Charged black holes have no hair. PTC, P.Tod, NI preprint NI05067-GMR

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1 Charged black holes have no hair PTC, P.Tod, NI preprint NI05067-GMR

2 Theorem I [PTC, P.Tod, NI preprint NI05067-GMR] Globally hyperbolic, electro-vacuum, static, regular black hole = Reissner-Nordström or Majumdar-Papapetrou in the exterior region

3 Theorem I [PTC, P.Tod, NI preprint NI05067-GMR] Globally hyperbolic, electro-vacuum, static, regular black hole = Reissner-Nordström or Majumdar-Papapetrou in the exterior region regular: contains a spacelike surface which is the union of a compact set and a finite number of asymptotically flat ends, with boundary lying on the event horizons

4 The standard Majumdar Papapetrou metrics: ds 2 = φ 2 dt 2 + φ 2 (dx 2 + dy 2 + dz 2 ) N m i φ = 1 + x x i, m i > 0 i=1 (N=1 degenerate Reissner Nordström in unusual coordinates) For every finite N this is a space time with N black holes, each having a degenerate event horizon.

5 The standard Majumdar Papapetrou metrics: ds 2 = φ 2 dt 2 + φ 2 (dx 2 + dy 2 + dz 2 ) N m i φ = 1 + x x i, m i > 0 i=1 (N=1 degenerate Reissner Nordström in unusual coordinates) For every finite N this is a space time with N black holes, each having a degenerate event horizon. if X is a Killing vector, Killing horizon H : null hypersurface on which X is null

6 The standard Majumdar Papapetrou metrics: ds 2 = φ 2 dt 2 + φ 2 (dx 2 + dy 2 + dz 2 ) N m i φ = 1 + x x i, m i > 0 i=1 (N=1 degenerate Reissner Nordström in unusual coordinates) For every finite N this is a space time with N black holes, each having a degenerate event horizon. if X is a Killing vector, Killing horizon H : null hypersurface on which X is null surface gravity κ : µ (g(x, X)) = 2 κ X µ

7 The standard Majumdar Papapetrou metrics: ds 2 = φ 2 dt 2 + φ 2 (dx 2 + dy 2 + dz 2 ) N m i φ = 1 + x x i, m i > 0 i=1 (N=1 degenerate Reissner Nordström in unusual coordinates) For every finite N this is a space time with N black holes, each having a degenerate event horizon. if X is a Killing vector, Killing horizon H : null hypersurface on which X is null surface gravity κ : µ (g(x, X)) = 2 κ X µ H degenerate or extreme if κ = 0

8 Previous versions of Theorem I Israel, Robinson, Masood-ul-Alam, Ruback, Simon: Assume moreover that all horizons non-degenerate or

9 Previous versions of Theorem I Israel, Robinson, Masood-ul-Alam, Ruback, Simon: Assume moreover that all horizons non-degenerate or Heusler: Assume instead all horizons degenerate Then Theorem I holds.

10 Previous versions of Theorem I Israel, Robinson, Masood-ul-Alam, Ruback, Simon: Assume moreover that all horizons non-degenerate or Heusler: Assume instead all horizons degenerate Then Theorem I holds. In this work we remove the spurious red hypotheses

11 Near horizon geometry has no hair (compare Reall hep-th/ )

12 Near horizon geometry has no hair (compare Reall hep-th/ ) Gauss coordinates near a degenerate horizon H = {r = 0} (see Isenberg, Moncrief, CMP 1983) Killing vector X = u g = A }{{} r 2 du 2 2du dr 2rh a dx a du h ab dx a dx b degenerate

13 Near horizon geometry has no hair (compare Reall hep-th/ ) Gauss coordinates near a degenerate horizon H = {r = 0} (see Isenberg, Moncrief, CMP 1983) Killing vector X = u g = A }{{} r 2 du 2 2du dr 2rh a dx a du h ab dx a dx b degenerate Theorem II [PTC, P.Tod] degenerate, static, elvac, S 2 crosssection implies: A = Åh ab = r }{{} Å +O(r), h a = O(r), constant>0 hab }{{} +O(r), unit round metric on S 2 ϕ }{{} electric potential = ± Å + O(r). First Prev Next Last Go Back Full Screen Close Quit

14 g Idea of the proof: }{{} År 2 du 2 2du dr 2r h a dx a du h ab dx a dx b leading order staticity: X dx = 0 = h a = a λ

15 g Idea of the proof: }{{} År 2 du 2 2du dr 2r h a dx a du h ab dx a dx b leading order staticity: X dx = 0 = h a = a λ Einstein-Maxwell equations: ( ) R ab = 1 2 a λ b λ a b λ + }{{} C e 2λ hab const.

16 g Idea of the proof: }{{} År 2 du 2 2du dr 2r h a dx a du h ab dx a dx b leading order staticity: X dx = 0 = h a = a λ Einstein-Maxwell equations: ( ) R ab = 1 2 a λ b λ a b λ + S 2 : λ = const, h a = 0, Rab = C h ab, }{{} C e 2λ hab const. and the values of Å and r ϕ follow from the remaining Einstein- Maxwell equations

17 g Idea of the proof: }{{} År 2 du 2 2du dr 2r h a dx a du h ab dx a dx b leading order staticity: X dx = 0 = h a = a λ Einstein-Maxwell equations: ( ) R ab = 1 2 a λ b λ a b λ + S 2 : λ = const, h a = 0, Rab = C h ab, }{{} C e 2λ hab const. and the values of Å and r ϕ follow from the remaining Einstein- Maxwell equations Remark: As emphasised by Reall in his September lecture at the NI, it would be of interest to understand the non-static equivalent of ( ) compare Lewandowski Paw lowski gr-qc/