Critical Phenomena in Gravitational CollapseCritical Phenomena in Gravitational CollapseMatthew W. ChoptuikMatthew W. Choptuik

CIAR Cosmology and Gravity ProgramCIAR Cosmology and Gravity ProgramDept. of Physics and Astronomy, UBC, Vancouver, BC, CanadaDept. of Physics and Astronomy, UBC, Vancouver, BC, Canada

Grand Challenge Problems in AstrophysicsGrand Challenge Problems in AstrophysicsRelativistic AstrophysicsRelativistic Astrophysics

IPAM, UCLA, Los Angeles CAIPAM, UCLA, Los Angeles CA

May 5, 2005May 5, 2005

Research supported by NSERC, CIAR and CFI

Photo: Monique Choptuik © 2004

2Wreck Beach, April 10, 2005

CollaboratorsCollaborators

Eric Hircshmann, BYUEric Hircshmann, BYU

Steve Liebling, Long Island University Steve Liebling, Long Island University

Inaki Olabarrieta, UBC Inaki Olabarrieta, UBC

Scott Noble, UT/UBC/UIUCScott Noble, UT/UBC/UIUC

Frans Pretorius, UBC/CaltechFrans Pretorius, UBC/Caltech

Bill Unruh, UBCBill Unruh, UBC

Jason Ventrella, UBC/LSUJason Ventrella, UBC/LSU

Photo: Monique Choptuik © 2004

4

Outline

• General Background– Context, general relativity and black holes

• Overview of black hole critical phenomena

• Specific model problems– Spherically symmetric (SS) massless scalar collapse– SS Perfect fluid collapse

• Understanding critical phenomena via perturbation theory

• Angular Momentum / Axisymmetric Collapse– Spherically symmetric collapse with angular momentum– Axisymmetric massless scalar collapse without and with angular

momentum

• Concluding remarks

5

Context

• Want to understand strong-field, dynamical, classical (i.e. non-quantum) gravity

• Phenomena: black holes (formation, interaction), gravitational radiation, singularities …

– Intrinsic physical and mathematical interest

– Astrophysical applications

– Hints for a quantum theory of gravity

• Direct numerical solution of Einstein’s equations (numerical relativity), allow us to probe dynamical, non-linear, strong-field regime; particularly important given inability to perform strong-field experiments

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General Relativity(G = c = 1)

• Einstein: Gravitational interaction can be interpreted in terms of geometry of spacetime, which is encoded in the spacetime metric

• Einstein field equations

• If matter fields are present, their equations of motion must be solved in concert with the Einstein equations

• In numerical relativity, typically solve coupled field equations as an initial value problem

– Specify initial data for matter and gravitational fields, subject to constraints– Specify prescription for laying down coordinates on spacetime– Evolve fields to future (or past) using some combination of evolution and

constraint equations

12 8R g gG TR

2 gds dx dx

7

Gravitational Collapse and Black Holes

• Black hole: Region of spacetime from which no physical signal can escape

• During collapse of matter and/or radiation, BH forms when gravitational field becomes strong enough to “trap” light rays

• Surface of black hole is known as the event horizon

• Singularities (infinite, crushing gravitational forces) apparently inevitable inside black holes

• Schwarzschild metric (spherically symmetric BH)

From Wald, General Relativity, 1984

• Cosmic censorship hypothesis (Penrose): Singularities resulting from gravitational collapse of “reasonable” matter are generically hidden inside black holes; collapse does not generically produce “naked singularities”

-12 2 2 2 22 2

1- 1-M M

ds dt dr r dr r

8

Black Hole Critical Phenomena

• Consider parameterized families of initial data representing gravitational collapse

• Family parameter, , controls strength of gravitational field during time evolution

• What is nature of no-BH to BH transition? (Christodoulou)

• Can one make BHs with arbitrarily small mass? (view mass as “order parameter”)

Type I

Type II

No black hole forms

Black hole forms

Threshold of black hole formation

p p

p p

p p

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Black Hole Critical Phenomena

• BH threshold generically characterized by exponential sensitivity to initial conditions

• For any collapse model, find (isolated) universal solutions with additional symmetry

• Find universal scaling laws for solutions near-criticality; for example, in Type-II collapse, black hole masses satisfy

where is a universal (independent of initial data family) exponent

• Mass can be viewed as an “order parameter”

Type II

Type I: static or periodic

Type II: continuously or discretely self-similar (CSS or DSS)

Natural control parameter: ln( )p p

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Spherically Symmetric Massless Scalar Collapse(MWC, 1993)

• Arguably simplest model for gravitational collapse using “reasonable matter”

– Choose spherical symmetry to minimize computational cost, complexity; best “1+1” approximation to generic collapse

– No independent dynamics of gravitational field in spherical symmety (Birkhoff theorem); need matter

• Matter field: massless scalar field

• Initial data:

• Dynamics: Imploding / exploding shells of radiation

• Fundamental non-linearity gives rise to competition

( , )r t

0 0( , ), ( , )r rt

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Competition

Where can energy in system end up?

