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About Stewart Walker: American musician (1974-) | Biography, Facts, Career, Life
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Stewart Walker
American musician

Stewart Walker

Stewart Walker
The basics

Quick Facts

Intro American musician
Is Musician Record producer Composer
From United States of America
Field Business Music
Gender male
Birth 1 January 1974, Charlotte, Mecklenburg County, North Carolina, U.S.A.
Age 48 years
The details (from wikipedia)

Biography

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 {\displaystyle \Delta \delta T=0} if and only if one of the following holds

1. T 0 = 0 {\displaystyle T_{0}=0}

2. T 0 {\displaystyle T_{0}} is a constant scalar field

3. T 0 {\displaystyle T_{0}} is a linear combination of products of delta functions δ a b {\displaystyle \delta _{a}^{b}}

Derivation

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

δ T = ϵ h ϵ ( L X T ϵ ) ϵ ( L X T ϵ ) 0 {\displaystyle \delta T=\epsilon h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })\equiv \epsilon ({\mathcal {L}}_{X}T_{\epsilon })_{0}} is the linear perturbation of T {\displaystyle T} . However, since the choice of X {\displaystyle 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 {\displaystyle \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 ) {\displaystyle X^{a}=(\xi ^{\mu },1)} and Y a = ( 0 , 1 ) {\displaystyle Y^{a}=(0,1)} then X a Y a = ( ξ μ , 0 ) {\displaystyle X^{a}-Y^{a}=(\xi ^{\mu },0)} which is a well defined vector in any M ϵ {\displaystyle {\mathcal {M}}_{\epsilon }} and gives the result

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

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