On Perturbation Theory and Critical Exponents for Self-Similar Systems
OOn Perturbation Theory and Critical Exponents for Self-SimilarSystems
Ehsan Hatefi ∗ and Adrien Kuntz † Scuola Normale Superiore and I.N.F.N,Piazza dei Cavalieri 7, 56126, Pisa, Italy Center for Theoretical Physics and College of Physics,Jilin University, Changchun, 130012, ChinaDecember 22, 2020
Abstract
Gravitational critical collapse in the Einstein-axion-dilaton system is known to lead to continuous self-similar solutions characterized by the Choptuik critical exponent γ . We complete the existing literatureon the subject by computing the linear perturbation equations in the case where the axion-dilaton systemassumes a parabolic form. Next, we solve the perturbation equations in a newly discovered self-similarsolution in the hyperbolic case, which allows us to extract the Choptuik exponent. Our main result isthat this exponent depends not only on the dimensions of spacetime but also the particular ansatz andthe critical solutions that one started with. ∗ ehsan.hatefi@sns.it † [email protected] a r X i v : . [ h e p - t h ] D ec Introduction
A well-known property of black holes as the end state of gravitational collapse is that they are completelydefined by only three numbers : their mass, angular momentum, and charge. Choptuik revealed in [1]that there may be a fourth universal quantity characterizing the collapse itself. Following the study ofChristodolou in [2] on the spherically symmetric collapse of scalar fields, Choptuik discovered a criticalbehavior illustrating some sort of discrete spacetime self-similarity. Expressing the amplitude of the scalarfield fluctuation by the number p , he found that p should exceed a critical value p crit in order to form ablack hole. Furthermore, for values of p above the threshold, the mass of the black hole M bh (equivalently,its Schwarzschild radius r S ) exhibits the scaling law r S ( p ) ∝ M bh ( p ) ∝ ( p − p crit ) γ , (1)where the Choptuik exponent was found to be γ (cid:39) .
37 [1, 3, 4] in 4d for a single real scalar field. Note thatin general dimensions ( d ≥
4) the definitions get changed [5, 6] r S ( p ) ∝ ( p − p crit ) γ , M bh ( p ) ∼ ( p − p crit ) ( D − γ . (2)Along the same lines, diverse numerical simulations with different matter fields have been carried out[7, 8,9, 10, 11, 12]. For example the critical collapse of a perfect fluid was performed in [13, 14, 5, 15]. In [14]the authors found γ (cid:39) .
36 and hence it was conjectured in [16] that γ may be universal for any matterfield that is coupled to four dimensional gravity. Later on, in [5, 15, 17] it was discovered that the Choptuikexponent can be explored by dealing with perturbations of the self-similar solutions. In order to do so, oneneeds to perturb any field h (be it the metric or the matter content) as follows h = h + ε h − κ (3)where the perturbation h − κ has scaling − κ ∈ C which labels the different modes. Among the allowed valuesof κ , we define the most relevant mode κ ∗ as the highest value of Re( κ ) . It was shown in [5, 15, 17] that κ ∗ is related to the Choptuik exponent by γ = 1Re κ ∗ . (4)In [18] the case of axial symmetry had been studied and the critical collapse in the presence of shock waveswas reviewed in [19]. The case of axion-dilaton critical collapse coupled to gravity in four dimension was firstexamined in [20] which found the value γ (cid:39) . γ in four dimensions.One motivation to study critical collapse in the axion-dilaton system is the AdS/CFT correspondence [21],relating Choptuik exponent, the imaginary part of quasinormal modes, and the dual conformal field theory[22]. Other motivations include the holographic description of black hole formation [6] as well as the physicsof black holes and its applications [23]. In type IIB string theory one is often interested in exploring thegravitational collapse on spaces that can asymptotically approach to AdS × S where the matter contentis described by the axion-dilaton system and the self-dual 5-form field. The minus sign indicates a growing mode near the black-hole formation time t → , R ) were recently explored in [24] thatgeneralized the previous efforts done in [25, 26]. Based on some robust analytic and numerical techniquesin [27], we did perturb critical solution of four-dimensional elliptic