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# 2 gravitational redshift δ f f κ m rc 2 3 redshift

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2. Gravitational redshift: Δ f f = κ M rc 2 . 3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation: λ 0 λ 1 = R 0 R 1 . 3.1.3 The stress-energy tensor and the field tensor The stress-energy tensor is given by: T μ ν = ( c 2 + p ) u μ u ν + pg μ ν + 1 c 2 ( F μ α F α ν + 1 4 g μ ν F αβ F αβ ) The conservation laws can than be written as: ν T μ ν = 0 . The electromagnetic field tensor is given by: F αβ = A β x α - A α x β with A μ := ( A, iV/c ) and J μ := ( J, ic ρ ) . The Maxwell equations can than be written as: ν F μ ν = μ 0 J μ , λ F μ ν + μ F νλ + ν F λ μ = 0 The equations of motion for a charged particle in an EM field become with the field tensor: dp α d τ = qF αβ u β 3.2 General relativity 3.2.1 Riemannian geometry, the Einstein tensor The basic principles of general relativity are: 1. The geodesic postulate: free falling particles move along geodesics of space-time with the proper time τ or arc length s as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds ds = 0 . From δ ds = 0 the equations of motion can be derived: d 2 x α ds 2 + Γ α βγ dx β ds dx γ ds = 0
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