r (cid:16) αa Q (cid:17) + αar w (1 − w ) − g αawϕ . (9)For the matter fields, the inner boundary condi-tions are ϕ = O ( r ) Φ = O (1) Π = O ( r ) w = 1 + O ( r ) Q = O ( r ) P = O ( r ) (10)and the outer boundary conditions are ϕ ( t, ∞ ) = ± v , with the rest of the matter fields vanishing atinfinity, where v is the vacuum value of the scalarfield. The inner boundary condition for a is a =1 + O ( r ). The inner boundary condition for α isgauge dependent and I shall fix α = 1 at the origin,which is a standard gauge choice in studies of typeII critical phenomena.To determine the scaling exponent from the blackhole mass scaling law (2), which I’ll label as γ m , Ineed to know the black hole mass at the moment ofcollapse. The total mass inside a radius r is givenby m ( t, r ) = r G (cid:20) − a ( t, r ) (cid:21) , (11)which I can use to determine the black hole massat collapse if I know the horizon radius at collapse.Since coordinates in radial-polar gauge do not pen-etrate apparent horizons, I cannot use an apparenthorizon finder to find the radius. As is standard,I take a spike in the metric function a to indicatecollapse and its position to be the horizon radius.The Ricci scalar scaling law (3) is not entirely use-ful for determining the scaling exponent, which whendetermined from (3) I’ll label as γ R . I mentionedthat the Ricci scalar vanishes in pure SU (2). Notsurprisingly, something similar happens in the grav-itating monopole system. Starting with the Einsteinfield equations, it is not hard to show that R µµ = − πGT µµ , where T µν is the energy-momentum ten-sor (its components are given in [18]). In the gravi- tating monopole system it can be shown that T µµ = Π − Φ a − w ϕ r = − + O ( r ) , (12)where I ignored the scalar potential and where thesecond equality is only valid near the origin. Wefind that the central value of T µµ , and hence also thecentral value of the Ricci scalar, only probes directlythe scalar field and not the gauge field. Below Ishall report the value of γ R , but we should not besurprised if it does not equal γ m .The value of the scaling exponent that followsfrom the R scaling law (4), which I’ll label as γ R ,is much better adapted for the gravitating monopolesystem, just as it is for pure SU (2) [17]. Start-ing again from the Einstein field equations, one canshow that R µν R µν = (8 πG ) T µν T µν and furtherthat T µν T µν depends explicitly on both ϕ and w near the origin. The formula for T µν T µν is compli-cated and I do not present it here, but the pointis that we should expect γ R to agree with γ m (atleast, for typical type II behavior).The code I use is the same code used in [18], butwith three changes. First, I use radial-polar gauge(instead of radial-maximal gauge) and second, I in-clude Kreiss-Oliger dissipation [28] to help with sta-bility. The third, and most important, change hasto do with the computational grid. Finding a typeII critical solution requires code that can probe veryclose to the origin. The usual best method for do-ing this is an adaptive mesh [1, 5], but this can bechallenging to implement. A simpler alternative isto use a fixed, but nonuniform computational grid(for examples, see [17, 29]). I use the nonuniformgrid used by Akbarian and Choptuik in [29]: r = e x − e x min + x max x max − x − x max x max − x min , (13)which maps the uniform computational domain x =( x min , x max ) to the nonuniform radial domain r =(0 , ∞ ). The results in this paper are for x min = − x max = 4, which shrinks the innermost gridpoint by 2 orders of magnitude compared to the uni-form grid, and 2011 grid points.To test the universality of the scaling and echo-ing exponents, I use various families of initial data.Some of the initial data I’ve used is ϕ (0 , r ) = v tanh ( r/s )+ c rr (cid:104) e − ( r − r ) /d + e − ( r + r ) /d (cid:105) (14a) ϕ (0 , r ) = v ( r/s ) − r/s ( r/s ) + c (14b) ϕ (0 , r ) = v (cid:40) − (cid:20) a (cid:18) brs (cid:19) e − r/s ) (cid:21) × tanh (cid:18) r − rs (cid:19)(cid:41) (14c)and w (0 , r ) = 1 − tanh ( r/s ) + c (cid:18) rr (cid:19) e − ( r − r ) /d (15a) w (0 , r ) = c − ( r/s ) c + ( r/s ) (15b) w (0 , r ) = 12 (cid:40) (cid:20) a (cid:18) brs (cid:19) e − r/s ) (cid:21) × tanh (cid:18) r − rs (cid:19)(cid:41) , (15c)along with ∂ t ϕ (0 , r ) = ∂ t w (0 , r ) = 0. In the aboveequations s , r , c , and d are constants and a and b are chosen such that the inner boundary conditionsare satisfied and are given by a = coth( r /s ) − b = coth( r /s ) + 1. Initial data (14a) and (15a) takesimple functions that satisfy the boundary condi-tions and add to them Gaussians. Initial data (14c)and (15c) are adaptations to the monopole systemof initial data used in [5, 7].The scalar potential for the monopole system is V = λ ϕ − v ) , (16)where λ is the scalar field self-coupling and v is thescalar field vacuum value. The constants λ , v , and g parametrize the gravitating monopole system. It ispossible to absorb g into a redefinition of the fieldsand parameters so that λ/g and v determine themodel and I see no reason not to expect the scalingand echoing exponents to be functions of them. Toreduce this parameter space I consider only λ = 0,which is not uncommon, and ¯ v ≡ √ πGv = 0 . v and found very small variation in the scalingand echoing exponents, but it would be interestingto look more closely at the dependence. An important check on the code is whether it canreproduce the accepted values for the scaling andechoing exponents for pure SU (2) [5, 14]. Setting λ = v = 0 and using initial data with ϕ = ∂ t ϕ = 0forces fields related to the scalar field ( ϕ , Φ, Π) tobe permanently zero throughout an evolution, re-ducing the evolution equations in (9) to those forpure SU (2) [5, 7]. This alone is not sufficient be-cause the outer boundary condition in the monopolesystem is w ( t, ∞ ) = 0, while for pure SU (2) it is w ( t, ∞ ) = ±
1, and so slightly different initial datafor w is needed. Using pure SU (2) initial data w (0 , r ) = 1 + pe − [( r − r ) /s ] , (17)with ( g/ √ πG ) r = 3 √ g/ √ πG ) s = √ / ∂ t w (0 , r ) = 0, I find γ m = 0 . ± . γ R = 0 . ± . ln r = 0 . ± . τ = 0 . ± . γ = 0 .
20 and∆ = 0 .
74 [5] as well as the more refined values γ = 0 . ± . . ± . r → ( g/ √ πG ) r , t → ( g/ √ πG ) t , ϕ → √ πGϕ , m BH → ( g (cid:112) G/ π ) m BH , R → ( √ πG/g ) R , and R → ( √ πG/g ) R . I note that w is already dimension-less. III. CRITICAL SOLUTION
To find a critical solution I start with initial data,such as that in (14) and (15), with one of the pa-rameters taken to be p . I then tune p toward itscritical value, p ∗ , through a bisectional search. Allinitial data I’ve tried such that p is a parameter inthe initial data for ϕ (and not w ), such as p beingone of the parameters in (14) (and not in (15)), haslead to (nearly) identical scaling and echoing expo-nents and hence the same critical solution. Withina limited sector of initial data, then, the scaling andechoing exponents are universal.Figure 1 is a diagram for the scaling expo-nent. It displays results using three different single-parameter families of initial data, which I’ll label as1- i (black), 1- ii (blue), and 1- iii (purple), the “1”indicating that this is initial data for the first crit-ical solution presented. Plot (a) shows results forthe mass scaling relation (2), (b) for the R scal-ing relation (4), and (c) for the R scaling relation(3). The best-fit lines are determined using a least-squares fit. Also in Fig. 1 is a table that gives thevalues for the scaling exponents γ m , γ R , and γ R , - - - - - - l n m B H ( a ) l n ℛ ( b ) - - - - ln | p - p * | l n ℛ ( c ) i.d. γ m γ ℛ γ ℛ - i ± ± ± - ii ± ± ± - iii ± ± ± FIG. 1. Scaling exponent results for three differentsingle-parameter families of initial data: Initial data 1- i (black points) is (14a) with p = c , r = 5, s = 10, and d = 0 . c = 0 and s = 5; initial data1- ii (blue points) is (14b) with p = c and s = 1 and(15b) with c = 2 and s = 5; and initial data 1- iii (purplepoints) is (14c) with p = s and r = 3 and (15c) with r = 7 and s = 10. (a) shows results for the mass scalingrelation (2), (b) shows results for the R scaling relation(4), and (c) shows results for the R scaling relation (3).The table gives the values of the