critical collapse and were able to recoverthe known value [20] of γ ∼ . , R ), andparticularly in the parabolic case which was not studied before. We extract the Choptuik exponent in a newbranch of the 4d hyperbolic class of solutions and find that its value is different from the other branches ofsolutions. Thus, our results cast doubts concerning the universality of the Choptuik exponent. The Einstein-axion-dilaton system that coupled to gravity in d dimensions is defined by the following action S = (cid:90) d d x √− g (cid:18) R − ∂ a τ ∂ a ¯ τ (Im τ ) (cid:19) . (5)that can be described by the effective action of type II string theory [28, 29] where the axion-dilaton isdefined by τ ≡ a + ie − φ . This action enjoys the SL(2 , R ) symmetry τ → M τ ≡ ατ + βγτ + δ , (6)where α , β , γ , δ are real parameters satisfying αδ − βγ = 1. It was known that once quantum effects aretaken into account the SL(2 , R ) symmetry does reduce to SL(2 , Z ) and this S-duality was also believed tobe a non-perturbative symmetry of IIB string theory [30, 31, 32]. Now from the above action one can readoff the equations of motion R ab = ˜ T ab ≡ τ ) ( ∂ a τ ∂ b ¯ τ + ∂ a ¯ τ ∂ b τ ) , (7) ∇ a ∇ a τ + i ∇ a τ ∇ a τ Im τ = 0 . (8)We assume spherical symmetry on both background and perturbations so that the general form of the metricin d dimensions is ds = (1 + u ( t, r ))( − b ( t, r ) dt + dr ) + r d Ω q , (9) τ = τ ( t, r ) , q ≡ d − , (10)where d Ω q is the angular part of the metric in d spacetime dimensions. A scale invariant solution is foundby requiring that under a spacetime dilation (or scale transformation), ( t, r ) → (Λ t, Λ r ), the line elementgets changed as ds → Λ ds . Thus, the metric functions should be scale invariant, i.e. u ( t, r ) = u ( z ), b ( t, r ) = b ( z ), z ≡ − r/t . Since the action (5) is invariant under the SL(2 , R ) transformation (6), τ only needsto be invariant and up to an SL(2 , R ) transformation, τ (Λ t, Λ r ) = M (Λ) τ ( t, r ) . (11)3e call a system of ( g, τ ) respecting the above properties a continuous self-similar (CSS) solution. Note thatdifferent cases do relate to various classes of d M dΛ (cid:12)(cid:12) Λ=1 [24], so that τ can take three different forms, τ ( t, r ) = i − ( − t ) iω f ( z )1 + ( − t ) iω f ( z ) , elliptic f ( z ) + ω log( − t ) , parabolic1 − ( − t ) ω f ( z )1 + ( − t ) ω f ( z ) , hyperbolic (12)where ω is an unknown real parameter and the function f ( z ) must satisfy | f ( z ) | < f ( z ) > τ ( t, r ) = ( − t ) ω f ( z )also leads to the same equations of motion for hyperbolic case (it is simply a conformal transformation of τ ). If we replace the CSS ans¨atze in the equations of motion we then get a differential system of equationsfor u ( z ), b ( z ), f ( z ). Due to spherical symmetry one can show that u ( z ) can be expressed in terms of b ( z )and f ( z ) so that eventually we are left out with some ordinary differential equations (ODEs) b (cid:48) ( z ) = B ( b ( z ) , f ( z ) , f (cid:48) ( z )) , (13) f (cid:48)(cid:48) ( z ) = F ( b ( z ) , f ( z ) , f (cid:48) ( z )) . (14)The above equations have five singularities [25] located at z = ± z = ∞ and z = z ± where the lastsingularities are defined by b ( z ± ) = ± z ± . The latter correspond to the homotetic horizon and it can beshown that z = z + is just a mere coordinate singularity [20, 25], hence τ is regular across it which translatesback to the finiteness of f (cid:48)(cid:48) ( z ) as z → z + . Now one may observe that the vanishing of the divergent part of f (cid:48)(cid:48) ( z ) gives us a complex valued constraint at z + which we denote by G ( b ( z + ) , f ( z + ) , f (cid:48) ( z + )) = 0 where theexplicit form of G was given in [24].Using regularity at z = 0 and some residual symmetries one obtains the initial conditions b (0) = 1 , f (cid:48) (0) = 0 f (0) = x elliptic (0 < x < ix parabolic (0 < x )1 + ix hyperbolic (0 < x ) (15)Here x is a real parameter. Hence, we have two constraints (the vanishing of the real and imaginary partsof G ) and two parameters ( ω, x ). The discrete solutions in four and five dimensions were found in [27].These solutions are constructed by integrating numerically the equations of motion. For instance, for thefour dimensional elliptic case just one solution is determined [33, 25] as ω = 1 . , | f (0) | = 0 . , z + = 2 .
