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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send feedback and suggestions to bombelli at olemiss.edu – modified 16 jan 2016