The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. Δ δ T = 0 if and only if one of the following holds
1. T 0 = 0
2. T 0 is a constant scalar field
3. T 0 is a linear combination of products of delta functions δ a b
A 1-parameter family of manifolds denoted by M ϵ with M 0 = M 4 has metric g i k = η i k + ϵ h i k . These manifolds can be put together to form a 5-manifold N . A smooth curve γ can be constructed through N with tangent 5-vector X , transverse to M ϵ . If X is defined so that if h t is the family of 1-parameter maps which map N → N and p 0 ∈ M 0 then a point p ϵ ∈ M ϵ can be written as h ϵ ( p 0 ) . This also defines a pull back h ϵ ∗ that maps a tensor field T ϵ ∈ M ϵ back onto M 0 . Given sufficient smoothness a Taylor expansion can be defined
h ϵ ∗ ( T ϵ ) = T 0 + ϵ h ϵ ∗ ( L X T ϵ ) + O ( ϵ 2 ) δ T = ϵ h ϵ ∗ ( L X T ϵ ) ≡ ϵ ( L X T ϵ ) 0 is the linear perturbation of T . However, since the choice of X is dependent on the choice of gauge another gauge can be taken. Therefore the differences in gauge become Δ δ T = ϵ ( L X T ϵ ) 0 − ϵ ( L Y T ϵ ) 0 = ϵ ( L X − Y T ϵ ) 0 . Picking a chart where X a = ( ξ μ , 1 ) and Y a = ( 0 , 1 ) then X a − Y a = ( ξ μ , 0 ) which is a well defined vector in any M ϵ and gives the result
Δ δ T = ϵ L ξ T 0 . The only three possible ways this can be satisfied are those of the lemma.
