Feynman Path Integral Formulation

8 1 Continuum Formulation∇ α R μνβγ + ∇ β R μνγα + ∇ γ R μναβ = 0 , (1.44)with ∇ μ the covariant derivative. It is known that these, in their contracted form,ensure the consistency of the field equations. From the expansion of the Einstein-Hilbert gravitational action in powers of the deviation of the metric from the flatmetric η μν ,usingR μν = 1 2 (∂ 2 h μν − ∂ α ∂ μ h α ν − ∂ α ∂ ν h α ν + ∂ μ ∂ ν h α α)+O(h 2 )R = ∂ 2 h μ μ − ∂ α ∂ μ h αμ + O(h 2 ) , (1.45)one has for the action contribution√ gR = −14∂ σ h μν ∂ σ h μν + 1 2 ∂ ν h μν ∂ σ h μσ− 1 2 ∂ ν h μν ∂ μ hσσ + 1 4 ∂ μ hν ν ∂ μ hσ σ + O(h 3 ) , (1.46)again up to total derivatives. This last expression is in fact the same as Eq. (1.7). Thecorrect relationship between the original graviton field h μν and the metric field g μνisg μν (x)=η μν + κ h μν (x) . (1.47)If, as is often customary, one rescales h μν in such a way that the κ factor does notappear on the r.h.s., then both the g and h fields are dimensionless.The weak field invariance properties of the gravitational action of Eq. (1.11)are replaced in the general theory by general coordinate transformations x μ → x ′μ ,under which the metric transforms as a covariant second rank tensorg ′ μν(x ′ )= ∂xρ ∂x σ∂x ′ μ∂x ′ ν g ρσ(x) , (1.48)which leaves the infinitesimal proper time interval dτ withdτ 2 = −g μν dx μ dx ν , (1.49)invariant. In their infinitesimal form, coordinate transformations are written asx ′μ = x μ + ε μ (x) , (1.50)under which the metric at the same point x then transforms asδg μν (x)=−g λν (x)∂ μ ε λ (x) − g λμ (x)∂ ν ε λ (x) − ε λ (x)∂ λ g μν (x) , (1.51)and which is usually referred to as the Lie derivative of g. The latter generalizes theweak field gauge invariance property of Eq. (1.11) to all orders in h μν .For infinitesimal coordinate transformations, one can gain some additional physicalinsight by decomposing the derivative of the small coordinate change ε μ inEq. (1.50) as∂ε μ∂x ν = s μν + a μν +t μν (1.52)