The Kay-Wald theorem and HHI-like states on black hole space-times - - PowerPoint PPT Presentation
The Kay-Wald theorem and HHI-like states on black hole space-times Elizabeth Winstanley Consortium for Fundamental Physics School of Mathematics and Statistics The University of Sheffield Elizabeth Winstanley (Sheffield) Kay-Wald theorem and
Consortium for Fundamental Physics School of Mathematics and Statistics The University of Sheffield
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 1 / 47
1
Introduction
2
HHI state on Schwarzschild space-time
3
HHI-like states on Kerr space-time Scalar field Fermion field
4
Conclusions
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 2 / 47
Introduction
Two fundamental results in QFT in curved space-time
Unruh effect
A uniformly accelerating observer in Minkowski space-time observes thermal radiation in the Minkowski vacuum
[ Fulling PRD 7 2850 (1973); Davies JPA 8 609 (1975); Unruh PRD 14 870 (1976) ]
Hawking effect
A black hole formed by gravitational collapse emits thermal radiation
[ Hawking CMP 43 199 (1975) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 3 / 47
Introduction
Minkowski space-time Kruskal space-time t = x t = −x
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 4 / 47
Introduction
Rindler wedge Exterior Schwarzschild t = x t = −x
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 4 / 47
Introduction
Rindler vacuum Boulware state t = x t = −x
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 4 / 47
Introduction
Minkowski vacuum Hartle-Hawking-Israel state t = x t = −x
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 4 / 47
HHI state on Schwarzschild space-time
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 5 / 47
HHI state on Schwarzschild space-time
ds2 = −
r
r −1 dr2 + r2 dθ2 + r2 sin2 θ dϕ2
I + I − i+ i− i0
r = 0 r = 0
H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 6 / 47
HHI state on Schwarzschild space-time
ds2 = −
r
r −1 dr2 + r2 dθ2 + r2 sin2 θ dϕ2
I + I − i+ i− i0
r = 0 r = 0
H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 6 / 47
HHI state on Schwarzschild space-time
Klein-Gordon equation
Φ = 0
Klein-Gordon inner product
(Φ1, Φ2)KG = i
2∇µΦ1 − Φ1∇µΦ∗ 2
Involves time derivative of Φ
I + I − i+ i− i0
r = 0 r = 0
H+ H− I II III IV
Σ Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 7 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes Φ = ∑
j
ajφ+
j + a† j φ− j
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes Φ = ∑
j
ajφ+
j + a† j φ− j
Positive frequency modes
φ+
j ∝ e−iωt
ω > 0 Positive KG “norm”
j , φ+ k
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes Φ = ∑
j
ajφ+
j + a† j φ− j
Negative frequency modes
φ−
j ∝ e−iωt
ω < 0 Negative KG “norm”
j , φ− k
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes ˆ Φ = ∑
j
ˆ ajφ+
j + ˆ
a†
j φ− j
Promote expansion coefficients to operators ˆ aj, ˆ a†
j with
aj, ˆ a†
k
ˆ aj, ˆ ak = 0
a†
j , ˆ
a†
k
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes ˆ Φ = ∑
j
ˆ ajφ+
j + ˆ
a†
j φ− j
Promote expansion coefficients to operators ˆ aj, ˆ a†
j with
aj, ˆ a†
k
ˆ aj, ˆ ak = 0
a†
j , ˆ
a†
k
ˆ aj - particle annihilation operators
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes ˆ Φ = ∑
j
ˆ ajφ+
j + ˆ
a†
j φ− j
Promote expansion coefficients to operators ˆ aj, ˆ a†
j with
aj, ˆ a†
k
ˆ aj, ˆ ak = 0
a†
j , ˆ
a†
k
ˆ aj - particle annihilation operators ˆ a†
j - particle creation operators
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
Expand classical field in terms of orthonormal basis of field modes ˆ Φ = ∑
j
ˆ ajφ+
j + ˆ
a†
j φ− j
Promote expansion coefficients to operators ˆ aj, ˆ a†
j with
aj, ˆ a†
k
ˆ aj, ˆ ak = 0
a†
j , ˆ
a†
k
ˆ aj - particle annihilation operators ˆ a†
j - particle creation operators
Define the vacuum state |0 ˆ aj |0 = 0
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 8 / 47
HHI state on Schwarzschild space-time
φωℓm(t, r, θ, ϕ) = 1 rN
e−iωteimϕYℓm(θ)Rωℓ(r) Yℓm(θ): spherical harmonics N : normalization constant independent of ω Positive frequency with respect to Schwarzschild time t: ω > 0
