Classical and Quantum Dynamics in a Black Hole

Background

Chris Doran

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Thanks etc.• Work in collaboration with

– Anthony Lasenby– Steve Gull– Jonathan Pritchard– Alejandro Caceres– Anthony Challinor– Ian Hinder

• Papers on www.mrao.cam.ac.uk/~Clifford– gr-qc/0106039– gr-qc/0209090

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Outline• 4 phenomena to give a classical and

quantum description for

Classical Quantum

xEmission

Bound states

Absorption

Scattering

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Classical Scattering• Main method of comparison is the differential

cross section

bpi

pfGM

θ

For r-1 potential get Rutherford formula

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Classical Dynamics• The Schwarzschild line element contains all

relativistic information (c=1)

• The geodesic equation for a radially infalling particle is essentially Newtonian

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Painlevé Coordinates• Necessary for later calculations to remove the

singularity at the horizon• Convert to time as measured by infalling

observers

• Find metric is now (no problem at horizon)

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Geodesic Equation• The geodesic equation can be written

• Vectors in 3-space• Overdots denote proper time derivatives• r is a local observable obtained from the

strength of the tidal force – not just a coordinate• Summarise in effective potential (per unit mass)

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Radial geodesics

From rest

From infinity

Light-like geodesics

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Geodesic Motion• Geodesics can be quite complicated• Write the geodesic equation in form (u=1/r)

• A cubic equation, so solution is an elliptic function

• For intermediate angular velocities, get spiralling

• Complicates the calculation of the cross section

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Sample Geodesics

=0.9cv=0.5c vSpiralling

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Cross-section• Analytic formula for the motion involves an

elliptic integral• Best evaluated numerically, for a range of

velocities • Collins et al. J. Phys A 6 (161), 1973• Result in a series of cross-section graphs• Can do small angle case analytically

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Numerical Results

Corresponds to v=0.995c

Rutherfordat small θ

Additional scattering as θ ≈ π

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Quantum Treatment• Concentrate on fermions.• These are described by the Dirac equation• Uses apparatus of spinors, Dirac matrices,

tetrads and spin connections• Typically neglected in black hole treatments –

favour massless scalar fields• But in fact, Dirac theory is easier

– First order– Simple, Hamiltonian form

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Dirac Equation• Standard notation, in full gruesome detail

• Of course, much easier using geometric algebra – which is how we do it!

Spin Connection

Dirac spinor

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Hamiltonian Form• Return to the metric

• Convert to Cartesians

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Hamiltonian Form• Return to the metric

• Now introduce the matrices / vectors

‘Flat’ Minkowski vectors

Gravitational interaction

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Hamiltonian Form II• Now insert matrices into Dirac equation

• Convert to Hamiltonian form• All interactions contained in the interaction

Hamiltonian

Flat space Interaction

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The Interaction Hamiltonian

• All gravitational effects in a single term• This is gauge dependent• In all gauge theories, trick is to

1. Find a sensible gauge2. Ensure that all physical predictions are

gauge invariant• Hamiltonian is scalar (no spin effects)• Independent of particle mass• Independent of c

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Non-relativistic limit• The non-relativistic limit of the Dirac equation

is the Pauli equation• No spin effects - insert directly into

Schrödinger equation

• Substitution

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Implications• Recovered Newtonian potential• With a Hamiltonian independent of mass!• Solutions are confluent hypergeometrics• Phase factor irrelevant to density, hence to

cross-section• Non-relativistic limit of cross-section must be

Rutherford formula (exact)• Also expect a bound state spectrum

equivalent to Hydrogen atom (later)

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Iterative Solution• Borrow technique from quantum field theory

• Has an iterative solution

+ + + …FeynmanDiagrams

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Amplitude• Convert to momentum space

Amplitude Plane wave spin states

Use amplitude to compute differential cross section

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Vertex Factor• Fourier transform of interaction term is

• Evaluates to

Energy conserved so this vanishes on shellProcess must be second order

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Vertex Factor II• Evaluate the second order diagram

pi pf

k

Result is

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Cross-section• Reinsert the asymptotic spinors. Get

differential cross-section

• q is the momentum transfer pf -pi

• Unpolarised version, after spin sums, is

Scattering angle θVelocity

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Comments

• Result is independent of particle mass• Equivalence principle holds to lowest order in

quantum theory• Small angle approximation agrees with point

particle dynamics• No boundary conditions specified at horizon• Can extend to higher order and include

radiation• Get terms violating equivalence principle

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Comments II• Massless limit well defined (v =1)

