Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole...
Transcript of Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole...
![Page 1: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/1.jpg)
Classical and Quantum Dynamics in a Black Hole
Background
Chris Doran
![Page 2: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/2.jpg)
Black Holes 2002 2
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
![Page 3: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/3.jpg)
Black Holes 2002 3
Outline• 4 phenomena to give a classical and
quantum description for
Classical Quantum
xEmission
Bound states
Absorption
Scattering
![Page 4: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/4.jpg)
Black Holes 2002 4
Classical Scattering• Main method of comparison is the differential
cross section
bpi
pfGM
θ
For r-1 potential get Rutherford formula
![Page 5: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/5.jpg)
Black Holes 2002 5
Classical Dynamics• The Schwarzschild line element contains all
relativistic information (c=1)
• The geodesic equation for a radially infalling particle is essentially Newtonian
![Page 6: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/6.jpg)
Black Holes 2002 6
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)
![Page 7: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/7.jpg)
Black Holes 2002 7
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)
![Page 8: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/8.jpg)
Black Holes 2002 8
Radial geodesics
From rest
From infinity
Light-like geodesics
![Page 9: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/9.jpg)
Black Holes 2002 9
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
![Page 10: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/10.jpg)
Black Holes 2002 10
Sample Geodesics
=0.9cv=0.5c vSpiralling
![Page 11: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/11.jpg)
Black Holes 2002 11
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
![Page 12: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/12.jpg)
Black Holes 2002 12
Numerical Results
Corresponds to v=0.995c
Rutherfordat small θ
Additional scattering as θ ≈ π
![Page 13: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/13.jpg)
Black Holes 2002 13
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
![Page 14: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/14.jpg)
Black Holes 2002 14
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
![Page 15: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/15.jpg)
Black Holes 2002 15
Hamiltonian Form• Return to the metric
• Convert to Cartesians
![Page 16: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/16.jpg)
Black Holes 2002 16
Hamiltonian Form• Return to the metric
• Now introduce the matrices / vectors
‘Flat’ Minkowski vectors
Gravitational interaction
![Page 17: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/17.jpg)
Black Holes 2002 17
Hamiltonian Form II• Now insert matrices into Dirac equation
• Convert to Hamiltonian form• All interactions contained in the interaction
Hamiltonian
Flat space Interaction
![Page 18: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/18.jpg)
Black Holes 2002 18
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
![Page 19: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/19.jpg)
Black Holes 2002 19
Non-relativistic limit• The non-relativistic limit of the Dirac equation
is the Pauli equation• No spin effects - insert directly into
Schrödinger equation
• Substitution
![Page 20: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/20.jpg)
Black Holes 2002 20
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)
![Page 21: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/21.jpg)
Black Holes 2002 21
Iterative Solution• Borrow technique from quantum field theory
• Has an iterative solution
+ + + …FeynmanDiagrams
![Page 22: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/22.jpg)
Black Holes 2002 22
Amplitude• Convert to momentum space
Amplitude Plane wave spin states
Use amplitude to compute differential cross section
![Page 23: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/23.jpg)
Black Holes 2002 23
Vertex Factor• Fourier transform of interaction term is
• Evaluates to
Energy conserved so this vanishes on shellProcess must be second order
![Page 24: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/24.jpg)
Black Holes 2002 24
Vertex Factor II• Evaluate the second order diagram
pi pf
k
Result is
![Page 25: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/25.jpg)
Black Holes 2002 25
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
![Page 26: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/26.jpg)
Black Holes 2002 26
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
![Page 27: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/27.jpg)
Black Holes 2002 27
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
![Page 28: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/28.jpg)
Black Holes 2002 28
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+
![Page 29: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/29.jpg)
Black Holes 2002 29
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
![Page 30: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/30.jpg)
Black Holes 2002 30
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
![Page 31: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/31.jpg)
Black Holes 2002 31
Absorption Cross-section II• Slightly more complicated calculation gives
Photon limit
![Page 32: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/32.jpg)
Black Holes 2002 32
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
![Page 33: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/33.jpg)
Black Holes 2002 33
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
![Page 34: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/34.jpg)
Black Holes 2002 34
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
![Page 35: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/35.jpg)
Black Holes 2002 35
Horizon• Series expansion about horizon η=(r-2M)
• Get indicial equation
• Roots are
Regular branch -physical
Singular branch -unphysical
Gauge invariant
![Page 36: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/36.jpg)
Black Holes 2002 36
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
![Page 37: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/37.jpg)
Black Holes 2002 37
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
![Page 38: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/38.jpg)
Black Holes 2002 38
Asymptotic Behaviour• At large r have
• Similar for u2
• Normalise such that• Absorption cross section is
![Page 39: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/39.jpg)
Black Holes 2002 39
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
![Page 40: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/40.jpg)
Black Holes 2002 40
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
![Page 41: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/41.jpg)
Black Holes 2002 41
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
![Page 42: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/42.jpg)
Black Holes 2002 42
Semi-Classical Model• Carry out a ‘Bohr’ quantisation L=n~• Find that energy is
Dimensionless coupling
Angular momentum of ground state increases with coupling
![Page 43: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/43.jpg)
Black Holes 2002 43
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
![Page 44: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/44.jpg)
Black Holes 2002 44
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
![Page 45: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/45.jpg)
![Page 46: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/46.jpg)
![Page 47: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/47.jpg)
Black Holes 2002 47
Probability Density α=0.1
![Page 48: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/48.jpg)
Black Holes 2002 48
Probability Density α=0.35
![Page 49: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/49.jpg)
Black Holes 2002 49
Probability Density α=0.5
![Page 50: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/50.jpg)
Black Holes 2002 50
Variation with κ
α=0.5First excited states with Increasing angular momentum
Further out, become Hydrogen-like
![Page 51: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/51.jpg)
Black Holes 2002 51
Expectation value h r i
1S1/2
2S1/2
3S1/2
Horizon
![Page 52: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/52.jpg)
Black Holes 2002 52
Imaginary Energy1S1/2
Decay rate increases with coupling constant α and decreases with κ
2P3/2
![Page 53: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/53.jpg)
Black Holes 2002 53
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
![Page 54: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/54.jpg)
Black Holes 2002 54
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
![Page 55: Classical and Quantum Dynamics in a Black Hole Backgroundcjld1/pages/blackhole.pdf · Black Hole Case • Black hole Hamiltonian includes derivative terms. Find that radial equations](https://reader034.fdocuments.us/reader034/viewer/2022042318/5f072def7e708231d41bb390/html5/thumbnails/55.jpg)
Black Holes 2002 55
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?