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Classical and Quantum Dynamics in a Black Hole Background
Chris Doran


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
Scattering Absorption Bound states Emission x
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Quantum


Classical Scattering
· Main method of comparison is the differential cross section pf GM p b
i



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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PainlevÈ 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
Light-like geodesics

From rest

From infinity
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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

v=0.5c

Spiralling
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v=0.9c
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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
Rutherford at small

Additional scattering as

Corresponds to v=0.995c
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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

Spin Connection

Dirac spinor

· Of course, much easier using geometric algebra ­ which is how we do it!
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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
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Gravitational interaction

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Hamiltonian Form II
· Now insert matrices into Dirac equation

Flat space

Interaction

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

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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 gauge 2. 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 SchrÆdinger 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

Feynman Diagrams

+

+

+...

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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 shell Process must be second order
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Vertex Factor II
· Evaluate the second order diagram p
i

p k

f

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

Velocity

Scattering angle
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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 First-order in M + · Calculation is a different order · But result is unchanged ­ a physical prediction
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Absorption
· Particles too close to the horizon end up captured · See this from the effective potential E too high get absorbed Higher J values are scattered Low J are absorbed
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Plot of increasing J


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 (r=u1)

· With ||=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 horizon, and at infinity (irregular) · Special function theory underdeveloped for this problem
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)

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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 and energy

· also ratio r0/ ­ horizon/Compton w/length · 1 corresponds to primordial black holes · Also have
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Horizon
· Series expansion about horizon =(r-2M)

· Get indicial equation

· Roots are Regular branch physical Singular branch unphysical

Gauge invariant

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

=0.1, l=1

=0.2, l=1

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

=0.1, l=1

=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
100 90 80 70 60 50 40 30 20 10 0 0 0.5 1 1.5 2 2.5

Photon limit

Momentum
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Massive Case
Coupling 0.03
100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1

=0.03

Coupl ing = 0.1
500 450 400 350 300 250 200 150 100 50 0 0 0.2 0.4 0.6 0.8

=0.1

1

Coupling = 0.5
500

=0.5

4000

=1

3000
400

300

2000

200

1000
100

0 0 0.2 0.4 0.6 0.8 1

0 0 0.2 0.4 0.6 0.8 1

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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 NewtonRaphson method
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Probability Density =0.1

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Probability Density =0.35

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Probability Density =0.5

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Variation with
First excited states with Increasing angular momentum

=0.5

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 Energy
1S

Decay rate increases with coupling constant and decreases with
2P

1/2

3/2

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Comments
· 1 is the scale appropriate to primordial black holes · Solar mass black holes have 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 Outgoing Ingoing · Form ratio of outgoing to total current 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?

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