HHI state on Schwarzschild space-time HHI state [ Hartle & Hawking PRD 13 2188 (1976), Israel PLA 57 107 (1976) ] i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes? H + “Up” and “down” modes are i 0 not orthogonal H − I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time HHI state [ Hartle & Hawking PRD 13 2188 (1976), Israel PLA 57 107 (1976) ] i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes? H + “Up” and “down” modes are i 0 not orthogonal Instead use “in” and “up” H − modes I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time HHI state [ Hartle & Hawking PRD 13 2188 (1976), Israel PLA 57 107 (1976) ] i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes? H + “Up” and “down” modes are i 0 not orthogonal Instead use “in” and “up” H − modes I − Resulting vacuum state is HHI state | H � i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 12 / 47
HHI state on Schwarzschild space-time Expectation values in the HHI-state | H � Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time Expectation values in the HHI-state | H � Stress-energy tensor operator T µν = 2 Φ − 1 Φ − 1 Φ ∇ λ ˆ ˆ 3 ∇ µ ˆ Φ ∇ ν ˆ Φ ∇ µ ∇ ν ˆ ˆ 6 g µν ∇ λ ˆ Φ 3 Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time Expectation values in the HHI-state | H � Stress-energy tensor operator T µν = 2 Φ − 1 Φ − 1 Φ ∇ λ ˆ ˆ 3 ∇ µ ˆ Φ ∇ ν ˆ Φ ∇ µ ∇ ν ˆ ˆ 6 g µν ∇ λ ˆ Φ 3 Unrenormalized stress-energy tensor expectation value � ω � � � ∞ ∞ ℓ � � + T µν � �� φ up � H | ˆ ∑ ∑ φ in T µν | H � = d ω coth T µν ω ℓ m ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ [ Candelas PRD 21 2185 (1980) ] Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 13 / 47
HHI state on Schwarzschild space-time Expectation values in the HHI-state | H � Stress-energy tensor operator T µν = 2 Φ − 1 Φ − 1 Φ ∇ λ ˆ ˆ 3 ∇ µ ˆ Φ ∇ ν ˆ Φ ∇ µ ∇ ν ˆ ˆ 6 g µν ∇ λ ˆ Φ 3 Unrenormalized stress-energy tensor expectation value � ω � � � ∞ ∞ ℓ � � + T µν � �� φ up � H | ˆ ∑ ∑ φ in T µν | H � = d ω coth T µν ω ℓ m ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ [ Candelas PRD 21 2185 (1980) ] 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 � H | ˆ T µν | H � for a massless scalar field [ 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 HHI state | H � 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 HHI state | H � For a quantum scalar field on Schwarzschild space-time Properties Thermal state in region I Regular on the horizons � H | ˆ T µν | 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 HHI state | H � For a quantum scalar field on Schwarzschild space-time Properties Thermal state in region I Regular on the horizons � H | ˆ T µν | 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 The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ] Globally hyperbolic space-time with a bifurcate Killing horizon II H + B IV I H − III 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) ] Globally hyperbolic space-time with a bifurcate Killing horizon Theorem II H + B IV I H − III 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) ] Globally hyperbolic space-time with a bifurcate Killing horizon Theorem On a large subalgebra of II observables, there can be at most one quasifree, H + isometry invariant, Hadamard state B IV I H − III 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) ] Globally hyperbolic space-time with a bifurcate Killing horizon Theorem On a large subalgebra of II observables, there can be at most one quasifree, H + isometry invariant, Hadamard state B IV I This state, if it exists, is a KMS state at the Hawking H − temperature T H on observables in the III subalgebra localized in region I 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 HHI state | H � on Schwarzschild The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ] | H � exists and is unique on Schwarzschild r = 0 i + I + II H + i 0 IV I H − I − III i − r = 0 Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 17 / 47
HHI state on Schwarzschild space-time Massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 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 � � ˆ � � � � �� T µν = i ∇ µ ˆ ∇ ν ˆ ˆ Ψ , γ µ ∇ ν ˆ Ψ , γ ν ∇ µ ˆ Ψ , γ ν ˆ Ψ , γ µ ˆ − − Ψ + Ψ Ψ Ψ 8 Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 18 / 47
