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

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

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