proof of a universal lower bound on η/s
TRANSCRIPT
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גוריואוניברסיטת ב
Ram Brustein+MEDVED 0908.1473======================+MEDVED
0808.3498,0810.2193, 0901.2191+GORBONOS, HADAD 0712.3206+G 0902.1553, +H 0903.0823+DVALI, VENEZIANO 0907.5516
Proof of a Universal Lower Bound on η/s
• 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
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• KSS bound:
• 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 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)
Outline/ 1/ 4sη π≥
Preferred direction
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Hydro of generalized gravity
( ), , , ,pS dtdrd x g L g Rµνµν ρσ φ φ= − ∇∫ …
,g g hµν µν µν φ φ ϕ= + = +
Expand to quadratic order in gauge invariant fields Zi and to quadratic order derivatives, diagonalize
(2) 22
1( )
( ( ))p
ieff i
S dtdrd x Zrκ
= ∇∫
~ i t iQziZ e Ω −
,2 2
Qq w
T Tπ πΩ
= =
Liu+Iqbal:0809.3808 Einstein gravity
R.B+MEDVED:generalized gravity
0808.3498,0810.2193, 0901.2191
1
16 corrE
L R LG
λπ
= +
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The effective gravitational coupling:Einstein gravity
2 32 EGκ π=
E E
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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
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The effective gravitational coupling:General action
Kinetic matrix, coupling constant matrix
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• KSS bound:
• 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)
Outline/ 1/ 4sη π≥
![Page 8: Proof of a Universal Lower Bound on η/s](https://reader033.vdocuments.mx/reader033/viewer/2022043002/626b1aebce5bc94b3c5f7d78/html5/thumbnails/8.jpg)
Wald’s entropy
• Stationary BH solutions with bifurcating Killing horizons
Wald ‘93
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1
4Weff
AS
G=
2
1
" "κ=
Wald’s prescription picks a specific polarization and location - horizon
Spherically symmetric, static BHsκrt
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Fixed geometryFixed chargesDifferent S (perhaps)Sufficient: Relation betweenrh
& temp. higher order in λ
Calibration of entropy density
Transform to (leading order) “Einstein frame”Can be done order by order in λ
To preserve the form of leading term rescale
X EXG G G→ =
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• Entropy invariant to field redefinitions S~=SX X
• Entropy density not invariant• Calibration possible: entropygeometry
Preferred direction determined by η only!
rh largest zero of |gtt/grr| not changed
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
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• KSS bound:
• 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)
Outline/ 1/ 4sη π≥
![Page 13: Proof of a Universal Lower Bound on η/s](https://reader033.vdocuments.mx/reader033/viewer/2022043002/626b1aebce5bc94b3c5f7d78/html5/thumbnails/13.jpg)
The graviton propagatorin Einstein gravity
Gravitons exchanged between two conserved sourcesOne graviton exchange approximation h’s << 1
In Einstein gravity: a single massless spin-2 graviton
Vectors do not contribute to exchanges between conserved sources
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First appearance of a preferred direction
2
2 2
( ) ( )E Eq sds
q q s
ρ ρ=
−∫Spectral decomposition
semiclassical unitarity ( ) 0E sρ ≥ρE(q2) can only increase towards the UVMass screening is not allowed
RB+ DVALI, VENEZIANO 0907.5516DVALI: MANY PAPERS
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The graviton propagator in generalized gravity
• massless spin-2 graviton• massive spin-2 graviton• scalar gravitons
Gravitons exchanged between two conserved sourcesOne graviton exchange approximation h’s << 1Vectors do not contribute to exchanges between conserved sources
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Gravitational couplings of generalized gravity can only increase compared to Einstein gravity
2
2 2
( ) ( )NE NEq sds
q q s
ρ ρ=
−∫Spectral decomposition
semiclassical unitarity ( ) 0NE sρ ≥ 2( ) 0NE qρ ≥
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( ), , , ,pS dtdrd x g L g Rµνµν ρσ φ φ= − ∇∫ …
xy xyg g hµν = +
(2) 22
1( )
( ( ))p y
xeff xy
S dtdrd x Zrκ
= ∇∫
1
16 corrE
L R LG
λπ
= +
The shear viscosity as a gravitational coupling
~y i t iQzxZ e Ω −
, 12 2
Qq w
T Tπ πΩ
= =
y xx yZ Z η∼
In the hydro limit
See also 0811.1665CAI, NIE, SUN
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The shear viscosity of generalized gravity (and its FT dual) can only increase compared to
its Einstein gravity value
Only spin-2 particles withcontribute to the sum
, , 12 2
Q mq w
T T Tπ πΩ
= =
(0) 1, (0) 0 1XE NE
E
ηρ ρ
η= ≥ ⇒ ≥
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KSS bound
1 1X X
E E
ηηη η
≥ ⇒ ≥
1
4X Es s
η ηπ
≥ =
True also for r infinity =AdS boundary valid for FT dual of bulk theory
Valid for FT by the gauge-gravity duality
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Example: KK modelEinstein 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
KK contribution dominates η∼TR ηE >> ηE
qR>>12 2E E
KK
G GD qR
q q∼
2( ) 0ik qρ ≥
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Example: KK modelEntropy density remains the same!
2 22
4 44 4 4
nn h h
n nn
A r R rs s
G G R G+
++
∼ ∼ ∼ ∼
KK E E
TRs s s
η η η
∼
For TR >> 1:
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Example: 5D GB gravity
On horizon (for example)
No contributions from rt gravitons => no corrections to entropy density
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Example: 5D GB gravity
xy gravitons on horizon(similarly elsewhere)
Entropy density remains the same as in EinsteinViolation of the KSS boundghost contribution
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Summary & conclusions
/ 1/ 4sη π≥ For extensions of Einstein gravity and their FT duals
• Unitarity• Entropy Geometry
• Is it possible to do better?