η η ( v , w ) = η η μ μ ν ν v μ μ w ν ν = v 0 w 0 + v 1 w 1 + v 2 w 2 + v 3 w 3 = v μ μ w μ μ = v μ μ w μ μ , and η η ( v , v ) = η η μ μ ν ν v μ μ v ν ν = v 0 v 0 + v 1 v 1 + v 2 v 2 + v 3 v 3 = v μ μ v μ μ .