New guises of the BTZ black hole and the entropy of 2d CFT

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsNew guises of AdS 3andthe entropy of two-dimensional CFTN. TetradisUniversity of AthensJune 21, 2011N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsOutlineFefferman-Graham parametrization of AdS 3 with variousboundary metricsStress-energy tensorEntropyCommentsP. Apostolopoulos, G. Siopsis, N. T. : arxiv:0809.3505[hep-th], Phys.Rev. Lett. 102 (2009) 151301N. T. : arxiv:0905.2763[hep-th], JHEP 1003 (2010) 040N. Lamprou, S. Nonis, N. T. : arXiv:1106.1533 [gr-qc]N. T. : arXiv:1106.2492 [hep-th]N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsThe BTZ black holeMetric in Schwarzschild coordinates (φ has a period equal to 2π):ds 2 = −f(r)dt 2 + dr 2f(r) + r 2 dφ 2 , f(r) = r 2 − µ. (1)Temperature, energy and entropy of the black hole (V = 2π):T = 1 √ Vµ, E = µ,2π16πG 3Metric in Fefferman-Graham coordinates:ds 2 = 1 ( [dz 2z 2 − 1 − µ 4 z2) 2dt 2 +S = V4G 3√ µ. (2)(1 + µ 4 z2) 2dφ2]. (3)withz = 2 µ(r ∓ √ )r 2 − µ , r = 1 z + µ z. (4)4z takes values 0 < z ≤ z e = 2/ √ µ and z e ≤ z < ∞, coveringtwice the region outside the event horizon.r takes values r e = √ µ ≤ r < ∞. Throat at z = 2/ √ µ.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsGeneral metric in Fefferman-Graham coordinates:ds 2 = 1 z 2 [dz 2 + g µν dx µ dx ν] , (5)whereg µν = g µν (0) + z 2 g µν (2) + z 4 g µν (4) . (6)Holographic stress-energy tensor of the dual CFT (Skenderis2000)〈T µν (CFT) 〉 = 1 [ (g (2) − tr g (2)) g (0)] . (7)8πG 3Apply this to the BTZ black hole with a flat boundaryEnergy density and pressure:ds 2 0 = g(0) µν dxµ dx ν = −dt 2 + dφ 2 . (8)ρ = E V = −〈T t µt〉 = ,16πG 3(9)p = 〈T φ φ 〉 = µ.16πG 3(10)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsAdS 3 with a Rindler boundaryFor the Rindler wedge (x > 0), the metric (1) with µ = 0 (AdS 3 inPoincare coordinates) can be put in the formds 2 = 1 z 2 [with a Rindler boundarydz 2 − a 2 x 2 (1 + z24x 2 ) 2dt 2 +(1 − z24x 2 ) 2dx 2 ],(11)ds 2 0 = g (0)µν dx µ dx ν = −a 2 x 2 dt 2 + dx 2 . (12)The coordinate transformation does not affect the timecoordinate:( xr(z, x) = az + z )(13)4xφ(z, x) = 1 a log [ax] − 8a(4 + z 2 /x 2 ) . (14)The transformation maps the whole region near negative infinityfor φ to the neighborhood of zero with x > 0 . φ is not periodic.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsFor fixed x, there is a minimal value for r(z, x) as a function of z.It is obtained forz m (x) = 2x (15)and is equal toThe corresponding value of φ isr m (x) = a. (16)φ m (x) ≡ φ m (z m (x), x) = (log[ax] − 1)/a. (17)Bridge connecting the two asymptotic regions (that are copies ofeach other) at z → 0 and z → ∞.The holographic stress-energy tensor isρ = −〈T t t〉 = − 1 116πG 3 x 2, (18)p = 〈T x x〉 = − 1 116πG 3 x 2. (19)It displays the expected singularity at x = 0. The conformalanomaly vanishes.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsGlobal coordinatesAdS 3 can be written asds 2 =1cos 2 (˜χ)[−d˜t 2 + d ˜χ 2 + sin 2 (˜χ) d ˜φ 2] , (20)with 0 ≤ φ ≤ 2π, 0 ≤ ˜χ < π/2. The boundary is approached for˜χ → π/2.The Schwarzschild (Poincare) and the global coordinates arerelated through[√r2− µ sinh( √ ]µ t)˜t(t, r, φ) = arctanr 2 cosh( √ (21)µ φ)√r˜χ(t, r, φ) = arctan2 µ sinh2 ( √ µφ) + r 2 − µcosh 2 ( √ µt)(22)µ[√r2− µ cosh( √ ]µ t)˜φ(t, r, φ) = arctanr 2 sinh( √ . (23)µ φ)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsΧ1.41.21.00.80.60.40.23 2 1 0 1 2 3ΦFigure: Lines of constant r.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsA different stateThe AdS 3 metric can also be put in the formds 2 = 1 z 2 [dz 2 − a 2 x 2 dη 2 + dx 2] , (24)with a Rindler boundary. The coordinate transformation thatachieves this is given byt(η, x) = x sinh(aη) (25)r(z) =1 z(26)φ(z, x) = x cosh(aη). (27)The corresponding stress-energy tensor vanishes.