605 (16)To deal with self-similar solutions for parabolic cases, the following remarks are in order. First, we have anadditional symmetry as follows ω → Kω , f ( z ) → Kf ( z ) , K ∈ R + (17)then τ also transforms as τ → Kτ , which means that if ( ω, Im f (0)) is a solution, so is ( Kω, K Im f (0)).The reason behind it is that all equations of motion and the constraint G ( ω, Im f (0)) are invariant underthis new scaling. Therefore, the only real unknown parameter for parabolic class is the ratio ω/ Im f (0). Wethen need to look for the zeroes of G ( ω, Im f (0)) for only a real parameter ω/ Im f (0). Hence we just set4 d hyperbolic ω Im f (0) z + α .
362 0 .
708 1 . β .
003 0 . . γ .
541 0 . . f (0) = 1. For the five dimensional parabolic case, we draw below the plot of the zeroes of the real andimaginary parts of G ( ω, - - ( ω )- - Figure 1: The plot of absolute value (blue), real and imaginary parts (orange) of G ( ω,
1) for the fivedimensional parabolic case.From Figure 1 in five dimensions we might note to the tiny value of | G | for a specific value of ω . This maybe related to the only possible solution ray, but numerical accuracy is insufficient to assess it with certainty.It is given by | G | ∼ . , ω ∼ . . (18)Note that remarks about the higher dimensional parabolic solutions are given in [34]. On the other handthree different solutions for the hyperbolic case in four dimensions were determined in [27]. These threesolutions denoted by α , β and γ are summarized in Table 1.Making use of the root-finding procedure, we also identify a fourth solution δ for the four-dimensionalhyperbolic class (with accuracy less than the other solutions and G ∼ − ) whose parameters are given by ω = 0 . , Im f (0) = 0 . , z + = 19 . Here we derive the perturbation equations in general dimensions. We will apply our method for the paraboliccase, but it could be taken as an extensive method which holds for all matter content as well. Note that wehave taken some of the steps from [35] while with some algebraic calculations we are able to remove u ( t, r )and its derivative from the actual computations . We perturb the exact solutions h (where h denotes either b , u or f ) found in Section 2 according to Similar perturbations of spherically symmetric background solutions for Horava Gravity were also explored in [36]. ( t, r ) = h ( z ) + ε h ( t, r ) (20)where ε is a small number. If we expand all the equations in powers of ε , then the zeroth order part givesrise to background equations already studied in Section 2 and the linearized equations for the perturbations h ( t, r ) are related to linear terms in ε . Let us consider the perturbations of the form h ( z, t ) = h ( z ) + ε ( − t ) − κ h ( z ) , (21)One finds the spectrum of κ by solving the equations for h ( z ) and indeed the general solution to the first-order equations is obtained with the linear combination of these modes. We want to find the mode κ ∗ withlargest real part (assuming a growing mode for t →
0, i.e Re κ >
