Feynman Path Integral Formulation

4.5 Intrinsic and Extrinsic Curvature, Hamiltonian 111
4.5 Intrinsic and Extrinsic Curvature, Hamiltonian
Some of the quantities introduced in the previous section (such as 3 R) describe intrinsic
properties of the spacelike hypersurface. Some others can be related to the extrinsic
properties of such a hypersurface when embedded in four-dimensional space.
Consider spacetime as sliced up (foliated) by a one-parameter family of spacelike
hypersurfaces x μ = x μ (x i ,t). One then has for the intrinsic metric within the spacelike
hypersurface
g ij = g μν X μ
i
X ν j with X μ
i
≡ ∂ i x μ , (4.46)
while the extrinsic curvature is given in terms of the unit normals to the spacelike
surface, U μ ,
K ij (x k ,t) =−∇ μ U ν Xi ν X μ j
. (4.47)
In this language, the lapse and shift functions appear in the expression
∂ t x μ
= NU μ + N i X μ
i
. (4.48)
Then the Einstein tensor G μν can be projected into directions normal (⊥) and tangential
(i) to the hypersurface, with the result
G ⊥⊥ ≡ G μν U μ U ν = − 1 2 (K ijK ij − K 2 − 3 R)
G i⊥ ≡−G μν X μ
i
U ν = ∇ j (K j
i − Kδ j
i )
G i j ≡ G μν X μi Xj
ν = −∂ t (K i j − K δ j)+KK i i j
− 1 2 (Km n Km n + K 2 )δ j i + 3 G i j , (4.49)
with K = g ij K ij = K i i the trace of the matrix K.
Then in the canonical formalism the momentum can be expressed in terms of the
extrinsic curvature as
π ij = − √ g(K ij − Kg ij ) . (4.50)
By inverting this last relationship, one can then replace the extrinsic curvatures K by
the six momenta π. Then the quantities G ⊥⊥ and G i⊥ in Eq. (4.49) can be expressed
entirely in terms of g ij and π ij within a given spacelike hypersurface,
−2 √ gG ⊥⊥ ≡ H = 2Kg −1/2 ( π ij π ij − 1 2 π2) − 1 √ g 3 R
2K
2 √ gG i⊥ ≡ H i = −2Kg −1/2 ∇ j π j
i
, (4.51)
with π = π i i . The last two statements are essentially equivalent to the definitions
in Eq. (4.35). On the other hand the expression involving G i j in Eq. (4.49) can be
rearranged to give
∂ t π ij (x,t) = { π ij }
(x,t),H N + H N + NG
ij
, (4.52)