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 9 / 47
HHI state on Schwarzschild space-time
φωℓm(t, r, θ, ϕ) = 1 rN
e−iωteimϕYℓm(θ)Rωℓ(r) Yℓm(θ): spherical harmonics N : normalization constant independent of ω Positive frequency with respect to Schwarzschild time t: ω > 0
Radial mode equation
0 = d2 dr2
∗
+ Vωℓm(r)
dr∗ dr =
r −1 Vωℓm(r) = ω2 as r → 2M, r∗ → −∞ ω2 as r → ∞, r∗ → ∞
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 9 / 47
HHI state on Schwarzschild space-time
“In” modes Rin
ωℓ
Bin
ωℓe−iωr∗
r∗ → −∞ e−iωr∗ + Ain
ωℓeiωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
“Up” modes Rup
ωℓ
eiωr∗ + Aup
ωℓe−iωr∗
r∗ → −∞ Bup
ωℓeiωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 10 / 47
HHI state on Schwarzschild space-time
“Out” modes Rout
ωℓ
Bout
ωℓ eiωr∗
r∗ → −∞ eiωr∗ + Aout
ωℓ e−iωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
“Down” modes Rdown
ωℓ
e−iωr∗ + Adown
ωℓ
eiωr∗ r∗ → −∞ Bdown
ωℓ
e−iωr∗ r∗ → ∞
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 11 / 47
HHI state on Schwarzschild space-time
Define positive frequency with respect to Kruskal time T
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes?
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes? “Up” and “down” modes are not orthogonal
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes Resulting vacuum state is HHI state |H
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time
Stress-energy tensor operator ˆ Tµν = 2 3∇µ ˆ Φ∇ν ˆ Φ − 1 3 ˆ Φ∇µ∇ν ˆ Φ − 1 6gµν∇λ ˆ Φ∇λ ˆ Φ
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time
Stress-energy tensor operator ˆ Tµν = 2 3∇µ ˆ Φ∇ν ˆ Φ − 1 3 ˆ Φ∇µ∇ν ˆ Φ − 1 6gµν∇λ ˆ Φ∇λ ˆ Φ
Unrenormalized stress-energy tensor expectation value
H| ˆ Tµν|H =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH Tµν
ωℓm
+ Tµν
ωℓm
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time
Stress-energy tensor operator ˆ Tµν = 2 3∇µ ˆ Φ∇ν ˆ Φ − 1 3 ˆ Φ∇µ∇ν ˆ Φ − 1 6gµν∇λ ˆ Φ∇λ ˆ Φ
Unrenormalized stress-energy tensor expectation value
H| ˆ Tµν|H =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH Tµν
ωℓm
+ Tµν
ωℓm
Compute renormalized expectation values using point-splitting
[ Howard PRD 30 2532 (1984) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time
[ Howard PRD 30 2532 (1984) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 14 / 47
HHI state on Schwarzschild space-time
For a quantum scalar field on Schwarzschild space-time
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 15 / 47
HHI state on Schwarzschild space-time
For a quantum scalar field on Schwarzschild space-time
Properties
Thermal state in region I Regular on the horizons H| ˆ Tµν|H finite everywhere in region I Time-reversal symmetric
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 15 / 47
HHI state on Schwarzschild space-time
For a quantum scalar field on Schwarzschild space-time
Properties
Thermal state in region I Regular on the horizons H| ˆ Tµν|H finite everywhere in region I Time-reversal symmetric
Rigorous results on the existence of |H
Kay CMP 100 57 (1985) HHI state in regions I & IV Jacobson PRD 50 R6031 (1994) HHI state on Euclidean section Sanders IJMPA 28 1330010 (2013) HHI state on Kruskal space-time Sanders Lett. Math. Phys. 105 575 (2015) HHI state across horizons
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 15 / 47
HHI state on Schwarzschild space-time
Globally hyperbolic space-time with a bifurcate Killing horizon
B H+ H− I II III IV
Wedge isometry maps I ↔ IV
[ Kay JMP 34 4519 (1993), Kay & Lupo CQG 33 215001 (2016) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 16 / 47
HHI state on Schwarzschild space-time
Theorem
Globally hyperbolic space-time with a bifurcate Killing horizon
B H+ H− I II III IV
Wedge isometry maps I ↔ IV
[ Kay JMP 34 4519 (1993), Kay & Lupo CQG 33 215001 (2016) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 16 / 47
HHI state on Schwarzschild space-time
Theorem
On a large subalgebra of
most one quasifree, isometry invariant, Hadamard state Globally hyperbolic space-time with a bifurcate Killing horizon
B H+ H− I II III IV
