Proof of a Universal Lower Bound on η/s
Ram Brustein
+MEDVED 0908.1473
======================
‫אוניברסיטת בגוריו‬
+MEDVED
0808.3498,0810.2193, 0901.2191
+GORBONOS, HADAD 0712.3206
+G 0902.1553, +H 0903.0823
+DVALI, VENEZIANO 0907.5516
• Quadratic, 2-derivatives effective action for perturbations
brane thermodynamics & hydrodynamics (linear) for
generalized theories of gravity
• Geometry entropy density calibrated to Einstein value
• Unitarity preferred direction on the space of couplings
Outline
• KSS bound:
η / s ≥ 1/ 4π
• Both η, s couplings of 2-derivatives effective action
no apparent reason for directionality of deviation
from Einstein gravity
Preferred direction
• Idea 1: Entropy is always a geometric quantity The
entropy density can be calibrated to its Einstein value
• Idea 2: (non-trivial) Extensions of Einstein can only
increase the number of gravitational DOF Unitarity forces increase of couplings (including η)
Hydro of generalized gravity
S = ∫ dtdrd x − g L ( g µν , R
p
1
L=
R + λ Lcorr
16π GE
S
(2)
µν
ρσ
, φ , ∇φ , …)
Expand to quadratic order in gauge
invariant fields Zi and to quadratic order
derivatives, diagonalize
g µν = g µν + hµν , φ = φ + ϕ
1
2
= ∫ dtdrd x 2
(∇Z i )
(κ eff (r ))i
Liu+Iqbal:0809.3808 Einstein gravity
Zi ~ e
p
iΩt −iQz
Q
Ω
q=
, w=
2π T
2π T
R.B+MEDVED:generalized gravity
0808.3498,0810.2193, 0901.2191
The effective gravitational coupling:
Einstein gravity
E
E
κ = 32π GE
2
The effective gravitational coupling:
Generalized gravity
Contributions to the graviton kinetic terms must appear
through factors of the Riemann tensor (or it derivatives)
in the action.
RB+GORBONOS, HADAD 0712.3206
The effective gravitational coupling:
General action
Kinetic matrix, coupling constant matrix
Outline
• KSS bound:
η / s ≥ 1/ 4π
• Both η, s couplings of 2-derivatives effective action
no apparent reason for directionality of deviation
from Einstein gravity
• Idea 1: Entropy is always a geometric quantity Entropy density can always be calibrated to its Einstein
value
• Idea 2: (non-trivial) Extensions of Einstein can only
increase the number of gravitational DOF Unitarity forces increase of couplings (including η)
Wald’s entropy
Wald ‘93
• Stationary BH solutions with bifurcating
Killing horizons
1 A
SW =
4 Geff
Wald’s prescription picks a
specific polarization and
location - horizon
=
1
"κ "
2
Spherically symmetric, static
BHs rt
κ
Calibration of entropy density
Fixed geometry
Fixed charges
Different S (perhaps)
Sufficient: Relation between rh
& temp. higher order in λ
Transform to (leading order) “Einstein frame”
Can be done order by order in λ
rescale
GX → GX = GE
To preserve the form of leading term
Change the units of GX such that its numerical value is equal to GE
The value of rh = largest zero of |gtt/grr| is not changed
rh largest zero of
|gtt/grr| not changed
Preferred direction determined by η only!
• Entropy invariant to field redefinitions SX~=SX
• Entropy density not invariant
• Calibration possible: entropygeometry
Outline
• KSS bound:
η / s ≥ 1/ 4π
• Both η, s couplings of 2-derivatives effective action
no apparent reason for directionality of deviation
from Einstein gravity
• Idea 1: Entropy is always a geometric quantity Entropy density can always be calibrated to its Einstein
Value
• Idea 2: (non-trivial) Extensions of Einstein can only
increase the number of gravitational DOF Unitarity forces increase of couplings (including η)
The graviton propagator
in Einstein gravity
Gravitons exchanged between two conserved sources
One graviton exchange approximation h’s << 1
In Einstein gravity: a single massless spin-2 graviton
Vectors do not contribute to exchanges between conserved sources
First appearance of a preferred direction
ρ E (q )
2
Spectral decomposition
q
semiclassical unitarity
2
=∫
ρ E (s) ≥ 0
ρ E ( s)
q −s
2
ds
RB+ DVALI, VENEZIANO 0907.5516
DVALI: MANY PAPERS
ρE(q2) can only increase towards the UV
Mass screening is not allowed
The graviton propagator in
generalized gravity
• massless spin-2 graviton
• massive spin-2 graviton
• scalar gravitons
Gravitons exchanged between two conserved sources
One graviton exchange approximation h’s << 1
Vectors do not contribute to exchanges between conserved sources
Gravitational couplings of generalized gravity can
only increase compared to Einstein gravity
ρ NE (q )
2
Spectral decomposition
semiclassical unitarity
q
2
=∫
ρ NE ( s ) ≥ 0
ρ NE ( s )
q −s
2
ds
ρ NE (q ) ≥ 0
2
The shear viscosity as a gravitational coupling
S = ∫ dtdrd p x − g L ( g µν , R µν ρσ , φ , ∇φ ,…)
g µν = g xy + hxy
S
(2)
1
L=
R + λ Lcorr
16π GE
1
= ∫ dtdrd x 2
(∇Z xy ) 2
(κ eff (r )) xy
p
In the hydro limit
Q
Ω
q=
, w=
1
2π T
2π T
y
Zx Z
x
y
∼η
See also 0811.1665
CAI, NIE, SUN
y
Zx ~ e
iΩt −iQz
The shear viscosity of generalized gravity
(and its FT dual) can only increase compared to
its Einstein gravity value
ηX
ρ E (0) = 1, ρ NE (0) ≥ 0 ⇒
≥1
ηE
Q
Ω m
q
=
,
w
=
, 1
Only spin-2 particles with
2π T
2π T T
contribute to the sum
KSS bound
η X
ηX
≥1⇒
≥1
ηE
ηE
True also for r infinity =AdS boundary
valid for FT dual of bulk theory
Valid for FT by the gauge-gravity duality
1
η 
η 
  ≥  =
 s  X  s  E 4π
Example: KK model
Einstein gravity D=4+n compactified on an n-torus of radius R.
Two regimes: Hawking thermal wavelength >> R T R << 1
Hawking thermal wavelength << R T R >> 1
ρ k (q ) ≥ 0
2
i
qR>>1
DKK
GE GE
∼ qR 2 2
q
q
KK contribution dominates η∼TR ηE >> ηE
Example: KK model
Entropy density
remains the same!
s4+ n
A2+ n rh 2 R n rh 2
∼
∼
∼
∼ s4
n
G4+ n G4 R
G4
For TR >> 1:
η 
η 
η 
  ∼ TR    
 s  KK
 s E  s E
Example: 5D GB gravity
On horizon (for example)
No contributions from rt gravitons => no corrections to entropy density
Example: 5D GB gravity
xy gravitons on horizon
(similarly elsewhere)
Entropy density remains the same as in Einstein
Violation of the KSS boundghost contribution
Summary & conclusions
η / s ≥ 1/ 4π
For extensions of Einstein
gravity and their FT duals
• Unitarity
• Entropy Geometry
• Is it possible to do better?