Greybody Factors for d -Dimensional Black Holes Jos e Nat ario - - PowerPoint PPT Presentation

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

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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

• n a d-dimensional spherically symmetric black hole background:

ds2 = −f (r)dt2 + f (r)−1dr2 + r2dΩd−22.

Here f (r) = 1 − 2µ rd−3 + θ2 r2d−6 − λr2 and M = (d − 2) Ωd−2 8πGd µ, Ωn = 2π

n+1 2

Γ “

n+1 2

” , Q2 = (d − 2) (d − 3) 8πGd θ2, Λ = 1 2 (d − 1) (d − 2) Λ. 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

• n a d-dimensional spherically symmetric black hole background:

ds2 = −f (r)dt2 + f (r)−1dr2 + r2dΩd−22.

Here f (r) = 1 − 2µ rd−3 + θ2 r2d−6 − λr2 and M = (d − 2) Ωd−2 8πGd µ, Ωn = 2π

n+1 2

Γ “

n+1 2

” , Q2 = (d − 2) (d − 3) 8πGd θ2, Λ = 1 2 (d − 1) (d − 2) Λ. 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) = eiωtΦω,ℓ(r)Yℓm(Ω) Define the tortoise coordinate x through dx = dr f (r) Then wave equation is written as d2 dx2 + ω2 − V (r) r

d−2 2 Φω,ℓ

• = 0

Here V (r) = f (r) ℓ (ℓ + d − 3) r2 + (d − 2) (d − 4) f (r) 4r2 + (d − 2) f ′(r) 2r ! 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) = eiωtΦω,ℓ(r)Yℓm(Ω) Define the tortoise coordinate x through dx = dr f (r) Then wave equation is written as d2 dx2 + ω2 − V (r) r

d−2 2 Φω,ℓ

• = 0

Here V (r) = f (r) ℓ (ℓ + d − 3) r2 + (d − 2) (d − 4) f (r) 4r2 + (d − 2) f ′(r) 2r ! 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) = eiωtΦω,ℓ(r)Yℓm(Ω) Define the tortoise coordinate x through dx = dr f (r) Then wave equation is written as d2 dx2 + ω2 − V (r) r

d−2 2 Φω,ℓ

• = 0

Here V (r) = f (r) ℓ (ℓ + d − 3) r2 + (d − 2) (d − 4) f (r) 4r2 + (d − 2) f ′(r) 2r ! 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) = eiωtΦω,ℓ(r)Yℓm(Ω) Define the tortoise coordinate x through dx = dr f (r) Then wave equation is written as d2 dx2 + ω2 − V (r) r

d−2 2 Φω,ℓ

• = 0

Here V (r) = f (r) ℓ (ℓ + d − 3) r2 + (d − 2) (d − 4) f (r) 4r2 + (d − 2) f ′(r) 2r ! 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

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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 d2 dx2 + ω2 − V (r(x))

• Ψω,ℓ = 0

subject to

• Ψω,ℓ ∼ eiωx + Re−iωx,

x → +∞ Ψω,ℓ ∼ Teiωx, 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 d2 dx2 + ω2 − V (r(x))

• Ψω,ℓ = 0

subject to

• Ψω,ℓ ∼ eiωx + Re−iωx,

x → +∞ Ψω,ℓ ∼ Teiωx, x → −∞ Then γ = |T|2.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 12 / 34

Introduction Why do we care?

Outline

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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

Introduction Why do we care?

Why do we care?

Hawking radiation: expectation value n(ω) for the number of particles of a given species, emitted in a mode with frequency ω, is given by n(ω) = γ(ω) e

ω TH ± 1

where TH is the Hawking temperature and the plus (minus) sign describes fermions (bosons). Carry information about quantum gravity?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 14 / 34

Introduction Why do we care?

Why do we care?

Hawking radiation: expectation value n(ω) for the number of particles of a given species, emitted in a mode with frequency ω, is given by n(ω) = γ(ω) e

ω TH ± 1

where TH is the Hawking temperature and the plus (minus) sign describes fermions (bosons). Carry information about quantum gravity?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 14 / 34

Low Frequency Method

Outline

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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 15 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Method

Method

Split up spacetime into 3 regions:

Region I: The region near the (outer) event horizon, defined by r ≃ RH and V (r) ≪ ω2. Region II: The intermediate region, defined by V (r) ≫ ω2. Region III: The asymptotic region, defined by r ≫ RH.

