Spherical dust collapse from the perspective ofloop quantum gravity
Tomohiro Harada
Department of Physics, Rikkyo University

Seminar at OCU, 16 October 2008

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Contents
1Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Introduction

Contents
1Introduction

2

Loop quantum gravity: minimum

3Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Introduction

Quantum gravity
We have yet no completed theory of quantum gravity. Quantum
gravity should answer the following questions.
Spacetime singularities (big bang and gravitational collapse)
Black hole thermodynamics, Hawking evaporation, information
loss problem
How does the law of gravity change from low-energy scale to thePlanck scale?
We here take loop quantum gravity. Loop quantum gravity isNonperturbative
Background independent
Canonical quantisation
No ultraviolet divergence
Not intended to be a unified theory
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Introduction

Spacetime singularity

Spacetime singularity: a place where quantities which are used
to measure the gravitational field become infinite Moreaccurately, a point which is in the boundary of the maximallyextended spacetime manifold and is not infinity.
Classical general relativity breaks down at the singularity.
Cosmic censorship (Penrose 1969) states that singularities are
hidden behind horizons in physical spacetimes. But we have
many naked singularities...
Censored or not, spacetime singularities should be dealt with in a
regular manner.

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Introduction

What is loop quantum gravity?
3 dimensional space + time
Einstein-Hilbert action
Canonical conjugate pair: tetrad Eia
and connection Aia
Hamiltonian H expressed in terms
iand volume Vof curvature FabQuantum state space constructed byspin-network statesholonomy ĥ, area operator Ŝ,volume operator V̂
Hamiltonian operator Ĥ quantisedin terms of ĥ and V̂
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j2=1/2

S=n1

j1=1

n2

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Loop quantum gravity: minimumContents
1

Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Loop quantum gravity: minimum

TriadTetrad:
eI = eIµ dxµ ;η IJ = eIµ eJν g µν ;gµν = eIµ eJν ηIJ .
We take e0a = 0 gauge. qab = gab (a, b = 1, 2, 3)
Densitized triad:1
Eia = ijk abc ejb ekc
2
orEia Ejb δ ij = (det qcd )q ab ,where i, j = 1, 2, 3.
Notation for an antisymmetric tensor AijAi = ijk Ajk
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Loop quantum gravity: minimum

ConnectionSpin connection:
deI + ωJI ∧ eJ = 0,
dei + Γij ∧ ej = 0.
Ashtekar (Barbero) connection:
Ai = ω i + γω 0i,where γ is the only one unspecified parameter in the theory.
This is called the Immirzi parameter.
Extrinsic curvature
K i ≡ Ai − Γ iPoisson bracket{Aia (x), Ejb (y)} = 8πγGδab δji δ 3 (x, y)
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Loop quantum gravity: minimum

Hamiltoniani= ∂a Aib − ∂b Aia + ijk Aja Akb
Curvature: FabHamiltonian (unsmeared form):√jiji)Eia Ejb /( det E)
Kb]
H = (Fab+ (γ 2 + 1)K[a√R
Volume: V = d3 x det E
Hamiltonian (K 2 term neglected):Z
H[N ] =N tr(F ∧ {V, A}),

where N is the lapse function.

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Loop quantum gravity: minimum

Spin network
Spin network (Γ, jl , in ):j2=1/2

S=

j1=1

n1

n2

j3=1/2Spin network stateΨS [A] =ORjl (H[A, γl ])l!

·Onin!

,where H[A, γl ] is a holonomy.
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Loop quantum gravity: minimum

Smeared operators
Aia (x) and Eia (x) = −i~ δAiδ(x) are not well defined.
aHolonomy H(A, γ) and Flux Ei (S)Z

H(A, γ) = P exp A,γZ
Ei (S) ≡ −i~dS 2 naS

δδAia (x(σ))

Area operator and volume operator
Xp
A(S)|Si = ~ji (ji + 1)|Sii

V(R)|Si = lim

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Loop quantum gravity: minimum

Hamiltonian
Discretized Hamiltoniani XNnXll0 l00~ n∈Sl,l0 ,l00
× tr hγx−1 hαxn ,l0 ,l00 [V(Rn ), hγxn ,l )] |Si,

H|Si = −n,lwhere Nn is the lapse, h denotes a holonomy.
l’αγxlTomohiro Harada (Rikkyo U)

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l’’

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Effects of loop quantisation

Contents
1

Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correctionholonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Effects of loop quantisation

LTB spacetime

Contents
1

Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Effects of loop quantisation

