Perturbations
of Metrics |

**In General** > s.a. gauge transformations.

* __Metric perturbation__: Denote by *γ*_{ab} an infinitesimal change
in a metric *g*_{ab}, i.e.,

*γ*_{ab}:=
(d/d*λ*) *g*_{ab}(*λ*)|_{λ = 0} ,

where *g*_{ab}(*λ*)
is a 1-parameter family of metrics such that *g*_{ab}(0)
= *g*_{ab}.

* __Linearized volume element__: If
g denotes the determinant of the unperturbed metric, then to first order in *γ*_{ab} the
change in the volume element is

δ|*g*|^{1/2} =
\(1\over2\)|*g*|^{1/2} g^{ab} *γ*_{ab} .

* __Linearized connection__:
If ∇_{a} is the covariant
derivative of the unperturbed metric, then to first order in *γ*_{ab} the
change in the connection coefficients for the perturbed metric is

δΓ^{m}_{ab} = \(1\over2\)*g*^{mn}
(∇_{a}*γ*_{nb}
+ ∇_{b}*γ*_{an} – ∇_{n}*γ*_{ab})
.

* __Linearized Ricci tensor__:
If ∇_{a} is the covariant
derivative of the unperturbed metric, then to first order in *γ*_{ab} the
change in the Ricci tensor of the perturbed metric is

δ*R*_{ab}
= ∇_{m }δΓ^{m}_{ab} – ∇_{a} δΓ^{m}_{bm} = 0 .

* __Linearized Einstein tensor__:
If ∇_{a} is the covariant
derivative of the unperturbed metric, then to first order in *γ*_{ab} the
change in the Einstein tensor of the perturbed metric is

*G*^{(1)}_{ab} = ∇^{m}∇_{(a}*γ*_{b) m} – \(1\over2\)∇^{m}∇_{m}*γ*_{ab}
– \(1\over2\)∇_{a}∇_{b}*γ* – \(1\over2\)g_{ab} (∇^{m}∇^{n}*γ*_{mn}
– ∇^{m}∇_{m} *γ*)
.

* __Stress-energy tensor__: For matter fields Φ (with perturbation *φ*)
we write *G*^{(1)}_{ab} =
8π*G* *T*^{ (1)}_{ab}, with

*T*^{ (1)}_{ab}:=
(d*T*_{ab} /d*λ*)|_{λ = 0}
= (∂*T*_{ab} /∂*g*_{mn})
*λ*_{mn} +
(∂*T*_{ab} /∂Φ) *φ* .

**Applications** > see black-hole
perturbations, cosmological
perturbations and perturbations in general relativity.

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jan 2016