Editing Relativistic rocket

{{Short description|Type of spacecraft}}
'''Relativistic rocket''' means any [[spacecraft]] that travels close enough to [[light speed]] for [[special relativity|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 (0.3''c'' to 0.5''c'') is used. At 30% c, the difference between relativistic mass and rest mass is only about 5%, while at 50% it is 15%, (at 0.75''c'' 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 0.1''c'' using current known technology, but would still require many engineering advances to achieve this. The relativistic [[gamma factor]] <math>\gamma</math> at 10% of light velocity is 1.005. A 0.1''c'' 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 experiment]]s such as the [[twin paradox]].

==Relativistic rocket equation==
As with the classical rocket equation, one wants to calculate the velocity change <math>\Delta v</math> that a rocket can achieve depending on the [[exhaust speed]] <math>v_e</math> and the mass ratio, i. e. the ratio of starting rest mass <math>m_0</math> and rest mass at the end of the acceleration phase (dry mass) <math>m_1</math>.

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 <math>v_e</math> is constant.

In the nonrelativistic case, one knows from the (classical) Tsiolkovsky rocket equation that
:<math>\Delta v = v_e \ln \frac {m_0}{m_1}.</math>
Assuming constant acceleration <math>a</math>, the time span <math>t</math> during which the acceleration takes place is
:<math>t = \frac {v_e}{a} \ln \frac {m_0}{m_1}.</math>
In the relativistic case, the equation is still valid if <math>a</math> is the acceleration in the rocket's reference frame and <math>t</math> is the rocket's proper time because at velocity 0 the [[Special relativity#Force|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
:<math>\frac{m_0}{m_1} = \exp\left[\frac{at}{v_e}\right].</math>
where "exp" is the [[exponential function]]. Another related equation<ref>Forward, Robert L. [http://www.relativitycalculator.com/images/rocket_equations/AIAA.pdf "A Transparent Derivation of the Relativistic Rocket Equation"] {{Webarchive|url=https://web.archive.org/web/20180906064549/http://www.relativitycalculator.com/images/rocket_equations/AIAA.pdf |date=2018-09-06 }} (see the right side of equation 15 on the last page, with R as the ratio of initial to final mass and w as the specific impulse)</ref> gives the mass ratio in terms of the end velocity <math>\Delta v</math> relative to the rest frame (i. e. the frame of the rocket before the acceleration phase):
:<math>\frac{m_0}{m_1} = \left[\frac{1 + {\frac{\Delta v}{c}}}{1 - {\frac{\Delta v}{c}}}\right]^{\frac{c}{2v_e}}.</math>
For constant acceleration, <math>\frac{\Delta v}{c} = \tanh\left[\frac{at}{c}\right]</math> (with a and t again measured on board the rocket),<ref>{{cite web|url=http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html |title=The Relativistic Rocket |publisher=Math.ucr.edu |date= |accessdate=2015-06-21}}</ref> so substituting this equation into the previous one and using the [[hyperbolic function]] [[Identity (mathematics)|identity]] <math>\tanh x = \frac{e^{2x} - 1} {e^{2x} + 1}</math> returns the earlier equation <math>\frac{m_0}{m_1} = \exp\left[\frac{at}{v_e}\right]</math>.

By applying the [[Lorentz transformation]], one can calculate the end velocity <math>\Delta v</math> as a function of the rocket frame acceleration and the rest frame time <math>t'</math>; the result is
:<math>\Delta v = \frac {a t'} {\sqrt{1 + \frac{(a t')^2}{c^2}}}.</math>
The time in the rest frame relates to the proper time by the [[Hyperbolic motion (relativity)|hyperbolic motion]] equation:
:<math>t' = \frac{c}{a} \sinh \left(\frac{a t}{c} \right).</math>
Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for <math>\Delta v</math>, one gets the desired formula:
:<math>\Delta v = c \tanh \left(\frac {v_e}{c} \ln \frac{m_0}{m_1} \right).</math>
The formula for the corresponding [[rapidity]] (the [[Inverse hyperbolic functions|inverse hyperbolic tangent]] of the velocity divided by the speed of light) is simpler:
:<math>\Delta r = \frac {v_e}{c} \ln \frac{m_0}{m_1}.</math>
Since rapidities, contrary to velocities, are additive, they are useful for computing the total <math>\Delta v</math> of a multistage rocket.

