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The '''Stewart–Walker lemma''' provides necessary and sufficient conditions for the [[linear]] [[wiktionary:perturbation|perturbation]] of a [[tensor]] field to be [[gauge theory|gauge]]-invariant. <math>\Delta \delta T = 0</math> [[if and only if]] one of the following holds
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1. <math>T_{0} = 0</math>
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2. <math>T_{0}</math> is a constant scalar field
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3. <math>T_{0}</math> is a linear combination of products of delta functions <math>\delta_{a}^{b}</math>
 
== Derivation ==
 
A 1-parameter family of manifolds denoted by <math>\mathcal{M}_{\epsilon}</math> with <math>\mathcal{M}_{0} = \mathcal{M}^{4}</math> has [[Metric (mathematics)|metric]] <math>g_{ik} = \eta_{ik} + \epsilon h_{ik}</math>. These manifolds can be put together to form a 5-manifold <math>\mathcal{N}</math>. A smooth curve <math>\gamma</math> can be constructed through <math>\mathcal{N}</math> with tangent 5-vector <math>X</math>, transverse to <math>\mathcal{M}_{\epsilon}</math>. If <math>X</math> is defined so that  if <math>h_{t}</math> is the family of 1-parameter maps which map <math>\mathcal{N} \to \mathcal{N}</math> and <math>p_{0} \in \mathcal{M}_{0}</math> then a point <math>p_{\epsilon} \in \mathcal{M}_{\epsilon}</math> can be written as <math>h_{\epsilon}(p_{0})</math>. This also defines a [[pullback (differential geometry)|pull back]] <math>h_{\epsilon}^{*}</math> that maps a tensor field <math>T_{\epsilon} \in \mathcal{M}_{\epsilon} </math> back onto <math>\mathcal{M}_{0}</math>. Given sufficient smoothness a Taylor expansion can be defined
 
:<math>h_{\epsilon}^{*}(T_{\epsilon}) = T_{0} + \epsilon \, h_{\epsilon}^{*}(\mathcal{L}_{X}T_{\epsilon}) + O(\epsilon^{2})</math>
 
<math>\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>T</math>. However, since the choice of <math>X</math> is dependent on the choice of [[Gauge theory|gauge]] another gauge can be taken. Therefore the differences in gauge become <math>\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 (topology)|chart]] where <math>X^{a} = (\xi^\mu,1)</math> and <math>Y^a = (0,1)</math> then <math>X^{a}-Y^{a} = (\xi^{\mu},0)</math> which is a well defined vector in any <math>\mathcal{M}_\epsilon</math> and gives the result
 
:<math>\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 ==
 
{{refbegin}}
* {{cite book | author=Stewart J. | title=Advanced General Relativity | location=Cambridge | publisher=Cambridge University Press | year=1991 | isbn=0-521-44946-4}}  Describes derivation of result in section on Lie derivatives
{{refend}}
 
{{DEFAULTSORT:Stewart-Walker lemma}}
[[Category:Tensors]]
[[Category:Lemmas]]

Revision as of 11:51, 23 February 2014

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