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DO EVAPORATING BLACK
HOLES FORM PHOTOSPHERES
AND/OR CHROMOSPHERES?
Jane H MacGibbon, B.J. Carr and Don N. Page
MacGibbon Carr & Page PRD 78, 064043 (2008); Page Carr & MacGibbon PRD 78, 064044 (2008)
JANE H MACGIBBON
UNIVERSITY OF NORTH FLORIDA
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
MOTIVATION
Heckler Model
A.F.Heckler PRD 55, 480 (1997); A.F.Heckler PRL 78, 3430 (1997)
• QED/QCD bremsstrahlung and pair-production
interactions between Hawking-radiated particles
form photosphere/chromosphere
Other Photosphere/Chromosphere Models
• Belyanin et al
• Bugaev et al
• D. Cline and Hong
• Kapusta and Daghigh
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
BLACK HOLE THERMODYNAMICS
HAWKING TEMPERATURE:
 M BH 
c

 1.06  13  GeV
8 GM BH
 10 g 
3
kTBH
Solar Mass BH TBH ~ 10 -7 K
MBH ~ 10 25 g TBH ~ 3 K CMB
HAWKING RADIATION FLUX:
d NS
 snl

dt dE n,l 2
2

2s 
 E  n   e 
  1 
exp 

 / 2 c 



MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
1
HAWKING RADIATION
Sources: Page, Elster, Simkins
STANDARD PICTURE
BH should directly Hawking evaporate those
particles which appear non-composite compared
to wavelength of the radiated energy (or
equivalently BH size) at given TBH
As TBH increases:
BH directly emits photons + gravitons  + neutrinoes  +
electrons  + muons  + pions
Once TBH >>ΛQCD:
BH directly emits quarks and gluons (not direct pions)
which shower and hadronize into astrophysically stable γ ,
ν, p, pbar, e-, e+
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
HAWKING RADIATION
Source: MacGibbon and Webber (1990)
TOTAL BLACK HOLE EMISSION
dM BH
2
25
1


5
x
10
M
/
g
f
M
g
s




MASS LOSS RATE:
BH
BH
dt
1
- 27
3




6
.
24
x10
M
f
M
s
BLACK HOLE LIFETIME: evap
i
i
Mass of PBH whose lifetime equals age of Universe
(MacGibbon, Carr & Page 2008):
M   5.00 0.04 10 gm
14
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
HECKLER MODEL
Number Density at radius r from BH of e- directly
Hawking radiated by BH n  r   104 where  k  G  c  1
0
M BH r 2
Two-body QED bremsstrahlung cross-section in
BH center-of-mass frame 8 3 2E
 brem 
Plasma mass correction
me
2
ln
m 'e  me  m pm
2
Total Number of Scatterings
N  R 
rmax  R

rmin  rBH
me
2
for e  e  e  e  
where m pm 
2
4 n  r 
N r
1
dr
3
where   r    n  r   brem vrel  and n  r    
 r 
2
Eav
n0  r 
QED Photosphere above TBH ~ 45 GeV. Similarly
QCD
 brem
8 s 3 2 E

ln
2
mq
mq
 Chromosphere above TBH ~ ΛQCD
IS THE HECKLER MODEL CORRECT?
QED 3-vertex Bremsstrahlung
IS THE HECKLER MODEL CORRECT?
√ Two-body bremsstrahlung cross-section
 brem
d brem
1
8 3 2 E
 
d  2 ln
E0
d
me
me
E
Average momentum exchanged is ~ me in centerof-momentum (CM) frame  particles must be
within ~ 1/ me of each other to interact
Average angle between final on-shell electron and
outgoing photon in CM frame is φav~ me / 2E
Average energy of final on-shell electron and
outgoing photon in CM frame is Ee~ ω ~ E / 2
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
√ Heckler assumes, even after photosphere /
chromosphere develops, that most particles are
moving radially out from BH
For random walk, particle emitted by TBH ~ 1-10
2
  