==Matter-antimatter annihilation rockets==
It is clear from the above calculations that a relativistic rocket would likely need to be antimatter-fired.{{or|date=April 2024}} Other antimatter rockets in addition to the photon rocket that can provide a 0.6''c'' specific impulse (studied for basic [[hydrogen]]-[[antimatter|antihydrogen]] annihilation, no [[ionization]], no recycling of the radiation<ref name="Analysis of relativistic rocketry; S. Westmoreland; Kansas State University">{{Cite journal|arxiv=0910.1965|last1=Westmoreland|first1=Shawn|title=A note on relativistic rocketry|journal=Acta Astronautica|volume=67|issue=9–10|pages=1248–1251|year=2009|doi=10.1016/j.actaastro.2010.06.050|bibcode=2010AcAau..67.1248W|s2cid=54735356 }}</ref>) needed for interstellar flight include the "beam core" [[pion]] rocket. In a pion rocket, frozen antihydrogen is stored inside electromagnetic bottles. Antihydrogen, like regular hydrogen, is [[diamagnetic]] which allows it to be [[Electromagnetic force|electromagnetically]] [[Magnetic levitation#Diamagnetic levitation|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 (hence several peta[[watt]]s when annihilated with equal amounts of matter). It is then ionized into [[antiproton]]s which can be electromagnetically accelerated into the reaction chamber. The [[positron]]s are usually discarded since their [[Electron–positron annihilation|annihilation]] only produces harmful [[gamma ray]]s with negligible effect on thrust. However, non-relativistic rockets may exclusively rely on these gamma rays for propulsion.<ref name="futureofthings; positron engines">{{cite web |url=http://thefutureofthings.com/3031-new-antimatter-engine-design/ |title=New Antimatter Engine Design|date=29 October 2006 }}</ref> 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.<ref name="science.nasa.gov examining antimatter as a propellent">{{cite web|url=https://science.nasa.gov/science-news/science-at-nasa/1999/prop12apr99_1/ |title=Reaching for the Stars - NASA Science |publisher=Science.nasa.gov |date= |accessdate=2015-06-21}}</ref>

===Design notes on a pion rocket===
The pion rocket has been studied independently by Robert Frisbee<ref name="R. Frisbee's comprehensive analysis of pion propulsion">{{cite web|url=http://www.relativitycalculator.com/images/relativistic_photon_rocket/ANTIMATTER_ROCKET_FOR_INTERSTELLAR_MISSIONS.pdf |title=How to Build an Anitmatter Rocket for Interstellar Missions |publisher=Relativitycalculator.com |accessdate=2015-06-21}}</ref> 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 into the magnetic confinement nozzle of a pion rocket engine, usually as part of hydrogen atoms. The resulting charged pions have a speed of 0.94''c'' (i.e. <math>\beta</math> = 0.94), and a [[Lorentz factor]] <math>\gamma</math> of 2.93 which extends their lifespan enough to travel 21 meters through the nozzle before decaying into [[muon]]s. 60% of the pions will have either a negative, or a positive electric charge. 40% of the pions will be neutral. The neutral pions decay immediately into gamma rays. These can't be reflected by any known material at the energies involved, though they can undergo [[Compton scattering]]. They can be absorbed efficiently by a shield of [[tungsten]] placed between the pion rocket engine reaction volume and the crew modules and various electromagnets to protect them from the gamma rays. The consequent heating of the shield will make it radiate visible light, which could then be collimated to increase the rocket's specific impulse.<ref name="Analysis of relativistic rocketry; S. Westmoreland; Kansas State University"/> The remaining heat will also require the shield to be refrigerated.<ref name="R. Frisbee's comprehensive analysis of pion propulsion"/> 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 moving at 0.94''c''. In realistic matter/antimatter reactions, this jet only represents a fraction of the reaction's mass-energy: over 60% of it is lost as [[gamma-rays]], collimation is not perfect, and some pions are not reflected backward by the nozzle. Thus, the effective exhaust speed for the entire reaction drops to just 0.58c.<ref name="Analysis of relativistic rocketry; S. Westmoreland; Kansas State University"/> Alternate propulsion schemes include physical confinement of hydrogen atoms in an antiproton and pion-transparent [[beryllium]] reaction chamber with collimation of the reaction products achieved with a single external electromagnet; see [[Project Valkyrie]].

==See also==
* The [[Bussard ramjet]]

==General references==
*''The star flight handbook,'' Matloff & Mallove, 1989. 
*''Mirror matter: pioneering antimatter physics,'' Dr. Robert L Forward, 1986

==References==
{{Reflist}}

==External links==
*[http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html Physics FAQs: The Relativistic Rocket]
*[https://web.archive.org/web/20011108173239/http://www.geocities.com/albmont/relroket.htm  Javascript that calculates the Relativistic Rocket Equation]
*''Spacetime Physics: Introduction to Special Relativity'' (1992). W. H. Freeman, {{ISBN|0-7167-2327-1}}
*[http://www.relativitycalculator.com/relativistic_photon_rocket.shtml The Relativistic Photon Rocket]

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[[Category:Interstellar travel]]
[[Category:Rocket propulsion]]