scaling exponents ex-tracted from the best-fit lines. The scaling exponentsappear to agree and be universal. as determined from the best-fit lines. From the tablewe can see that the different methods for comput-ing the scaling exponent agree with another. Wecan also see the universality of the scaling exponent.Interestingly, γ R is in agreement with the scalingexponents found using the other methods. As I ex-plained above, this is not necessarily expected. Itseems to imply that the scalar field dominates overthe gauge field in determining the geometry of space-time at the origin for this critical solution.Even from visual inspection of Fig. 1, one can seethat all curves have a periodic wiggle around theirbest-fit line and that the period is roughly equal to 2. - - - - - - - - - ln | p - p * | l n m B H r e s i d i u a l s FIG. 2. Residuals for initial data 1- ii in Fig. 1(a). Eachpoint is found by subtracting from the point in Fig. 1(a)the corresponding value of the best-fit line. The period-icity is clearly seen, with a period right around 2. Thisis consistent with ∆ / (2 γ ) = 1 .
93, as computed using theaverage values of γ and ∆ from the tables in Fig. 1 andFig. 4 below. (Note that the lines connecting the pointsare simple straight lines and are not from any sort of fit.) In Fig. 2 I show a plot of the residuals for one of thecurves in Fig. 1(a). A period of right around 2 is eas-ily seen. (The Fourier transform of the residuals hasa peak at 2, but unfortunately there is not enoughdata for the Fourier transform to give a more accu-rate answer.) In looking at residuals I have foundthat all scaling data has a period of about 2. Sucha period is consistent with ∆ / (2 γ ) = 1 .
93, where Iused the average values of γ and ∆ from the tablesin Figs. 1 and 4.Figure 3 displays a near-critical evolution, withln | p − p ∗ | ≈ −
32 (or | p − p ∗ | ≈ − ), using initialdata 1- i , and is plotted at moments in time whenthe spacetime is on the verge of collapse. The topthree figures, (a)–(c), plot fields associated with thescalar field and the echoing is readily seen to be typ-ical of a type II critical solution. The bottom twofigures, (d) and (e), plot fields associated with thegauge field, but the echoing has a somewhat differentappearance.In Fig. 4 I show diagrams whose purpose is to ex-hibit the discrete self-similarity of the solutions, ifit exists. I shall refer to these diagrams, and theanalogous ones below in Fig. 8, as self-similarity di-agrams. In such diagrams I plot a near-critical scaleinvariant function, Z , at some central proper time, T , which, in terms of coordinate time t , is given by T ( t ) = (cid:90) t α ( t (cid:48) , dt (cid:48) . (18)I then search for values of ∆ ln r and ∆ τ such that Z ( τ + n ∆ τ , ln r + n ∆ ln r ) and Z ( τ, ln r ) overlap, - φ ( a ) - - - r Φ ( b ) - - - r Π ( c ) - - - - Q ( d ) - - - - - - ln r P ( e ) FIG. 3. Values of five fields for a near-critical evolu-tion at moments in time when the spacetime is on theverge of collapse is shown for initial data 1- i . In (a)–(c),which displays fields associated with the scalar field, wesee echoing typical of a type II critical solution. In (d)and (e), which shows fields associated with the gaugefield, we see echoes, but they have a somewhat differentappearance. where n is a positive integer, τ = ln( T ∗ − T ), and T ∗ is a constant called the accumulation time. Theexpectation in type II collapse is that ∆ ln r = ∆ τ .Figure 4(a) displays the discrete self-similarity ofthe field r Π. Though not shown, the other fields as-sociated with the scalar field ( ϕ and r Φ) also exhibitself-similarity analogously to Fig. 4(a). The table inFig. 