LTB spacetimeSpherially symmetric spacetime
Metric dΩ2 = dϑ2 + sin2 ϑdϕ2
ds2 = −N 2 dt2 + L2 (dx + N x dt)2 + R2 dΩ2
Spatial metric and densitised triad
ds2 =E ϕ2 2
dx + |E x |dΩ2x|E |Canonical pairs (γ: Immirzi parameter){Ax (x), E x (y)} = 2γGδ(x, y),{γKϕ (x), E ϕ (y)} = γGδ(x, y)
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Effects of loop quantisation

LTB spacetime

The LTB system interms of triad and connection
Classical LTB metric (marginally bound)ds2 = −dt2 + R02 dx2 + R2 dΩ2“LTB relation”
1 x|E (x)|02
Hamiltonian (unsmeared)
E ϕ (x) =Hgrav

1= −2G

Hdust =F 0 (x)and Kϕ0 = Kx sgnE x!
p
Kϕ2 E ϕp+ 2Kϕ Kx |E x | ,|E x |,2Gwhere F (x)/2 is the conserved dust mass.
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Effects of loop quantisation

Inverse triad correction

Contents
1Introduction

2Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Effects of loop quantisation

Inverse triad correction

Spherically symmetric loop quantum gravity
Tg,k,µ : An orthonormal basis of gauge-invariant states
YY10
Tg,k,µ =exp 2 ike ∫ (Ax + η )dxexp(iµv γKϕ (v))
e∈gev∈gwith integer labels ke and positive real labels µv on edges e andvertices v, respectively, forming a finite graph g in the
1-dimensional radial line.
Eigenvalues
ke+ (x) + ke− (x)Ê x (x)Tg,k,µ = γ`2PTg,k,µ ,2ZXµv Tg,k,µ ,Ê ϕ Tg,k,µ = γ`2PIv∈I

where where `2P = G~ is the Planck length squared and e± (x)denote the neighboring edges to a point x.

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Effects of loop quantisation

Inverse triad correction

Inverse triad effects

Inverse triad can be replaced asZ

ϕ sgn(E x )E\−ip=tr(τ3 hx [h−1x , V̂ ])x2πγG~|E |I

Eigenvalues
pp∼|ke+ + ke− + 1| − |ke+ + ke− − 1|

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Effects of loop quantisation

Inverse triad correction

Inverse triad correction
Correction functionα(E x ) :== 2pp|E x + γ`2P /2| − |E x − γ`2P /2| p x|E |γ`2P

Corrected Hamiltonian(I)Hgrav(II)Hgrav

1= −2Gpα(E x ) 2 ϕp
Kϕ E + 2Kϕ Kx |E x ||E x |!

.

1 α(E x )p= −(Kϕ2 E ϕ + 2Kϕ Kx E x ) .
x
2G |E |Tomohiro Harada (Rikkyo U)

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Effects of loop quantisation

Inverse triad correction

Consistent formulation ver. 1
Corrected LTB relation:(E x )0 = 2f (E x )E ϕ,

Kϕ0 = f (E x )Kx

Hamiltonian constraint and evolution
Ṙ2 R0 (α(R) − 1) + 2RṘṘ0 + Ṙ2 R0 = f (R)F 02RR̈ + Ṙ2 + (α(R) − 1)Ṙ2 = 0 .
Correction functions: α(R) and f (R)
df (R)
dR
Tomohiro Harada (Rikkyo U)= (1 − α(R))
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fR
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Effects of loop quantisation

Inverse triad correction

Correction functions

Figure: α(R) (solid) and f (R) (dashed), where R ∗ :=α(R) = 2
Tomohiro Harada (Rikkyo U)q|R2 + γ`2P /2| −qγ`2P

LTB in LQG

p|R2 − γ`2P /2|

γ/2`P .

R
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Effects of loop quantisation

Inverse triad correction

Consistent formulation ver. 2Corrected LTB relation:
Eϕ =12α(E x )0,αKx = Kϕ0

Hamiltonian constraintṘ2 Rα2!0− F0 = 0

Evolution equation
2RR̈ + Ṙ2 = 2

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Effects of loop quantisation

holonomy correction

Contents
1Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5SummaryTomohiro Harada (Rikkyo U)

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Effects of loop quantisation

holonomy correction

holonomy effectHolonomy is a periodic funtion of the connection. The effects
may be included by the replacements:
Kϕ → (γδ)−1 sin(γδKϕ ),Kx → (γ`0 )−1 sin(γKx `0 ),where `0 and δ parametrize the discreteness of the state along
the radial direction and on the sphere, respectively.
Corrected Hamiltoniansin2 (γδKϕ ) E ϕp2G
γ 2δ2|E x |sin(γδKϕ ) sin(γKx `0 ) p x+2|E | .
γδγ`0(III)= −Hgrav

