Stewart–Walker lemma

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

{\displaystyle \Delta \delta T=0}

if and only if one of the following holds

1.

T

0

=

0

{\displaystyle T_{0}=0}

2.

T

0

{\displaystyle T_{0}}

is a constant scalar field

3.

T

0

{\displaystyle T_{0}}

is a linear combination of products of delta functions

δ

a

b

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

Derivation

A 1-parameter family of manifolds denoted by

M

ϵ

{\displaystyle {\mathcal {M}}_{\epsilon }}

with

M

0

=

M

4

{\displaystyle {\mathcal {M}}_{0}={\mathcal {M}}^{4}}

has metric

g

i

k

=

η

i

k

+

ϵ

h

i

k

{\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

M

ϵ

{\displaystyle {\mathcal {M}}_{\epsilon }}

. If

X

{\displaystyle X}

is defined so that if

h

t

{\displaystyle h_{t}}

is the family of 1-parameter maps which map

N

→

N

{\displaystyle {\mathcal {N}}\to {\mathcal {N}}}

and

p

0

∈

M

0

{\displaystyle p_{0}\in {\mathcal {M}}_{0}}

then a point

p

ϵ

∈

M

ϵ

{\displaystyle p_{\epsilon }\in {\mathcal {M}}_{\epsilon }}

can be written as

h

ϵ

(

p

0

)

{\displaystyle h_{\epsilon }(p_{0})}

. This also defines a pull back

h

ϵ

∗

{\displaystyle h_{\epsilon }^{*}}

that maps a tensor field

T

ϵ

∈

M

ϵ

{\displaystyle T_{\epsilon }\in {\mathcal {M}}_{\epsilon }}

back onto

M

0

{\displaystyle {\mathcal {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

)

{\displaystyle h_{\epsilon }^{*}(T_{\epsilon })=T_{0}+\epsilon \,h_{\epsilon }^{*}({\mathcal {L}}_{X}T_{\epsilon })+O(\epsilon ^{2})}

δ

T

=

ϵ

h

ϵ

∗

(

L

X

T

ϵ

)

≡

ϵ

(

L

X

T

ϵ

)

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}

.