Gravitational Waves

(a) gμνA ν = (b) If ∂ν is the contravariant gradient operator, η ∂ν = (c) ggjk = (d) ggAγδ = (e) If n̂ is a unit vector, then nni = (f) In spatial coordinates, δ i = 2. Start with a metric of the form gμν = ημν +hμν . From the definition of the Christoffel symbols Γμν and the Riemann tensor R μ νρσ, show that to linear order in hμν , the Riemann tensor becomes Rμνρσ = 1 2 (∂ν∂ρhμσ + ∂μ∂σhνρ − ∂μ∂ρhνσ − ∂ν∂σhμρ) . (1)