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

• 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

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

= +

The effective gravitational coupling:Einstein gravity

2 32 EGκ π=

E E

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

• 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≥

Wald’s entropy

• Stationary BH solutions with bifurcating Killing horizons

Wald ‘93

1

4Weff

AS

G=

2

1

" "κ=

Wald’s prescription picks a specific polarization and location - horizon

Spherically symmetric, static BHsκrt

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→ =

• 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

• 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≥

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

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

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

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 ≥

( ), , , ,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

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= ≥ ⇒ ≥

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

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 ≥

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:

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 EinsteinViolation of the KSS boundghost contribution

Summary & conclusions

/ 1/ 4sη π≥ For extensions of Einstein gravity and their FT duals

• Unitarity• Entropy Geometry

• Is it possible to do better?