New guises of the BTZ black hole and the entropy of 2d CFT
New guises of the BTZ black hole and the entropy of 2d CFT 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 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
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<strong>BTZ</strong> <strong>black</strong> <strong>hole</strong> AdS 3 with Rindler boundary AdS 3 with de Sitter boundary Time-dependent boundary Entropy CommentsFor fixed x, <strong>the</strong>re is a minimal value for r(z, x) as a function <strong>of</strong> z.It is obtained forz m (x) = 2x (15)<strong>and</strong> is equal toThe corresponding value <strong>of</strong> φ isr m (x) = a. (16)φ m (x) ≡ φ m (z m (x), x) = (log[ax] − 1)/a. (17)Bridge connecting <strong>the</strong> two asymptotic regions (that are copies <strong>of</strong>each o<strong>the</strong>r) at z → 0 <strong>and</strong> 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 <strong>the</strong> expected singularity at x = 0. The conformalanomaly vanishes.N. Tetradis University <strong>of</strong> A<strong>the</strong>ns
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