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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. Δ δ T = 0 {\displaystyle \Delta \delta T=0} if and only if one of the following holds

1. T 0 = 0 {\displaystyle T_{0}=0}

2. T 0 {\displaystyle T_{0}} is a constant scalar field

3. T 0 {\displaystyle T_{0}} is a linear combination of products of delta functions δ a b {\displaystyle \delta _{a}^{b}}

Derivation

A 1-parameter family of manifolds denoted by M ϵ {\displaystyle {\mathcal {M}}_{\epsilon }} with M 0 = M 4 {\displaystyle {\mathcal {M}}_{0}={\mathcal {M}}^{4}} has metric g i k = η i k + ϵ h i k {\displaystyle g_{ik}=\eta _{ik}+\epsilon h_{ik}} . These manifolds can be put together to form a 5-manifold N {\displaystyle {\mathcal {N}}} . A smooth curve γ {\displaystyle \gamma } can be constructed through N {\displaystyle {\mathcal {N}}} with tangent 5-vector X {\displaystyle X} , transverse to M ϵ {\displaystyle {\mathcal {M}}_{\epsilon }} . If X {\displaystyle X} is defined so that if h t {\displaystyle h_{t}} is the family of 1-parameter maps which map N → N {\displaystyle {\mathcal {N}}\to {\mathcal {N}}} and p 0 ∈ M 0 {\displaystyle p_{0}\in {\mathcal {M}}_{0}} then a point p ϵ ∈ M ϵ {\displaystyle p_{\epsilon }\in {\mathcal {M}}_{\epsilon }} can be written as h ϵ ( p 0 ) {\displaystyle h_{\epsilon }(p_{0})} . This also defines a pull back h ϵ ∗ {\displaystyle h_{\epsilon }^{*}} that maps a tensor field T ϵ ∈ M ϵ {\displaystyle T_{\epsilon }\in {\mathcal {M}}_{\epsilon }} back onto M 0 {\displaystyle {\mathcal {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 ) {\displaystyle h_{\epsilon }^{*}(T_{\epsilon })=T_{0}+\epsilon \,h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })+O(\epsilon ^{2})}

δ T = ϵ h ϵ ∗ ( L X T ϵ ) ≡ ϵ ( L X T ϵ ) 0 {\displaystyle \delta T=\epsilon h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })\equiv \epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}} is the linear perturbation of T {\displaystyle T} . However, since the choice of X {\displaystyle 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 {\displaystyle \Delta \delta T=\epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}-\epsilon ({\mathcal {L}}_{Y}T_{\epsilon })_{0}=\epsilon ({\mathcal {L}}_{X-Y}T_{\epsilon })_{0}} . Picking a chart where X a = ( ξ μ , 1 ) {\displaystyle X^{a}=(\xi ^{\mu },1)} and Y a = ( 0 , 1 ) {\displaystyle Y^{a}=(0,1)} then X a − Y a = ( ξ μ , 0 ) {\displaystyle X^{a}-Y^{a}=(\xi ^{\mu },0)} which is a well defined vector in any M ϵ {\displaystyle {\mathcal {M}}_{\epsilon }} and gives the result

Δ δ T = ϵ L ξ T 0 . {\displaystyle \Delta \delta T=\epsilon {\mathcal {L}}_{\xi }T_{0}.}

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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