Tsiolkovsky rocket equation under relativistic speed

As widely known, Tsiolkovsky rocket equation only works with speed under 14%c

How do we fix it?

The relationship between the final mass M and velocity V of a rocket is derived as follows:



Let the initial mass of the rocket be M0 and the velocity of the ejected material be U, initially at rest. We can use the classical conservation of momentum equation:



d(Mv) - dM(v-u) = 0



Therefore, Mdv + udM = 0



Hence, dv = -u*(dM)/M



Integrating both sides, we get:



v = u*ln(M0/M)



In an upgraded derivation, let's assume the "dynamic mass" of the material ejected by the rocket in the inertial frame to be dm. By energy conservation, dm = -dM. Therefore, we have:



Mdv' = -udM



Using the Lorentz transformation, we have dv' = (dv)/(1-v^2/c^2), where c is the speed of light. Thus,



-dm/M = (c^2/u)*(dv/(c^2-u^2))



Taking the natural logarithm, we get:



ln(M0/M) = (c/2u)*ln((c+v)/(c-v))



Simplifying further, we have:



M0/M = ((c+v)/(c-v))^(c-2u)



Continuing to simplify, we obtain:



v (or ΔV) = c*(1-(M/M0)^(2u/c))/(1+(M/M0)^(2u/c)) (or Ackley Equation? idk its official translation)


quite simple, isn't it?