Stewart–Walker lemma
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The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gaugeinvariant. if and only if one of the following holds
1.
2. is a constant scalar field
3. is a linear combination of products of delta functions
Derivation
A 1parameter family of manifolds denoted by with has metric . These manifolds can be put together to form a 5manifold . A smooth curve can be constructed through with tangent 5vector , transverse to . If is defined so that if is the family of 1parameter maps which map and then a point can be written as . This also defines a pull back that maps a tensor field back onto . Given sufficient smoothness a Taylor expansion can be defined
is the linear perturbation of . However, since the choice of is dependent on the choice of gauge another gauge can be taken. Therefore the differences in gauge become . Picking a chart where and then which is a well defined vector in any and gives the result
The only three possible ways this can be satisfied are those of the lemma.
Sources
 Stewart J. (1991). Advanced General Relativity. Cambridge: Cambridge University Press. ISBN 0521449464. Describes derivation of result in section on Lie derivatives