0) that is related to the Choptuik exponentby [5, 15, 17] γ = 1Re κ ∗ (22)Note that just like the four dimensional elliptic case, for simplicity we consider only real modes κ ∗ . It canbe shown that the values κ = 0 and κ = 1 are gauge modes with respect to global U (1) re-definitions of f and time translations respectively, see Section 3.1.1 of [27]. These modes should be excluded from thecomputations. Let us apply this program to the parabolic case and explore all the linearized perturbations in arbitrarydimension d = q + 2 ≥
4. One applies the perturbation ansatz (21) to all the functions u , b , τ as u ( t, r ) = u ( z ) + ε ( − t ) − κ u ( z ) , (23) b ( t, r ) = b ( z ) + ε ( − t ) − κ b ( z ) , (24) τ ( t, r ) = f ( z ) + ω log( − t ) , (25) f ( t, r ) ≡ f ( z ) + ε ( − t ) − κ f ( z ) . (26)One calculates the Ricci tensor for the following metric ds = (1 + u )( − b dt + dr ) + r d Ω q , (27)where b and u should be replaced by the perturbed quantities (23), (24). The zeroth-order and first-orderparts of the Ricci tensor are obtained from R (0) ab = lim ε → R ab ( ε ) , (28) R (1) ab = lim ε → d R ab ( ε )d ε . (29)Likewise one does the same for the matter content, applying the axion-dilaton perturbations (25), (26) so6hat ˜ T (0) ab = lim ε → ˜ T ab ( ε ) , (30)˜ T (1) ab = lim ε → d ˜ T ab ( ε )d ε . (31)The Einstein field equations should be held order by order hence R (0) ab = ˜ T (0) ab , R (1) ab = ˜ T (1) ab . (32)We now use some of the above equations to remove u ( t, r ) and its derivatives from the other equations.Indeed by using R (0) tr = ˜ T (0) tr we remove u (cid:48) ( z ), R (0) ij = ˜ T (0) ij = 0 eliminates u ( z ) (where i, j denote indiceson the ( d − R (1) tr = ˜ T (1) tr also removes u (cid:48) ( z ), where eventually R (1) ij = ˜ T (1) ij = 0 is used to actuallyremove u ( z ). From now on we also remove the z argument of all functions, so that qu (cid:48) u ) = ω ( f (cid:48) + ¯ f (cid:48) ) − z ¯ f (cid:48) f (cid:48) ( f − ¯ f ) , (33) u = zb (cid:48) ( q − b , (34) u = − ( q − b u − zb (cid:48) ( q − b . (35)Since the final form of u (cid:48) ( z ) is complicated we will not write it here. Therefore, u , u (cid:48) , u , u (cid:48) are completelyexpressed for all equations in terms of other functions. Using the following combination of temporal andradial equations of motion C ( ε ) ≡ R tt + b R rr − ˜ T tt − b ˜ T rr = 0 , (36)we also remove the first derivative terms in b ( t, r ). Indeed we recover the zeroth-order equation as follows b (cid:48) = − (cid:16)(cid:0) z − b (cid:1) f (cid:48) (cid:16) z ¯ f (cid:48) − ω (cid:17) + ω (cid:16)(cid:0) b − z (cid:1) ¯ f (cid:48) + ωz (cid:17)(cid:17) qb ( f − ¯ f ) , (37)where the overbar on f denotes complex conjugation. In the same way the first correction is defined byd C ( ε )d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = 0 , (38)which is an equation relating b (cid:48) to b , b (cid:48) f , f (cid:48) , f (cid:48)(cid:48) , and to the other perturbations b , f , f (cid:48) , and which isreally linear in all perturbations. In the parabolic case this equation takes the following form7 L ) b (cid:48) = r (( t − qt ) b + rb (cid:48) ) (cid:18) − qt b b (cid:48) r + qκt b b b (cid:48) r ( t − qt ) b + r b (cid:48) + 4 t b b f (cid:48) ¯ f (cid:48) s + 4 t b f (cid:48) ¯ f (cid:48) ( − f + ¯ f ) s + 4 t b ( f − ¯ f )( tω + rf (cid:48) ) ¯ f (cid:48) rs + 2 t b f (cid:48) ¯ f (cid:48) s + 2 t b ¯ f (cid:48) ( κtf − rf (cid:48) ) rs + 4 t b ( f − ¯ f )( tω + r ¯ f (cid:48) ) f (cid:48) rs − f − ¯ f )( tω + rf (cid:48) )( tω + r ¯ f (cid:48) ) s − t b f (cid:48) ( tω + r ¯ f (cid:48) ) rs + 2( − κtf + rf (cid:48) )( tω + r ¯ f (cid:48) ) s − t b b ( tω ¯ f (cid:48) + f (cid:48) ( tω + 2 r ¯ f (cid:48) )) rs + 2 t b f (cid:48) ¯ f (cid:48) s − t b ( tω + rf (cid:48) ) ¯ f (cid:48) rs + 2 t b f (cid:48) ( κt ¯ f − r ¯ f (cid:48) ) rs + 2( − κt ¯ f + r ¯ f (cid:48) )( tω + rf (cid:48) ) s (cid:19) , (39)with L = qt b (( κ − q + 1) tb + rb (cid:48) ) , s = ( f − ¯ f ) . (40)The perturbations are also scale invariant, thus making the coordinate change ( t, r ) → ( t, z ), the factors of t cancels out. We now introduce the perturbation ans¨atze in the τ equation of motion (8). Replacing b (cid:48) according to (37) and solving for f (cid:48)(cid:48) , one recovers the second order background equation for f , qz (cid:0) z − b (cid:1) ( f − ¯ f ) f (cid:48)(cid:48) = b f (cid:48) (cid:0) zf (cid:48) (cid:16) z ¯ f (cid:48) − ω (cid:17) − qf (cid:0) zf (cid:48) + q ¯ f (cid:1) + 2 qz ¯ f f (cid:48) + q f − ωz ¯ f (cid:48) + q ¯ f (cid:1) + z (cid:16) ωz ¯ f (cid:48) ( ω − zf (cid:48) ) + 2 qf (cid:0) ( ω − zf (cid:48) ) − ¯ f ( ω − zf (cid:48) ) (cid:1) − q ¯ f ( ω − zf (cid:48) ) + q ¯ f ( ω − zf (cid:48) ) + qf ( ω − zf (cid:48) ) (cid:17) + 2 z b ( ω − zf (cid:48) ) (cid:16) ω − z ¯ f (cid:48) (cid:17) . (41)Going to first order, the linearized equation for f (cid:48)(cid:48) is( L ) f (cid:48)(cid:48) = (cid:18) t κ (1 + κ ) b f − t b b b (cid:48) f (cid:48) − t b b (cid:48) f (cid:48) + ( κtb − rb (cid:48) )( tω + rf (cid:48) ) − b ( t ω + 2 rtωf (cid:48) + ( r − t b ) f (cid:48) ) m − b ( f − ¯ f )( − t ω − rtωf (cid:48) + ( − r + t b ) f (cid:48) ) m − rt (1 + κ ) b f (cid:48) + qt b f (cid:48) r − t b b (cid:48) f (cid:48) − rb (cid:48) ( − tκf + rf (cid:48) )+ m b (cid:0) t b b f (cid:48) + κtf ( tω + rf (cid:48) ) + t b f (cid:48) f (cid:48) − r ( tω + rf (cid:48) ) f (cid:48) (cid:1) + 3 t b b ( qtf (cid:48) − rf (cid:48)(cid:48) ) r − b ( t ω + 2 rtf (cid:48) − r f (cid:48)(cid:48) ) (cid:19) , (42)where L = − r b + t b and m = f − ¯ f . This equation is also scale-invariant. By integrating numericallythe unperturbed equations, and also substituting b (cid:48) from eq (39) , we derive the ordinary linear differentialequations as follows b (cid:48) = B ( b , f , f (cid:48) ) , (43) f (cid:48)(cid:48) = F ( b , f , f (cid:48) ) . (44)8 and F are indeed functions linear in the perturbations that have however non-linear dependence on theunperturbed solution. The perturbed equations are also singular at z = 0 and b ( z ) = z . The perturbationequations for hyperbolic case were derived in [27] where the modes are explored by finding the κ values thatare related to smooth solutions of the perturbed equations (43), (44) which need to satisfy the appropriateboundary conditions, which we will now discuss. We now turn to boundary conditions needed to solve Eqs. (43), (44). First of all at z = 0 we rescale thetime coordinate, so that b (0) = 0, and also using the regularity condition for the axion-dilaton at z = 0 wefind that f (cid:48) (0) = 0 so that the freedom in f is reduced to f (0) which is an unknown complex parameter.We also demand that at z + (we recall that z + is defined by the equation b ( z + ) = z + ) all equations