We observe critical phenomena in spherical collapse of radiation fluid. A
sequence of spacetimes $\cal{S}[\eta]$ is numerically computed, containing
models ($\eta\ll 1$) that adiabatically disperse and models ($\eta\gg 1$) that
form a black hole. Near the critical point ($\eta_c$), evolutions develop a
self-similar region within which collapse is balanced by a strong,
inward-moving rarefaction wave that holds $m(r)/r$ constant as a function of a
self-similar coordinate $\xi$. The self-similar solution is known and we show
near-critical evolutions asymptotically approaching it. A critical exponent
$\beta \simeq 0.36$ is found for supercritical ($\eta>\eta_c$) models.Comment: 10 pages (LaTeX) (to appear in Phys. Rev. Lett.), TAR-039-UN

A. M. Abrahams

A. Ori

C. Gundlach

Charles R. Evans

D. M. Eardley

G. B. Whitham

G. Guderley

G. I. Barenblatt

J. M. Bardeen

Jason S. Coleman

L. I. Petrich

M. W. Choptuik

R. F. Stark

English

arXiv

Crossref

Physical Review Letters

Critical phenomena and self-similarity in the gravitational collapse of radiation fluid

We observe critical phenomena in spherical collapse of radiation fluid. A sequence of spacetimes S[eta] is numerically computed, containing models (eta1) that form a black hole. Near the critical point (etac), evolutions develop a self-similar region within which collapse is balanced by a strong, inward-moving rarefaction wave that holds m(r)/r constant as a function of a self-similar coordinate xi. The self-similar solution is known and we show near-critical evolutions asymptotically approaching it. A critical exponent beta~=0.36 is found for supercritical (eta&gt;etac) models

Evans, Charles R.

Coleman, Jason S.

