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.