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

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.