Wedge isometry maps I ↔ IV
[ Kay JMP 34 4519 (1993), Kay & Lupo CQG 33 215001 (2016) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 16 / 47
HHI state on Schwarzschild space-time
Theorem
On a large subalgebra of
most one quasifree, isometry invariant, Hadamard state This state, if it exists, is a KMS state at the Hawking temperature TH on
subalgebra localized in region I Globally hyperbolic space-time with a bifurcate Killing horizon
B H+ H− I II III IV
Wedge isometry maps I ↔ IV
[ Kay JMP 34 4519 (1993), Kay & Lupo CQG 33 215001 (2016) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 16 / 47
HHI state on Schwarzschild space-time
The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ]
|H exists and is unique on Schwarzschild I + I − i+ i− i0
r = 0 r = 0
H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 17 / 47
HHI state on Schwarzschild space-time
Dirac equation
γµ∇µΨ = 0
Canonical quantization
Expansion of classical field in orthonormal basis of field modes “in” and “up” modes Positive frequency with respect to Kruskal time T
Stress-energy tensor
ˆ Tµν = i 8 ˆ Ψ, γµ∇ν ˆ Ψ
ˆ Ψ, γν∇µ ˆ Ψ
Ψ, γν ˆ Ψ
Ψ, γµ ˆ Ψ
Kay-Wald theorem and HHI-like states York, April 2017 18 / 47
HHI state on Schwarzschild space-time
[ Carlson et al PRL 91 051301 (2003) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 19 / 47
HHI-like states on Kerr space-time
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 20 / 47
HHI-like states on Kerr space-time
ds2 = −∆ Σ
2 + Σ ∆dr2 + Σdθ2 + sin2 θ Σ
dϕ − adt 2 ∆ = r2 − 2Mr + a2 Σ = r2 + a2 cos2 θ
I + I − i+ i− i0 H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 21 / 47
HHI-like states on Kerr space-time
ds2 = −∆ Σ
2 + Σ ∆dr2 + Σdθ2 + sin2 θ Σ
dϕ − adt 2 ∆ = r2 − 2Mr + a2 Σ = r2 + a2 cos2 θ
I + I − i+ i− i0 H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 21 / 47
HHI-like states on Kerr space-time
Event horizon
rH = M +
ΩH = a r2
H + a2
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 22 / 47
HHI-like states on Kerr space-time
Event horizon
rH = M +
ΩH = a r2
H + a2
Stationary limit surface
rS = M +
For rH < r < rS an observer cannot remain at rest relative to infinity and must have a non-zero angular velocity
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 22 / 47
HHI-like states on Kerr space-time
Event horizon
rH = M +
ΩH = a r2
H + a2
Stationary limit surface
rS = M +
For rH < r < rS an observer cannot remain at rest relative to infinity and must have a non-zero angular velocity
Speed-of-light surface
An observer can have the same angular velocity as the event horizon between r = rH and the speed-of-light surface SL
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 22 / 47
HHI-like states on Kerr space-time
[ Casals et al PRD 87 064027 (2013) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 23 / 47
HHI-like states on Kerr space-time Scalar field
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 24 / 47
HHI-like states on Kerr space-time Scalar field
Properties of |H on Schwarzschild
Regular on and outside horizon Time-reversal symmetric Thermal state in region I
I + I − i+ i− i0 H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field
Theorem
There does not exist any Hadamard state on Kerr which is invariant under the isometries generating the event horizon
I + I − i+ i− i0 H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field
Theorem
No HHI state exists for a quantum scalar field on Kerr
I + I − i+ i− i0 H+ H− I II III IV
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field
Scalar field modes
φωℓm(t, r, θ, ϕ) = 1 N 1 (r2 + a2)
1 2
e−iωteimϕSωℓm(cos θ)Rωℓm(r) Sωℓm(cos θ): spheroidal harmonics
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 26 / 47
HHI-like states on Kerr space-time Scalar field
Scalar field modes
φωℓm(t, r, θ, ϕ) = 1 N 1 (r2 + a2)
1 2
e−iωteimϕSωℓm(cos θ)Rωℓm(r) Sωℓm(cos θ): spheroidal harmonics
Radial mode equation
0 = d2 dr2
∗
+ Vωℓm(r)
dr∗ dr = r2 + a2 ∆ Vωℓm(r) =
as r → rH, r∗ → −∞ ω2 as r → ∞, r∗ → ∞
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 26 / 47