Solve wave equation for each region separately and match. Assumed ℓ = 0.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 16 / 34

Low Frequency Results

Outline

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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 17 / 34

Low Frequency Results

Results

Asymptotically Flat Solutions

γ(ω) =

4πωd−2Rd−2

H

2d−2[Γ( d−1

2 )]2 (RH = radius of the (outer) horizon).

Results best described in terms of the absorption cross section: σ(ω) = γ(ω)|α|2 where α is the coefficient of the ℓ = 0 term in the decomposition of a plane wave into ingoing spherical harmonic waves. We have the universal result σ(ω) = AH where AH is the area of the (outer) horizon. First done in [Das, Gibbons and Mathur].

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 18 / 34

Low Frequency Results

Results

Asymptotically Flat Solutions

γ(ω) =

4πωd−2Rd−2

H

2d−2[Γ( d−1

2 )]2 (RH = radius of the (outer) horizon).

Results best described in terms of the absorption cross section: σ(ω) = γ(ω)|α|2 where α is the coefficient of the ℓ = 0 term in the decomposition of a plane wave into ingoing spherical harmonic waves. We have the universal result σ(ω) = AH where AH is the area of the (outer) horizon. First done in [Das, Gibbons and Mathur].

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 18 / 34

Low Frequency Results

Results

Asymptotically Flat Solutions

γ(ω) =

4πωd−2Rd−2

H

2d−2[Γ( d−1

2 )]2 (RH = radius of the (outer) horizon).

Results best described in terms of the absorption cross section: σ(ω) = γ(ω)|α|2 where α is the coefficient of the ℓ = 0 term in the decomposition of a plane wave into ingoing spherical harmonic waves. We have the universal result σ(ω) = AH where AH is the area of the (outer) horizon. First done in [Das, Gibbons and Mathur].

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 18 / 34

Low Frequency Results

Results

Asymptotically Flat Solutions

γ(ω) =

4πωd−2Rd−2

H

2d−2[Γ( d−1

2 )]2 (RH = radius of the (outer) horizon).

Results best described in terms of the absorption cross section: σ(ω) = γ(ω)|α|2 where α is the coefficient of the ℓ = 0 term in the decomposition of a plane wave into ingoing spherical harmonic waves. We have the universal result σ(ω) = AH where AH is the area of the (outer) horizon. First done in [Das, Gibbons and Mathur].

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 18 / 34

Low Frequency Results

Results

Asymptotically Flat Solutions

For comparison, in the high frequency limit we have the geometric

• ptics result

σ(ω) = ν(d)AH where

ν(d) = 1 d − 2 „ d − 1 2 « d−2

d−3 „ d − 1

d − 3 « d−2

2

Ωd−3 Ωd−2 „ ν(4) = 27 8 , ν(5) = 16 3π , . . . « Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 19 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

γ(ω) = 4h(ˆ ω)AH

AC .

AH = area of the black hole horizon AC = area of the cosmological horizon ˆ ω = frequency normalized by the cosmological radius h(ˆ ω) given by

h(ˆ ω) =

d−2 2

Y

n=1

1 + ˆ ω2 (2n − 1)2 ! for even d ≥ 4 and h(ˆ ω) = π ˆ ω 2 coth ` π ˆ ω 2 ´

d−3 2

Y

n=1

1 + ˆ ω2 (2n)2 ! for odd d ≥ 5. Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 20 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

γ(ω) = 4h(ˆ ω)AH

AC .

AH = area of the black hole horizon AC = area of the cosmological horizon ˆ ω = frequency normalized by the cosmological radius h(ˆ ω) given by

h(ˆ ω) =

d−2 2

Y

n=1

1 + ˆ ω2 (2n − 1)2 ! for even d ≥ 4 and h(ˆ ω) = π ˆ ω 2 coth ` π ˆ ω 2 ´

d−3 2

Y

n=1

1 + ˆ ω2 (2n)2 ! for odd d ≥ 5. Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 20 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Again universal result. h(ˆ ω) = 1 as ˆ ω → 0, implying a nonzero greybody factor in this limit (cosmological horizon at a finite distance).

