Stewart–Walker lemma

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

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

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

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

Derivation

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

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

${\displaystyle \delta T=\epsilon h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })\equiv \epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}}$ is the linear perturbation of $T$. However, since the choice of $X$ is dependent on the choice of gauge another gauge can be taken. Therefore the differences in gauge become ${\displaystyle \Delta \delta T=\epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}-\epsilon ({\mathcal {L}}_{Y}T_{\epsilon })_{0}=\epsilon ({\mathcal {L}}_{X-Y}T_{\epsilon })_{0}}$. Picking a chart where ${\displaystyle X^{a}=(\xi ^{\mu },1)}$ and ${\displaystyle Y^{a}=(0,1)}$ then ${\displaystyle X^{a}-Y^{a}=(\xi ^{\mu },0)}$ which is a well defined vector in any ${\displaystyle {\mathcal {M}}_{\epsilon }}$ and gives the result

${\displaystyle \Delta \delta T=\epsilon {\mathcal {L}}_{\xi }T_{0}.\,}$

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