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?