Kiefer C. Quantum gravity

THE SEVENTH ROUTE TO GEOMETRODYNAMICS 103
Using the result (4.21) for Ha g , one can construct from (3.90) the explicit
expression for H g ⊥
. A rather lengthy but straightforward calculation leads to
(Hojman et al. 1976; Teitelboim 1980)
H g ⊥ =16πG G abcdp ab p cd + V [h ab ] , (4.22)
with
G abcd = 1
2 √ h (h ach bd + h ad h bc − h ab h cd ) (4.23)
as the (inverse) ‘DeWitt metric’, 5 h denoting the determinant of h ab ,and
V = σ√ h
16πG ( (3) R − 2Λ) , (4.24)
where (3) R is the three-dimensional Ricci scalar. 6 The inverse of (4.23) is called
‘DeWitt’ metric because it plays the role of a metric in the space of all metrics
(DeWitt 1967a); cf. Section 4.2.5. Due to this it is often referred to as ‘supermetric’.
The explict expression reads
√
h
G abcd =
2 (hac h bd + h ad h bc − 2h ab h cd ) (4.25)
(the last term here is the same in all space dimensions), obeying
G abcd (
G cdef = 1 2 δ
a
e δf b + ) δa f δb e . (4.26)
We recall that the Poisson-bracket relation (3.91) states that H ⊥ transforms as
a scalar density under coordinate transformations; this is explicitly fulfilled by
(4.22) (G abcd has weight −1, p ab and V have weight 1, so H g ⊥
has weight 1). We
thus have
∫
δH g ⊥ (x) = dy {H g ⊥ (x), Hg a (y)}δNa (y) =
∂
∂x a (Hg ⊥ (x)δNa (x)) . (4.27)
It will be shown in Section 4.2 that H g ⊥ and Hg a uniquely characterize GR, that
is, they follow from the Einstein–Hilbert action (1.1). Finally, we want to remark
that the uniqueness of the construction presented here ceases to hold in space
dimensions greater than three (Teitelboim and Zanelli 1987).
4.1.3 Geometrodynamics and gauge theories
We have seen that for vector fields H a is of such a form that the condition
of ultralocality for H ⊥ would be violated, see (4.16). Since vector fields are an
important ingredient in the description of nature, the question arises whether a
5 In d space dimensions, the last term reads −2/(d − 1)h ab h cd .
6 G and Λ are at this stage just free parameters. They will later be identified with the
gravitational constant and the cosmological constant, respectively.