Carolina Digital Repository

arXiv:gr-qc/9402041v1  23 Feb 1994Observation of Critical Phenomena and Self-Similarityin the Gravitational Collapse of Radiation FluidCharles R. Evans and Jason S. ColemanDepartment of Physics and AstronomyUniversity of North Carolina, Chapel Hill, North Carolina 27599(January 30, 2018)AbstractWe observe critical phenomena in spherical collapse of radiation fluid.A sequence of spacetimes S[η] is numerically computed, containing models(η ≪ 1) that adiabatically disperse and models (η ≫ 1) that form a black hole.Near the critical point (ηc), evolutions develop a self-similar region withinwhich collapse is balanced by a strong, inward-moving rarefaction wave thatholds m(r)/r constant as a function of a self-similar coordinate ξ. The self-similar solution is known and we show near-critical evolutions asymptoticallyapproaching it. A critical exponent β ≃ 0.36 is found for supercritical (η > ηc)models.PACS numbers: 04.20.Dw, 04.25.Dm, 04.40.Nr, 04.70.Bw1New solutions of Einstein’s equations at the threshold of black hole formation have re-cently been discovered. These solutions exhibit nonlinear dynamical behavior closely anal-ogous to critical phenomena, and include power-law behavior, discrete scaling relations anda form of universality. All of these features were discovered in Choptuik’s [1] studies ofspherical collapse of massless scalar field. Abrahams and Evans [2] found strikingly similarbehavior in axisymmetric collapse of gravitational waves, indicating the behavior is a genericfeature of gravity. Confirmation has recently been made [3] of some of Choptuik’s results.In this Letter, we report observation of critical phenomena in spherical collapse of radiationfluid and show the asymptotic approach of near-critical spacetimes to a self-similar solutionnear the center of collapse. This new result is fundamental in two ways; the self-similarsolution has been separately and precisely calculated [4], and it exhibits local self-similarity.The process for finding the threshold of black hole formation and critical phenomena ingravitational collapse has been described elsewhere [1,2,5,6]. Adopting a physical model withmetric g, matter fields φA, stress-energy T(φA), and symmetries, the future development iscomputed of elements of a family of Cauchy data, distinguished by a dimensionless strength-parameter η and a length (or mass) scale r0. Subcritical models (η ≪ 1) have smooth futuredevelopment, dispersing all mass-energy to infinity, while a black hole appears in supercriticalmodels (η ≫ 1) with only some mass-energy escaping. As η is tuned in a search for theonset (at ηc) of black hole formation, black hole mass (for η > ηc) diminishes as a power law:MBH = K(η − ηc)β, with β > 0. In both scalar field collapse [1] and vacuum collapse [2,5],simulations give β ≃ 0.37, results that appear to be universal (i.e., independent of theparameterization of the initial data).By implication, tuning the parameter to just the right value (η = ηc) creates arbitrarilysmall black holes and regions of arbitrarily high spacetime curvature from smooth Cauchydata. Hence, it is argued [6] that the threshold of black hole formation is also the thresholdof naked singularity formation. Spacetimes with η = ηc are termed precisely critical.In any physical model, many families of Cauchy data may exist with a critical point anda precisely-critical spacetime. Each family of data will have a characteristic proper length2scale r0. Related to this scale is the limiting interval, lim |T | = |T0|, as η−ηc → 0+, of propertime T of the observer O centered in the collapse. Let T = −T0 correspond to the initialdata. Further, let r be a proper spatial distance from O. The central region of collapse R(in spacetime) exists for r ≪ r0, |T | ≪ |T0|. Different precisely-critical spacetimes can benormalized, using scale invariance, to a common length scale r0. These spacetimes will stilldiffer on scales T ∼ −T0 and r ∼ r0 but approach a (physical-model-specific but otherwiseunique) self-similar solution within R. Within R, all knowledge of the initial data is lost upto a single, dimensional constant.Previous critical systems have displayed discrete self-similarities. Let q denote dimensionand let length have q = 1. Allow the metric g to carry dimension q = 2 [7]. For η = ηc, withinR a map will exist under which g → g′ = e−2∆g, for a fixed value of ∆. These are echoesresulting purely from nonlinearities. The scale factors ∆ are pure numbers that