BH

KE

PE

r

r R

Play two effects off each other to generate critical solutions

COMPLETE DISPERSAL

BLACK HOLE FORMATION

(Heuristic picture has been rigorously established by Christodoulou; see e.g., Classical & Quantum Gravity, 16, A23-A25, (1999))

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Scalar Collapse: Weak Field Regime

• Metric (flat space-time)

• Scalar field equation of motion

• General solution: ingoing & outgoing waves

• Can choose data to be purely ingoing at initial time

2 2 2 2 2 2 2( sin )ds dt dr r d d

2 2

2 20 ( ) ( )r rt r

( , ) ( ) ( )r r t u r t v r t

13

Scalar Collapse: Weak Field Regime

• Initial data: purely ingoing pulse 0

0

( ,0) ( )

( ,0)

r r f r

dfr r

t dr

End state of evolution is complete dispersal of scalar field

14

Scalar Collapse: Strong Field Regime

• Metric

• Auxiliary scalar field variables

• Mass function, conserved “mass-at-infinity”

• Coordinates cannot penetrate apparent horizons (or event horizons for the most part); in practice does not preclude identification of spacetimes containing black holes

2 2 2 2 2 22 2 2( si( , ) ( , ) n )r t a r tds dt dr r d d

( , ) ( , )r t ra

r tt

12 ( , )2( , ) 1

m ra r

rt

t

0

dmdr

drM

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Scalar Collapse: Strong Field Regime

• Equations of motion

2

22

2

2

2

1

1 1 10

1 12 0

t

t

d

r

rr r

da aa ddr

dadr

r ra

ra r

a

a

Strong-field evolution, scalar field still completely disperses, but non-linear self-gravitational effects are apparent in trailing edge of waveform.

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Generating a Critical Solution

• Choose arbitrary one parameter family of initial data

• Locate initial bracket – no black hole– black hole

• Refine bracket via, e.g. bisection search

• Can tune to machine precision, , provided code has enough dynamic range (adaptive mesh refinement) 0( ,0) ( ) ( )r f r f rp

-1610

p p

p p

[ , ]p p

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Sample Bracketing Solutions (Mass aspect plotted)

Dispersal

BH formation

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Approach to the Critical Solution

0ln( )r c

dmdr

3

4 2

34

3

ln( )p p

19

Approach to the Critical Solution

1ln( )r c

dmdr

2

34

34

ln( )p p

20

Approach to the Critical Solution

2ln( )r c

dmdr

4

34

3

ln( )p p

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Scalar Collapse: Near-Critical Regime

• Tune control parameter, , to machine precision

• As , unique critical solution emerges (details of initial data “washed away”)

• Critical solution is discretely self-similar

• Curvature grows without bound in precisely-critical limit, and no black hole forms:

naked singularity

p

p p

Marginally sub-critical evolution; each oscillation represents same strong-field dynamics playing out on a spatio-temporal scale some 30 times smaller than its predecessor.

Logarithmic time, radial coordinates used

ln , lnr c t t

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Generic Features of Type II Collapse

• Arbitrarily small black holes possible (by definition)

• Critical solution is self-similar, possibly discretely

• Precisely critical solution contains a naked singularity (but formation clearly not generic)

• Scaling laws for dimensionful physical quantities, such as the black hole mass

• Universality: Same results independent of specifics of initial data

| |fBHM c p p

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Self-Similarity of Type II Solutions

• central proper time (measured from “accumulation event”)

• physical (e.g. “areal” radius)

• Similarity variable:

• Naked singularity at (0,0)

T

R

/R T

naked singularity

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Self-Similarity of Type II Solutions

• Continuously self-similar (CSS)

• Discretely self-similar (DSS)

( , ) ( , ) , arbitraryg g

( , ) ( , ) 0, 1, 2,g g n n

Model-dependent “echoing” exponent

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Type I Collapse: Critical “Neutron Stars”(Scott Noble, UT Austin PhD, 2003)

• Perfect fluid coupled to gravity, spherically symmetry

• Ideal-gas (Synge) equation of state

• Family of static solutions, “stars”, parametrized by central density

• Stars to right of maximum are unstable

( 1)

1 2

restP

Stable branch Unstable branch

2

2 : extremely stiff fluid

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Critical “Neutron Stars”

• “Perturb” stable, static, star via imploding pulse of scalar field

• Fluid/scalar interact only through gravitational field.

