The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant.
δ
if and only if one of the following holds 1.
is a constant scalar field 3.
is a linear combination of products of delta functions
δ
A 1-parameter family of manifolds denoted by
has metric
{\displaystyle g_{ik}=\eta _{ik}+\epsilon h_{ik}}
These manifolds can be put together to form a 5-manifold
A smooth curve
γ
can be constructed through
, transverse to
is the family of 1-parameter maps which map
This also defines a pull back
that maps a tensor field
back onto
Given sufficient smoothness a Taylor expansion can be defined
δ
is the linear perturbation of
However, since the choice of
is dependent on the choice of gauge another gauge can be taken.
Therefore the differences in gauge become
δ
Picking a chart where
μ
μ
which is a well defined vector in any
and gives the result The only three possible ways this can be satisfied are those of the lemma.