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 21 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Again universal result. h(ˆ ω) = 1 as ˆ ω → 0, implying a nonzero greybody factor in this limit (cosmological horizon at a finite distance).

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 21 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Small black holes (RH much smaller than the cosmological scale). γ(ˆ ω) = 1 −

• 1−z(ˆ

ω) 1+z(ˆ ω)

• 2

where z(ˆ ω) = π 2d−2 [Γ(d−1

2 )]2

[Γ(d−1+ˆ

ω 2

)Γ(d−1−ˆ

ω 2

)]2 ˆ ωd−2 (κRH)d−2 Again universal result. γ(ˆ ω) = 0 for ˆ ω = d − 1 + 2n with n ∈ {0, 1, 2, ...}. These are exactly the normal frequencies for pure AdS. γ(ˆ ω) = 1 for ˆ ω ≃ d − 1 + 2n with n ∈ {0, 1, 2, ...}.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 22 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Small black holes (RH much smaller than the cosmological scale). γ(ˆ ω) = 1 −

• 1−z(ˆ

ω) 1+z(ˆ ω)

• 2

where z(ˆ ω) = π 2d−2 [Γ(d−1

2 )]2

[Γ(d−1+ˆ

ω 2

)Γ(d−1−ˆ

ω 2

)]2 ˆ ωd−2 (κRH)d−2 Again universal result. γ(ˆ ω) = 0 for ˆ ω = d − 1 + 2n with n ∈ {0, 1, 2, ...}. These are exactly the normal frequencies for pure AdS. γ(ˆ ω) = 1 for ˆ ω ≃ d − 1 + 2n with n ∈ {0, 1, 2, ...}.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 22 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Small black holes (RH much smaller than the cosmological scale). γ(ˆ ω) = 1 −

• 1−z(ˆ

ω) 1+z(ˆ ω)

• 2

where z(ˆ ω) = π 2d−2 [Γ(d−1

2 )]2

[Γ(d−1+ˆ

ω 2

)Γ(d−1−ˆ

ω 2

)]2 ˆ ωd−2 (κRH)d−2 Again universal result. γ(ˆ ω) = 0 for ˆ ω = d − 1 + 2n with n ∈ {0, 1, 2, ...}. These are exactly the normal frequencies for pure AdS. γ(ˆ ω) = 1 for ˆ ω ≃ d − 1 + 2n with n ∈ {0, 1, 2, ...}.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 22 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Small black holes (RH much smaller than the cosmological scale). γ(ˆ ω) = 1 −

• 1−z(ˆ

ω) 1+z(ˆ ω)

• 2

where z(ˆ ω) = π 2d−2 [Γ(d−1

2 )]2

[Γ(d−1+ˆ

ω 2

)Γ(d−1−ˆ

ω 2

)]2 ˆ ωd−2 (κRH)d−2 Again universal result. γ(ˆ ω) = 0 for ˆ ω = d − 1 + 2n with n ∈ {0, 1, 2, ...}. These are exactly the normal frequencies for pure AdS. γ(ˆ ω) = 1 for ˆ ω ≃ d − 1 + 2n with n ∈ {0, 1, 2, ...}.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 22 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

Small black holes (RH much smaller than the cosmological scale). γ(ˆ ω) = 1 −

• 1−z(ˆ

ω) 1+z(ˆ ω)

• 2

where z(ˆ ω) = π 2d−2 [Γ(d−1

2 )]2

[Γ(d−1+ˆ

ω 2

)Γ(d−1−ˆ

ω 2

)]2 ˆ ωd−2 (κRH)d−2 Again universal result. γ(ˆ ω) = 0 for ˆ ω = d − 1 + 2n with n ∈ {0, 1, 2, ...}. These are exactly the normal frequencies for pure AdS. γ(ˆ ω) = 1 for ˆ ω ≃ d − 1 + 2n with n ∈ {0, 1, 2, ...}.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 22 / 34

Low Frequency Results

Results

Asymptotically de Sitter Solutions

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 23 / 34

High Imaginary Frequency Method

Outline

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency 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 24 / 34

High Imaginary Frequency Method

Method

Complex ω:

Φ∗

ω is replaced by Φ−ω.

