Wald’s Entropy, Area & Entanglement

1. Ram Brustein אוניברסיטת בן-גוריון Wald’s Entropy, Area & Entanglement R.B., MERAV HADAD ===================== R.B, Einhorn, Yarom, 0508217, 0609075 Series of papers with Yarom, (also David Oaknin) • 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

2. Plan • 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 ? 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

3. 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 Rabcdand gab are independent

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

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

6. The simplest example: .

7. A more complicate example , .

8. 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 sufficientlyrapidly

9. A more2 complicated Example: For a1=0 Weyl transformation

10. is the metric in the subspace normal to the horizon

11. 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” )

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

13. All |↓22↓| elements 1 2 Entanglement S=0 S1=-Tr (r1lnr1)=ln2 S2=-Trace (r2lnr2)=ln2

14. Entanglement If : thermal & time translation invariancethen TFD: purification

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

16. r = 0 t r = rs x x “Kruskal” extension

17. The vacuum state r=0 t r = rs x |0

18. r = rs r = rs t = const. t = const. t = 0 t = 0 r = const. r = const. Two ways of calculating rin R.B., M. Einhorn and A.Yarom out in out in Construct the HH vacuum: the invariant regular state Kabat & Strassler (flat space) Jacobson

19. Results*: * Method works for more general cases If • The boundary conditions are the same • The actions are equal • The measures are equal Then Heff – generator of (Imt) time translations

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

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

22. 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

23. 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).

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

28. 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

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

30. 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

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