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 Black Hole surrounded by Thermal Atmosphere How much entropy does the atmosphere contribute? How much entropy does the atmosphere contribute? s ~ T 3 local, T local ~ 1/L L = dist to horizon S ~ s dV ~ A L -3 dL ~ A/L c 2 ~ A L -3 dL ~ A/L c 2

For L c ~ l p, large correction to Bekenstein- Hawking entropy! Black hole thermo? Black hole thermo? Counting of states? Counting of states? S atm ~ A/L c 2, but S BH = A/4l p 2 Results: Note: essentially same as “entanglement” calculation.

Perhaps S atm is not a correction; perhaps it is the Bekenstein-Hawking entropy. (Entanglement Entropy) Shouldn’t WKB approx. contain classical Euclidean term + fluctuations? Shouldn’t WKB approx. contain classical Euclidean term + fluctuations? Species problem, etc. Species problem, etc. Resolutions: But I’m confused:

Renormalization of G? Alternate Solution? Perhaps L c >> l p. Depends on field content. Depends on field content. Can choose cases (e.g., w/ enough SUSY) w/ no renormalization but similar S from thermal atmosphere. Can choose cases (e.g., w/ enough SUSY) w/ no renormalization but similar S from thermal atmosphere. My understanding:

Why would L c be large? Proposal: Horizon has finite quantum width, larger than the Planck scale. Sorkin (mid-1990’s): Fluctuations in T  can deform horizon gravitationally – “Newtonian treatment” gives L c ~ (l p 2 R) 1/3 Our goal: Relativistic treatment, again find L c > l p

Which modes dominate S atm ? Work within  R and  R /2 of the horizon, w/  R << R. R +  R Each mode w/  < T H contributes (roughly) one bit of entropy:  S =  E/T But the sphere gives ~ R 2 / 2 modes of wavelength, so shortest dominates. and E = N  = T for  < T H. Thus S ~ 1.

What is shortest w/  < T H ? Redshift =T H /T Rindler ~ LT H Redshift =T H /T Rindler ~ LT H So,  = LT H /  or,  = LT H /  > L. So,  = LT H /  or,  = LT H /  > L. Thus, entropy is dominated by modes of wavelength ~ L. Thus, entropy is dominated by modes of wavelength ~ L. Recall: L ~ (R  R) 1/2. Recall: L ~ (R  R) 1/2. L

First estimate of L c Look for self-consistent cut-off. Look for self-consistent cut-off. Assume no modes below R +  R/2. Assume no modes below R +  R/2. Estimate  M betwteen R+  R/2 and R+  R due to modes with ~ L. Estimate  M betwteen R+  R/2 and R+  R due to modes with ~ L. # of modes: A/L 2, Energy of each fluct: T H, so  M ~ [A/L 2 ] 1/2 T H. # of modes: A/L 2, Energy of each fluct: T H, so  M ~ [A/L 2 ] 1/2 T H. _ _ _ R +  R _ _ _ _ _ _ _ Averages + and – : Conservative!

Forbid forming a larger BH requires:  M +  M < 2G(R +  R); i.e.,  M ~ [A/L 2 ] 1/2 T H < L 2 / Rl p 2. L > (R 2 T H l p 2 ) 1/3 ~ (Rl p 2 ) 1/3 L C ~ (Rl p 2 ) 1/3

This calc first done by Casher, Englert, Itzhaki, Massar, and Parentani for different purpose. This calc first done by Casher, Englert, Itzhaki, Massar, and Parentani for different purpose. Matches Sorkin! Matches Sorkin! Conservative lower bound on Quantum width of horizon Conservative lower bound on Quantum width of horizon Suffices in d < 5+1. (=l p in d=6) Suffices in d < 5+1. (=l p in d=6) But we can do better! But we can do better!

More complete estimate:  Recall that + and – flucts largely cancel in  M shell.  But, flucts only last a (locally measured) time L.  Can only receive info from distance L away.  Fate of one is independent of rest! R +  R _ _ _ _ _ _ _ Of course, horizon not causal…

Result Works for all d, all T H ! L C is large for T H ~ 0

 Fluctuations in BH thermal atmosphere lead to horizon fluctuations on scale L w/  Works for any spherical BH if conjecture of independent fluctuations holds. Summary l p << L << R