where for the duration of this section all indices shall be raised and lowered using the "flat space" metric η η μ μ ν ν , the "mass" m ( u ) is an arbitrary function of the proper-time u along the mass's world line as measured using the "flat" metric, d u 2 = η η μ μ ν ν d X μ μ d X ν ν , and X μ μ ( u ) describes the arbitrary world line of the mass, λ λ μ μ ( u ) = d X μ μ ( u ) / d u is then the four-velocity of the mass, σ σ μ μ ( x ) is a "flat metric" null-vector field implicitly defined by Eqn. (20), and u ( x ) implicitly extends the proper-time parameter to a scalar field throughout spacetime by viewing it as constant on the outgoing light cone of the "flat" metric that emerges from the event X μ μ ( u ) , and satisfies the identity λ λ μ μ ( u ( x ) ) ∂ ∂ μ μ u ( x ) = 1. Grinding out the Einstein tensor for the metric g μ μ ν ν and integrating the outgoing energy–momentum flux "at infinity," one finds that the metric g μ μ ν ν describes a mass with proper-time dependent four-momentum P μ μ = m ( u ) λ λ μ μ ( u ) that emits a net <<link:0>> at a proper rate of − − d P μ μ / d u ; as viewed from the mass's instantaneous rest-frame, the radiation flux has an angular distribution A ( u ) + B ( u ) cos ⁡ ⁡ ( θ θ ( u ) ) , where A ( u ) and B ( u ) are complicated scalar functions of m ( u ) , λ λ μ μ ( u ) , σ σ μ μ ( u ) , and their derivatives, and θ θ ( u ) is the instantaneous rest-frame angle between the 3-acceleration and the outgoing null-vector. The Kinnersley metric may therefore be viewed as describing the gravitational field of an accelerating photon rocket with a very badly collimated exhaust.