| Intro | American musician |
| Is |
Musician Record producer Composer |
| From | United States of America |
| Type |
Business Music |
| Gender | male |
| Birth | 1 January 1974, Charlotte, Mecklenburg County, North Carolina, U.S.A. |
| Age: | 46 years |
The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. ΔδT=0{displaystyle Delta delta T=0} if and only if one of the following holds
1. T0=0{displaystyle T_{0}=0}
2. T0{displaystyle T_{0}} is a constant scalar field
3. T0{displaystyle T_{0}} is a linear combination of products of delta functions δab{displaystyle delta _{a}^{b}}
Derivation
A 1-parameter family of manifolds denoted by Mϵ{displaystyle {mathcal {M}}_{epsilon }} with M0=M4{displaystyle {mathcal {M}}_{0}={mathcal {M}}^{4}} has metric gik=ηik+ϵhik{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 Mϵ{displaystyle {mathcal {M}}_{epsilon }}. If X{displaystyle X} is defined so that if ht{displaystyle h_{t}} is the family of 1-parameter maps which map N→N{displaystyle {mathcal {N}}to {mathcal {N}}} and p0∈M0{displaystyle p_{0}in {mathcal {M}}_{0}} then a point pϵ∈Mϵ{displaystyle p_{epsilon }in {mathcal {M}}_{epsilon }} can be written as hϵ(p0){displaystyle h_{epsilon }(p_{0})}. This also defines a pull back hϵ∗{displaystyle h_{epsilon }^{*}} that maps a tensor field Tϵ∈Mϵ{displaystyle T_{epsilon }in {mathcal {M}}_{epsilon }} back onto M0{displaystyle {mathcal {M}}_{0}}. Given sufficient smoothness a Taylor expansion can be defined
δT=ϵhϵ∗(LXTϵ)≡ϵ(LXTϵ)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 ΔδT=ϵ(LXTϵ)0−ϵ(LYTϵ)0=ϵ(LX−YTϵ)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 Xa=(ξμ,1){displaystyle X^{a}=(xi ^{mu },1)} and Ya=(0,1){displaystyle Y^{a}=(0,1)} then Xa−Ya=(ξμ,0){displaystyle X^{a}-Y^{a}=(xi ^{mu },0)} which is a well defined vector in any Mϵ{displaystyle {mathcal {M}}_{epsilon }} and gives the result
The only three possible ways this can be satisfied are those of the lemma.