The Quantum Width of a
Black Hole Horizon
Donald Marolf
UCSB
Quantum Theory of Black Holes
OCTS, Sep. 17, 2004
Motivation: BH Entropy
• Black Hole surrounded by Thermal Atmosphere
• How much entropy does the atmosphere
contribute?
s ~ T3local, Tlocal ~ 1/L
L = dist to horizon
S~
~A
s dV
L-3 dL ~ A/Lc2
Results:
Satm ~ A/Lc2, but SBH = A/4lp2
For Lc ~ lp, large correction to BekensteinHawking entropy!
• Black hole thermo?
• Counting of states?
Note: essentially same as
“entanglement” calculation.
Resolutions:
Perhaps Satm is not a correction; perhaps it
is the Bekenstein-Hawking entropy.
(Entanglement Entropy)
But I’m confused:
• Shouldn’t WKB approx. contain
classical Euclidean term +
fluctuations?
• Species problem, etc.
Renormalization of G?
My understanding:
• Depends on field content.
• Can choose cases (e.g., w/ enough
SUSY) w/ no renormalization but
similar S from thermal atmosphere.
Alternate Solution? Perhaps Lc >> lp.
Why would Lc be large?
Proposal: Horizon has finite quantum width,
larger than the Planck scale.
Sorkin (mid-1990’s): Fluctuations in Tmn can
deform horizon gravitationally –
“Newtonian treatment” gives Lc ~ (lp2R)1/3
Our goal: Relativistic treatment, again find
Lc << R, and Lc >> lp
Which modes dominate Satm?
Work within dR and dR /2 of the horizon,
w/ dR << R .
Each mode w/ w< TH
contributes (roughly)
one bit of entropy:
R + dR
dS = dE/T
and E = Nw = T for w < TH.
Thus S ~ 1.
But the sphere gives ~ R2/l2 modes of
wavelength l, so shortest l dominates.
What is shortest l w/ w < TH?
• Redshift =TH/TRindler
~ LTH
• So, w = LTH / l,
or, l = LTH/ w > L.
• Thus, entropy is
dominated by modes of
wavelength l ~ L.
• Recall: L ~ (RdR)1/2.
L
First estimate of Lc
• Look for self-consistent cut-off.
• Assume no modes below R + dR/2.
• Estimate dM betwteen R+dR/2 and R+dR
due to modes with l ~ L.
• # of modes: A/L2,
Energy of each fluct: TH,
so dM ~ [A/L2]1/2TH.
Averages + and – :
Conservative!
_ _
_
+
_
_
+
+
+
+
_
+
R + dR
+_
_
+
++
_
_
+ +
+
+
+
+
+
Forbid forming a larger BH
requires:
 M + dM < 2G(R + dR); i.e.,
dM ~ [A/L2]1/2TH < L2 / Rlp2.
L > (R2 TH lp2)1/3 ~ (Rlp2)1/3
LC ~ (Rlp2)1/3
LC ~ (Rlp2)1/3
• This calc first done by Casher, Englert,
Itzhaki, Massar, and Parentani for
different purpose.
• Matches Sorkin!
• Conservative lower bound on
Quantum width of horizon
• Suffices in d < 5+1. (=lp in d=6)
• But we can do better!
More complete estimate:
+
 Recall that + and – flucts
+
+
_
_
_
+
+
largely cancel in dMshell.
+
+
R + dR
+
 But, flucts only last
+_
+
_
+
a (locally measured)
++
+
_
+
time L.
_
+ +
 Can only receive info from distance L
away.

Fate of one is independent of rest!
Of course, horizon not causal…
Result
Works for all d, all TH!
LC is large for TH ~ 0
Summary
 Fluctuations in BH thermal atmosphere lead
to horizon fluctuations on scale L w/

lp << L << R
 Works for any spherical BH if conjecture of
independent fluctuations holds.