HHI state on Schwarzschild space-time � H | ˆ T µν | H � for a massless fermion field Ψ [ 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 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 Kerr space-time ∆ dr 2 + Σ d θ 2 + sin 2 θ � � 2 + Σ �� r 2 + a 2 � � 2 ds 2 = − ∆ dt − a sin 2 θ d ϕ d ϕ − adt Σ Σ ∆ = r 2 − 2 Mr + a 2 Σ = r 2 + a 2 cos 2 θ i + I + II H + i 0 IV I H − I − III i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 21 / 47
HHI-like states on Kerr space-time Kerr space-time ∆ dr 2 + Σ d θ 2 + sin 2 θ � � 2 + Σ �� r 2 + a 2 � � 2 ds 2 = − ∆ dt − a sin 2 θ d ϕ d ϕ − adt Σ Σ ∆ = r 2 − 2 Mr + a 2 Σ = r 2 + a 2 cos 2 θ i + I + II H + i 0 IV I H − I − III i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 21 / 47
HHI-like states on Kerr space-time Features of Kerr space-time Event horizon � a M 2 − a 2 r H = M + Ω H = r 2 H + a 2 Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 22 / 47
HHI-like states on Kerr space-time Features of Kerr space-time Event horizon � a M 2 − a 2 r H = M + Ω H = r 2 H + a 2 Stationary limit surface � M 2 − a 2 cos 2 θ r S = M + For r H < r < r S 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 Features of Kerr space-time Event horizon � a M 2 − a 2 r H = M + Ω H = r 2 H + a 2 Stationary limit surface � M 2 − a 2 cos 2 θ r S = M + For r H < r < r S 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 = r H and the speed-of-light surface S L Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 22 / 47
HHI-like states on Kerr space-time Location of stationary limit surface and speed-of-light surface [ 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 HHI state on Kerr space-time Quantum 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 The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ] Properties of | H � on Schwarzschild Regular on and outside horizon Time-reversal symmetric Thermal state in region I i + I + II H + i 0 IV I H − I − III i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ] Theorem There does not exist any Hadamard state on Kerr which is invariant under the isometries generating the event horizon i + I + II H + i 0 IV I H − I − III i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field The Kay-Wald theorem [ Kay & Wald Phys. Rept. 207 49 (1991) ] Theorem No HHI state exists for a quantum scalar field on Kerr i + I + II H + i 0 IV I H − I − III i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 25 / 47
HHI-like states on Kerr space-time Scalar field Massless scalar field on Kerr space-time Scalar field modes φ ω ℓ m ( t , r , θ , ϕ ) = 1 1 e − i ω t e im ϕ S ω ℓ m ( cos θ ) R ω ℓ m ( r ) N 1 ( r 2 + a 2 ) 2 S ω ℓ m ( 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 Massless scalar field on Kerr space-time Scalar field modes φ ω ℓ m ( t , r , θ , ϕ ) = 1 1 e − i ω t e im ϕ S ω ℓ m ( cos θ ) R ω ℓ m ( r ) N 1 ( r 2 + a 2 ) 2 S ω ℓ m ( cos θ ) : spheroidal harmonics Radial mode equation � d 2 � dr = r 2 + a 2 dr ∗ 0 = + V ω ℓ m ( r ) R ω ℓ m ( r ) dr 2 ∆ ∗ � ω 2 = ( ω − m Ω H ) 2 as r → r H , r ∗ → − ∞ � V ω ℓ m ( r ) = ω 2 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” and “Up” modes “Up” modes R up “In” modes R in ω ℓ m ω ℓ m � B in � e i � ω r ∗ + A up ω ℓ m e − i � ω ℓ m e − i � ω r ∗ ω r ∗ r ∗ → − ∞ r ∗ → − ∞ e − i ω r ∗ + A in B up ω ℓ m e i ω r ∗ r ∗ → ∞ ω ℓ m e i ω r ∗ r ∗ → ∞ i + i + I + I + H + H + i 0 i 0 H − H − I − I − i − i − 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” and “Down” modes “Out” modes R out “Down” modes R down ω ℓ ω ℓ � B out � e − i � ω r ∗ + A down ω ℓ e i � ω r ∗ r ∗ → − ∞ e i � ω r ∗ r ∗ → − ∞ ω ℓ e i ω r ∗ + A out ω ℓ e − i ω r ∗ e − i ω r ∗ r ∗ → ∞ B down r ∗ → ∞ ω ℓ i + i + I + I + H + H + i 0 i 0 H − H − I − I − i − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 28 / 47
HHI-like states on Kerr space-time Scalar field Modes with positive KG “norm” 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 Modes with positive KG “norm” Positive frequency scalar modes must have positive KG “norm” “In” and “out” modes “In” and “out” modes have positive KG “norm” for i + ω > 0 i + I + I + H + H + i 0 i 0 H − H − I − I − i − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 29 / 47
HHI-like states on Kerr space-time Scalar field Modes with positive KG “norm” Positive frequency scalar modes must have positive KG “norm” “Up” and “down” modes “Up” and “down” modes have positive KG “norm” for ω = ω − m Ω H > 0 � i + i + I + I + H + H + i 0 i 0 H − H − I − I − i − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 29 / 47