• By fine-tuning amplitude of scalar pulse, can drive fluid to unstable stellar state ( and perturbations)

• End state is either black hole or perturbed stable star

This near-critical evolution exhibits Type I behaviour (finite BH at threshold); Type II behaviour also possible, with different initial conditions.

Radial coordinate is logarithmic.

24dm

rdr

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Generic Features of Type I Collapse

• Mass-gap at black hole threshold (by definition)

• Critical solution is static, or possibly periodic

• Scaling law for lifetime of critical solution in near-critical regime

• Universality: Same results independent of specifics of initial data

• Well known in pre-critical-phenomena literature, though black-hole threshold nature perhaps not appreciated ln| |T p p

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Perturbation Theory(Evans & Coleman; Koike et al; Maison; Eardley & Hirshcmann; Gundlach)

• For given model (matter content, couplings, symmetry), can generically find one or more isolated critical (threshold) solutions, , via dynamical evolution of 1-parameter families of initial data

• can also be generally constructed “directly” from ansatz that reflects particular symmetry (e.g. continuous self-symmetry, discrete self-symmetry, time-independence, periodicity) exhibited by solution

• Consider CSS case, and use as a background solution; can find eigenmodes , eigenvalues , then as

where the are constants (expansion coefficients)with .

g

g

g

(if p p

1

( ) ( ; ) ( ) ( ) expi i in

g p g p g C p f

( )iC p ( ) 0iC p

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Perturbation Theory

• Observed critical behaviour strongly suggests that only one eigenmode is unstable

• (Assumes no marginally stable / relevant modes, ) Black hole formation

Dispersal

Critical Solution

Re( ) 0

single unstable mode

Re( ) 0, 2,3,

- all other modes damp as

i i

p p

Re( ) 0i

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Perturbation Theory

• Assuming , then as

• Independent of scale, black hole formation or (“peel-off” to dispersal, differs only in sign of ) is signaled by

and the black hole will form on the scale

• Then

which is just the scaling law

1 1Re( ) p p

11 1 1 1( ) ( ) exp( ( ( )( ) exp( ) ( )

dCg p C p f p p p f

dp

( )p p

1 (( ) exp( ) .)pp p K const

ln ( ) ln( ( ) / 2) ( )BH BHM p R p p k

1*exp(log( )) exp( ln ( ))BHp p M p K

1

1( ) with

Re( )BH fM p c p p

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Perturbation Theory

• Picture of critical solutions as “intermediate attractors” (Barenblatt), with 1-dimensional unstable manifolds (“minimally unstable solutions”), also explains observed universality of critical phenomena

• Validated for essentially every spherically symmetric model in which critical phenomena have been observed, including

– Perfect fluid (Koike et al, Maison, Evans)– Non-linear sigma field (Eardley & Hirschmann)– Massless scalar (DSS) (Gundlach)– SU(2) Einstein-YM (Gundlach)– Scalar electrodynamics (Gundlach & Martin-Garcia)

• No equivalent “theory” yet for “echoing exponents” ( ) that arise is DSS solutions

• Consideration of general admissible DSS scaling solution predicts “universal wiggle” in mass scaling law (Gundlach); subsequently observed (Hod & Piran)

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Critical Scalar Collapse with Angular Momentum in Spherical Symmetry

(MWC, Olabarrieta, Unruh, Ventrella (in progress))

• IDEA: For given value of principal angular momentum “quantum number”, L, consider superposition of stress tensors for 2L + 1 scalar fields, each of which has non-trivial angular dependence, but which in sum produce spherical matter distribution

where are normalized real eigenfunctions of the angular part of the flat-space Laplacian with eigenvalues L(L+1)

• Stress tensor of each component field:

• Total stress tensor:

( , , , ) ( , ) ( , ) , , ( 1) 1,M ML Lt r t r M L L L L

12

LM M M M ML L L LT g

,

1sin( )

4L LM LM LM

M M M

T T T T d d

( , )ML

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Critical Scalar Collapse with Angular Momentum in Spherical Symmetry

• Individual fields are non-spherical, but superposition yields spherical geometry; can investigate some of the effects of angular momentum on critical collapse in spherically symmetric setting

• Again choose, Schwarzschild-like coordinates

• Scalar field equation of motion (from vanishing of divergence of total stress tensor)