4 gives the echoing exponents found for r Π for - - r Π ( a ) - - - - ln r + n Δ ln r P ( b ) initial data Δ ln r Δ τ - i ± ± - ii ± ± - iii ± ± FIG. 4. (a) and (b) are self-similarity diagrams for thesame evolution shown in Fig. 3, which uses initial data1- i . r Π is a field associated with the scalar field and in(a) we see that it exhibits self-similarity typical of typeII behavior, with n = 0 (solid green), n = 1 (dashedblue), and n = 2 (dotted black). Though not shown, theother fields associated with the scalar field ( ϕ and r Φ)also exhibit self-similarity. P is a field associated withthe gauge field and in (b) we see that it does not exhibitself-similarity (nor does the other field associated withthe gauge field, Q , which is not shown). The table givesthe echoing exponents found for r Π for the three familiesof initial data listed in the caption to Fig. 1. The echoingexponents appear to agree and be universal. the three families of initial data listed in the captionto Fig. 1. The table suggests the echoing exponentsare universal and that ∆ ln r and ∆ τ agree.Figure 4(b) shows a self-similarity diagram for P ,a field associated with the gauge field. The curvesin Fig. 4(b) are for the same times and the same∆ ln r used in Fig. 4(a). It is clear that the field P is not exhibiting self-similarity. Though not shown,neither does Q . In general, for the critical solution ofthis section, there does not exist values for ∆ ln r and∆ τ such that the fields associated with the gaugefield ( Q and P ) exhibit self-similarity. IV. SECOND CRITICAL SOLUTION
Almost all initial data I’ve tried such that p is aparameter in the initial data for w (and not ϕ ) haslead to, by all appearances, a second critical solu-tion, with scaling and echoing exponents differentfrom those in the previous section. However, thescaling and echoing exponents are not as universaland the self-similarity is not as precise as in the pre-vious section.Figure 5 is a diagram for the scaling expo-nent. It displays results using three different single-parameter families of initial data (which are differentthan those used in the previous section), which I’lllabel as 2- i (black), 2- ii (blue), and 2- iii (purple),the “2” indicating that this is initial data for thesecond critical solution presented. Plot (a) showsresults for the mass scaling relation (2), (b) for the R scaling relation (4), and (c) for the R scalingrelation (3). The best-fit lines are determined us-ing a least-squares fit. Also in Fig. 5 is a table thatgives the values for the scaling exponents γ m , γ R ,and γ R , as determined from the best fit lines. Fromthe table we can see that the scaling exponent is notas universal for the critical solution of this sectionas the scaling exponent is for the critical solution ofthe previous section.For the critical solution of this section I find that γ R does not agree with the scaling exponent foundusing the other methods. As I explained above, thisis not unexpected. This implies that the gauge fieldis playing a much more important role in the crit-ical solution of this section, something that is alsonot unexpected. Interestingly, however, the value of γ R , though not in agreement with γ m and γ R , isas universal in its value as they are in their values.A periodic wiggle, though small, can be seen inFig. 5. In Fig. 6 I show a plot of the residuals forone of the curves in Fig. 1(b) and a period of rightaround 2 is easily seen. (The Fourier transform ofthe residuals has a peak at 2, but unfortunately thereis not enough data for the Fourier transform to givea more accurate answer.) In looking at residuals Ihave found that all scaling data has a period of about2. Such a period is consistent with ∆ / (2 γ ) = 1 . γ and ∆ from thetables in Figs. 5 and 8.Figure 7 displays a near-critical evolution, withln | p − p ∗ | ≈ −
32 (or | p − p ∗ | ≈ − ), using initialdata 2- iii , and is plotted at moments in time whenthe spacetime is on the verge of collapse. The topthree figures, (a)–(c), plot fields associated with thescalar field and the bottom two figures, (d) and (e),plot fields associated with the gauge field. Compar-ing with Fig. 3, we see that for the critical solutionof this section, it is instead the fields associated with - - - - - l n m B H ( a ) l n ℛ ( b ) - - - - ln | p - p * | l n ℛ ( c ) i.d. γ m γ ℛ γ ℛ - i ± ± ± - ii ± ± ± - iii ± ± ± FIG. 5. Scaling exponent results for three differentsingle-parameter families of initial data: Initial data 2- i (black points) is (14a) with c = 0 and s = 5 and (15a)with p = c , s = 10, and r = 5, and d = 0 .