HHI-like states on Kerr space-time Scalar field A HHI-like state for a scalar field on Kerr? i + Define positive frequency with respect to Kruskal time T I + H + i 0 H − I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field A HHI-like state for a scalar field on Kerr? i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes H + with � ω > 0? i 0 H − I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field A HHI-like state for a scalar field on Kerr? i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes H + with � ω > 0? “Up” and “down” modes are i 0 not orthogonal H − I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field A HHI-like state for a scalar field on Kerr? i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes H + with � ω > 0? “Up” and “down” modes are i 0 not orthogonal H − Instead use “in” and “up” modes I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field A HHI-like state for a scalar field on Kerr? i + Define positive frequency with respect to Kruskal time T I + Use “up” and “down” modes H + with � ω > 0? “Up” and “down” modes are i 0 not orthogonal H − Instead use “in” and “up” modes I − But “in” modes have positive “norm” for ω > 0 i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 30 / 47
HHI-like states on Kerr space-time Scalar field Attempts at defining a HHI-like state for Kerr | CCH � [ Candelas, Chrzanowski & Howard PRD 24 297 (1981) ] � ω � � ∞ ∞ ℓ � � � CCH | ˆ φ in ∑ ∑ T µν | CCH � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 31 / 47
HHI-like states on Kerr space-time Scalar field Attempts at defining a HHI-like state for Kerr | CCH � [ Candelas, Chrzanowski & Howard PRD 24 297 (1981) ] � ω � � ∞ ∞ ℓ � � � CCH | ˆ φ in ∑ ∑ T µν | CCH � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Does not represent an equilibrium state [ 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 Attempts at defining a HHI-like state for Kerr | CCH � [ Candelas, Chrzanowski & Howard PRD 24 297 (1981) ] � ω � � ∞ ∞ ℓ � � � CCH | ˆ φ in ∑ ∑ T µν | CCH � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Does not represent an equilibrium state 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 Renormalized expectation values on Kerr space-time 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 � CCH | ˆ T µν | CCH � for an electromagnetic 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 Attempts at defining a Hartle-Hawking state for Kerr | FT � [ Frolov & Thorne PRD 39 2125 (1989) ] � � � � ∞ ∞ ℓ � � ω � FT | ˆ φ in ∑ ∑ T µν | FT � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 34 / 47
HHI-like states on Kerr space-time Scalar field Attempts at defining a Hartle-Hawking state for Kerr | FT � [ Frolov & Thorne PRD 39 2125 (1989) ] � � � � ∞ ∞ ℓ � � ω � FT | ˆ φ in ∑ ∑ T µν | FT � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Potentially an equilibrium state [ 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 Attempts at defining a Hartle-Hawking state for Kerr | FT � [ Frolov & Thorne PRD 39 2125 (1989) ] � � � � ∞ ∞ ℓ � � ω � FT | ˆ φ in ∑ ∑ T µν | FT � = d ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ � � � � ∞ ∞ ℓ � � ω φ up ∑ ∑ + d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Potentially an equilibrium state 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 Kerr space-time with a mirror i + Mirror M at fixed r = r 0 inside S L H + M H − [ Duffy & Ottewill PRD 77 024007 (2008) ] i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field Kerr space-time with a mirror i + Mirror M at fixed r = r 0 inside S L Modes H + ω ℓ m − R up ω ℓ m ( r 0 ) φ up ω ℓ m ( r 0 ) φ in ω > 0 R in ω ℓ m φ M M ω ℓ m = R up ω ℓ m ( r 0 ) φ up − ω ℓ − m ( r 0 ) φ in ∗ ω ℓ m − ω < 0 R in ∗ − ω ℓ − m H − [ Duffy & Ottewill PRD 77 024007 (2008) ] i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field Kerr space-time with a mirror i + Mirror M at fixed r = r 0 inside S L Modes H + ω ℓ m − R up ω ℓ m ( r 0 ) φ up ω ℓ m ( r 0 ) φ in ω > 0 R in ω ℓ m φ M M ω ℓ m = R up ω ℓ m ( r 0 ) φ up − ω ℓ − m ( r 0 ) φ in ∗ ω ℓ m − ω < 0 R in ∗ − ω ℓ − m H − Positive KG “norm” for � ω > 0 [ Duffy & Ottewill PRD 77 024007 (2008) ] i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 35 / 47