R∗, T ∗ are replaced by R, T. R R + T T = 1.

ω → i∞: solve equation in the complex plane at origin, horizons, infinity; match solutions along (anti-)Stokes lines Re x = 0 and use monodromies of Φω, Φ−ω at horizons to implement boundary conditions.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 25 / 34

High Imaginary Frequency Method

Method

Complex ω:

Φ∗

ω is replaced by Φ−ω.

R∗, T ∗ are replaced by R, T. R R + T T = 1.

ω → i∞: solve equation in the complex plane at origin, horizons, infinity; match solutions along (anti-)Stokes lines Re x = 0 and use monodromies of Φω, Φ−ω at horizons to implement boundary conditions.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 25 / 34

High Imaginary Frequency Method

Method

Complex ω:

Φ∗

ω is replaced by Φ−ω.

R∗, T ∗ are replaced by R, T. R R + T T = 1.

ω → i∞: solve equation in the complex plane at origin, horizons, infinity; match solutions along (anti-)Stokes lines Re x = 0 and use monodromies of Φω, Φ−ω at horizons to implement boundary conditions.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 25 / 34

High Imaginary Frequency Method

Method

Complex ω:

Φ∗

ω is replaced by Φ−ω.

R∗, T ∗ are replaced by R, T. R R + T T = 1.

ω → i∞: solve equation in the complex plane at origin, horizons, infinity; match solutions along (anti-)Stokes lines Re x = 0 and use monodromies of Φω, Φ−ω at horizons to implement boundary conditions.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 25 / 34

High Imaginary Frequency Method

Method

Complex ω:

Φ∗

ω is replaced by Φ−ω.

R∗, T ∗ are replaced by R, T. R R + T T = 1.

ω → i∞: solve equation in the complex plane at origin, horizons, infinity; match solutions along (anti-)Stokes lines Re x = 0 and use monodromies of Φω, Φ−ω at horizons to implement boundary conditions.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 25 / 34

High Imaginary Frequency Method

Method

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 26 / 34

High Imaginary Frequency Method

Method

Considered gravitational perturbations (tensor, vector, scalar) and any ℓ (includes massless scalar field).

Potentials are slightly more complicated: for example VS(r) = f (r)U(r) 16r2H2(r) , where H(r) = ℓ (ℓ + d − 3) − (d − 2) + (d − 1) (d − 2) µ rd−3 , and U(r) = − " 4d (d − 1)2 (d − 2)3 µ2 r2d−6 − 24 (d − 1) (d − 2)2 (d − 4) n ℓ (ℓ + d − 3) − (d − 2)

• µ

rd−3 + +4 (d − 4) (d − 6) n ℓ (ℓ + d − 3) − (d − 2)

• 2

# λr2 + 8 (d − 1)2 (d − 2)4 µ3 r3d−9 + 4 (d − 1) (d − 2) · · " 4 “ 2d2 − 11d + 18 ” n ℓ (ℓ + d − 3) − (d − 2)

• + (d − 1) (d − 2) (d − 4) (d − 6)

# µ2 r2d−6 − 24 (d − · " (d − 6) n ℓ (ℓ + d − 3) − (d − 2)

• + (d − 1) (d − 2) (d − 4)

#n ℓ (ℓ + d − 3) − (d − 2)

• µ

rd−3 + +16 n ℓ (ℓ + d − 3) − (d − 2)

• 3 + 4d (d − 2)

n ℓ (ℓ + d − 3) − (d − 2)

• 2.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 27 / 34

High Imaginary Frequency Method

Method

Considered gravitational perturbations (tensor, vector, scalar) and any ℓ (includes massless scalar field).

