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/dλ) 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

δR_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 ∇_mγ_ab − \(1\over2\)∇_a ∇_bγ − \(1\over2\)g_ab (∇^m ∇ⁿ γ_mn − ∇^m ∇_m γ) .

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

T ⁽¹⁾_ab:= (dT_ab / dλ)|_{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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