Samiksha Jaiswal (Editor)

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. 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 N and p 0 M 0 then a point p ϵ 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 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 ) 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.

References

Stewart–Walker lemma Wikipedia


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