emerge fromthe dynamics, with e∆ ≃ 30 in scalar wave collapse [1] and e∆ ≃ 1.8 in gravitational wavecollapse [2,5].An analytic explanation of these phenomena would be important. This has been soughtfor scalar field collapse but the discrete self-similarity proved to be an impediment. The effortmade obvious, though, that a local self-similarity would be more powerful, and suggestedexamining another system: perfect fluid collapse.Imagine a spherical configuration of radiation fluid confined within areal radius r0. Letthe total gravitational mass be M . Define a dimensionless control parameter η = 2M/r0 [8].When η ≪ 1, pressure dominates and should cause adiabatic expansion and dispersal of thefluid. When η >∼ 1, gravity should overwhelm pressure and spur the formation of a blackhole. The onset of black hole formation should occur at some ηc ∼ 1. One might anticipate,with a fluid, that any self-similarity emerging near ηc will be local.A successful attempt was made to find a self-similar solution representing the thresholdof black hole formation in perfect fluid collapse [4]. Details of this solution will be givenelsewhere; here we only describe the self-similar ansatz and a few numerical results. Weassume a perfect fluid T µν = ρUµUν + pgµν , with total energy density ρ, isotropic pressure3p and four-velocity Uµ satisfying UµUµ = −1. A relativistic equation of state p = (γ − 1)ρis adopted. In this paper, we specialize to radiation fluid p = 13ρ. Adopting (dynamical)Schwarzschild coordinates, the line element isds2 = −α2dt2 + a2dr2 + r2dΩ2, (1)with α(r, t) the lapse function and a(r, t) the radial metric function. Constant propersound speed cs =√p/ρ =√γ − 1 suggests the self-similar ansatz: a(ξ), U r(ξ), ρ(r, t) =Ω(ξ)/4πr2, α(r, t) = nrN(ξ)/t, with self-similar coordinate ξ = r/C(−t)n. Geometrically,w = (1/n)t∂/∂t+r∂/∂r is the infinitesimal generator of the homothetic motion: £wg = 2g,£wT = 0 . The similarity exponent n is initially unknown, making the problem (in these co-ordinates) a type-2 self-similarity [9]. With this ansatz, the fluid and gravitational equationsreduce to ordinary differential equations. These are solved by demanding regularity at ξ = 0(r = 0, t < 0), ξ = ∞ (r > 0, t = 0), and along the limiting, ingoing acoustic characteristic(sonic point) at ξ = 1, across which the fluid might otherwise be discontinuous. The systemis over-determined unless n is taken as an eigenvalue of the nonlinear system. An analogoustype-2 similarity problem is Guderley’s shock implosion problem [10].The similarity exponent was found to be n ≃ 1.1485 (γ = 4/3). The solution is depictedin Fig. 1. A physical singularity exists at r = 0, t = 0, since RµνRµν = 2563π2ρ2 ∼ T−4 andcentral proper time T ∼ (−t)n. Notice that a does not approach unity as ξ → ∞, but ratherapproaches a ≃ 1.07. Likewise, mass grows linearly with distance, with m(r)/r → 0.0596and Ω → 9.56 × 10−3, so the solution is not asymptotically flat. An asymptotically-flatspacetime would require truncating the self-similarity at some large r [11]. Nonetheless, theself-similar solution is anticipated to represent the asymptotic behavior within R (i.e., asr → 0, t → 0).To confirm these expectations, we compute a series of models of spherically-symmetriccollapse of radiation fluid, searching for a critical point and critical behavior. We adoptpolar time slicing and radial gauge, retaining the form (1) of the line element, and assumea radiation fluid. We define W ≡ αU t and U ≡ aU r, with which the velocity normalization4becomes W 2 = 1 + U2. Then, the equations of motion of the fluid, ∇νT µν = 0, are∂∂t(aρW ) +1r2∂∂r(r2αρU)+ p[∂∂t(aW ) +1r2∂∂r(r2αU)]= 0, (2)∂∂t[a2(ρ+ p)WU]+1r2∂∂r[r2α a (ρ+ p)U2]+α a (ρ+ p)(W 21α∂α∂r− U2 1a∂a∂r)+ α a∂p∂r= 0, (3)and the pressure is p = 13ρ. In these coordinates, the gravitational field equations Grr =8πT rr and Gtt = 8πTtt become1a∂a∂r+12r(a2 − 1)= 4πra2ρ[γW 2 − γ + 1], (4)1α∂α∂r− 12r(a2 − 1)= 4πra2ρ[γU2 + γ − 1], (5)with γ = 4/3 adopted in the models we discuss here.To construct a parameterized family of evolutions, we choose Cauchy data by specifyingthat the fluid be instantaneously at rest V ≡ U/W = 0 and have an energy density profileρ =12π−3/2 r−20 η exp(−r2/r20), (6)parameterized by η and r0. The total gravitational mass is M =12ηr0, so that η = 2M/r0is a dimensionless measure of the strength of the initial gravitational field, while r0 is thetypical initial length scale. Equations (4) and (5) are solved to complete the Cauchy data.The