Rindler Holography Daniel Grumiller Institute for Theoretical - PowerPoint PPT Presentation

Rindler Holography Daniel Grumiller Institute for Theoretical Physics TU Wien Workshop on Topics in Three Dimensional Gravity Trieste, March 2016 based on work w. H. Afshar, S. Detournay, W. Merbis, (B. Oblak), A. Perez, D. Tempo, R. Troncoso

Outline Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments Daniel Grumiller — Rindler Holography 2/23

Outline Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments Daniel Grumiller — Rindler Holography Motivation 3/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10 172 microstates Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10 172 microstates ( S Go ≈ 396 ) Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10 172 microstates ( S Go ≈ 396 ) → black holes more complicated! Daniel Grumiller — Rindler Holography Motivation 4/23

Black hole microstates Bekenstein–Hawking A [for M ⊙ : e S BH ∼ O ( e 10 76 ) ∼ e chess microstates ] S BH = 4 G N ◮ Motivation: microscopic understanding of generic black hole entropy Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking A S BH = 4 G N ◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT 2 symmetries (Carlip, Strominger, ...) using Cardy formula Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking A S BH = 4 G N ◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT 2 symmetries (Carlip, Strominger, ...) using Cardy formula ◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT, ...) → see talk by Stephane Detournay! Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking A S BH = 4 G N ◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT 2 symmetries (Carlip, Strominger, ...) using Cardy formula ◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT, ...) → see talk by Stephane Detournay! ◮ Main idea: consider near horizon symmetries for non-extremal horizons Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking A S BH = 4 G N ◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT 2 symmetries (Carlip, Strominger, ...) using Cardy formula ◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT, ...) → see talk by Stephane Detournay! ◮ Main idea: consider near horizon symmetries for non-extremal horizons ◮ Near horizon line-element with Rindler acceleration a : d s 2 = − 2 aρ d v 2 + 2 d v d ρ + γ 2 d ϕ 2 + . . . Meaning of coordinates: ◮ ρ : radial direction ( ρ = 0 is horizon) ◮ ϕ ∼ ϕ + 2 π : angular direction ◮ v : (advanced) time Daniel Grumiller — Rindler Holography Motivation 5/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? Daniel Grumiller — Rindler Holography Motivation 6/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale Recall scale invariance a → λa ρ → λρ v → v/λ of Rindler metric d s 2 = − 2 aρ d v 2 + 2 d v d ρ + γ 2 d ϕ 2 Daniel Grumiller — Rindler Holography Motivation 6/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in 1512.08233 : v ∼ v + 2 πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult Recall scale invariance a → λa ρ → λρ v → v/λ of Rindler metric d s 2 = − 2 aρ d v 2 + 2 d v d ρ + γ 2 d ϕ 2 Daniel Grumiller — Rindler Holography Motivation 6/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in 1512.08233 : v ∼ v + 2 πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult ◮ If chemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino) T U = a 2 π Daniel Grumiller — Rindler Holography Motivation 6/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in 1512.08233 : v ∼ v + 2 πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult ◮ If chemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino) T U = a 2 π We make this choice in this talk! Daniel Grumiller — Rindler Holography Motivation 6/23

Choices ◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in 1512.08233 : v ∼ v + 2 πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult ◮ If chemical potential: all states in theory have same (Unruh-)temperature (see talk by Miguel Pino) T U = a 2 π ◮ Work in 3d Einstein gravity in Chern–Simons formulation (see talk by Jorge Zanelli!) k � � A ± ∧ d A ± + 2 3 A ± ∧ A ± ∧ A ± � � I CS = ± 4 π ± with sl (2) connections A ± and k = ℓ/ (4 G N ) with AdS radius ℓ = 1 Daniel Grumiller — Rindler Holography Motivation 6/23

Outline Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments Daniel Grumiller — Rindler Holography Near horizon boundary conditions 7/23

Diagonal gauge Standard trick: partially fix gauge A ± = b − 1 d+ a ± ( x 0 , x 1 ) � � ± ( ρ ) b ± ( ρ ) with some group element b ∈ SL (2) depending on radius ρ Drop ± decorations in most of talk Manifold topologically a cylinder or torus, with radial coordinate ρ and boundary coordinates ( x 0 , x 1 ) ∼ ( v, ϕ ) Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b − 1 ( ρ ) � d+ a ( x 0 , x 1 ) � b ( ρ ) with some group element b ∈ SL (2) depending on radius ρ ◮ Standard AdS 3 approach: highest weight gauge a ∼ L + + L ( x 0 , x 1 ) L − b ( ρ ) = exp ( ρL 0 ) sl (2) : [ L n , L m ] = ( n − m ) L n + m , n, m = − 1 , 0 , 1 Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b − 1 ( ρ ) � d+ a ( x 0 , x 1 ) � b ( ρ ) with some group element b ∈ SL (2) depending on radius ρ ◮ Standard AdS 3 approach: highest weight gauge a ∼ L + + L ( x 0 , x 1 ) L − b ( ρ ) = exp ( ρL 0 ) sl (2) : [ L n , L m ] = ( n − m ) L n + m , n, m = − 1 , 0 , 1 ◮ For near horizon purposes diagonal gauge useful: a ∼ J ( x 0 , x 1 ) L 0 Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b − 1 ( ρ ) � d+ a ( x 0 , x 1 ) � b ( ρ ) with some group element b ∈ SL (2) depending on radius ρ ◮ Standard AdS 3 approach: highest weight gauge a ∼ L + + L ( x 0 , x 1 ) L − b ( ρ ) = exp ( ρL 0 ) sl (2) : [ L n , L m ] = ( n − m ) L n + m , n, m = − 1 , 0 , 1 ◮ For near horizon purposes diagonal gauge useful: a ∼ J ( x 0 , x 1 ) L 0 ◮ Precise boundary conditions ( ζ : chemical potential): a = ( J d ϕ + ζ d v ) L 0 ζ L 1 ) · exp ( ρ and b = exp ( 1 2 L − 1 ) . (assume constant ζ for simplicity) Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Near horizon metric Using A + µ − A − A + ν − A − g µν = 1 �� � � �� µ ν 2 Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23

Near horizon metric Using A + µ − A − A + ν − A − g µν = 1 �� � � �� µ ν 2 yields ( f := 1 + ρ/ (2 a ) ) d s 2 = − 2 aρf d v 2 + 2 d v d ρ − 2 ωa − 1 d ϕ d ρ γ 2 + 2 ρ a f ( γ 2 − ω 2 ) d ϕ 2 � � + 4 ωρf d v d ϕ + state-dependent functions J ± = γ ± ω , chemical potentials ζ ± = − a ± Ω For simplicity set Ω = 0 and a = const . in metric above EOM imply ∂ v J ± = ± ∂ ϕ ζ ± ; in this case ∂ v J ± = 0 Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23