1. Unitarity constraints on h/s
אוניברסיטת בן-גוריון
Ram Brustein
+MEDVED
======================
+MEDVED
, ,
+GORBONOS, HADAD
+G , +H  
+DVALI, VENEZIANO
+MEDVED  
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

2. 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

3. 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

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

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

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

7. 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

8. 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

9. 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)

10. 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

11. 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
DVALI: MANY PAPERS

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

13. 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

14. Gravitational couplings of generalized gravity can only increase compared to Einstein gravity
Spectral decomposition
semiclassical unitarity

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

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

17. 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

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

19. 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

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

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

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

23. Calculating h: previous results
e.g
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

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

25. 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

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

27. 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
horizon
boundary
} h0
} h1

28. 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

29. 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