But from the line element we have ε ε = − − x 2 t ˙ ˙ 2 + x ˙ ˙ 2 + y ˙ ˙ 2 + z ˙ ˙ 2 where ε ε ∈ ∈ { − − 1 , 0 , 1 } for timelike, null, and spacelike geodesics, respectively. This gives the fourth first integral, namely