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsAdS 3 with de Sitter boundaryds 2 = 1 ([dz 2z 2 − (1 − H 2 ρ 2 ) 1 + 1 [H 2 ] ) 24 1 − H 2 ρ 2 − H2 z 2 dt 2+φ(z, ρ) = 1 [ 1 + Hρ2H log 1 − Hρ(1 − 1 [H 2 ] ) 24 1 − H 2 ρ 2 + H2 z 2 dρ 21 − H 2 ρ 2 ](28) ,with a de Sitter boundary at z = 0.The coordinate transformation is (−1/H < ρ < 1/H)r(z, ρ) =√1 − H2 ρ 2 H 4 ρ 2+z 4 √ 1 − H 2 ρ z 2](29)−H 2 ρz 22(1 − H 2 ρ 2 + H 4 ρ 2 z 2 /4) .(30)The transformation maps the region near negative infinity for φ tothe vicinity of −1/H for ρ > −1/H, and the region near positiveinfinity for φ to the vicinity of 1/H for ρ < 1/H.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsFor fixed ρ, there is a minimal value for r(z, ρ), obtained for√1 − Hz m (ρ) = 22 ρ 2H 4 ρ 2 . (31)It is given byr m (ρ) = H 2 |ρ|. (32)The corresponding value of φ isφ m (ρ) ≡ φ(z m (ρ), ρ) = 1 [ ] 1 + Hρ2H log − 11 − Hρ H 2 ρ . (33)Bridge connecting the asymptotic regions at z → 0 and z → ∞.Stress-energy tensor:ρ = −〈T t t 〉 = − 1 (H 2 )16πG 3 1 − H 2 ρ 2 + H2 (34)p = 〈T ρ ρ 〉 = − 1 (H 2 )16πG 3 1 − H 2 ρ 2 − H2 . (35)The conformal anomaly is 〈T µ (CFT)µ 〉 = H 2 /(8πG 3 ).N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsΧ1.41.21.00.80.60.40.23 2 1 0 1 2 3ΦFigure: Global coordinates.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsA different stateThe metricds 2 =[dz 1 2z 2 +(1 − 1 ) 2 )4 H2 z(−(1 ] 2 − H 2 ρ 2 )dη 2 + dρ21 − H 2 ρ 2 ,also has a de Sitter boundary.The stress-energy tensor is(36)ρ = −〈T t t 〉 = − H216πG 3(37)p = 〈T ρ ρ〉 = H216πG 3. (38)The conformal anomaly is the same as before.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsAdS 3 with FRW boundaryThe BTZ metric can be expressed as (φ is now periodic)withds 2 = 1 z 2 [dz 2 − N 2 (τ, z)dτ 2 + A 2 (τ, z)dφ 2] , (39)A(τ, z)N(τ, z)The boundary has the form(= a(τ) 1 + µ − )ȧ2 (τ)4a(τ) 2 z 2(40)= 1 − µ − ȧ2 + 2aä4a 2 z 2 A(τ,= ˙ z). (41)ȧds 2 0 = g (0)µν dx µ dx ν = −dτ 2 + a 2 (τ)dφ 2 , (42)with a(τ) an arbitrary function.N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsThe transformation of the time coordinate for z < z e1 or forz > z e2 is[t(τ, z) =ɛ 4a 22 √ µ log − ( √ ) 2]µ + ȧ z24a 2 − ( √ ) 2+ ɛ c(τ), (47)µ − ȧ z2where the function c(τ) satisfies ċ = 1/a(τ) and ɛ = ±1.For z e1 < z < z e2 the transformation is[t(τ, z) =ɛ −4a 22 √ µ log + ( √ ) 2]µ + ȧ z24a 2 − ( √ ) 2µ − ȧ z2The transformation is singular on the event horizons.+ ɛ c(τ). (48)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsThe coordinate φ remains unaffected by the transformation. It isperiodic, with periodicity 2π.Dual picture: thermalized CFT on an expanding background, witha scale factor a(τ).Stress-energy tensor:Casimir energy ∼ ȧ 2 /a 2 .Conformal anomaly:ρ = E V = −〈T τ τ 〉 = 116πG 3µ − ȧ 2a 2 (49)P = 〈T φ φ 〉 = 116πG 3µ − ȧ 2 + 2aäa 2 , (50)〈T µ (CFT)µ 〉 = 18πG 3äa . (51)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsObservation: The entropy is proportional to the narrowest partof the throat or bridge.This line defines the boundary of the part of the bulk geometrythat is not covered by the Fefferman-Graham parametrization. Ina sense, it determines the part of the bulk that is not included inthe construction of the dual theory.In quantitative terms:S = 14G 3A, (52)with A the length of the narrowest part of the throat or bridge at agiven time.BTZ black hole with a flat boundary:A = 2π √ µ,S th = π2G 3√ µ. (53)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsAdS 3 with Rindler boundaryThe throat is located at:in Fefferman-Graham coordinates.Equivalently, it is located atz m (x) = 2x (54)r m (x) = a, φ m (x) ≡ φ m (z m (x), x) = (log[ax] − 1)/a (55)in Schwarzschild (Poincare) coordinates.The entropy isS = 14G 3∫ ∞−∞adφ = 14G 3∫ ∞0dxx = 14G 3∫ ∞0dzz . (56)The infinities come from the endpoints, where the lineapproaches the boundary.Regulate !S = 24G 3∫ 1/ɛds, (57)N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsFor a (d + 1)-dimensional bulk metricds 2 = 1 u 2 (du 2 − dt 2 + d⃗x 2) , (58)the effective Newton’s constant G d is1= 1 ∫ ∞duG d G d+1 ud−1. (59)For d = 2 and u = 1/r01= 1 ∫ ∞G 2 G 30drr . (60)A line integral in AdS 3 , starting and ending at the boundary.G 2 = 0. But, regulate !This gives1G 2= 2 G 3∫ 1/ɛds (61)S R = 14G 2. (62)N. Tetradis University of Athens

