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 generally speaking a velocity of at least 50% of the speed of light (0.5c) is required. The Lorentz factor - also known as the "gamma" factor, γ, and present in the equations for time dilation, relativistic mass, and length contraction - is equal to 1.15 at 0.5c. Above this speed Einsteinian physics are required to describe motion. Below this speed, motion is approximately described by Newtonian physics and the Tsiolkovsky rocket equation can be used.
Contents
- Relativistic rocket equation
- Specific impulse
- Formula for v
- Matter antimatter annihilation rockets
- Design notes on a pion rocket
- References
In this context, a rocket is defined as an object carrying all of its reaction mass, energy, and engines with it.
Achieving relativistic velocities is difficult, requiring advanced forms of spacecraft propulsion that have not yet been adequately developed. 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 (
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
Specific impulse
The specific impulse of relativistic rockets is the same as the effective exhaust velocity, despite the fact that the nonlinear relationship of velocity and momentum as well as the conversion of matter to energy have to be taken into account; the two effects cancel each other. I.e.
Of course this is only valid if the rocket does not have an external energy source (e. g. a laser beam from a space station; in this case the momentum carried by the laser beam also has to be taken into account). If all the energy to accelerate the fuel comes from an external source (and there is no additional momentum transfer), then the relationship between effective exhaust velocity and specific impulse is as follows:
where
In the case of no external energy source, the relationship between
The inverse relation is
Here are some examples of fuels, the energy conversion fractions and the corresponding specific impulses (assuming no losses):
In actual rocket engines, there will be losses, lowering the specific impulse. In electron-positron annihilation, the gamma rays are emitted in a spherically symmetric fashion, and they almost cannot be reflected with current technology. Therefore they cannot be directed towards the rear. A simple solution would be to have a gamma ray absorber absorbing all the gamma rays moving in the forward direction, delivering part of the thrust; and letting the rest be emitted without any deflection (therefore with an angle of divergence of 180°), which cuts in half the (average) useful momentum of the gamma rays, resulting in the specific impulse being less of what it would be in the idealized case.
Formula for Δv
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
In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that
Assuming constant acceleration
In the relativistic case, the equation still valid if
with "exp" denoting the exponential function. Another related equation gives the mass ratio in terms of the end velocity
For constant acceleration,
By applying the Lorentz transformation on the acceleration, one can calculate the end velocity
The time in the rest frame relates to the proper time by the following equation:
Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for
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
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.
Design notes on a pion rocket
The pion rocket has been studied independently by Robert Frisbee and Ulrich Walter, with similar results. Pions, short for pi-mesons, are produced by proton-antiproton annihilation. The antihydrogen or the antiprotons extracted from it will be mixed with a mass of regular protons pumped inside the magnetic confinement nozzle of a pion rocket engine, usually as part of hydrogen atoms. The resulting charged pions will have a velocity of 0.94c (i.e.