Kiefer C. Quantum gravity

THE 3+1 DECOMPOSITION OF GENERAL RELATIVITY 113
√
H g h
⊥ =16πG G abcdp ab p cd −
16πG ( (3) R − 2Λ) ≈ 0 , (4.69)
Ha g = −2D bpa b (4.70)
In fact, the constraint (4.69) is equivalent to (4.57), and (4.70) is equivalent to
(4.55)—they are called Hamiltonian constraint and diffeomorphism (or momentum)
constraint, respectively. From its structure, (4.69) has some similarity to the
constraint for the relativistic particle, Equation (3.24), while (4.70) is similar to
(4.30). It can now be seen explicitly that these constraints are equivalent to the
results from the ‘seventh route to geometrodynamics’, see (4.21) and (4.22). The
total Hamiltonian is thus constrained to vanish, a result that is in accordance
with our general discussion of reparametrization invariance of Section 3.1. In the
case of non-compact space, boundary terms are present in the Hamiltonian; see
Section 4.2.4.
In addition to the constraints, one has the six dynamical equations, the
Hamiltonian equations of motion. The first half, ḣ ab = {h ab ,H g },justgives
(4.65). The second half, ṗ ab = {p ab ,H g }, yields a lengthy expression (see e.g.
Wald 1984) that is not needed for canonical quantization. It is, of course, needed
for applications of the classical canonical formalism such as gravitational-wave
emission from compact binary objects.
If non-gravitational fields are coupled, the constraints acquire extra terms.
In (4.56) one has to use that
2G µν n µ n ν =16πGT µν n µ n ν ≡ 16πGρ .
Instead of (4.69) one now has the following expression for the Hamiltonian constraint,
√
h
H ⊥ =16πG G abcd p ab p cd −
16πG ( (3) R − 2Λ) + √ hρ ≈ 0 . (4.71)
Similarly, one has instead of (4.70) for the diffeomorphism constraints,
H a = −2D b p b
a + √ hJ a ≈ 0 , (4.72)
where J a ≡ h µ
a T µν n ν is the ‘Poynting vector’. Consider as special examples the
cases of a scalar field and the electromagnetic field. With the Lagrange density
L = √ −g ( − 1 2 gµν φ ,µ φ ,ν − 1 2 m2 φ 2) (4.73)
for the scalar field one finds for its Hamiltonian
∫
H φ = d 3 xN
∫
+
( √ )
p
2
φ h
2 √ h + 2 hab φ ,a φ ,b + 1 √
hm 2 φ 2
2
d 3 xN a p φ φ ,a . (4.74)