Relativistic rocket equation ($m{\gamma }^{2}dv+udm=0$)

For collinear motion without external forces, a rocket ejecting material backwards at constant speed $u$ in its instantaneous rest frame obeys $m{\gamma }^{2}dv+udm=0$. Here $m$ is the rocket's rest mass, including stored internal energy, and $\gamma$ is its Lorentz factor in the chosen inertial frame. The equation follows from momentum conservation, conservation of relativistic energy, and the relativistic velocity-addition formula; expelled rest mass is not simply the decrease in rocket rest mass.

A photon rocket converts stored mass-energy into a collimated beam of photons. In its instantaneous rest frame, energy conservation gives $c^{2}dM=-d{E}_{\gamma }$, and momentum conservation gives $Mcd\eta =d{E}_{\gamma }/c$. Therefore a monotone change in rapidity costs $M_f/M_i=e^{-|\Delta \eta |}$. The rocket's rest mass includes its fuel and stored internal energy; inefficiency or an angular spread of the exhaust increases the fuel requirement.

Accelerating from rest to rapidity $\eta$ and braking to rest requires two separate photon-rocket burns. Their mass ratios multiply, giving $M_f/M_i=e^{-2\eta }$. For ideal equal-mass matter–antimatter annihilation, the initial antimatter mass fraction is $(1-e^{-2\eta })/2$. Braking fuel must be carried and accelerated during the first burn.