HHI-like states on Kerr space-time Scalar field
“In” modes Rin
ωℓm
Bin
ωℓme−i ωr∗
r∗ → −∞ e−iωr∗ + Ain
ωℓmeiωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
“Up” modes Rup
ωℓm
ei
ωr∗ + Aup ωℓme−i ωr∗
r∗ → −∞ Bup
ωℓmeiωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 27 / 47
HHI-like states on Kerr space-time Scalar field
“Out” modes Rout
ωℓ
Bout
ωℓ ei ωr∗
r∗ → −∞ eiωr∗ + Aout
ωℓ e−iωr∗
r∗ → ∞
I + I − i+ i− i0 H+ H−
“Down” modes Rdown
ωℓ
e−i
ωr∗ + Adown ωℓ
ei
ωr∗
r∗ → −∞ Bdown
ωℓ
e−iωr∗ r∗ → ∞
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 28 / 47
HHI-like states on Kerr space-time Scalar field
Positive frequency scalar modes must have positive KG “norm”
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 29 / 47
HHI-like states on Kerr space-time Scalar field
Positive frequency scalar modes must have positive KG “norm”
“In” and “out” modes
“In” and “out” modes have positive KG “norm” for ω > 0
I + I − i+ i− i0 H+ H− I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 29 / 47
HHI-like states on Kerr space-time Scalar field
Positive frequency scalar modes must have positive KG “norm”
“Up” and “down” modes
“Up” and “down” modes have positive KG “norm” for
I + I − i+ i− i0 H+ H− I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 29 / 47
HHI-like states on Kerr space-time Scalar field
Define positive frequency with respect to Kruskal time T
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0?
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes But “in” modes have positive “norm” for ω > 0
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field
CCH| ˆ Tµν|CCH =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Kay-Wald theorem and HHI-like states York, April 2017 31 / 47
HHI-like states on Kerr space-time Scalar field
CCH| ˆ Tµν|CCH =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
[ Ottewill & Winstanley PRD 62 084018 (2000) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 31 / 47
HHI-like states on Kerr space-time Scalar field
CCH| ˆ Tµν|CCH =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Regular outside the event horizon
[ Ottewill & Winstanley PRD 62 084018 (2000) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 31 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization Plots for Kerr are relative to a fixed reference state |B−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization Plots for Kerr are relative to a fixed reference state |B−
“Past” Boulware state |B− [ Unruh PRD 10 3194 (1974) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization Plots for Kerr are relative to a fixed reference state |B−
“Past” Boulware state |B− [ Unruh PRD 10 3194 (1974) ]
“In” modes with positive frequency with respect to t near I − “Up” modes with positive frequency with respect to t near H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization Plots for Kerr are relative to a fixed reference state |B−
“Past” Boulware state |B− [ Unruh PRD 10 3194 (1974) ]
“In” modes with positive frequency with respect to t near I − “Up” modes with positive frequency with respect to t near H− Diverges on the event horizon
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
Method for computing renormalized expectation values on Kerr has been elusive until recently
[ Levi et al arXiv:1610.04848 [gr-qc]]
Differences in expectation values between two quantum states do not require renormalization Plots for Kerr are relative to a fixed reference state |B−
“Past” Boulware state |B− [ Unruh PRD 10 3194 (1974) ]
“In” modes with positive frequency with respect to t near I − “Up” modes with positive frequency with respect to t near H− Diverges on the event horizon Regular everywhere outside the event horizon in region I
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 32 / 47
HHI-like states on Kerr space-time Scalar field
[ Casals & Ottewill PRD 71 124016 (2005) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 33 / 47
HHI-like states on Kerr space-time Scalar field
FT| ˆ Tµν|FT =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Kay-Wald theorem and HHI-like states York, April 2017 34 / 47