andperturbations be regular so that all the second derivatives ∂ r f ( t, r ), ∂ r ∂ t f ( t, r ), ∂ t f ( t, r ) should be finite as z → z + . Hence, f (cid:48)(cid:48) ( z ) and f (cid:48)(cid:48) ( z ) are also finite as z → z + . For brevity, we introduce β = b ( z ) − z andrewrite eqs (42)-(41) as follows f (cid:48)(cid:48) ( β ) = 1 β G ( h ) + O (1) , (45) f (cid:48)(cid:48) ( β ) = 1 β ¯ G ( h ) + 1 β H ( h , h | κ ) + O (1) , (46)where it is understood that h = ( b ( z + ) , f ( z + ) , f (cid:48) ( z + )), h = ( b ( z + ) , f ( z + ) , f (cid:48) ( z + )). The vanishingunperturbed complex constraint is given by G ( h ) = 0 at z + , and we checked that it implies ¯ G ( h ) = 0at z + . Hence we are left just with the complex-valued constraint H ( h , h | κ ) = 0. Finally we solve thisconstraint for f (cid:48) ( z + ) in terms of f ( z + ), b ( z + ), κ and h . Thus this condition does reduce the free parametersat z + to just a real number b ( z + ) and a complex f ( z + ). Finally we will have 6 unknowns including κ andthe following five-component vector: X = (Re f (0) , Im f (0) , Re f ( z + ) , Im f ( z + ) , b ( z + )) (47)We also have the linear ODE’s eqs. (43), (44) whose total real order is five. Let us now briefly explain thenumerical procedure. Given a set of boundary conditions X , we integrate from z = 0 to an intermediatepoint z mid and similarly we integrate backwards from z + to z mid . Finally we collect the values of all functions( b , Re f , Im f , Re f (cid:48) , Im f (cid:48) ) at z mid and encode the difference between the two integrations in a “differencefunction” D ( κ ; X ). By definition, D ( κ ; X ) is linear in X thus it has a representation as a matrix form D ( κ ; X ) = A ( κ ) X (48)where A ( κ ) is a 5 × κ . So we need to just find the zeroes of D ( κ ; X ) and thiscan be achieved by evaluating det A ( κ ) = 0. We carry out the root search for the determinant as a functionof κ where the root with the biggest value will be related to the Choptuik exponent through eq. (22). Itis worth highlighting one last point: the perturbed equations of motion are singular whenever the factor W = (cid:16) κ + 1 − q − z b (cid:48) b (cid:17) in the denominator vanishes, so that the numerical procedure fails at particularpoint. We can get an estimate for the values of κ giving rise to this singular behaviour as follows,0 = W (cid:12)(cid:12)(cid:12)(cid:12) z = z + = κ sing + 1 − q − b (cid:48) ( z + ) ⇒ κ sing = q − b (cid:48) ( z + ) . (49)However, this apparent problem does not affect our evaluation of the critical exponent because in most casesthe most relevant mode κ ∗ lies outside that particular failure region.9 .0 1.5 2.0 2.5 3.0 3.5 4.0 κ - - ( det A ) Figure 2: Behaviour of arcsinh(det A ( κ )) as a function of κ for the unique four-dimensional elliptic criticalsolution. For clarity, we plot arcsinh(det A ( κ )) in order to limit the range of values. In [27] we had already tested the above techniques and were able to derive the critical exponent for theunique four-dimensional elliptic solution. For completeness we have drawn the behaviours of det A ( κ ) nearthe last crossing of the horizontal axis in Figure 2. The position of the crossing is found to be κ ∗ E ≈ . γ E ≈ . κ sing = 1 . (cid:46) κ (cid:46) .