Potentials are slightly more complicated: for example VS(r) = f (r)U(r) 16r2H2(r) , where H(r) = ℓ (ℓ + d − 3) − (d − 2) + (d − 1) (d − 2) µ rd−3 , and U(r) = − " 4d (d − 1)2 (d − 2)3 µ2 r2d−6 − 24 (d − 1) (d − 2)2 (d − 4) n ℓ (ℓ + d − 3) − (d − 2)

• µ

rd−3 + +4 (d − 4) (d − 6) n ℓ (ℓ + d − 3) − (d − 2)

• 2

# λr2 + 8 (d − 1)2 (d − 2)4 µ3 r3d−9 + 4 (d − 1) (d − 2) · · " 4 “ 2d2 − 11d + 18 ” n ℓ (ℓ + d − 3) − (d − 2)

• + (d − 1) (d − 2) (d − 4) (d − 6)

# µ2 r2d−6 − 24 (d − · " (d − 6) n ℓ (ℓ + d − 3) − (d − 2)

• + (d − 1) (d − 2) (d − 4)

#n ℓ (ℓ + d − 3) − (d − 2)

• µ

rd−3 + +16 n ℓ (ℓ + d − 3) − (d − 2)

• 3 + 4d (d − 2)

n ℓ (ℓ + d − 3) − (d − 2)

• 2.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 27 / 34

High Imaginary Frequency Results

Outline

1

Introduction What are Greybody Factors? How to compute them? Why do we care?

2

Low Frequency Method Results

Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

3

High Imaginary Frequency Method Results

Asymptotically Flat Solutions Asymptotically Flat Solutions Asymptotically de Sitter Solutions Asymptotically Anti de Sitter Solutions

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High Imaginary Frequency Results

Results

Asymptotically Flat Solutions

Schwarzschild: γ(ω) = T(ω) T(ω) = e

ω TH − 1

e

ω TH + 3

where TH is the Hawking temperature of the event horizon. Poles are the quasinormal frequencies. First done in [Neitzke]. Exotic statistics?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 29 / 34

High Imaginary Frequency Results

Results

Asymptotically Flat Solutions

Schwarzschild: γ(ω) = T(ω) T(ω) = e

ω TH − 1

e

ω TH + 3

where TH is the Hawking temperature of the event horizon. Poles are the quasinormal frequencies. First done in [Neitzke]. Exotic statistics?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 29 / 34

High Imaginary Frequency Results

Results

Asymptotically Flat Solutions

Schwarzschild: γ(ω) = T(ω) T(ω) = e

ω TH − 1

e

ω TH + 3

where TH is the Hawking temperature of the event horizon. Poles are the quasinormal frequencies. First done in [Neitzke]. Exotic statistics?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 29 / 34

High Imaginary Frequency Results

Results

Asymptotically Flat Solutions

Schwarzschild: γ(ω) = T(ω) T(ω) = e

ω TH − 1

e

ω TH + 3

where TH is the Hawking temperature of the event horizon. Poles are the quasinormal frequencies. First done in [Neitzke]. Exotic statistics?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 29 / 34

High Imaginary Frequency Results

Results

Asymptotically Flat Solutions

Reissner-Nordstrom: γ(ω) = T(ω) T(ω) = e

ω T+ H − 1

e

ω T+ H + (1 + 2 cos(πj)) + (2 + 2 cos(πj))e

− ω

T− H

where j = d−3

2d−5 for tensor and scalar type perturbations, and

j = 3d−7

2d−5 for vector type perturbations.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 30 / 34

High Imaginary Frequency Results

Results

Asymptotically de Sitter Solutions

Schwarzschild-de Sitter: γ(ω) = −2 sinh

• πω

kH

• sinh
• πω

kC

• 3 cosh
• πω

kH + πω kC

• + cosh
• πω

kH − πω kC

• where kH, kC are the surface gravities at the horizons (kC < 0).

