Entanglement Equilibrium First Law of Diamond Mechanics Extension - PowerPoint PPT Presentation

E NTANGLEMENT EQUILIBRIUM IN HIGHER CURVATURE GRAVITY Vincent Min P. Bueno, A. J. Speranza, M. R. Visser arXiv:1612.XXXXX 18 November 1

Entanglement Equilibrium First Law of Diamond Mechanics Extension to Higher Curvature Gravity First Law of Thermodynamics 2

E NTROPY = G RAVITY ✤ Black holes obey thermodynamic laws. Hawking, Bekenstein, ... ✤ Assuming thermodynamic laws implies gravity. Jacobson ✤ Bulk reconstruction using entanglement entropy in AdS/CFT van Raamsdonk, Ryu, Takayanagi, ... ✤ AdS is a tensor networks Swingle, ... ✤ Emergent gravity Verlinde ✤ Entanglement equilibrium implies the Einstein equation. Jacobson 3

E NTANGLEMENT ENTROPY OF A GEODESIC BALL ✤ We start in the vacuum of a QFT d . ✤ Consider the S EE of a geodesic ball. Σ ✤ Claim: S EE is maximal for the ∂ Σ vacuum. 4

M AXIMAL VACUUM ENTANGLEMENT HYPOTHESIS The Maximal Vacuum Entanglement Hypothesis (MVEH) for Einstein gravity states Jacobson δS EE | V = 0 . The variation splits in a UV and IR part δS EE = δS UV + δS IR . One can show that for Einstein gravity δS UV | V ∝ − G ab δS IR | V ∝ 2 πT ab . 4 G , Thus the MVEH implies the Einstein equation G ab = 8 πGT ab 5

W HY SHOULD ONE BELIEVE IN THE MVEH? MVEH is a microscopic interpretation of the First Law of Diamond Mechanics (FLDM), the classical first order identity: Iyer,Wald � δH ζ = δ Q ζ , ∂ Σ where H ζ is the Hamiltonian generating evolution along the flow of the conformal killing vector ζ and Q ζ is the Noether charge ( d − 2) -form. δH ζ receives contributions from changes in the state | ψ � , and the geometry g ab , such that the FLDM reads � δH ψ ζ + δH g ζ − δ Q ζ = 0 . ∂ Σ 6

W HY KEEP THE VOLUME FIXED ? For pure Einstein gravity the expressions in the FLDM read 1 d − 2 1 ∂A δH g ζ = − δV = − ∂V δV , 8 πG ℓ 8 πG 1 � δ Q ζ = − 8 πGδA . ∂ Σ The FLDM then reads � 1 δH ψ ζ + δH g Q ζ = δH ψ ζ − δ ζ + 8 πG δA | V = 0 . ∂ Σ Identifying 2 πδH ψ ζ = δS IR and δA 4 G = δS UV , we find the MVEH. δS EE | V = 0 7

H IGHER CURVATURE GRAVITY What would the generalization to higher curvature gravity look like? δS EE | V = 0 ⇒ δX | Y = 0 A natural guess for the entropy is the Wald entropy δS Wald | Y = 0 . But what is the generalization of volume? 8

B ACK TO F IRST L AW OF D IAMOND M ECHANICS What would the generalization to higher curvature gravity look like? � δH ψ ζ + δH g ζ − δ Q ζ = 0 . ∂ Σ Wald’s formalism provides the extension to higher curvature gravity � Q ζ = − 1 2 πS Wald ∂ Σ � ∂ L E abcd ≡ dA E abcd n ab n cd , S Wald = − 2 π , ∂R abcd ∂ Σ ζ = − 4 � ≡ − d − 2 � � δH g E abcd u a u d h bc − E 0 ℓ δ dV 8 πGℓδW Σ The generalization of volume is 1 � dV 1 ( E abcd u a u d h bc − E 0 ) , W = d − 2 E 0 Σ 9

B ACK TO E NTANGLEMENT E QUILIBRIUM Can we still interpret the FLDM as a generalized MVEH? � δH ψ ζ + δH g ζ − Q ζ = ∂ Σ ζ + 1 � δS Wald − ∂S Wald � δH ψ = 0 . δW 2 π ∂W The generalized MVEH reads δS | W = 0 10

R IEMANN N ORMAL C OORDINATES We can extract the equations of motion from the FLDM using Riemann Normal Coordinates (RNC) g ab ( x ) = η ab + 1 6 R acbd (0) x c x d + O � x 3 � Therefore a small variation around flat space is δg ab ( x ) = 1 6 R acbd (0) x c x d + O x 3 � � In a ball of radius ℓ < L c every spacetime looks locally like a variation around flat space. 11

E XTRACTING THE EQUATIONS OF MOTION Evaluating the FLDM with RNC in small geodesic balls leads to � δH ψ ζ + δH g Q ζ ∝ u a u b δ E ab (0) + O ℓ 2 � � ζ − . ∂ Σ We can extract the linear equations of motion for higher curvature gravity. Note that Jacobson found the non-linear Einstein equations in pure Einstein gravity. 12

L INEARIZING EQUATIONS OF MOTION WITH RNC Remember that the metric perturbation in RNC reads δg ab ( x ) = 1 6 R acbd (0) x c x d + O x 3 � � However, curvatures do not vanish for small balls δR ( x ) = R (0) + O ( x ) Linearizing the Einstein tensor around flat space reads � R ab [0] − 1 � δG ab ( x ) = 2 η ab R [0] + O ( x ) = G ab [0] + O ( x ) , while linearization of higher curvature terms vanishes � R 2 � = 0 + O ( x ) ⇒ δ E ab ( x ) � = E ab [0] + O ( x ) . δ 13

F IRST L AW OF T HERMODYNAMICS The first law of thermodynamics reads dU = − PdV + TdS , where � ∂S �� �� �� � ∂U � ∂U � � � T ≡ , P ≡ − = T . � � � ∂S ∂V ∂V � � � V S U Thus the first law of thermodynamics can also be rewritten as dU = T dS | V , which should remind you of the FLDM ζ + 1 δH ψ 2 π δS Wald | W = 0 . The FLDM can be interpreted as a first law, identifying 1 H ψ ζ = − U , 2 π = T , S Wald = S , W = V . 14

C ONCLUSION ✤ The Maximal Vacuum Entanglement Hypothesis (MVEH) provides new insights into the emergence of gravity. ✤ The MVEH is an interpretation of the First Law of Diamond Mechanics (FLDM). ✤ The FLDM and MVEH can be generalized to higher curvature gravity. ✤ Extracting the non-linear e.o.m. from the MVEH is special to pure Einstein gravity. ✤ The FLDM can alternatively be interpreted as a first law of thermodynamics. ✤ We propose a generalized volume for higher curvature gravity. 15

O UTLOOK ✤ Can our generalized volume be applied to Complexity/Fidelity Susceptibility Susskind, Brown/ Miyajia, Numasawaa, Shiba ✤ Can we include non-conformal matter? ✤ What is the role of the cosmological constant? ✤ Include higher order corrections. Can this lead to the non-linear e.o.m. for higher curvature gravity? 16