Editing Relativistic rocket (section)

==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.