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. $Δδ T = 0$ 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 $\delta_{a}^{b}$

[edit] Derivation

A 1-parameter family of manifolds denoted by $\mathcal{M}_{\epsilon}$ with $\mathcal{M}_{0} = \mathcal{M}^{4}$ has metric $g i k = η i k + ε h i k$ . These manifolds can be put together to form a 5-manifold $\mathcal{N}$ . A smooth curve $γ$ 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 $h ε (p 0)$ . 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 $X a = (ξ μ,1)$ and $Y a = (0,1)$ then $X a - Y a = (ξ μ,0)$ 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.

[edit] 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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Categories: Tensors | Lemmas

Stewart-Walker lemma

From Wikipedia, the free encyclopedia

[edit] Derivation

[edit] Sources

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