Walds Entropy Area Entanglement Ram Brustein Introduction R
Wald’s Entropy, Area & Entanglement Ram Brustein גוריון - אוניברסיטת בן • Introduction: R. B. , MERAV HADAD =========== R. B, Einhorn, Yarom, 0508217, 0609075 Series of papers with Yarom, (also David Oaknin) – 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
Plan • What is Wald’s entropy ? • How to evaluate Wald’s entropy – The Noether charge Method (W ‘ 93, Liv. Rev 2001+…) – The field redefinition method (JKM, ‘ 93) Result: a class of theories both? depend • Whatfor is entanglement entropy on the geometry in the same way, ? and can – How is it related to BH entropy be made by a choice of scale – Howequal to evaluate entanglement entropy ? • How are the two entropies related ?
Wald’s entropy • S – Bifurcating Killing Horizon: d-1 space-like surface @ intersection of two KH’s (d = D-1=# of space dimensions) – Killing vector vanishes on the surface • The binormal vector eab : normal to the tangent & normal of S • Functional derivative as if Rabcd and gab 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 = rs
The simplest example: .
A more complicate example , .
The field redefinition method for evaluating Wald’s entropy • The idea (Jacobson, Kang, Myers, gr-qc/9312023) – 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 Dab is constructed from the original metric and matter fields Lc Dab = 0 and Dab vanishes sufficiently rapidly
A 2 more 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 All |↓ 22 ↓| S 1=-Trelements (r 1 lnr 1)=ln 2 S 2=-Trace (r 2 lnr 2)=ln 2 1 2
Entanglement If : thermal & time translation invariance then TFD: purification
“Kruskal”inextension Entanglement space-time r= rs Examples: Minkowski, de Sitter, t = const Schwarzschild, non-rotating BTZ. BH, can be extended to rotating, charged, non-extremal BHs t=0 r = const.
“Kruskal” extension t r=0 r = rs x x
The vacuum state |0 t r=0 r = rs x
Two ways of calculating rin R. B. , M. Einhorn and A. Yarom out r = rs out in t = const. t=0 r = const. in r = rs t = const. t=0 r = const. Construct the HH vacuum: the invariant regular state Kabat & Strassler (flat space) Jacobson
Results*: If * Method works for more general cases 1. The boundary conditions are the same 2. The actions are equal 3. The measures are equal Then Heff – generator of (Im t) time translations
Entanglement entropy Emparan de Alwis & Ohta d– proper length short distance cutoff in optical metric S is divergent Naïve origin: divergence of the optical volume near the horizon, *not* brick wall. Choice of d S=A/4 G EXPLAIN d !!!!
Extensions, Consequences 1. Works for Eternal Ad. S BH’s, consistent with Ad. S-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 d 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 l) 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!)
- Slides: 31