PRESENTS DANNY TERNO & ETERA LIVINE With contributions from Asher Peres, Viqar Hussain and Oliver Winkler PRODUCTION

when met

n Noncovariance of reduced density matrices n Noninvariance of entropy n Implications to holography and thermodynamics Outline n Entanglement and black hole entropy n Entanglement and Hawking radiation Volume 1: new properties Volume 2: old applications

Volume 1: new properties

Noncovariance (1) spin momentum classical info Example: Lorentz transform of a single massive particle transform: v along the z - axis important parameter:

Partial trace is not Lorentz covariant Spin entropy is not scalar Distinguishability depends on motion Entropy Peres and Terno, Rev. Mod. Phys. 76, 93 (2004)

here there trace out “there” Bombelli et al, Phys. Rev. D34, 373 (1986) Holzhey, Larsen and Wilczek, Nucl. Phys. B424, 443 (1994) Callan and Wilczek, Phys. Lett. B333, 55 (1994). Noncovariance (2) Geometric entropy

= ? no correlations no Bell-type violations not irreducible Transformations do not split into here and there spaces Decomposition of Lorentz transformations Terno, Phys. Rev. Lett. 93, (2004) trivial 1D rep irrep of 1-particle states

Noninvariance (1) Yurtsever, Phys. Rev. Lett. 91, (2003) Bekenstein, Lett. Nuovo Cim. 4, 737 (1972) …. Busso, Rev. Mod. Phys. 74,825 (2002) Boundary conditions & cut-offs Model Number of degrees of freedom N is frame-dependent

Lorentz boost: factors 1/γ Spacelike holographic bound both area and entropy change Saved ? v Terno, Phys. Rev. Lett. 93, (2004)

Black holes: invariance Hawking’s area theorem Model 1+1 calculations: the same crossing point, relative boost Two observers with a relative boost Fiola, Preskill, Strominger, Trivedi, Phys. Rev. D 59, 3987 (1994) -

Noninvariance (2) Accelerated cavity Moore, J. Math. Phys. 11, 2679 (1970) Levin, Peleg, Peres, J.Phys.A 25, 6471 (1992) Accelerated observers & matter beyond the horizon Terno, Phys. Rev. Lett. 93, (2004)

Volume 2: old applications

Entanglement on the horizon Requirement: SU(2) invariance of the horizon states Object: static black hole States: spin network that crosses the horizon Qubit BH

density matrix Standard counting story area constraint 2n spins number of states entropy Fancy counting story entropy

Entanglement Measure: entanglement of formation 2 vs 2n-2 States of the minimal decomposition degeneracy indices Alternative decomposition: linear combinations Its reduced density matrices: mixtures Entropy: concavity

unentangled fraction entanglement n vs n Entropy of the whole vs. sum of its parts reduced density matrices BH is not made from independent qubits, but… Livine and Terno, gr-qc/0412xxx

Entanglement and Hawking radiation Hussein,Terno and Winkler, in preparation

Summary when met n Reduced density matrices are not covariant n Entropy (and the # of degrees of freedom) are observer-dependent n Entanglement is responsible for the logarithmic corrections of BH entropy n Entropy of the BH radiation = entanglement entropy between gravity and matter

Thanks to Jacob Bekenstein Ivette Fuentes-Schuller Florian Girelli Netanel Lindner Rob Myers Johnathan Oppenheim David Poulin Terry Rudolph Frederic Schuller Lee Smolin Rafael Sorkin Rowan Thomson

Technique Technique: Unruh effect Entropy usually diverges General: cut-off renormalization of entropy

Unruh+ Unruh + Audretsch and Müller, Phys. Rev. D 49, 4056 (1994) Matter outside the horizon n particles in the mode (k,m) Splitting: usual + super

Special case renormalized quantities temperature Of what? two subsystems General case: temperature is undefined Two observers: the same acceleration, different velocities