dynamical behavior of these models is consistent with expectations. For γ = 4/3and this family of Cauchy data, the critical point is found in a bisecting search to be nearηc ≃ 1.0188. For η > ηc, the impending formation of a black hole is observed. As iswell-known, polar time slices avoid penetrating regions of trapped surfaces [12] and asymp-totically approach the apparent horizon in collapsing spacetimes. When this occurs we haveclear evidence of a black hole forming. Fig. 2 illustrates the form of the mass function5m(r) = 12r(1 − a−2) late in a supercritical evolution. The “kink” where r ≃ 2m(r) locatesthe surface of the black hole and determines the hole’s mass MBH. Immediately beyond thekink, m(r) is nearly constant indicating an evacuated region that forms around the blackhole. At larger radii, additional mass is evident, representing the fluid which did not collapseand is expanding outward.Black-hole mass MBH is found to be well described by a power law (see Fig. 3)MBH ≃ K(η − ηc)β, (7)with a critical exponent β ≃ 0.36. This critical exponent value is numerically indistin-guishable from those found in scalar wave collapse [1] and gravitational wave collapse [2],doubtlessly reflecting a deep property of the gravitational field equations.The fluid is initially everywhere out of equilibrium. In near-critical models, the fluid inthe inner region collapses while that in the outer region accelerates and expands outward.The collapsing central region is chased by a strong, ingoing rarefaction wave. As the radiusR(t) of the inner region diminishes, the rarefaction wave causes the mass to diminish also,keeping m(R(t))/R(t) nearly constant. The radius R(t) of transition between collapse andexpansion (edge of the rarefaction wave) decreases by several orders of magnitude in very-near-critical models. On fine scales, r ≪ r0, the fluid and gravitational field assume aself-similar form (see Fig. 4). Furthermore, the self-similar form being approached is thelocally self-similar solution outlined above. As Fig. 4 shows, ultimately the self-similarityof the collapse is broken if η 6= ηc. Only a precisely-critical model would approach the self-similar solution and retain this dependence as r → 0, t → 0. Near- but not precisely-criticalspacetimes develop a self-similar region on scales r ≪ r0 but then lose self-similarity on ascale determined by proximity in parameter space to the critical point: r1 ≃ K|η − ηc|β.Self-similarity will only be apparent if r1 ≪ r0.Approach to self-similarity within a region spanning two disparate scales is a familiarconcept in hydrodynamics termed “intermediate asymptotics” [13]. The cardinal example isthe modified Sedov-Taylor blast wave problem. Besides the usual Sedov-Taylor parameters6of E0, the blast wave energy, and ρ0, the ambient gas density, the modified problem has alength scale r0, within which the energy E0 is arbitrarily distributed initially, and has a smallpressure p0 in the ambient gas. This small pressure introduces a second, new length scale:r1 = (E0/p0)1/3 ≫ r0. These two additional dimensional parameters break the self-similarityof the blast wave on length scales r <∼ r0 and r >∼ r1 but on intermediate scales the solutionasymptotically approaches the Sedov-Taylor self-similar form. There is a direct analogue inthe behavior of near-critical spacetimes on scales between r0 and r1.The self-similar solution determines the asymptotic properties of precisely-critical evolu-tions. Linear stability analysis should reveal the perturbative response of solutions nearbyηc in parameter space and thereby perhaps provide an estimate of the critical exponent β.The self-similar solution, and therefore any precisely-critical evolution, apparently containsa naked singularity. However, the precisely-critical evolutions are likely to be a set of mea-sure zero as there is extreme sensitivity to the initial conditions near ηc. Furthermore, thedependence of the lapse along ξ = ∞ (t = 0) is α ∼ r1−1/n ∼ r0.129, indicating the singularityto be shrouded in infinite redshift.C.R.E. acknowledges helpful conversations with A. Abrahams, M. Choptuik, R. Priceand J. York. This research was supported by NSF grants PHY 90-01645, PHY 90-57865and ASC 93-18152, which includes support from ARPA. C.R.E. thanks the Alfred P. SloanFoundation for research support and thanks the Aspen Center for Physics for hospitalityduring summer 1993 where key aspects of this work were initiated. Computations wereperformed at the North Carolina Supercomputing Center.7REFERENCES[1] M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993).