Perturbations of Metrics  

In General > s.a. gauge transformations.
* Metric perturbation: Denote by γab an infinitesimal change in a metric gab, i.e.,

γab:= (d/dλ) gab(λ)|λ = 0 ,

where gab(λ) is a 1-parameter family of metrics such that gab(0) = gab.
* Linearized volume element: If g denotes the determinant of the unperturbed metric, then to first order in γab the change in the volume element is

δ|g|1/2 = \(1\over2\)|g|1/2 gab γab .

* Linearized connection: If ∇a is the covariant derivative of the unperturbed metric, then to first order in γab the change in the connection coefficients for the perturbed metric is

δΓmab = \(1\over2\)gmn (∇aγnb + ∇bγan – ∇nγab) .

* Linearized Ricci tensor: If ∇a is the covariant derivative of the unperturbed metric, then to first order in γab the change in the Ricci tensor of the perturbed metric is

δRab = ∇m δΓmab – ∇a δΓmbm = 0 .

* Linearized Einstein tensor: If ∇a is the covariant derivative of the unperturbed metric, then to first order in γab the change in the Einstein tensor of the perturbed metric is

G(1)ab = ∇m(aγb) m – \(1\over2\)∇mmγab – \(1\over2\)∇abγ – \(1\over2\)gab (∇mnγmn – ∇mm γ) .

* Stress-energy tensor: For matter fields Φ (with perturbation φ) we write G(1)ab = 8πG T (1)ab, with

T (1)ab:= (dTab /dλ)|λ = 0 = (∂Tab /∂gmn) λmn + (∂Tab /∂Φ) φ .

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


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