Relativistic rocket refers to any spacecraft that travels at a velocity 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% of 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 that above this range of speeds special relativity is required to accurately describe motion, whereas below this range sufficient accuracy is usually provided by Newtonian physics and the Tsiolkovsky rocket equation.
In this context, a rocket is defined as an object carrying all of its reaction mass, energy, and engines with it.
There is no known technology capable of accelerating a rocket to relativistic velocities. Relativistic rockets require enormous advances in spacecraft propulsion, energy storage, and engine efficiency which may or may not ever be possible. Nuclear pulse propulsion could theoretically achieve 0.1c using current known technologies, but would still require many engineering advances to achieve this. The relativistic gamma factor () at 10% of light velocity is 1.005. The time dilation factor of 1.005 which occurs at 10% of light velocity is too small to be of major significance. A 0.1c velocity interstellar rocket is thus considered to be a non-relativistic rocket because its motion is quite accurately described by Newtonian physics alone.
Relativistic rockets are usually seen discussed in the context of interstellar travel, since most would require a great deal of space to accelerate up to those velocities. 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 that a rocket can achieve depending on the exhaust velocity and the mass ratio, i. e. the ratio of starting rest mass and rest mass at the end of the acceleration phase (dry mass) .
In order to make the calculations simpler, we assume that the acceleration is constant (in the rocket's reference frame) during the acceleration phase; however, the result is nonetheless valid if the acceleration varies, as long as exhaust velocity is constant.
In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that
Assuming constant acceleration , the time span during which the acceleration takes place is
In the relativistic case, the equation is still valid if is the acceleration in the rocket's reference frame and 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
where "exp" is the exponential function. Another related equation gives the mass ratio in terms of the end velocity relative to the rest frame (i. e. the frame of the rocket before the acceleration phase):
For constant acceleration, (with a and t again measured on board the rocket), so substituting this equation into the previous one and using the hyperbolic function identity returns the earlier equation .
By applying the Lorentz transformation, one can calculate the end velocity as a function of the rocket frame acceleration and the rest frame time ; the result is
The time in the rest frame relates to the proper time by the hyperbolic motion equation:
Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for , one gets the desired formula:
The formula for the corresponding rapidity (the inverse hyperbolic tangent of the velocity divided by the speed of light) is simpler:
Since rapidities, contrary to velocities, are additive, they are useful for computing the total of a multistage rocket.
Matter-antimatter annihilation rockets
It is clear on the basis of the above calculations that a relativistic rocket would likely need to be a rocket that is fueled by antimatter. Other antimatter rockets in addition to the photon rocket that can provide a 0.6c specific impulse (studied for basic hydrogen-antihydrogen annihilation, no ionization, no recycling of the radiation) needed for interstellar space flight include the "beam core" pion rocket. In a pion rocket, antimatter is stored inside electromagnetic bottles in the form of frozen antihydrogen. Antihydrogen, like regular hydrogen, is diamagnetic which allows it to be electromagnetically levitated when refrigerated. Temperature control of the storage volume is used to determine the rate of vaporization of the frozen antihydrogen, up to a few grams per second (amounting to several petawatts of power when annihilated with equal amounts of matter). It is then ionized into antiprotons which can be electromagnetically accelerated into the reaction chamber. The positrons are usually discarded since their annihilation only produces harmful gamma rays with negligible effect on thrust. However, non-relativistic rockets may exclusively rely on these gamma rays for propulsion. This process is necessary because un-neutralized antiprotons repel one another, limiting the number that may be stored with current technology to less than a trillion.