7
GeV BH would have to undergo N  

10

 av 
scatterings to deviate   O  0.1 1 from the radial
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
√ BH is center-of-momentum frame for most pairs
of emitted charged particles
BUT two particles moving in similar direction will
not interact near BH (because their center-ofmomentum frame is highly Lorentz-boosted) 
 ‘Exclusion cone’ around emitted particle  once
particle is a distance d from BH the transverse
distance to nearest particle for interaction is xT ~ d
 particles must be within ~1/me of BH to interact
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
 For radial emission  r   n r  bremvrel  is not correct
 must be replaced by radial description
1
 particles are Hawking emitted near BH so
particles do not travelling in from minus ∞, past the
BH and each other, then out to plus ∞
BUT bremsstrahlung cross-section assumes
interacting particles travel in from minus ∞
 interaction cross-section is decreased
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
 Causality Constraint
Two particles must be in casual contact to interact
BUT negligible fraction of Hawking emitted
particles are in causal contact with each other
Time between subsequent Hawking emissions is
Δte ~ 200 / Epeak
For causal contact within ~ 1 / me of BH require
Δte < Δtc ~1 / γ me where γ ~ Epeak / me
 Δtc << Δte for almost all emitted particles
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
 Scale for Completion of Interaction
Heckler assumed distance required for formation
of final on-shell electron and outgoing photon is
dform ~1 / me in CM frame
BUT average angle between final on-shell electron
and photon is φav~ me / 2E so dform ~E / me2 in CM
frame  Electron must travel dform ~E / me2 before it
can undergo next on-shell interaction
 Any multiple interactions of electron within ~1 / me
of BH are off-shell interactions and so strongly
suppressed by LPM effect
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
IS THE HECKLER MODEL CORRECT?
The Heckler QED photosphere model does not
work because it neglects the requirement that the
emitted particles must be in causal contact to
interact and neglects LPM effects in any (very
rare) multiple scatterings
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
QCD CHROMOSPHERE?
 when TBH >>ΛQCD the causality constraint (Δte ~
20 / Epeak ) and LPM suppression in any (rare)
multiple scatterings also prevent QCD
chromosphere formation for 4D BHs
BUT could a QCD chromosphere form when
TBH ~ ΛQCD?
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
QCD CHROMOSPHERE
WHEN TBH~ ΛQCD?
 Hawking emission damped (lower flux and
greater Δt between emissions) near rest mass
threshold (eg ΛQCD) + low multiplicity per jet
near ΛQCD
 Δt between consecutive Hawking emissions
increases around ΛQCD  causality constraint is
stronger
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
QCD CHROMOSPHERE
WHEN TBH~ ΛQCD?
 e+e- accelerator collisions – smooth transition
around ΛQCD from direct π regime to
quark/gluon mediated regime – sets in when π
relativistic i.e. sets in when constituent quarks
relativistic
when BH goes from directly emitting π to
directly emitting quarks and gluons, initial
quarks and gluons are relativistic
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
QCD CHROMOSPHERE
WHEN TBH~ ΛQCD?
 number of final states from hadronization is
limited by available energy (E ~ ΛQCD per
Hawking emitted particle) + conservation laws
 decays produce mainly π (only a couple of π)
around ΛQCD and soft gluon bremsstrahlung is
insignificant (because lowest colourless state
from g is π)
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
QCD CHROMOSPHERE
WHEN TBH~ ΛQCD?
 TBH ~ ΛQCD BH no simultaneous
production of ultrahigh density QCD
particles
 TBH ~ ΛQCD BH can NOT form quark-gluon
plasma
No analogy to RHIC quark-gluon plasma (RHIC
~ 200 GeV per nucleon, gluon-saturated, high
baryon/antibaryon asymmetry)
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
OTHER PHOTOSPHERE/
CHROMOSPHERE MODELS
• Kapusta and Daghigh – assumes plasma thermalized by
QED and QCD bremsstrahlung and pair-production of Heckler
model
• Belyanin et al – ‘collisionless’ QED plasma – omits Lorentz
factors  no self-induced MHD photosphere but strong
ambient magnetic field may induce (weak) photosphere
• Bugaev et al – ‘Stretched Horizon’ Tpl region just outside
horizon  neglects LPM suppression (and thermalization
scales)
• D. Cline and Hong – Hagedorn-type emission of remaining
BH mass into exponentially growing number of states at TBH ~
ΛQCD  state occupancy should be determined by available
energy E ~ ΛQCD  model would require direct coupling of BH
mass to Hagedorn states (but TBH increases as 1/MBH 2)
BREMSSTRAHLUNG EFFECTS
(Page, Carr and MacGibbon 2008)
Inner Bremsstrahlung
2-vertex Bremsstrahlung
3-vertex Bremsstrahlung
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
INNER BREMSSTRAHLUNG
Number flux of inner bremsstrahlung photons radiated
by charged particles of mass m and γav~ 4.20TBH / m
emitted by BH with spectrum dN/dt:
d 2 N b
2
dN
ln  2 av   1

dtd 
dt
 Nearly flat power spectrum up to ω ~ E – m cut-off
Total power in inner bremsstrahlung photons radiated
by charged particles emitted by BH with power dE/dt:
dEb
2
dE
ln  2 av   1

dt

dt
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
INNER BREMSSTRAHLUNG
Total power in inner bremsstrahlung photons radiated
by charged particle emitted by BH with power dE/dt:
dEb
2
dE
ln  2 av   1

dt

dt
Compare with power in direct photons:
dEd
dt
4
 0.3364 x 10 M BH
For MBH = 5x1014 g BH,
2
At low   0,
d 2 Eb
d 2 Ed
dtd
 1.73 x 1019 s 1

8
3 2
M 3 4
dtd
 inner bremsstrahlung photons dominate the directly
Hawking emitted photons below 57 MeV
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
CONCLUSIONS (FOR 4D BLACK HOLES)
MacGibbon, Carr and Page 2008:
 None of the photosphere/chromosphere models
work because they neglect the requirement that
the emitted particles must be in causal contact to
interact and/or neglect LPM effects in any
multiple scatterings and/or energy constraints
 Energy and quantum conservation laws prevent
significant increase in particle states near TBH ~
ΛQCD  no quark-gluon plasma near TBH ~ ΛQCD
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009
Predicted Black Hole Spectra
Source: MacGibbon and Webber (1990)
Astrophysical Spectra from Uniformly
Distributed PBHs with dn/dMi α Mi-2.5
Source: MacGibbon and Carr (1991)
HIGHER-DIMENSIONAL BLACK HOLES?
Even one interaction (ie N ~ 1) could
modify expected signal compared with
experimental precision
MacGibbon, Carr & Page "Black Hole Chromospheres/Photospheres" MG12 Paris July 12 - 18 2009