5; initialdata 2- ii (blue points) is (14b) with c = 3 and s = 1and (15b) with p = c and s = 4; and initial data 2- iii (purple points) is (14c) with r = 7 and s = 10 and (15c)with s = p and r = 3. (a) shows results for the massscaling relation (2), (b) shows results for the R scal-ing relation (4), and (c) shows results for the R scalingrelation (3). The table gives the values of the scalingexponents extracted from the best-fit lines. Comparingthis to Fig. 1, we see that the scaling exponent for thecritical solution of this section is not as universal as thescaling exponent for the critical solution of the previoussection. I note that γ R does not equal γ as found fromthe other methods, which, as explained in the main text,is not unexpected. the gauge field that exhibit echoing typical of a typeII critical solution and it is the fields associated withthe scalar field that have the somewhat different ap-pearance.Figure 8 displays discrete self-similarity diagramsfor r Π and P , and may be compared with Fig. 4. Wefind now that it is the field associated with the gaugefield, P in Fig. 8(b), which exhibits self-similarity - - - - - - - - - - ln | p - p * | l n ℛ r e s i d i u a l s FIG. 6. Residuals for initial data 2- i in Fig. 5(b). Eachpoint is found by subtracting from the point in Fig. 5(b)the corresponding value of the best-fit line. The period-icity is clearly seen, with a period right around 2. Thisis consistent with ∆ / (2 γ ) = 1 .
89, as computed using theaverage values of γ and ∆ from the tables in Fig. 5 andFig. 8 below. (Note that the lines connecting the pointsare simple straight lines and are not from any sort of fit.) (though not shown, Q exhibits it as well), and itis the field associated with the scalar field, r Π inFig. 8(a), which does not (though not shown, neitherdoes ϕ nor r Φ). There does not exist values for∆ ln r and ∆ τ such that the fields associated with thescalar field ( ϕ , r Φ, and r Π) exhibit self-similarity.The table in Fig. 8 gives echoing exponents for thethree families of initial data listed in the caption ofFig. 5. Just as with the scaling exponent, the tableshows us that the echoing exponent, ∆, is not asuniversal for the critical solution of this section asit is for the critical solution of the previous section.Further, close inspection of Fig. 8(b) and analogousdiagrams made with different initial data shows thatself-similarity is not as exact for the critical solutionof this section as it is for the critical solution of theprevious section.It would be elegant if all initial data such that p is a parameter in the initial data for w led to thecritical solution of this section. Though this is thecase for nearly all initial data I’ve tried, I have foundexceptions. For example, initial data (14b) with c =2 and s = 5 and (15b) with c = p and s = 1 leads tothe critical solution of the previous section.It is unclear why self-similarity is less exact andthe scaling and echoing exponents are less universalfor the critical solution of this section compared tothe critical solution of the previous section. It is im-possible to completely rule out this being a numer-ical artifact, but I have found no evidence for this.It may very well be, that for the critical solutionof this section, exact self-similarity and universality - - φ ( a ) - r Φ ( b ) - - r Π ( c ) - - Q ( d ) - - - - - - ln r P ( e ) FIG. 7. Values of five fields for a near-critical evolutionat moments in time when the spacetime is on the vergeof collapse is shown for initial data 2- iii . Comparingwith Fig. 3, we see that for the critical solution of thissection, it is instead the fields associated with the gaugefield in (d) and (e) that exhibit echoing typical of a typeII critical solution and it is the fields associated withthe scalar field in (a)–(c) that have a somewhat differentappearance. are lost. Intriguingly, something