HHI-like states on Kerr space-time Scalar field
FT| ˆ Tµν|FT =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
[ Ottewill & Winstanley PRD 62 084018 (2000) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 34 / 47
HHI-like states on Kerr space-time Scalar field
FT| ˆ Tµν|FT =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω coth ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Divergent everywhere except on the axis of rotation
[ Ottewill & Winstanley PRD 62 084018 (2000) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 34 / 47
HHI-like states on Kerr space-time Scalar field
Mirror M at fixed r = r0 inside SL
[ Duffy & Ottewill PRD 77 024007 (2008) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field
Mirror M at fixed r = r0 inside SL
Modes
φM
ωℓm =
φup
ωℓm − Rup
ωℓm(r0)
Rin
ωℓm(r0)φin
ωℓm
ω > 0 φup
ωℓm − Rup
ωℓm(r0)
Rin∗
−ωℓ−m(r0)φin∗
−ωℓ−m
ω < 0
[ Duffy & Ottewill PRD 77 024007 (2008) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field
Mirror M at fixed r = r0 inside SL
Modes
φM
ωℓm =
φup
ωℓm − Rup
ωℓm(r0)
Rin
ωℓm(r0)φin
ωℓm
ω > 0 φup
ωℓm − Rup
ωℓm(r0)
Rin∗
−ωℓ−m(r0)φin∗
−ωℓ−m
ω < 0 Positive KG “norm” for ω > 0
[ Duffy & Ottewill PRD 77 024007 (2008) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field
Modes with positive frequency with respect to Kruskal time T
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field
Modes with positive frequency with respect to Kruskal time T
HM | ˆ Tµν|HM =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field
Modes with positive frequency with respect to Kruskal time T
HM | ˆ Tµν|HM =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field
Modes with positive frequency with respect to Kruskal time T
HM | ˆ Tµν|HM =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
|BM defined by taking modes to have positive frequency with respect to t
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field
Modes with positive frequency with respect to Kruskal time T
HM | ˆ Tµν|HM =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω coth ω 2TH
ωℓm
|BM defined by taking modes to have positive frequency with respect to t |BM diverges on H±
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field
[ Duffy & Ottewill PRD 77 024007 (2008) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 37 / 47
HHI-like states on Kerr space-time Scalar field
Non-existence of HHI-state on Kruskal space-time with a single mirror
I + I − i+ i− i0 H+ H− I II III IV M
[ Kay & Lupo CQG 33 215001 (2016) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 38 / 47
HHI-like states on Kerr space-time Scalar field
Existence of HHI-state on Kruskal space-time with two mirrors
I + I − i+ i− i0 H+ H− I II III IV M M
[ Kay GRG 47 31 (2015) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 39 / 47
HHI-like states on Kerr space-time Fermion field
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 40 / 47
HHI-like states on Kerr space-time Fermion field
Dirac equation
γµ∇µΨ = 0
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 41 / 47
HHI-like states on Kerr space-time Fermion field
Dirac equation
γµ∇µΨ = 0
Dirac inner product
(Ψ1, Ψ2)D =
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 41 / 47
HHI-like states on Kerr space-time Fermion field
Dirac equation
γµ∇µΨ = 0
Dirac inner product
(Ψ1, Ψ2)D =
Positivity of the Dirac norm
All modes have positive Dirac norm
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 41 / 47
HHI-like states on Kerr space-time Fermion field
Dirac equation
γµ∇µΨ = 0
Dirac inner product
(Ψ1, Ψ2)D =
Positivity of the Dirac norm
All modes have positive Dirac norm Both positive frequency and negative frequency modes have positive Dirac norm
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 41 / 47
HHI-like states on Kerr space-time Fermion field
Dirac equation
γµ∇µΨ = 0
Dirac inner product
(Ψ1, Ψ2)D =
Positivity of the Dirac norm
All modes have positive Dirac norm Both positive frequency and negative frequency modes have positive Dirac norm More freedom in the choice of positive frequency?
