Savart: Difference between revisions

Revision as of 18:58, 1 June 2013

The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. $Δ δ T = 0$ 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 $δ_{a}^{b}$

Derivation

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

h_{ϵ}^{*} (T_{ϵ}) = T_{0} + ϵ h_{ϵ}^{*} (ℒ_{X} T_{ϵ}) + O (ϵ^{2})

$δ T = ϵ h_{ϵ}^{*} (ℒ_{X} T_{ϵ}) \equiv ϵ (ℒ_{X} T_{ϵ})_{0}$ 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 $Δ δ T = ϵ (ℒ_{X} T_{ϵ})_{0} - ϵ (ℒ_{Y} T_{ϵ})_{0} = ϵ (ℒ_{X - Y} T_{ϵ})_{0}$ . 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 $ℳ_{ϵ}$ and gives the result

Δ δ T = ϵ ℒ_{ξ} T_{0} .

The only three possible ways this can be satisfied are those of the lemma.

Sources

Template:Refbegin

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Describes derivation of result in section on Lie derivatives

Template:Refend

@@ Line 1: / Line 1: @@
-year-old Power Era Seed Operator Houston Nordahl from Fort Saskatchewan, has lots of interests that include physical exercise (aerobics weights), [http://www.comoganhardinheiro101.com/perguntas-frequentes/ como ganhar dinheiro] na internet and traveling. Preceding year just made a trip Old Walled City of Shibam.
+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
+. <math>T_{0} = 0</math>
+. <math>T_{0}</math> is a constant scalar field
+. <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]]

Savart: Difference between revisions

Revision as of 18:58, 1 June 2013

Derivation

Sources

Navigation menu

Search