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 gauge-invariant. 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

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A 1-parameter family of manifolds denoted by   with   has metric  . These manifolds can be put together to form a 5-manifold  . A smooth curve   can be constructed through   with tangent 5-vector  , transverse to  . If   is defined so that if   is the family of 1-parameter 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

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  • Stewart J. (1991). Advanced General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-44946-4. Describes derivation of result in section on Lie derivatives