Kiefer C. Quantum gravity

110 HAMILTONIAN FORMULATION OF GENERAL RELATIVITY
For the ‘time–time component’ one has
0=G µν n µ n ν = R µν n µ n ν + R 2 . (4.56)
From (4.49) one finds upon contraction of indices,
(3) R + K µ
µ K ν
ν − K µν K µν = h µµ′ h ν′
ν h λ′
µ h ν ρ ′R ρ ′
µ ′ ν ′ λ ′ .
The right-hand side is equal to
R +2R µν n µ n ν =2G µν n µ n ν ,
and so the ‘time–time component’ of Einstein’s equations reads
K 2 − K ab K ab + (3) R =0. (4.57)
This is the (3+1)-dimensional version of the theorema egregium. Both (4.55) and
(4.57) are constraints—they only contain first-order time derivatives. These constraints
play a crucial role in the initial value formulation of classical GR, see for
example, Choquet-Bruhat and York (1980) for a review. There, one can specify
initial data (h ab ,K cd )onΣ,whereh ab and K cd satisfy the constraints (4.55) and
(4.57). One can then prove that there exists one globally hyperbolic space–time
obeying Einstein’s equations (i.e. a unique solution for the four-metric up to
diffeomorphisms), which has a Cauchy surface on which the induced metric and
the extrinsic curvature are just h ab and K cd , respectively.
In electrodynamics, for comparison, one has to specify A and E on Σ satisfying
the constraint (Gauss’ law (4.30)) ∇E = 0. One then gets in space–time a
solution of Maxwell’s equations that is unique up to gauge transformation. The
important point is that the space–time is fixed in Maxwell’s theory, whereas in
the gravitational case it is part of the solution.
That the dynamical laws follow from the laws of the instant can be inferred
from the validity of the following ‘interconnection theorems’ (cf. Kuchař (1981))
1. If the constraints are valid on an initial hypersurface and if G ab =0(pure
spatial components of the vacuum Einstein equations) on space–time, the
constraints hold on every hypersurface.
2. If the constraints hold on every hypersurface, the equations G ab =0hold
on space–time.
Similar properties hold in Maxwell’s theory, cf. Giulini and Kiefer (2007). In
the presence of non-gravitational fields, ∇ µ T µν = 0 is needed as an integrability
condition (analogously to ∂ µ j µ = 0 for Maxwell’s equations).
In order to reformulate the Einstein–Hilbert action (1.1), one has to express
the volume element and the Ricci scalar in terms of h ab and K cd . For the volume
element one finds
√ −g = N
√
h. (4.58)