Stewart–Walker lemma

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

1. [math]\displaystyle{ T_{0} = 0 }[/math]

2. [math]\displaystyle{ T_{0} }[/math] is a constant scalar field

3. [math]\displaystyle{ T_{0} }[/math] is a linear combination of products of delta functions [math]\displaystyle{ \delta_{a}^{b} }[/math]

Derivation

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

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

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

[math]\displaystyle{ \Delta \delta T = \epsilon \mathcal{L}_{\xi}T_0. }[/math]

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