• Reproduces photon deflection formula at small angles

• Zero in backward direction – a neutrino diffraction effect

• Can apply to scalar fields as well

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Gauge Invariance• Important issue to address• Do not have a general proof, but can

reproduce calculation in another gauge• In Kerr-Schild gauge set

• Calculation is a different order• But result is unchanged – a physical prediction

First-order in M+

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Absorption• Particles too close to the horizon end up

captured• See this from the effective potential

Plot of increasing J

Higher J values are scattered

E too high get absorbed

Low J are absorbed

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Absorption Cross-section• Impact parameter b is critical distance from

hole for fixed velocity and angular momentum• Total absorption cross-section is

• For photons find that b2=27(GM)2

• Hole appears of a disk of radius b

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Absorption Cross-section II• Slightly more complicated calculation gives

Photon limit

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Quantum Equations• Radial Schrodinger equation is

• Convert to first-order form (rψ=u1)

• With |κ|=l+1 recover the correct Dirac radial separation

• Energy term tells us how to add in interaction

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Black Hole Case• Black hole Hamiltonian includes derivative

terms. Find that radial equations are (G=1)

• See that singular points exist at the origin (r-3/4) horizon, and at infinity (irregular)

• Special function theory underdeveloped for this problem

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Units and Dimensions• Convert to dimensionless form by introducing

distance function x=2r/r0• Dirac equation controlled by dimensionless

coupling constant α and energy ε

• α also ratio πr0/λ – horizon/Compton w/length• α ≈ 1 corresponds to primordial black holes• Also have

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Horizon• Series expansion about horizon η=(r-2M)

• Get indicial equation

• Roots are

Regular branch -physical

Singular branch -unphysical

Gauge invariant

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Regular Solutions (α=0.01)ε=0.1, l=0 ε =0.2, l=0

ε =0.1, l=1 ε =0.2, l=1

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Singular modes (α=0.01)ε =0.1, l=0 ε =0.2, l=0

ε =0.1, l=1 ε =0.2, l=1

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Asymptotic Behaviour• At large r have

• Similar for u2

• Normalise such that• Absorption cross section is

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Massless Case

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5

Momentum

Photon limit

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Massive Case

0

1000

2000

3000

4000

0 0.2 0.4 0.6 0.8 1

α=1

Coupling 0.03

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

α=0.03 Coupling = 0.1

050

100150200250300

350400450500

0 0.2 0.4 0.6 0.8 1

α=0.1

Coupling = 0.5

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8 1

α=0.5

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Classical Bound States• Can have stable, classical orbits outside a

black hole

Precessing ellipse

Find minimum bound state energy 0.95mc2

No stable orbits within 6M

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Semi-Classical Model• Carry out a ‘Bohr’ quantisation L=n~• Find that energy is

Dimensionless coupling

Angular momentum of ground state increases with coupling

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Quantum Bound States• Hamiltonian is not Hermitian

• Origin acts as a sink • Dirac current is future-pointing, timelike• Inside horizon, all current streamlines are

swept onto the singularity• Any normalizable states must have an

imaginary component to E – resonance mode

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Method• Start with regular solution at horizon and

integrate outwards• Simultaneously, integrate in from infinity,

assuming exponential fall-off• If both u1 and u2 meet at a fixed distance,

have a solution• Four terms to vary – real and imaginary

energy and normalisation • Four terms to set to zero – use a Newton-

Raphson method

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Probability Density α=0.1

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Probability Density α=0.35

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Probability Density α=0.5

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Variation with κ

α=0.5First excited states with Increasing angular momentum

Further out, become Hydrogen-like

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Expectation value h r i

1S1/2

2S1/2

3S1/2

Horizon

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Imaginary Energy1S1/2

Decay rate increases with coupling constant α and decreases with κ

2P3/2

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Comments• α ≈ 1 is the scale appropriate to primordial

black holes• Solar mass black holes have α ≈ 1,000• Corresponding spectrum of antiparticle states

also all have decay factors• Decay rates can be extremely slow for orbits

a long way from horizon• Binding energies much larger than classical

predictions

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Emission• Return to singular branch at horizon and

compute radial currents

• Form ratio of outgoing to total current

Outgoing

Ingoing

Fermi-Dirac distribution at the Hawking temperature

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Future Work• Carry our scattering work to higher order• Include radiation effects• Partial wave analysis of cross-section• Find bound state spectrum for larger coupling• Repeat analysis for Kerr states• Investigate QFT description of unstable states

(quasi-normal modes)• Contribution to Hawking radiation?