Quantum Field Theory and Gravity: Conceptual and Mathematical ...

Hawking Effect for Rotating Black Holes 133
where U(0,t) is the propagator defined in Proposition 4.6. The quantum spin
field is defined by:
Ψ col :Φ∈ ( C0 ∞ (M col ) ) 4
↦→ Ψ col (Φ) := Ψ 0 (S col Φ) ∈L(F(H 0 )).
Here F(H 0 ) is the Dirac-Fermi-Fock space associated to H 0 . For an arbitrary
set O ⊂ M col , we introduce U col (O), the C ∗ -algebra generated by
ψcol ∗ (Φ 1)Ψ col (Φ 2 ), supp Φ j ⊂O, j =1, 2. Eventually, we have:
U col (M col )=
⋃
U col (O).
O⊂M col
Then we define the fundamental state on U col (M col ) as follows:
(
ω col Ψ
∗
col (Φ 1 )Ψ col (Φ 2 ) ) (
:= ω vac Ψ
∗
0 (S col Φ 1 )Ψ 0 (S col Φ 2 ) )
= 〈 〉
1 [0,∞) (H 0 )S col Φ 1 ,S col Φ 2 .
Let us now consider the future black hole. The algebra U BH (M BH )andthe
vacuum state ω vac are constructed as before working now with the group e itH
rather than the evolution system U(t, s). We also define the thermal Hawking
state (S is analogous to S col , Ψ BH to Ψ col , and Ψ to Ψ 0 ):
(
Ψ
∗
BH (Φ 1 )Ψ BH (Φ 2 ) ) = 〈 μe σH (1 + μe σH ) −1 〉
SΦ 1 ,SΦ 2
ω η,σ
Haw
=: ω η,σ (
KMS Ψ ∗ (SΦ 1 )Ψ(SΦ 2 ) )
with
T Haw = σ −1 , μ = e ση , σ > 0,
where T Haw is the Hawking temperature and μ is the chemical potential.
5.2. The Hawking effect
In this subsection we formulate the main result of this work.
Let Φ ∈ (C0 ∞ (M col )) 4 . We put
Φ T (t, ˆr, ω) =Φ(t − T, ˆr, ω). (5.2)
Theorem 5.1 (Hawking effect). Let
Φ j ∈ (C0 ∞ (M col )) 4 , j =1, 2.
Then we have
lim ω (
col Ψ
∗
col (Φ T 1 )Ψ col (Φ T 2 ) ) = ω η,σ (
Haw Ψ
∗
BH (1 R +(P − )Φ 1 )Ψ BH (1 R +(P − )Φ 2 ) )
T →∞
(
+ ω vac Ψ
∗
BH (1 R −(P − )Φ 1 )Ψ BH (1 R −(P − )Φ 2 ) ) ,
(5.3)
T Haw =1/σ = κ + /2π, μ = e ση , η = qQr +
r+ 2 + + aD ϕ
a2 r+ 2 + .
a2
In the above theorem P ± is the asymptotic velocity introduced in Section
4. The projections 1 R ±(P ± ) separate outgoing and incoming solutions.
Remark 5.2. The result is independent of the choices of coordinate system,
tetrad and chiral angle in the boundary condition.
H