Relativistic rocket means any spacecraft that travels close enough to light speed for relativistic effects to become significant. The meaning of "significant" is a matter of context, but often a threshold velocity of 30% to 50% of the speed of light (0.3c to 0.5c) is used. At 30% c, the difference between relativistic mass and rest mass is only about 5%, while at 50% it is 15%, (at 0.75c the difference is over 50%); so above such speeds special relativity is needed to accurately describe motion, while below this range Newtonian physics and the Tsiolkovsky rocket equation usually give sufficient accuracy. In this context, a rocket is defined as an object carrying all of its reaction mass, energy, and engines with it. No known technology can bring a rocket to relativistic speed. Relativistic rockets require huge advances in spacecraft propulsion, energy storage, and engine efficiency which may or may not ever be possible. Nuclear pulse propulsion could theoretically reach 0.1c using current known technology, but would still require many engineering advances to achieve this. The relativistic gamma factor γ {\displaystyle \gamma } at 10% of light velocity is 1.005. A 0.1c speed rocket is thus considered non-relativistic since its motion is still quite accurately described by Newtonian physics alone. Relativistic rockets are usually seen discussed in the context of interstellar travel, since most would need a lot of space to reach such speed. They are also found in some thought experiments such as the twin paradox.
Relativistic rocket equation As with the classical rocket equation, one wants to calculate the velocity change Δ v {\displaystyle \Delta v} that a rocket can achieve depending on the exhaust speed v e {\displaystyle v_{e}} and the mass ratio, i. e. the ratio of starting rest mass m 0 {\displaystyle m_{0}} and rest mass at the end of the acceleration phase (dry mass) m 1 {\displaystyle m_{1}} . In order to make calculations simpler, we assume that the acceleration is constant (in the rocket's reference frame) during the acceleration phase; still, the result is nonetheless valid if the acceleration varies, as long as exhaust velocity v e {\displaystyle v_{e}} is constant. In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that
Δ v = v e ln m 0 m 1 . {\displaystyle \Delta v=v_{e}\ln {\frac {m_{0}}{m_{1}}}.}
Assuming constant acceleration a {\displaystyle a} , the time span t {\displaystyle t} during which the acceleration takes place is
t = v e a ln m 0 m 1 . {\displaystyle t={\frac {v_{e}}{a}}\ln {\frac {m_{0}}{m_{1}}}.}
In the relativistic case, the equation is still valid if a {\displaystyle a} is the acceleration in the rocket's reference frame and t {\displaystyle t} is the rocket's proper time because at velocity 0 the relationship between force and acceleration is the same as in the classical case. Solving this equation for the ratio of initial mass to final mass gives
m 0 m 1 = exp [ a t v e ] . {\displaystyle {\frac {m_{0}}{m_{1}}}=\exp \left[{\frac {at}{v_{e}}}\right].}
where "exp" is the exponential function. Another related equation gives the mass ratio in terms of the end velocity Δ v {\displaystyle \Delta v} relative to the rest frame (i. e. the frame of the rocket before the acceleration phase):
… excerpt ends here. Continue reading the full article.
