coarse-graining, thermodynamics
and quantum gravitation
Daniel F Litim
Department of Physics and Astronomy
U Sussex, Brighton, U.K.

Yukawa Institute for Theoretical Physics
Kyoto, Sep 2, 2011

Friday, 2 September 2011

coarse-graining

thermodynamics

gravitation

Friday, 2 September 2011

black hole thermodynamics
black holes in general relativity
solutions to Einsteinʼs classical equations
event horizon with area A
most general static BH solution for long-ranged forces:
Kerr Newman black holes
uniqueness theorem:

A = A(M, J, q)
singularities / break-down of predictivitycosmic censorship ?

Friday, 2 September 2011

black hole thermodynamics
laws of black hole mechanics(Bardeen, Carter, Hawking ’73)

infinitesimal amount of matter crossing the horizonκδA + Ω δJ + Φ δq
δM =8π GN
first law of thermodynamics

δU = δQ +!

µi δNiiwithU ↔M

κ
δA
δQ ↔
8πGN
Friday, 2 September 2011

µi ↔ {Ω, Φ}Ni ↔ {J, q}black hole thermodynamics
laws of black hole mechanics(Bardeen, Carter, Hawking ’73)

infinitesimal amount of matter crossing the horizonκδA + Ω δJ + Φ δq
δM =
8π GN
reversible thermodynamical process at temperature TδQ= δSTsecond law of thermodynamics

S∝A

κ
T =
2πFriday, 2 September 2011

(Bekenstein ’73)

(Hawking ’75)

black hole thermodynamics
laws of black hole mechanics(Bardeen, Carter, Hawking ’73)

infinitesimal amount of matter crossing the horizonκδA + Ω δJ + Φ δq
δM =
8π GN
identify the heat flow at temperature T as

δQ = δM − Ω δJ − Φ δq
to find that the first law of thermodynamics leads toA
SBH =4 GN
derivation from euclidean path integral
statistical entropy
Friday, 2 September 2011

(Gibbons, Hawking ’77)

coarse-graining(Falls, Litim ’11)

quantum gravitation
challenge for any UV completion of gravity:
identify the underlying coarse-grained degrees of freedom

flowing effective action

Γk =IR limit!

√d x −g4"(Wetterich ’93)

#

1
1 µν
R+F Fµν + Lm .
8πGk
4αk

G0 ≈ 6.674 × 10

−11

N (m/kg)2

family of Kerr-Newman BH solutions

A = A(M, J, q; k)Friday, 2 September 2011A
S=4Gk

α0 ≈ 1/137

coarse-graining(Falls, Litim ’11)

choice of scale“not too coarse- and not too fine-grained”

k = kopt (M, J, q)

RG thermodynamics
infinitesimal amount of matter crossing the horizon, with heat flow

δA
δQ=T4GkBH settles in a new state

M → M + δM J → J + δJq → q + δq kopt → kopt + δkopt

total change of horizon area

Friday, 2 September 2011!
2π∂A(M, J, q; k) !!
δA =T δA +!
κ∂kδkoptk=kopt

coarse-graining(Falls, Litim ’11)

choice of scale“not too coarse- and not too fine-grained”

k = kopt (M, J, q)

results
optimal scalemass function

temperature

entropy

Friday, 2 September 201124πξ22kopt (M, J, q) ≡ kopt (A) =A!"
$#
2
2 2
4π
A + 4πG(A)e (A)q
22M ≡+JA
8πG(A)∂M
T = 4G(A)∂AA
S=with4Gk

k = kopt

asymptotically safe black holes
RG thermodynamics

1k1=+GkGNg∗

2

asymptotic safety

T0.035(Falls, Litim ’11)

0.030

(Weinberg ’79)
(Kawai, Ninomiya ’90, Kawai, Kitazawa,Ninomiya ’92)
(Aida, Kitazawa, Ninomiya ’94, Aida, Kitazawa,’96)(Reuter ’96)
(Souma ’99)

0.025
0.020
0.015
0.010
0.005
02

outer/inner horizon

Friday, 2 September 2011

4A± = 4πGN

6!

810"

1±2
2GN M −
2g∗ ξ2M

2GMN
G2N M 4 − J 2 −g∗ ξ 2#

RG improved metrics
basic idea
replace

GN → G(r, · · · )usingKerr space-time!

2G(r)M r2
ds = − 1 −ρ2"k → k(r, · · · ) (Bonanno, Reuter ’01)

(Reuter, Tiuran ’10)
(Litim, Nikolakopoulos, ERG ’10)

2G(r)Mars
dt2 −dtdφ2
ρ2 !
" 2ρ2 2s2222 2
22+ dr + ρ dθ + 2 (r + a ) − a ∆ s dφ∆ρwith∆(r) = r2 − 2G(r)M r + a2

ρ2 = r2 + a2 cos2 θhorizons
solve

Friday, 2 September 2011∆(rh ) = 0

RG improved metrics
thermodynamical equivalence
provided if(Falls, Litim ’11)2ξk (r) = 2r + (J/M )22statistical entropy“off-shell” conical singularity method(Soludkhin ’96)

Schwarzschild black hole, free energy

Friday, 2 September 2011

improved metricAr+
F =−T
2GN
4GN (Bonanno, Reuter ’00)

here

F =M −STconclusion
black hole thermodynamics
RG puts black holes under a microscope
BH thermodynamics consistent and meaningful
beyond the semi-classical approximation

predictions
thermodynamical consistency + asymptotic safety implies
smallest black holes
maximal temperatureextensions + challenges
high-energy degrees of freedom?
cosmological constant, higher curvature invariants
non-equilibrium thermodynamics

Friday, 2 September 2011