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Effects of loop quantisation

holonomy correction

Consistent formulation ver. 3
Corrected LTB relation(E x )0 = 2g(Kϕ )E ϕ,

Kϕ0 = g(Kϕ )Kx ,where g(Kϕ ) = cos4 (γδKϕ /2) is required.
Hamiltonian constraintq
4Ṙ R 1 − γ 2 δ 2 Ṙ2 + 8RṘṘ0
2 q
q
0222
= F 1 + 1 − γ δ Ṙ1 − γ 2 δ 2 Ṙ2 .
20Evolution equation

q2RR̈ + Ṙ 1 − γ 2 δ 2 Ṙ2 = 0 .
2

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Application of the inverse triad effects

Contents
1

Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Application of the inverse triad effects

Analytical properties

Contents
1Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Application of the inverse triad effects

Analytical properties

Results (1)A bounce does not occur because Ṙ = 0 is impossible.
The vacuum static solution is nontrivial.

ds2(I) = −dt2 +1
dR2 + R2 dΩ2f (R)2

No Friedmann or Oppenheimer-Snyder solution because thehomogenous expansion R(t, x) = a(t)x is not a solution.
No self-similar (or homothetic) solution is expected because of`P in the equation.

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Application of the inverse triad effects

Analytical properties

Results (2)Effective massm(I) =

1
1R(1 − f 2 + Ṙ2 ) = mclass − R(f 2 − 1)2
2

Effective density
(I)

m(I)0
=
4πGR2 R0

Small x expansion:
If F (x) = F3 x3 + · · · , R(t, x) = R1 (t)x + · · · , we haveṘ1 = 0. The central density is constant as long as expanded
regularly.
We might allow for F (x) = F2 x2 + · · · for a finite effective
density. Then, we have a central singularity in a finite time.
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Application of the inverse triad effects

Numertical analysis

Contents
1

Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Application of the inverse triad effects

Numertical analysis

Classical collapse

Uniform model

Quadratic model

100000

1e+06

10000

10000010000
DensityDensity1000
100101000
100
101

10.10.1
00.02 0.04 0.06 0.08 0.1 0.12 0.14
Areal RadiusFigure: Oppenheimer-Snyder

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00.02 0.04 0.06 0.08 0.1 0.12 0.14
Areal RadiusFigure: LTB: naked singularity

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Application of the inverse triad effects

Numertical analysis

Inverse triad correction (1)1e+06100000

Density

10000
1000
100
1010.100.02 0.04 0.06 0.08 0.1 0.12 0.14
Areal RadiusFigure: Conserved density (dashed) and effective densityp (solid):
Quadratic model with Rs = 0.1, M = 0.01 (R∗ = γ/2`P ∼ 0.25)
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Application of the inverse triad effects

Numertical analysis

Inverse triad correction (2)0Coordinate Velocity

-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
00.1

0.2

0.3

0.4

0.5

Areal RadiusFigure: pVelocity profile: Quadratic model with R s = 0.1, M = 0.01
(R∗ = γ/2`P ∼ 0.25)
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Application of the inverse triad effects

Numertical analysis

Summary of numerical results

The gravitational collapse is strongly slowed down for R . R ∗ .
There appears a density spike for 0 < R . R∗ . This willcorrespond to shell-crossing singularity and hence gravitationallyweak.
The slow-down implies repulsive effects of quantum gravity but
this is not so strong that the collapse turns to bounce.
The qualitative feature does not change for the second version
formulation or even if we allow for F (x) = F2 x2 + · · · .

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Summary

Contents
1Introduction

2

Loop quantum gravity: minimum

3

Effects of loop quantisation
LTB spacetime
Inverse triad correction
holonomy correction

4

Application of the inverse triad effects
Analytical properties
Numertical analysis

5

Summary
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Summary

Summary
Loop quantum gravity is a promising candidate for successful
quantum gravity.
Inverse triad correction and honolomy correction are taken into
account in the marginally bound LTB spacetime.
Formulations are obtained for both effects, where theHamiltonian constraint is consistent with the evolution equation.
The dynamics with the inverse triad correction is studied.
No bounce, no homogenous soloution, nontrivial vacuum
solution and no self-similar solution
The centre will be frozen or collapse to singularity depending onwhether the conserved or effective density is physical.
The collapse only near the centre is slowed down, which resultsin shell-crossing.
This case gives a caution to loop quantum cosmology.
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