) Las prestaciones de desempleo;c) El acceso a los programas de obras públicas destinados a combatir el desempleo;d) El acceso a otro empleo en caso de quedar sin trabajo o darse término a otra actividadremunerada, con sujeción a lo dispuesto en el artículo 52 de la presente Convención.2. Si un trabajador migratorio alega que su empleador ha violado las condiciones de su contrato detrabajo, tendrá derecho a recurrir ante las autoridades competentes del Estado de empleo, segúnlo dispuesto en el párrafo 1 del artículo 18 de la presente Convención.Artículo 55Los trabajadores migratorios que hayan obtenido permiso para ejercer una actividad remunerada,con sujeción a las condiciones adscritas a dicho permiso, tendrán derecho a igualdad de tratorespecto de los nacionales del Estado de empleo en el ejercicio de esa actividad remunerada.Artículo 561. Los trabajadores migratorios y sus familiares a los que se refiere la presente parte de laConvención no podrán ser expulsados de un Estado de empleo salvo por razones definidas en lalegislación nacional de ese Estado y con sujeción a las salvaguardias establecidas en la parte III.2. No se podrá recurrir a la expulsión como medio de privar a un trabajador migratorio o a unfamiliar suyo de los derechos emanados de la autorización de residencia y el permiso de trabajo.3. Al considerar si se va a expulsar a un trabajador migratorio o a un familiar suyo, deben tenerseen cuenta consideraciones de carácter humanitario y también el tiempo que la persona de que setrate lleve residiendo en el Estado de empleo.PARTE VDisposiciones aplicables a categorías particulares de trabajadores migratorios ysus familiaresArtículo 57Los trabajadores migratorios y sus familiares incluidos en las categorías particulares enumeradasen la presente Parte de la Convención que estén documentados o en situación regular gozarán delos derechos establecidos en la parte III, y, con sujeción a las modificaciones que se especifican acontinuación, de los derechos establecidos en la parte IV.Artículo 581. Los trabajadores fronterizos, definidos en el inciso a) del párrafo 2 del artículo 2 de la presenteConvención, gozarán de los derechos reconocidos en la parte IV que puedan corresponderles envirtud de su presencia y su trabajo en el territorio del Estado de empleo, teniendo en cuenta que nohan establecido su residencia habitual en dicho Estado.21

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsThe shift of the mass term by 1 is a result of the influence of theCasimir energy on the entropy.Our result gives a generalization of the Cardy formula for atime-dependent background, with Casimir energy ∼ ȧ 2 /a 2 .N. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsAdS 3 with a Milne boundary: zero entropyAdS 3 with a steady-state boundary: zero entropyN. Tetradis University of Athens

BTZ black hole AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsCommentsThe de Sitter entropy was calculated through holographic meansin:Hawking, Maldacena, Strominger 2001Iwashita, Kobayashi, Shiromizu, Yoshiho 2006The Randall-Sundrum construction was used.A general framework for the calculation of entanglement entropyfor a flat boundary through holography was given in:Ryu,Takayanagi 2006 Hubeny,Rangamani,Takayanagi 2007Connection with minimal surfaces ?Higher dimensions ?N. Tetradis University of Athens