Wald’s Entropy, Area & Entanglement Introduction: –Wald’s Entropy –Entanglement entropy in space-time Wald’s entropy is (sometimes) an area ( of some metric) or related to the area by a multiplicative factor Relating Wald’s entropy to Entanglement entropy אוניברסיטת בן - גוריון Ram Brustein R.B., MERAV HADAD ===================== R.B, Einhorn, Yarom, , Series of papers with Yarom, (also David Oaknin)

What is Wald’s entropy ? How to evaluate Wald’s entropy –The Noether charge Method (W ‘93, LivRev 2001+…) –The field redefinition method (JKM, ‘93) What is entanglement entropy ? –How is it related to BH entropy ? –How to evaluate entanglement entropy ? How are the two entropies related ? Plan Result: for a class of theories both depend on the geometry in the same way, and can be made equal by a choice of scale

Wald’s entropy  – Bifurcating Killing Horizon: d-1 space-like intersection of two KH’s (d = D-1=# of space dimensions) –Killing vector vanishes on the surface The binormal vector  ab : normal to the tangent & normal of  Functional derivative as if R abcd and g ab are independent

Wald’s entropy Properties: Satisfies the first law Linear in the “correction terms” Seems to agree with string theory counting

. Wald’s entropy: the simplest example The bifurcation surface t =0, r = r s

. The simplest example:

. A more complicate example,

The field redefinition method for evaluating Wald’s entropy The idea ( Jacobson, Kang, Myers, gr-qc/ ) –Make a field redifinition –Simplify the action (for example to Einstein’s GR) Conditions for validity –The Killing horizons, bifurcation surface, and asymptotic structure are the same before and after –Guaranteed when  ab is constructed from the original metric and matter fields L   ab  = 0 and  ab vanishes sufficiently rapidly

A more 2 complicated Example: For a 1 =0 Weyl transformation

is the metric in the subspace normal to the horizon

The entanglement interpretation: The statistical properties of space-times with causal boundaries arise because classical observers in them have access only to a part of the whole quantum state  trace over the classically inaccessible DOF ( “Microstates are due to entanglement” ) The fundamental physical objects describing the physics of space-times with causal boundaries are their global quantum state and the unitary evolution operator. ( “Entropy is in the eyes of the beholder” )

The entanglement interpretation: Properties: –Observer dependent –Area scaling –UV sensitive –Depends on the matter content, # of fields …,

Entanglement S=0 S 1 =-Tr (  1 ln  1 )=ln2 S 2 =-Trace (  2 ln  2 )=ln2 All |↓  22  ↓| elements 1 2

Entanglement If : thermal & time translation invariance then TFD: purification

r = r s  = 0  = const. r = const. Entanglement in space-time Examples: Minkowski, de Sitter, Schwarzschild, non-rotating BTZ BH, can be extended to rotating, charged, non-extremal BHs “Kruskal” extension

t x r = r s r = 0 x

The vacuum state |0  t x r=0 r = r s

 = 0  = const. r = const. r = r s  = 0  = const. r = const. Two ways of calculating  in Kabat & Strassler (flat space) Jacobson Construct the HH vacuum: the invariant regular state inout inout R.B., M. Einhorn and A.Yarom

1.The boundary conditions are the same 2.The actions are equal 3.The measures are equal Results*: If Then H eff – generator of (Im  t) time translations * Method works for more general cases

S  is divergent Naïve origin: divergence of the optical volume near the horizon, *not* brick wall. Choice of   S=A/4G Entanglement entropy  – proper length short distance cutoff in optical metric Emparan de Alwis & Ohta EXPLAIN  !!!!

Extensions, Consequences 1.Works for Eternal AdS BH’s, consistent with AdS-CFT, RB, Einhorn, Yarom 2.Rotating and charged BHs, RB, Einhorn, Yarom 3.Extremal BHs (on FT side): Marolf and Yarom 4. Non-unitary evolution : RB, Einhorn, Yarom

Relating Wald’s entropy to Entanglement entropy Wald’s entropy is an area for some metric or related to the area by a multiplicative factor –So far: have been able to show this for theories that can be brought to Einstein’s by a metric redefinition equivalent to a conformal rescaling in the r-t plane on the horizon. Entanglement entropy scales as the area Changes in the minimal length  account for the differences

Relating Wald’s entropy to Entanglement entropy Example : more complicated matter action –Changes in the matter action do not change Wald’s entropy –Changes in the matter action do not change the entanglement entropy (as long as the matter kinetic terms start with a canonical term).

Example : theories that can be related by a Weyl transformation to Einstein + (conformal) matter

Relating Wald’s entropy to Entanglement entropy Example : theories that can be related by a Weyl transformation to Einstein + (conformal) matter By a consistent choice of make

JKM: It is always possible to find (to first order in ) a function

Relating Wald’s entropy to Entanglement entropy Example: –More complicated –The transformation is not conformal –The transformation is only conformal on r-t part of the metric, and only on the horizon –Works in a similar way to the fully conformal transformation

Summary 1.Wald’s entropy is consistent with entanglement entropy 2.Wald’s entropy is (sometimes) an area (for some metric) or related to the area by a multiplicative factor 3.BH Entropy can be interpreted as entanglement entropy (not a correction!)