Kiefer C. Quantum gravity

QUANTUM THEORY OF COLLAPSING DUST SHELLS 225
on a fixed background manifold M. Here, the corresponding functions A and R
have the form
A(η, M, w; U, V ) , R(η, M, w; U, V ) , (7.76)
and the trajectory of the shell on the background manifold is simply U = u for
η =+1andV = v for η = −1.
The next step is the explicit transformation to embedding variables (the
Kuchař decomposition). The standard (ADM) formulation of the shell was studied
in Louko et al. (1998); see also Kraus and Wilczek (1995). The spherically
symmetric metric is written in the form
ds 2 = −N 2 dτ 2 + L 2 (dρ + N ρ dτ) 2 + R 2 dΩ 2 , (7.77)
and the shell is described by its radial coordinate ρ = r. The action reads
∫ [ ∫
]
S 0 = dτ pṙ + dρ(P L ˙L + PR Ṙ − H 0 ) , (7.78)
and the Hamiltonian is
H 0 = NH ⊥ + N ρ H ρ + N ∞ E ∞ ,
where N ∞ := lim ρ→∞ N(ρ), E ∞ is the ADM mass, and N and N ρ are the lapse
and shift functions. The constraints read
H ⊥ = LP L
2
2R 2 − P LP R
R
+ RR′′
L − RR′ L ′
L 2 + R′2
2L − L 2 + ηp δ(ρ − r) ≈ 0 (7.79)
L
H ρ = P R R ′ − P LL ′ − pδ(ρ − r) ≈ 0 , (7.80)
where the prime (dot) denotes the derivative with respect to ρ (τ). These are
the same constraints as in (7.22) and (7.24), except for the contribution of the
shell.
ThetaskistotransformthevariablesintheactionS 0 . This transformation
will be split into two steps. The first step is a transformation of the canonical
coordinates r, p, L, P L , R, andP R at the constraint surface Γ defined by the
constraints (7.79) and (7.80). The new coordinates are u and p u = −M for
η =+1,v and p v = −M for η = −1, and the embedding variables U(ρ) and
V (ρ).
The second step is an extension of the functions u, v, p u , p v , U(ρ), P U (ρ),
V (ρ), and P V (ρ) out of the constraint surface, where the functions u, v, p u ,
p v , U(ρ), and V (ρ) are defined by the above transformation, and P U (ρ), P V (ρ)
by P U (ρ)| Γ = P V (ρ)| Γ = 0. The extension must satisfy the condition that the
functions form a canonical chart in a neighbourhood of Γ. That such an extension
exists was shown in Hájíček and Kijowski (2000). The details of the calculation
can be found in Hájíček and Kiefer (2001a). The result is the action
∫
∫ ∫ ∞
S = dτ (p u ˙u + p v ˙v − np u p v )+ dτ dρ(P U ˙U + PV ˙V − H) , (7.81)
where H = N U P U +N V P V ,andn, N U (ρ), and N V (ρ) are Lagrange multipliers.
The first term in (7.81) contains the physical variables (observables), while the
0