• Regularity condition:

2 2 2 2 2 2 2( , ) ( , )ds r t dt a r t dr r d

2

( 1)L Lr

a

0lim ( , ) L

rr t r

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Critical Scalar Collapse with Angular Momentum in Spherical Symmetry

• For each L look for black hole threshold solutions using variety of initial data families

• Find distinct critical solution for each L; at least for small values of L, solutions are apparently all discretely self similar

• As L increases, echoing exponent decreases; threshold solutions become more and more “periodic”, and increasingly long-lived

• Quite reminiscent of geon idea (Wheeler and Brill) whereby configurations of massless field with large values of L were conjectured to be able to stabilize themselves against gravitational collapse

35

Critical Scalar Collapse in Spherical Symmetry with Angular Momentum

ln( )r c

dmdr

ln( )r c

dmdr

dmdr

dmdr

36

Critical Scalar Collapse in Spherical Symmetry with Angular Momentum

L

0 3.44 0.37

1 0.46 0.12

2 0.12 0.04

3 0.039 0.02

5 0.0044 0.003

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Critical Scalar Collapse in Spherical Symmetry with Angular Momentum

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Axisymmetric Scalar Collapse (MWC, Pretorius, Hirschmann, Liebling, Pretorius, (2003))

• Want to study critical phenomena in more general context (as well as develop infrastructure for more astrophysically relevant calculations in numerical relativity)

• Adopt cylindrical coordinates:

• Metric

where all metric functions are functions of

• Again, couple a real, massless scalar field to gravitational field, use partially constrained evolution (resolve momentum constraints at each time step)

• Perturbative results due to Gundlach and Martin-Garcia (1999), suggest that there are no non-spherical unstable modes

2 2 2 4 2 2 2 2 2[( ) ( ) ]zds dt d dt dz dt e d

( , , )t z

( , )t z

39

Near-critical Scalar Field CollapseSpherically Symmetric Initial Data

Solution is good match to result from spherically-symmetric code

40

Near-critical Scalar Collapse z-Antisymmetric Initial Data

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Increasingly Prolate Collapse (p = “prolateness”)

• Intriguing suggestion of non-spherical unstable mode, and possible cascade of “echoers”

• Further study needed to determine whether this result holds in continuum limit

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Axisymmetric Scalar Collapse With Angular Momentum(MWC, Hirschmann, Liebling, Pretorius, PRL, 93:131101 (2004))

• Real massless scalar field cannot carry angular momentum in axisymmetry, thus consider complex scalar field with ansatz

where m is an integer (required for regularity)

• Similar in spirit to spherically-symmetric case with angular momentum just discussed

• Metric

where is the axial Killing vector

( , , , ) ( , , ) imt z t z e

2 2 2 4 2 2 2 2 2

4 2 2

[( ) ( ) ]

( ) 2

z

zz

ds dt d dt dz dt e d

d dzd dz d

e

( , , , ) (0, , ,0)t z z

43

Axisymmetric Scalar Collapse With Angular Momentum

1mRe( )

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Axisymmetric Scalar Collapse With Angular Momentum

• Comparison with spherical results

so spherical calculation apparently capture gross features of axisymmetric computation remarkably well

Model

Spherical 0.46 0.12

Axisymmetric 0.42 0.11

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Axisymmetric Scalar Collapse With Angular Momentum

2 6AH AH AHIn critical limit, 0 significantly faster than , possibly as fast as J M M

ln(Apparent horizon mass)

Ln(A

ppare

nt

hori

zon a

ngula

r m

om

entu

m)

46

Summary

• Parameterized black hole formation exhibits many of the same features associated with phase transitions in statistical mechanical systems

– Both “first order” and “second order” transitions are observed

– Black hole mass can be identified as an order parameter, exhibits scaling with “universal” scaling exponent for any given model

– Critical solutions have small number (frequently one) of unstable modes in perturbation theory; this observation explains existence and uniqueness of scaling exponents; mathematics is precisely the same as for “ordinary”critical behavior

• Angular momentum studies suggest existence of “ladder” of critical solutions, parameterized e.g. via behaviour of solution in vicinity of center of symmetry

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Concluding Remarks

• Many open questions

– To what extent are critical solutions truly one-mode unstable; what happens when all symmetry restrictions are relaxed?

– How generic is this type of behaviour (including discrete self-similarity) in nonlinear PDEs of evolution type? (Bizon et al, Isenberg et al)

– Astrophysical implications?

• Primordial black hole formation? (Niemeyer & Jedamzik)

• Core collapse?