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. ΔδT = 0 if and only if one of the following holds
1. T0 = 0
2. T0 is a constant scalar field
3. T0 is a linear combination of products of delta functions
[edit] Derivation
A 1-parameter family of manifolds denoted by with has metric gik = ηik + εhik. These manifolds can be put together to form a 5-manifold . A smooth curve γ can be constructed through with tangent 5-vector X, transverse to . If X is defined so that if ht is the family of 1-parameter maps which map and then a point can be written as hε(p0). 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 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 . Picking a chart where Xa = (ξμ,1) and Ya = (0,1) then Xa − Ya = (ξμ,0) 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.
[edit] 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