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 41 / 47
HHI-like states on Kerr space-time Fermion field
Expand classical field in terms of orthonormal basis of field modes Ψ = ∑
j
bjψ+
j + c† j ψ− j
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 42 / 47
HHI-like states on Kerr space-time Fermion field
Expand classical field in terms of orthonormal basis of field modes ˆ Ψ = ∑
j
ˆ bjψ+
j + ˆ
c†
j ψ− j
Promote expansion coefficients to operators ˆ bj, ˆ cj with
bj, ˆ b†
k
cj, ˆ c†
k
bj, ˆ bk
b†
j , ˆ
b†
k
ˆ cj, ˆ ck =
c†
j , ˆ
c†
k
Kay-Wald theorem and HHI-like states York, April 2017 42 / 47
HHI-like states on Kerr space-time Fermion field
Expand classical field in terms of orthonormal basis of field modes ˆ Ψ = ∑
j
ˆ bjψ+
j + ˆ
c†
j ψ− j
Promote expansion coefficients to operators ˆ bj, ˆ cj with
bj, ˆ b†
k
cj, ˆ c†
k
bj, ˆ bk
b†
j , ˆ
b†
k
ˆ cj, ˆ ck =
c†
j , ˆ
c†
k
ˆ bj |0 = 0 = ˆ cj |0
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 42 / 47
HHI-like states on Kerr space-time Fermion field
Define positive frequency with respect to Kruskal time T
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
HHI-like states on Kerr space-time Fermion field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0?
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
HHI-like states on Kerr space-time Fermion field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
HHI-like states on Kerr space-time Fermion field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes with ω > 0
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
HHI-like states on Kerr space-time Fermion field
Define positive frequency with respect to Kruskal time T Use “up” and “down” modes with ω > 0? “Up” and “down” modes are not orthogonal Instead use “in” and “up” modes with ω > 0 Call the resulting vacuum state |H
I + I − i+ i− i0 H+ H−
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
HHI-like states on Kerr space-time Fermion field
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 44 / 47
HHI-like states on Kerr space-time Fermion field
H| ˆ Tµν|H =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω tanh ω 2TH Tµν
ωℓm
+ Tµν
ωℓm
Kay-Wald theorem and HHI-like states York, April 2017 44 / 47
HHI-like states on Kerr space-time Fermion field
H| ˆ Tµν|H =
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω tanh ω 2TH Tµν
ωℓm
+ Tµν
ωℓm
CCH| ˆ Tµν|CCH =
∞
ℓ=0 ℓ
m=−ℓ
∞
dω tanh ω 2TH
ωℓm
∞
ℓ=0 ℓ
m=−ℓ
∞
d ω tanh ω 2TH
ωℓm
Kay-Wald theorem and HHI-like states York, April 2017 44 / 47
HHI-like states on Kerr space-time Fermion field
[ Casals et al PRD 87 064027 (2013) ]
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 45 / 47
HHI-like states on Kerr space-time Fermion field
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 46 / 47
Conclusions
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 47 / 47
Conclusions
Schwarzschild
Kay CMP 100 57 (1985) Existence of HHI state Kay & Wald Phys. Rept. 207 49 (1991) Uniqueness of HHI state
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 47 / 47
Conclusions
Schwarzschild
Kay CMP 100 57 (1985) Existence of HHI state Kay & Wald Phys. Rept. 207 49 (1991) Uniqueness of HHI state
Kerr
Kay & Wald Phys. Rept. 207 49 (1991) No HHI state
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 47 / 47
Conclusions
Schwarzschild
Kay CMP 100 57 (1985) Existence of HHI state Kay & Wald Phys. Rept. 207 49 (1991) Uniqueness of HHI state
Kerr
Kay & Wald Phys. Rept. 207 49 (1991) No HHI state
HHI-like states for scalars on Kerr
Nonequilibrium state Enclose horizon inside a mirror
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 47 / 47
Conclusions
Schwarzschild
Kay CMP 100 57 (1985) Existence of HHI state Kay & Wald Phys. Rept. 207 49 (1991) Uniqueness of HHI state
Kerr
Kay & Wald Phys. Rept. 207 49 (1991) No HHI state
HHI-like states for scalars on Kerr
Nonequilibrium state Enclose horizon inside a mirror
HHI-like states for fermions on Kerr
Equilibrium state diverges on and outside SL Kay-Wald theorem extends to fermions?
Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 47 / 47