Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems General Relativity: An Introduction for Physicists · Problem 17.3
Linearised general relativity — Hobson, Efstathiou & Lasenby
Physics · linearised-riemann-ricci-einstein-equations
Problem If g μ ν = η μ ν + h μ ν g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} g μν = η μν + h μν ∣ h μ ν ∣ ≪ 1 |h_{\mu\nu}| \ll 1 ∣ h μν ∣ ≪ 1 h μ ν h_{\mu\nu} h μν R σ μ ν ρ = 1 2 ( ∂ ν ∂ μ h ρ σ + ∂ ρ ∂ σ h μ ν − ∂ ν ∂ σ h μ ρ − ∂ ρ ∂ μ h ν σ ) , R^{\sigma}{}_{\mu\nu\rho} = \tfrac{1}{2}(\partial_{\nu}\partial_{\mu}h^{\sigma}_{\rho} + \partial_{\rho}\partial^{\sigma}h_{\mu\nu} - \partial_{\nu}\partial^{\sigma}h_{\mu\rho} - \partial_{\rho}\partial_{\mu}h^{\sigma}_{\nu}), R σ μν ρ = 2 1 ( ∂ ν ∂ μ h ρ σ + ∂ ρ ∂ σ h μν − ∂ ν ∂ σ h μ ρ − ∂ ρ ∂ μ h ν σ ) , R μ ν = 1 2 ( ∂ ν ∂ μ h + □ 2 h μ ν − ∂ ν ∂ ρ h μ ρ − ∂ ρ ∂ μ h ν ρ ) , R_{\mu\nu} = \tfrac{1}{2}(\partial_{\nu}\partial_{\mu}h + \Box^2 h_{\mu\nu} - \partial_{\nu}\partial_{\rho}h^{\rho}_{\mu} - \partial_{\rho}\partial_{\mu}h^{\rho}_{\nu}), R μν = 2 1 ( ∂ ν ∂ μ h + □ 2 h μν − ∂ ν ∂ ρ h μ ρ − ∂ ρ ∂ μ h ν ρ ) , R = □ 2 h − ∂ ρ ∂ μ h μ ρ . R = \Box^2 h - \partial_{\rho}\partial_{\mu}h^{\mu\rho}. R = □ 2 h − ∂ ρ ∂ μ h μ ρ . ∂ ν ∂ μ h + □ 2 h μ ν − ∂ ν ∂ ρ h μ ρ − ∂ ρ ∂ μ h ν ρ − η μ ν ( □ 2 h − ∂ ρ ∂ σ h σ ρ ) = − 2 κ T μ ν . \partial_{\nu}\partial_{\mu}h + \Box^2 h_{\mu\nu} - \partial_{\nu}\partial_{\rho}h^{\rho}_{\mu} - \partial_{\rho}\partial_{\mu}h^{\rho}_{\nu} - \eta_{\mu\nu}(\Box^2 h - \partial_{\rho}\partial_{\sigma}h^{\sigma\rho}) = -2\kappa T_{\mu\nu}. ∂ ν ∂ μ h + □ 2 h μν − ∂ ν ∂ ρ h μ ρ − ∂ ρ ∂ μ h ν ρ − η μν ( □ 2 h − ∂ ρ ∂ σ h σ ρ ) = − 2 κ T μν . Browse the problem library