4, which is well below the location ofthe most relevant mode as it is seen in a range of κ values in Figure 2, where Mathematica was not able tocomplete the computation of det A .In the 4d hyperbolic case, there are four branches of solutions that we denote by 4 Hα , 4 Hβ , 4 Hγ and 4 Hδ respectively. The Choptuik exponent was not known in the 4 Hδ case, which is one of the new results of thisarticle. In Figure 3 we plot the behaviour of det A ( κ ) near the last crossing which defines the most relevantmode, κ ∗ Hδ ≈ . , (50)so that the Choptuik exponent is γ Hδ ≈ .
393 (51)which is different from the Choptuik exponent for the third critical solution γ Hγ = 0 .
436 (already foundin [27]) that is illustrated in 5. We collect these results in table 2.For completeness, we include some other Choptuik exponents in table 3. We refer the reader to [27] for acomplete discussion of these other cases. 10 .5 1.0 1.5 2.0 2.5 3.0 3.5 κ - - - ( det A ) Figure 3: Behaviour of arcsinh(det A ( κ )) as a function of κ for the fourth soltion of 4 dimensional hyperboliccase, denoted 4 Hδ . For clarity, we plot arcsinh(det A ( κ )) in order to limit the range of values. κ - ( det A ) Figure 4: A zoom on the last crossing of the plot of Figure 3 κ - - - - ( det A ) Figure 5: The behaviour of arcsinh(det A ( κ )) for the 4H γ solution near the last crossing.11olution κ ∗ γ failureregion κ sing comments4H δ . .
393 1 − .
58 1 . γ .
293 0 .
436 1 − . .
32 see Figure 5Table 2: Choptuik exponents for the two last branches of solutions of the 4 dimensional hyperbolic class.solution κ ∗ γ failureregion κ sing comments4E 3 . . − . .
224 see Figures 24H α − . . − − . . κ ∗ likely inside failure region4H β − .
55 0 . − − .
55 1 . κ ∗ inside failure region5E α .
186 0 .
843 2 − .
25 2 . α .
546 0 .
647 2 . − . . In this article, we have obtained the linear perturbation equations in all classes of solutions of the self-similarcollapse solution to the Einstein-axion-dilaton system, including the parabolic case which was not studiedpreviously. The method which we employ is quite generic and could be applied to any matter content inarbitrary dimensions as well. This is certainly a path that we intend to follow in the future.Through a numerical procedure, we have obtained the fastest growing mode of the perturbations thatdetermine the Choptuik exponent. We have applied this methodology to a particular branch of solutionswhose Choptuik exponent was still unknown. Interestingly, we revealed that not only the Choptuik exponentdoes depend on the spacetime dimension but also it depends on matter content (which is composed of anaxion-dilaton system in this case) as well as the different branches of solutions of self-similar critical collpase.Hence, one may conclude that the original conjecture about the universality of Choptuik exponent is notsatisfied. However, there might actually exist some universal behaviours hidden in combinations of criticalexponents and various other parameters of the theory which have not been taken into account by our currentefforts.
Acknowledgments
EH would like to thank R. Antonelli, E. Hirschmann, L. ´Alvarez-Gaum´e and A. Sagnotti for useful conversa-tions. This work is supported by INFN (ISCSN4-GSS-PI), by Scuola Normale Superiore, and by MIUR-PRINcontract 2017CC72MK003. 12 eferences [1] M.W. Choptuik, Universality and Scaling in Gravitational Collapse of a Massless Scalar Field,
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