Stewart–Walker lemma explained

The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant.

\Delta \delta 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 \delta ^{b} _{a}

Derivation

A 1-parameter family of manifolds denoted by

l{M} _{\epsilon} with l{M} _{0} = l{M} ^{4} has metric g _{ik} = η _{ik} + \epsilon h _{ik} . These manifolds can be put together to form a 5-manifold l{N} . A smooth curve \gamma can be constructed through l{N} with tangent 5-vector X , transverse to l{M} _{\epsilon} . If X is defined so that if h _{t} is the family of 1-parameter maps which map l{N} \to l{N} and p _{0} \in l{M} _{0} then a point p _{\epsilon} \in l{M} _{\epsilon} can be written as h _{\epsilon} (p _{0} ) . This also defines a pull back h ^{*} _{\epsilon} that maps a tensor field T _{\epsilon} \in l{M} _{\epsilon} back onto l{M} _{0} . Given sufficient smoothness a Taylor expansion can be definedh ^{*} _{\epsilon} (T _{\epsilon} ) = T _{0} + \epsilon h ^{*} _{\epsilon} ( l{L} _{X} T _{\epsilon} ) + O(\epsilon ^{2} ) \delta T = \epsilon h ^{*} _{\epsilon} ( l{L} _{X} T _{\epsilon} ) \equiv \epsilon ( l{L} _{X} T _{\epsilon} ) _{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 \Delta \delta T = \epsilon( l{L} _{X} T _{\epsilon} ) _{0} - \epsilon( l{L} _{Y} T _{\epsilon} ) _{0} = \epsilon( l{L} _{X-Y} T _{\epsilon)} _{0} . Picking a chart where X ^{a} = (\xi ^{\mu,1)} and Y ^{a} = ( 0 , 1 ) then X ^{a} - Y ^{a} = (\xi ^{\mu} , 0 ) which is a well defined vector in any l{M} _{\epsilon} and gives the result\Delta \delta T = \epsilon l{L} _{\xi} T _{0.}

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

Sources

• Book: Stewart J. . Advanced General Relativity . Cambridge . Cambridge University Press . 1991 . 0-521-44946-4. Describes derivation of result in section on Lie derivatives