Relativistic rocket
Encyclopedia
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 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.
We define a rocket as carrying all of its reaction mass, energy, and engines with it. Bussard ramjet
s, 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 (
) 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 experiment
s such as the twin paradox
.
that a rocket can achieve depending on the specific impulse
and the mass ratio, i. e. the ratio of starting mass 
and mass at the end of the acceleration phase (dry mass) 
. 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).
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
is the Lorentz factor
.
In the case of no external energy source, the relationship between
and the fraction of the fuel mass 
which is converted into energy might also be of interest; assuming no losses, is
The inverse relation is
Here are some examples of fuels, the energy conversion fractions and the corresponding specific impulses (assuming no losses if not specified otherwise):
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 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.
By applying the Lorentz transformation
on the acceleration, one can calculate the end velocity
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 
; the result is
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
, one gets the desired formula:
The formula for the corresponding rapidity
(the area 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.
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.
rocket has been studied independently by Robert Frisbee and Ulrich Walter, with similar results. Pion
s, 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.
= 0.94) and a Lorentz factor
of 2.93 which extends their lifespan enough to travel 2.6 meters through the nozzle, before decaying into muon
s. 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.
Spacecraft
A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....
that is travelling at a velocity close enough to light speed for relativistic
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
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
Lorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
) are 1.15 at 0.5c. Above this speed Einsteinian
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
physics are required to describe motion. Below this speed, motion is approximately described by Newtonian physics and the Tsiolkovsky rocket equation
Tsiolkovsky rocket equation
The Tsiolkovsky rocket equation, or ideal rocket equation is an equation that is useful for considering vehicles that follow the basic principle of a rocket: where a device that can apply acceleration to itself by expelling part of its mass with high speed and moving due to the conservation of...
can be used.
We define a rocket as carrying all of its reaction mass, energy, and engines with it. Bussard ramjet
Bussard ramjet
The Bussard ramjet is a theoretical method of spacecraft propulsion proposed in 1960 by the physicist Robert W. Bussard, popularized by Larry Niven in his Known Space series of books, and referred to by Carl Sagan in the television series and book Cosmos....
s, RAIRs, light sails, and maser
Maser
A maser is a device that produces coherent electromagnetic waves through amplification by stimulated emission. Historically, “maser” derives from the original, upper-case acronym MASER, which stands for "Microwave Amplification by Stimulated Emission of Radiation"...
or laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
-electric vehicles are not rockets.
Achieving relativistic velocities is difficult, requiring advanced forms of spacecraft propulsion
Spacecraft propulsion
Spacecraft propulsion is any method used to accelerate spacecraft and artificial satellites. There are many different methods. Each method has drawbacks and advantages, and spacecraft propulsion is an active area of research. However, most spacecraft today are propelled by forcing a gas from the...
that have not yet been adequately developed. Nuclear pulse propulsion
Nuclear pulse propulsion
Nuclear pulse propulsion is a proposed method of spacecraft propulsion that uses nuclear explosions for thrust. It was first developed as Project Orion by DARPA, after a suggestion by Stanislaw Ulam in 1947...
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 0.10c 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
Interstellar travel
Interstellar space travel is manned or unmanned travel between stars. The concept of interstellar travel in starships is a staple of science fiction. Interstellar travel is much more difficult than interplanetary travel. Intergalactic travel, or travel between different galaxies, is even more...
, since most would require a great deal of space to accelerate up to those velocities. They are also found in some thought experiment
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
s such as the twin paradox
Twin paradox
In physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...
.
Relativistic rocket equation
As with the classical rocket equation, one wants to calculate the velocity change that a rocket can achieve depending on the specific impulseSpecific impulse
Specific impulse is a way to describe the efficiency of rocket and jet engines. It represents the derivative of the impulse with respect to amount of propellant used, i.e., the thrust divided by the amount of propellant used per unit time. If the "amount" of propellant is given in terms of mass ,...
and the mass ratio, i. e. the ratio of starting mass and mass at the end of the acceleration phase (dry mass) . 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. 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
is the Lorentz factorLorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
.
In the case of no external energy source, the relationship between
and the fraction of the fuel mass which is converted into energy might also be of interest; assuming no losses, isThe inverse relation is
Here are some examples of fuels, the energy conversion fractions and the corresponding specific impulses (assuming no losses if not specified otherwise):
| Fuel |  |
 |
|---|---|---|
| electron Electron The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton... -positron Positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron... annihilation |
1 | 1 |
| proton Proton The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number.... -antiproton Antiproton The antiproton is the antiparticle of the proton. Antiprotons are stable, but they are typically short-lived since any collision with a proton will cause both particles to be annihilated in a burst of energy.... annihilation, using only charged pion Pion In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force.... s |
0.56 | 0.60 |
| electron Electron The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton... -positron Positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron... annihilation with simple hemispherical absorption of gamma rays |
1 | 0.25 |
| electron Electron The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton... -positron Positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron... annihilation with hemispherical Compton scattering |
1 | >0.25 |
| nuclear fusion Nuclear fusion Nuclear fusion is the process by which two or more atomic nuclei join together, or "fuse", to form a single heavier nucleus. This is usually accompanied by the release or absorption of large quantities of energy... : H Hydrogen Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly... to He Helium Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table... |
0.00712 | 0.119 |
| nuclear fission Nuclear fission In nuclear physics and nuclear chemistry, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts , often producing free neutrons and photons , and releasing a tremendous amount of energy... : 235U Uranium Uranium is a silvery-white metallic chemical element in the actinide series of the periodic table, with atomic number 92. It is assigned the chemical symbol U. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons... |
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 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 isIn the relativistic case, the equation 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.By applying the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
on the acceleration, one can calculate the end velocity
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 ; the result isThe 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
, one gets the desired formula:The formula for the corresponding rapidity
Rapidity
In relativity, rapidity is an alternative to speed as a framework for measuring motion. On parallel velocities rapidities are simply additive, unlike speeds at relativistic velocities. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The...
(the area 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 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 PionPion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
rocket has been studied independently by Robert Frisbee and Ulrich Walter, with similar results. Pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
s, 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.
= 0.94) and a Lorentz factorLorentz factor
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ . It gets its name from its earlier...
of 2.93 which extends their lifespan enough to travel 2.6 meters through the nozzle, before decaying into muonMuon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
s. 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
Compton scattering
In physics, Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The inelastic scattering of photons in matter results in a decrease in energy of an X-ray or gamma ray photon, called the Compton effect...
. They can be absorbed efficiently by a shield of tungsten
Tungsten
Tungsten , also known as wolfram , is a chemical element with the chemical symbol W and atomic number 74.A hard, rare metal under standard conditions when uncombined, tungsten is found naturally on Earth only in chemical compounds. It was identified as a new element in 1781, and first isolated as...
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
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...
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
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
to avoid sublimation and annihilation on the vessel's walls.
Sources
- The star flight handbook, Matloff & Mallove, 1989
- Mirror matter: pioneering antimatter physics, Dr. Robert L Forward, 1986
External links
- Physics FAQs: The Relativistic Rocket
- Javascript that calculates the Relativistic Rocket Equation
- Spacetime Physics: Introduction to Special Relativity (1992). W. H. Freeman, ISBN 0-7167-2327-1