Unitarity constraints on h/s אוניברסיטת בן-גוריון Ram Brustein +MEDVED 0908.1473 1005.5274 ====================== +MEDVED 0808.3498,0810.2193, 0901.2191 +GORBONOS, HADAD 0712.3206 +G 0902.1553, +H 0903.0823  +DVALI, VENEZIANO 0907.5516 +MEDVED 1003.2850  Linear equations (Quadratic effective action) for perturbations  (linear) thermodynamics & hydrodynamics for higher-derivative gravity Unitarity  preferred direction for h Explicit example: 4 derivative gravity Unitarity  specific choice of BC, h/s=1/4p Geometry  s calibrated to Einstein value

Outline Preferred direction KSS bound: Both h, 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 h) Preferred direction

Outline Calculating h for four-derivative gravity Results Significance Derive e.o.m. Solve e.o.m (boundary conditions, low energy approx., hydro limit) Calculate h from solution Results Exposing the ghost modes To lowest order : h/s=1/4p Significance Relationship to general arguments Expectations for higher-derivatives Validity of the KSS bound

Hydro of generalized gravity Expand to quadratic order in gauge invariant fields Zi and to quadratic order derivatives, diagonalize Liu+Iqbal:0809.3808 Einstein gravity R.B+MEDVED:generalized gravity 0808.3498,0810.2193, 0901.2191 Banerjee+Dutta, Cai, Nie,Sun

Wald’s entropy Wald ‘93 Stationary BH solutions with bifurcating Killing horizons

Spherically symmetric, static BHs krt RB+GORBONOS, HADAD 0712.3206 Wald’s prescription picks a specific polarization and location - horizon Spherically symmetric, static BHs krt

Calibration of entropy density Fixed geometry Fixed charges Different S (perhaps) Sufficient: Relation between rh & temp. higher order in l Transform to (leading order) “Einstein frame” Can be done order by order in l To preserve the form of leading term rescale

Preferred direction determined by h only! 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 h only! Entropy invariant to field redefinitions S~=SX X Entropy density not invariant Calibration possible: entropygeometry

Outline KSS bound: Both h, 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 h)

1PI graviton propagator in Einstein gravity Gravitons exchanged between two conserved sources* One graviton exchange approximation h << 1 In Einstein gravity: a single massless spin-2 graviton q2 = - qmqm : flat-spacetime, positive, (spacelike) momentum, Propagator evaluated in vacuum GE is the D-dimensional Newton's constant * Vectors do not contribute

First appearance of a preferred direction Spectral decomposition semiclassical unitarity MP(q2) can only decrease towards the UV Mass screening is not allowed RB+ DVALI, VENEZIANO 0907.5516 DVALI: MANY PAPERS

Graviton propagator near a bifurcating horizon s=1 D-dimensional propagator, internal states have 2D momentum

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 Spectral decomposition semiclassical unitarity

Gravitational couplings of generalized gravity near a horizon can only increase compared to Einstein gravity D-dimensional propagator, internal states have 2D momentum

The shear viscosity as a gravitational coupling In the hydro limit Liu+Iqbal,RB+Medved, Banerjee+Dutta, … CAI, NIE, SUN 0811.1665 explict 4-der

The shear viscosity of generalized gravity (and its FT dual) can only increase compared to its Einstein gravity value Only spin-2 particles with contribute to the sum

KSS bound True also for r infinity =AdS boundary  valid for FT dual of bulk theory Valid for FT by the gauge-gravity duality

Outline Calculating h for four-derivative gravity Results Significance Derive e.o.m. Solve e.o.m (boundary conditions, low energy approx., hydro limit) Calculate h from solution Results Exposing the ghost modes To lowest order : h/s=1/4p Significance Relationship to general arguments Expectations for higher-derivatives Validity of the KSS bound

Four-derivative gravity Einstein gravity in AdS Black brane metric Four-derivative gravity Use *same* solution to 1st order in a, b, g

Equations of motion:Einstein gravity Iyer+Wald ‘94 Need expressions like: Sometimes On horizon:

Equations of motion: Four-derivative gravity perturbative mass terms suppressed * On horizon Total e.o.m

Calculating h: previous results e.g. 0811.1665 CAI, NIE, SUN Myers et al. RB+Medved Agreementh/s = 1-8g Tension with unitarity argument No relevant contribution from the Einstein term Changes to the leading order solution are not important Agreement does not continue for higher derivatives and is not expected to continue Banerjee+Dutta

Contribution of the c 2 terms On horizon Correct results Fix: On-shell

Ghosts Ghosts  c 2 terms which contribute to h Coefficient of c 2 is a constant  c 2 survives all the way to the AdS boundary Existence of ghosts independent on the sign or magnitude of a Ghosts have Planck scale mass expected to decouple but manage to contribute

An Exorcizing BC } H0 } H1 a –constant, b-radial function, unique up to perturbative mass terms Incoming @ horizon Dirichlet @ boundary Dirichlet @ horizon } H0 } H1 Correction to h ambiguos at order e, h /s unambigous

An Exorcizing BC: compariso to the “standard choice” e’s suppressed, a –constant ~ e, b-radial function ~ e perturbative mass terms suppressed split the modes order by order in e Incoming @ horizon Dirichlet @ boundary } h0 } h1

An Exorcizing BC : General prescription Re-invert sum of the non-ghost propagators Convert equation into local form by expanding in c /M 2 Reduce order of equation to two-derivatives following effective-action method (Banerjee+Dutta) Translate reduced equation into a two-derivative effective action Read off h from the gravitational coupling

Summary & conclusions For unitary weakly coupled extensions of Einstein gravity and their FT duals Unitarity Entropy Geometry Demonstrated explicitly for Four-derivatives example to leading order in the corrections Difference from previous results - BC