Greybody Factors for d -Dimensional Black Holes Jos´ e Nat´ ario (based on work with Troels Harmark and Ricardo Schiappa) CAMGSD, Department of Mathematics Instituto Superior T´ ecnico Talk at Universidade do Porto, 2008 Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 1 / 34
Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 2 / 34
Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 2 / 34
Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 2 / 34
Introduction What are Greybody Factors? Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 3 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Consider the massless wave equation � Φ = 0 on a d -dimensional spherically symmetric black hole background: ds 2 = − f ( r ) dt 2 + f ( r ) − 1 dr 2 + r 2 d Ω d − 22 . Here θ 2 2 µ r 2 d − 6 − λ r 2 f ( r ) = 1 − r d − 3 + and n +1 ( d − 2) Ω d − 2 2 π 2 = Ω n = M µ, ” , “ n +1 8 π G d Γ 2 ( d − 2) ( d − 3) Q 2 θ 2 , = 8 π G d 1 Λ = ( d − 1) ( d − 2) Λ . 2 Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 4 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Consider the massless wave equation � Φ = 0 on a d -dimensional spherically symmetric black hole background: ds 2 = − f ( r ) dt 2 + f ( r ) − 1 dr 2 + r 2 d Ω d − 22 . Here θ 2 2 µ r 2 d − 6 − λ r 2 f ( r ) = 1 − r d − 3 + and n +1 ( d − 2) Ω d − 2 2 π 2 = Ω n = M µ, ” , “ n +1 8 π G d Γ 2 ( d − 2) ( d − 3) Q 2 θ 2 , = 8 π G d 1 Λ = ( d − 1) ( d − 2) Λ . 2 Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 4 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Decompose Φ( t , r , Ω) = e i ω t Φ ω,ℓ ( r ) Y ℓ m (Ω) Define the tortoise coordinate x through dx = dr f ( r ) Then wave equation is written as � d 2 � � � dx 2 + ω 2 − V ( r ) d − 2 2 Φ ω,ℓ r = 0 Here ( d − 2) f ′ ( r ) ! ℓ ( ℓ + d − 3) ( d − 2) ( d − 4) f ( r ) V ( r ) = f ( r ) + + r 2 4 r 2 2 r Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 5 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Decompose Φ( t , r , Ω) = e i ω t Φ ω,ℓ ( r ) Y ℓ m (Ω) Define the tortoise coordinate x through dx = dr f ( r ) Then wave equation is written as � d 2 � � � dx 2 + ω 2 − V ( r ) d − 2 2 Φ ω,ℓ r = 0 Here ( d − 2) f ′ ( r ) ! ℓ ( ℓ + d − 3) ( d − 2) ( d − 4) f ( r ) V ( r ) = f ( r ) + + r 2 4 r 2 2 r Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 5 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Decompose Φ( t , r , Ω) = e i ω t Φ ω,ℓ ( r ) Y ℓ m (Ω) Define the tortoise coordinate x through dx = dr f ( r ) Then wave equation is written as � d 2 � � � dx 2 + ω 2 − V ( r ) d − 2 2 Φ ω,ℓ r = 0 Here ( d − 2) f ′ ( r ) ! ℓ ( ℓ + d − 3) ( d − 2) ( d − 4) f ( r ) V ( r ) = f ( r ) + + r 2 4 r 2 2 r Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 5 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Decompose Φ( t , r , Ω) = e i ω t Φ ω,ℓ ( r ) Y ℓ m (Ω) Define the tortoise coordinate x through dx = dr f ( r ) Then wave equation is written as � d 2 � � � dx 2 + ω 2 − V ( r ) d − 2 2 Φ ω,ℓ r = 0 Here ( d − 2) f ′ ( r ) ! ℓ ( ℓ + d − 3) ( d − 2) ( d − 4) f ( r ) V ( r ) = f ( r ) + + r 2 4 r 2 2 r Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 5 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Potentials for d = 6 and ℓ = 0 in Schwarzschild, Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter: Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 6 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Choose a solution Φ representing an incoming wave at infinity. Then the greybody factor is γ ( ω, ℓ ) = Total flux of Φ at the horizon Total flux of Φ at infinity Can interchange “ingoing” ↔ “outgoing” and “horizon” ↔ “infinity”. Interpretation: γ ( ω, ℓ ) represents the probability for an outgoing wave, in the ( ω, ℓ )–mode, to reach infinity. Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 7 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Choose a solution Φ representing an incoming wave at infinity. Then the greybody factor is γ ( ω, ℓ ) = Total flux of Φ at the horizon Total flux of Φ at infinity Can interchange “ingoing” ↔ “outgoing” and “horizon” ↔ “infinity”. Interpretation: γ ( ω, ℓ ) represents the probability for an outgoing wave, in the ( ω, ℓ )–mode, to reach infinity. Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 7 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Choose a solution Φ representing an incoming wave at infinity. Then the greybody factor is γ ( ω, ℓ ) = Total flux of Φ at the horizon Total flux of Φ at infinity Can interchange “ingoing” ↔ “outgoing” and “horizon” ↔ “infinity”. Interpretation: γ ( ω, ℓ ) represents the probability for an outgoing wave, in the ( ω, ℓ )–mode, to reach infinity. Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 7 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 8 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 9 / 34
Introduction What are Greybody Factors? What are Greybody Factors? Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 10 / 34
Introduction How to compute them? Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 11 / 34
Introduction How to compute them? How to compute them? Solve � d 2 � dx 2 + ω 2 − V ( r ( x )) Ψ ω,ℓ = 0 subject to � Ψ ω,ℓ ∼ e i ω x + Re − i ω x , x → + ∞ Ψ ω,ℓ ∼ Te i ω x , x → −∞ Then γ = | T | 2 . Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 12 / 34
Introduction How to compute them? How to compute them? Solve � d 2 � dx 2 + ω 2 − V ( r ( x )) Ψ ω,ℓ = 0 subject to � Ψ ω,ℓ ∼ e i ω x + Re − i ω x , x → + ∞ Ψ ω,ℓ ∼ Te i ω x , x → −∞ Then γ = | T | 2 . Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 12 / 34
Introduction Why do we care? Outline Introduction 1 What are Greybody Factors? How to compute them? Why do we care? Low Frequency 2 Method Results Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions High Imaginary Frequency 3 Method Results Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 13 / 34
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