HHI-like states on Kerr space-time Scalar field HHI-like state | H M � 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 HHI-like state | H M � Modes with positive frequency with respect to Kruskal time T | H M � [ Duffy & Ottewill PRD 77 024007 (2008) ] � � � � ∞ � � ∞ ℓ ω � H M | ˆ φ M ∑ ∑ T µν | H M � = d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field HHI-like state | H M � Modes with positive frequency with respect to Kruskal time T | H M � [ Duffy & Ottewill PRD 77 024007 (2008) ] � � � � ∞ � � ∞ ℓ ω � H M | ˆ φ M ∑ ∑ T µν | H M � = d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Compute expectation values relative to | B M � Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 36 / 47
HHI-like states on Kerr space-time Scalar field HHI-like state | H M � Modes with positive frequency with respect to Kruskal time T | H M � [ Duffy & Ottewill PRD 77 024007 (2008) ] � � � � ∞ � � ∞ ℓ ω � H M | ˆ φ M ∑ ∑ T µν | H M � = d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Compute expectation values relative to | B M � | B M � 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 HHI-like state | H M � Modes with positive frequency with respect to Kruskal time T | H M � [ Duffy & Ottewill PRD 77 024007 (2008) ] � � � � ∞ � � ∞ ℓ ω � H M | ˆ φ M ∑ ∑ T µν | H M � = d � ω coth T µν ω ℓ m 2 T H 0 ℓ = 0 m = − ℓ Compute expectation values relative to | B M � | B M � defined by taking modes to have positive frequency with respect to t | B M � 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 � H M | ˆ T µν | H M � [ 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 HHI-states on space-times with enclosed horizons Non-existence of HHI-state on Kruskal space-time with a single mirror i + I + II H + I i 0 M IV H − I − III i − [ 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 HHI-states on space-times with enclosed horizons Existence of HHI-state on Kruskal space-time with two mirrors i + I + II H + IV I i 0 M M H − I − III i − [ 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 HHI state on Kerr space-time Quantum 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 Canonical quantization of a massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 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 Canonical quantization of a massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 0 Dirac inner product � Σ Ψ 1 γ µ Ψ 2 d Σ µ ( Ψ 1 , Ψ 2 ) 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 Canonical quantization of a massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 0 Dirac inner product � Σ Ψ 1 γ µ Ψ 2 d Σ µ ( Ψ 1 , Ψ 2 ) 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 Canonical quantization of a massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 0 Dirac inner product � Σ Ψ 1 γ µ Ψ 2 d Σ µ ( Ψ 1 , Ψ 2 ) 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 Canonical quantization of a massless fermion field Ψ Dirac equation γ µ ∇ µ Ψ = 0 Dirac inner product � Σ Ψ 1 γ µ Ψ 2 d Σ µ ( Ψ 1 , Ψ 2 ) 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 Mode expansion of the massless fermion field Ψ Expand classical field in terms of orthonormal basis of field modes Ψ = ∑ b j ψ + j + c † j ψ − j 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 Mode expansion of the massless fermion field Ψ Expand classical field in terms of orthonormal basis of field modes Ψ = ∑ ˆ ˆ b j ψ + c † j ψ − j + ˆ j j Promote expansion coefficients to operators ˆ b j , ˆ c j with � � � � b j , ˆ ˆ b † c † = δ jk = c j , ˆ ˆ k k � � � � � � � = � ˆ b j , ˆ ˆ b † ˆ j , ˆ b † c † c † = = 0 = b k c j , ˆ c k ˆ j , ˆ k k Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 42 / 47
HHI-like states on Kerr space-time Fermion field Mode expansion of the massless fermion field Ψ Expand classical field in terms of orthonormal basis of field modes Ψ = ∑ ˆ b j ψ + ˆ c † j ψ − j + ˆ j j Promote expansion coefficients to operators ˆ b j , ˆ c j with � � � � b j , ˆ ˆ b † c † = δ jk = c j , ˆ ˆ k k � � � � � � � = � ˆ b j , ˆ ˆ b † ˆ j , ˆ b † c † c † = = 0 = b k c j , ˆ c k ˆ j , ˆ k k Define the vacuum state | 0 � ˆ b j | 0 � = 0 = ˆ c j | 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 A HHI-like state for a fermion field on Kerr? i + Define positive frequency with respect to Kruskal time T I + H + i 0 H − I − i − Elizabeth Winstanley (Sheffield) Kay-Wald theorem and HHI-like states York, April 2017 43 / 47
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