Stewart–Walker lemma

The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. $\Delta \delta T=0$ if and only if one of the following holds

1. $T_{0}=0$

2. $T_{0}$ is a constant scalar field

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

Derivation

A 1-parameter family of manifolds denoted by ${\mathcal {M}}_{\epsilon }$ with ${\mathcal {M}}_{0}={\mathcal {M}}^{4}$ has metric $g_{ik}=\eta _{ik}+\epsilon h_{ik}$ . These manifolds can be put together to form a 5-manifold ${\mathcal {N}}$ . A smooth curve $\gamma$ can be constructed through ${\mathcal {N}}$ with tangent 5-vector $X$ , transverse to ${\mathcal {M}}_{\epsilon }$ . If $X$ is defined so that if $h_{t}$ is the family of 1-parameter maps which map ${\mathcal {N}}\to {\mathcal {N}}$ and $p_{0}\in {\mathcal {M}}_{0}$ then a point $p_{\epsilon }\in {\mathcal {M}}_{\epsilon }$ can be written as $h_{\epsilon }(p_{0})$ . This also defines a pull back $h_{\epsilon }^{*}$ that maps a tensor field $T_{\epsilon }\in {\mathcal {M}}_{\epsilon }$ back onto ${\mathcal {M}}_{0}$ . Given sufficient smoothness a Taylor expansion can be defined

h_{\epsilon }^{*}(T_{\epsilon })=T_{0}+\epsilon \,h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })+O(\epsilon ^{2})

$\delta T=\epsilon h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })\equiv \epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}$ 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 $\Delta \delta T=\epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}-\epsilon ({\mathcal {L}}_{Y}T_{\epsilon })_{0}=\epsilon ({\mathcal {L}}_{X-Y}T_{\epsilon })_{0}$ . Picking a chart where $X^{a}=(\xi ^{\mu },1)$ and $Y^{a}=(0,1)$ then $X^{a}-Y^{a}=(\xi ^{\mu },0)$ which is a well defined vector in any ${\mathcal {M}}_{\epsilon }$ and gives the result

\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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