similar was seenrecently by Maliborski and Rinne in their study oftype II critical behavior in pure SU (2) [17]. It maybe useful to touch on the similarities of their systemand the system studied here. Most numerical studiesof gravitational SU (2), including the original studies[5, 7] , work within the magnetic ansatz (the present - - - - r Π ( a ) - - - - - ln r + n Δ ln r P ( b ) initial data Δ ln r Δ τ - i ± ± - ii ± ± - iii ± ± FIG. 8. (a) and (b) are self-similarity diagrams for thesame evolution shown in Fig. 7, which uses initial data2- iii . P is a field associated with the gauge field and in(b) we see that it exhibits self-similarity typical of type IIbehavior with n = 0 (solid green), n = 1 (dashed blue),and n = 2 (dotted black). Though not shown, the otherfield associated with the gauge field ( Q ) also exhibits self-similarity. r Π is a field associated with the scalar fieldand in (a) we see that it does not exhibit self-similarity(nor do the other fields associated with the scalar field, ϕ and r Φ, which are not shown). The table gives theechoing exponents for P for the three families of initialdata listed in the caption of Fig. 5. From the table wesee that the echoing exponent is not as universal for thecritical solution of this section as it is for the criticalsolution of the previous section. work is also within the magnetic ansatz), which re-duces the four fields parametrizing the sphericallysymmetric SU (2) gauge field down to a single field.Maliborski and Rinne [17] are the first to study criti-cal behavior in SU (2) without making the magneticansatz. The particular SU (2) gauge they work inreduces the four gauge fields down to effectively two fields (there is a third field, but it obeys a constraintequation instead of an equation of motion). Beyondthe fact that both the system studied in [17] andthe present system are part of SU (2), an obvioussimilarity is that both systems have multiple matterfields. It would be interesting to know what role,if any, this plays in the possible loss of universalityand self-similarity. V. CONCLUSION
In this work I studied type II critical behav-ior in the gravitating magnetic monopole system.This system is characterized by two matter fields: Areal scalar field, which parametrizes the scalar fieldgauged under SU (2), and what is effectively a realscalar field, which parametrizes the gauge field. Thissystem offers some differences compared to othersystems. For example, on the non-black hole side ofthe critical solution, the matter fields do not com-pletely disperse, but instead settle down to a stableand static configuration. More interesting, however,is that the gravitating monopole system appears tohave two critical solutions.All initial data I tried in which the scalar fieldis tuned toward a critical value led to the criticalsolution I presented first in Sec. III. This critical so-lution exhibits precise self-similarity and universalscaling and echoing exponents. In Sec. IV I pre-sented a second critical solution, which most of theinitial data I tried in which the gauge field is tunedtoward a critical value led to, but I did find excep-tions. Though this second critical solution has dif-ferent scaling and echoing exponents than the firstcritical solution, the self-similarity is less exact andthe scaling and echoing exponents are less universal.Indeed, exact self-similarity and universality of thescaling and echoing exponents may be lost, a possi-bility that was recently seen elsewhere [17].It is interesting that, in the first critical solution ofSec. III, which is obtained by tuning the scalar fieldtoward a critical value, the fields associated with thescalar field exhibit self-similarity, while the fields as-sociated with the gauge field do not. And on theother hand, in the second critical solution of Sec. 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