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. \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[edit]

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[edit]

  • Stewart J. (1991). Advanced General Relativity. Cambridge: Cambridge University Press. ISBN 0-521-44946-4.  Describes derivation of result in section on Lie derivatives