[2] A. M. Abrahams and C. R. Evans, Phys. Rev. Lett. 70, 2980 (1993).[3] C. Gundlach, R. H. Price and J. Pullin, submitted to Phys. Rev. D (1993).[4] C. R. Evans, to be submitted to Phys. Rev. D (1993).[5] A. M. Abrahams and C. R. Evans, submitted to Phys. Rev. D (1993).[6] D. M. Eardley, in Texas/PASCOS 92: Relativistic Astrophysics and Particle Cosmology,eds. C. W. Akerlof and M. A. Srednicki (N. Y. Acad. Sci., New York), 1993.[7] D. M. Eardley, Commun. Math. Phys. 37, 287 (1974).[8] We use natural units in which G = c = 1.[9] L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 10th Edition (CRCPress: Boca Raton), 1993. The self-similar solution described here may be equivalentto the ‘exploding’ models of Ref. [11].[10] G. Guderley, Luftfahrtforschung 19, 302 (1942). See also the discussion of this prob-lem in G. B. Whitham, Linear and Nonlinear Waves, (Wiley: New York), 1974. Aspherically-symmetric imploding shock wave asymptotically approaches a self-similarform and “remembers” about the initial conditions only a single number related to thestrength of the initiation of the shock.[11] A. Ori and T. Piran, Phys. Rev. D 42, 1068 (1990).[12] D. M. Eardley, talk at Numerical Astrophysics: A Symposium in Honor of James R. Wil-son, at the University of Illinois, 1982 (unpublished); See e.g., J. M. Bardeen and T.Piran, Phys. Rep. 96, 205 (1983); L. I. Petrich, S. L. Shapiro and S. A. Teukolsky, Phys.Rev. D 33, 2100 (1986); R. F. Stark and T. Piran, Phys. Rev. Lett. 55, 891 (1985).8[13] G. I. Barenblatt and Ya. B. Zel’dovich, in Ann. Rev. Fluid Mech., 4, 285 (1972). Seetheir discussion of the modified blast wave problem on p. 287.9FIGURESFIG. 1. Self-similar radiation fluid collapse. Fluid and metric variables are plotted versusself-similarity coordinate ξ = r/C(−t)n (with n ≃ 1.1485): (1) radial metric function a(ξ)(solid), (2) velocity V (ξ) = aU r/αU t (dotted), (3) Ω(ξ) = 4πr2ρ (dashed), and (4) lapse func-tion α = nrN(ξ)/t (dot-dashed). Note the collapsing interior and expanding exterior regions.FIG. 2. The mass function m(r) (solid) at a late time in a supercritical evolution. Polar timeslices limit outside the horizon producing the kink near where the function approaches the curve(dotted) r = 2m(r), locating the surface of the black hole and determining the hole’s mass. At latetimes, the fluid bifurcates and forms an evacuated region immediately beyond the hole, indicatedby the decrease of Ω = 4πr2ρ (dashed) outside the hole. At larger radii, escaping fluid expandsoutward.FIG. 3. Critical behavior of black hole mass. For η > ηc, MBH is determined and fit to a powerlaw of the form (8). Here log of MBH is plotted versus log of the critical separation η − ηc. Thepower-law fit is indicated by the solid line and is determined by ηc = 1.018828234, K = 3.27, andβ = 0.36. We obtain black holes with masses down to MBH ≃ 0.001MADM and the scaling law(8) is seen to hold over two orders of magnitude in mass. Simulations with discretization scales of3.3× 10−3 and 1.0× 10−3 are included in the plot, using open and filled circles, respectively.FIG. 4. Intermediate asymptotic approach of a near-critical evolution to self-similarity. Themetric function a(r) (light curves) at a series of times, equally spaced in logarithm of centralproper time τ = log(−T ), obtained from a near-critical evolution. The rightmost curve is nearthe beginning of the simulation and resembles a(r) in the Cauchy data. As collapse proceeds, thepeak of a(r) shifts to progressively smaller scales. The metric and fluid approach the self-similarsolution depicted in Fig. 1. The self-similar form of a(ξ) (see Fig. 1) is overlaid at τ = −0.5, clearlyindicating approach to self-similarity. At later times, self-similarity is broken as a black hole beginsto form and a → ∞ on the horizon.10This figure "fig1-1.png" is available in "png" format from:http://arxiv.org/ps/gr-qc/9402041v1This figure "fig1-2.png" is available in "png" format from:http://arxiv.org/ps/gr-qc/9402041v1This figure "fig1-3.png" is available in "png" format from:http://arxiv.org/ps/gr-qc/9402041v1This figure "fig1-4.png" is available in "png" format from:http://arxiv.org/ps/gr-qc/9402041v1

Observation of critical phenomena and self-similarity in the
  gravitational collapse of radiation fluid

Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid

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