Relativistic rocket

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A relativistic rocket is any spacecraft that is travelling at a velocity close enough to light speed for relativistic effects to become significant. What "significant" means is a matter of context, but generally speaking a velocity of at least 50% of the speed of light (0.5c) is required. The time dilation factor, mass factor, and length contraction factor (all these factors equal the Lorentz factor) are 1.15 at 0.5c. Above this speed Einstein's physics are required to describe motion. Below this speed, motion is approximately described by Newtonian physics and the Tsiolkovsky rocket equation can be used.

We define a rocket as carrying all of its reaction mass, energy, and engines with it. Bussard ramjets, RAIRs, light sails, and maser or laser-electric vehicles are not rockets.

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 (γ) 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 0.10c 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 Δv that a rocket can achieve depending on the specific impulse Isp and the mass ratio, i. e. the ratio of starting mass m0 and mass at the end of the acceleration phase (dry mass) m1. It should be noted that subsequently specific impulse means the momentum produced by the exhaust of a certain amount of rocket fuel divided by the mass of that amount of rocket fuel. Thus specific impulse is a velocity, as opposed to the common usage of the word as the ratio of momentum and weight (weight would not make much sense in this context).

### 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. 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:

$I_{sp} = \frac {v_e}{\sqrt{1 - \frac{v_e^2}{c^2}}} = \gamma \ v_e$

where γ is the Lorentz factor. In the case of no external energy source, the relationship between Isp and the fraction of the fuel mass η which is converted into energy might also be of interest; assuming no losses, it is

$\eta = 1 - \sqrt{1 - \frac{I_{sp}^2}{c^2}} = 1 - \frac{1}{\gamma}$

The inverse relation is

$I_{sp} = c \cdot \sqrt{2 \eta - \eta^2}$

Here are some examples of fuels, the energy conversion fractions, and the specific impulses (assuming no losses if not specified otherwise):

Fuel η Isp / c
electron-positron annihilation 1 1
proton-antiproton annihilation, using only charged pions 0.56 0.60
electron-positron annihilation with simple hemispherical absorption of gamma rays 1 0.25
nuclear fusion: H to He 0.00712 0.119
nuclear fission: 235U 0.001 0.04

### Delta-v

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 Isp is constant.

In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that

$\Delta v = I_{sp} \ln \frac {m_0}{m_1}$

Assuming constant acceleration a, the time span t during which the acceleration takes place is

$t = \frac {I_{sp}}{a} \ln \frac {m_0}{m_1}$

In the relativistic case, the equation still valid if a is the acceleration in the rocket's reference frame and 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.

By applying the Lorentz transformation on the acceleration, one can calculate the end velocity Δv relative to the rest frame (i. e. the frame of the rocket before the acceleration phase) as a function of the rocket frame acceleration and the rest frame time t'; the result is

$\Delta v = \frac {a \cdot t'} {\sqrt{1 + \frac{(a \cdot t')^2}{c^2}}}$

The time in the rest frame relates to the proper time by the following equation:

$t' = \frac{c}{a} \sinh \left(\frac{a \cdot t}{c} \right)$

Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for Δv, one gets the desired formula:

$\Delta v = c \cdot \tanh \left(\frac {I_{sp}}{c} \ln \frac{m_0}{m_1} \right)$

The formula for the corresponding rapidity (the area hyperbolic tangent of the velocity divided by the speed of light) is simpler:

$\Delta r = \frac {I_{sp}}{c} \ln \frac{m_0}{m_1}$

Since rapidities, contrary to velocities, are additive, they are useful for computing the total Δv 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 0.5c needed for interstellar space flight include the "beam core" pion rocket. In a pion rocket antimatter is stored inside superconducting electromagnetic bottles in the form of electromagnetically levitated frozen antihydrogen. Laser beams vaporize and ionize the antihydrogen a rate of a few grams per second.

The pion rocket might need a superconducting nozzle with electromagnets of 10 teslas or more. Antihydrogen and regular hydrogen are diamagnetic which allows them to be electromagnetically levitated.

### Design notes on a pion rocket

The Pion rocket has been studied independently by Robert Frisbee and Ulrich Walter, with similar results. Pions, also known as pi-mesons, are produced by proton-antiproton annihilation. The antiprotons, in the form of solid anti-hydrogen, will be mixed with an exactly equal mass of regular protons pumped inside the magnetic confinement nozzle of a pion rocket engine. The resulting charged pions will have a velocity of 0.94c (i.e. β = 0.94) and a Lorentz factor γ of 2.93 which extends their lifespan enough to travel 2.6 meters through the nozzle, before decaying into muons. Sixty percent of the pions will have either a negative, or a positive electric charge. Forty percent of the pions will be neutral. The neutral pions will decay immediately into gamma rays. Gamma rays can't be reflected by any known material at these energies, although they can undergo Compton scattering. They can be absorbed efficiently by a shield of tungsten placed between the pion rocket engine nozzle and the crew modules and anti-hydrogen tanks to protect them from the gamma rays. The charged pions would travel in helical spirals around the axial electromagnetic field lines inside the nozzle and in this way the charged pions could be collimated into an exhaust jet that is moving at 0.94c. If 1 kg of these pions were expelled per second the pion engine would have a thrust of 282,000,000 newtons, but in realistic matter/antimatter reactions 78% of the propellant mass-energy is lost as gamma-rays and thus the effective exhaust velocity drops to just 0.33c. The property of diamagnetism can be used to store the antimatter in the form of anti-hydrogen ice magnetically levitated in the center of a superconducting electromagnetic vacuum bottle. It would need to be kept below 1 kelvin to avoid sublimation and annihilation on the vessel's walls.

## Sources

1. The star flight handbook, Matloff & Mallove, 1989
2. Mirror matter: pioneering antimatter physics, Dr. Robert L Forward, 1986

Published - July 2009