Feynman Path Integral Formulation

Chapter 4
Hamiltonian and Wheeler-DeWitt Equation
4.1 Classical Initial Value Problem
In formulating the gravitational analogue of the classical initial value (or Cauchy)
problem one needs to specify initial value data at some given time; one should then
be able to determine the field configurations at some later time, by making appropriate
use of the field equations. In such a program the first requirement is therefore
a knowledge of g μν and ∂ 0 g μν everywhere on some spatial hypersurface at a given
initial time x 0 = t. If one can extract from the field equations the quantities ∂ 2 0 g μν,
then these can be used to fix g μν and ∂ 0 g μν at a later time x 0 = t + δt. The process
could then be iterated, and one would eventually obtain a full solution for the metric
g μν valid at all subsequent times.
It would seem at first that the above procedure should work, as there are ten
second time derivatives of the metric, and ten field equations. But this is not so,
since the field equations are not all independent due to the Bianchi identity,
(
R μν − 1 2 gμν R ) ;ν = 0 , (4.1)
which implies that the Einstein tensor G μν = R μν − 1 2 gμν R satisfies
∂ 0 G μ0 = −∂ i G μi − Γ μ
νλ Gλν − Γνλ ν Gμλ . (4.2)
One can verify that the right-hand side only contains first and second time derivatives
of the metric, which implies that G μ0 does not contain second time derivatives
of the metric. As a result, one cannot use the four field equations
G μ0 = 8πGT μ0 , (4.3)
as time evolution equations for the metric. Instead, these should be regarded as
constraints to be imposed on the initial conditions, that is on the quantities g μν
and ∂ 0 g μν at an initial time x 0 = t.
The remaining six field equations
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