Reissner-Nordstrom-de Sitter:

γ(ω) = −2 sinh “

πω k+

” sinh “

πω kC

” cosh “

πω k+ − πω kC

” + (1 + 2 cos(πj)) cosh “

πω k+ + πω kC

” + (2 + 2 cos(πj)) cosh “

2πω k− + πω k+ + πω kC

” Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 31 / 34

High Imaginary Frequency Results

Results

Asymptotically de Sitter Solutions

Schwarzschild-de Sitter: γ(ω) = −2 sinh

• πω

kH

• sinh
• πω

kC

• 3 cosh
• πω

kH + πω kC

• + cosh
• πω

kH − πω kC

• where kH, kC are the surface gravities at the horizons (kC < 0).

Reissner-Nordstrom-de Sitter:

γ(ω) = −2 sinh “

πω k+

” sinh “

πω kC

” cosh “

πω k+ − πω kC

” + (1 + 2 cos(πj)) cosh “

πω k+ + πω kC

” + (2 + 2 cos(πj)) cosh “

2πω k− + πω k+ + πω kC

” Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 31 / 34

High Imaginary Frequency Results

Results

Asymptotically Anti de Sitter Solutions

Schwarzschild-Anti de Sitter: γ(ω) = 1. Reissner-Nordstrom-Anti de Sitter: γ(ω) = 1. In this case there are no poles because of reflecting boundary conditions are imposed when computing quasinormal modes.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 32 / 34

High Imaginary Frequency Results

Results

Asymptotically Anti de Sitter Solutions

Schwarzschild-Anti de Sitter: γ(ω) = 1. Reissner-Nordstrom-Anti de Sitter: γ(ω) = 1. In this case there are no poles because of reflecting boundary conditions are imposed when computing quasinormal modes.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 32 / 34

High Imaginary Frequency Results

Results

Asymptotically Anti de Sitter Solutions

Schwarzschild-Anti de Sitter: γ(ω) = 1. Reissner-Nordstrom-Anti de Sitter: γ(ω) = 1. In this case there are no poles because of reflecting boundary conditions are imposed when computing quasinormal modes.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 32 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Summary and Outlook

Summary and Outlook

Computed greybody factors for spherically symmetric bacgrounds in the two regimes:

Massless scalar waves, ℓ = 0, at low (real) frequency; Gravitational perturbations (includes massless scalar waves), any ℓ, at large imaginary frequencies.

Found universal behaviour at low frequencies. Hints of exotic statistics at high imaginary frequencies do not seem to extend beyond Schwarzschild. Can one compute the greybody factors in the full complex plane?

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 33 / 34

Appendix

Bibliography

Troels Harmark, JN and Ricardo Schiappa, Greybody Factors for d-Dimensional Black Holes, Advances in Theoretical and Mathematical Physics (in press), arXiv:0708.0017 [hep-th].

• S. Das, G. Gibbons and S. Mathur,

Universality of the Low Energy Absorption Cross-Sections for Black Holes, Physical Review Letters 78 (1997) 417, arXiv:hep-th/9609052.

• A. Neitzke,

Greybody Factors at Large Imaginary Frequencies, arXiv:hep-th/0304080.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 34 / 34

Appendix

Bibliography

Troels Harmark, JN and Ricardo Schiappa, Greybody Factors for d-Dimensional Black Holes, Advances in Theoretical and Mathematical Physics (in press), arXiv:0708.0017 [hep-th].

• S. Das, G. Gibbons and S. Mathur,

Universality of the Low Energy Absorption Cross-Sections for Black Holes, Physical Review Letters 78 (1997) 417, arXiv:hep-th/9609052.

• A. Neitzke,

Greybody Factors at Large Imaginary Frequencies, arXiv:hep-th/0304080.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 34 / 34

Appendix

Bibliography

Troels Harmark, JN and Ricardo Schiappa, Greybody Factors for d-Dimensional Black Holes, Advances in Theoretical and Mathematical Physics (in press), arXiv:0708.0017 [hep-th].

• S. Das, G. Gibbons and S. Mathur,

Universality of the Low Energy Absorption Cross-Sections for Black Holes, Physical Review Letters 78 (1997) 417, arXiv:hep-th/9609052.

• A. Neitzke,

Greybody Factors at Large Imaginary Frequencies, arXiv:hep-th/0304080.

Jos´ e Nat´ ario (IST, Lisbon) Greybody Factors UP 2008 34 / 34