Relativistic rocket

As with the classical rocket equation, one wants to calculate the velocity change ${\displaystyle \Delta v}$ that a rocket can achieve depending on the exhaust speed ${\displaystyle v_{e}}$ and the mass ratio, i. e. the ratio of starting rest mass ${\displaystyle m_{0}}$ and rest mass at the end of the acceleration phase (dry mass) ${\displaystyle m_{1}}$.

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 ${\displaystyle v_{e}}$ is constant.

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

${\displaystyle \Delta v=v_{e}\ln {\frac {m_{0}}{m_{1}}}.}$

Assuming constant acceleration ${\displaystyle a}$, the time span ${\displaystyle t}$ during which the acceleration takes place is

${\displaystyle t={\frac {v_{e}}{a}}\ln {\frac {m_{0}}{m_{1}}}.}$

In the relativistic case, the equation is still valid if ${\displaystyle a}$ is the acceleration in the rocket's reference frame and ${\displaystyle t}$ is the rocket's proper time because at velocity 0 the relationship between force and acceleration is the same as in the classical case. Solving this equation for the ratio of initial mass to final mass gives

${\displaystyle {\frac {m_{0}}{m_{1}}}=\exp \left[{\frac {at}{v_{e}}}\right].}$

where "exp" is the exponential function. Another related equation[1] gives the mass ratio in terms of the end velocity ${\displaystyle \Delta v}$ relative to the rest frame (i. e. the frame of the rocket before the acceleration phase):

${\displaystyle {\frac {m_{0}}{m_{1}}}=\left[{\frac {1+{\frac {\Delta v}{c}}}{1-{\frac {\Delta v}{c}}}}\right]^{\frac {c}{2v_{e}}}.}$

For constant acceleration, ${\displaystyle {\frac {\Delta v}{c}}=\tanh \left[{\frac {at}{c}}\right]}$ (with a and t again measured on board the rocket),[2] so substituting this equation into the previous one and using the hyperbolic function identity ${\displaystyle \tanh x={\frac {e^{2x}-1}{e^{2x}+1}}}$ returns the earlier equation ${\displaystyle {\frac {m_{0}}{m_{1}}}=\exp \left[{\frac {at}{v_{e}}}\right]}$.

By applying the Lorentz transformation, one can calculate the end velocity ${\displaystyle \Delta v}$ as a function of the rocket frame acceleration and the rest frame time ${\displaystyle t'}$; the result is

${\displaystyle \Delta v={\frac {at'}{\sqrt {1+{\frac {(at')^{2}}{c^{2}}}}}}.}$

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

${\displaystyle t'={\frac {c}{a}}\sinh \left({\frac {at}{c}}\right).}$

Substituting the proper time from the Tsiolkovsky equation and substituting the resulting rest frame time in the expression for ${\displaystyle \Delta v}$, one gets the desired formula:

${\displaystyle \Delta v=c\tanh \left({\frac {v_{e}}{c}}\ln {\frac {m_{0}}{m_{1}}}\right).}$

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

${\displaystyle \Delta r={\frac {v_{e}}{c}}\ln {\frac {m_{0}}{m_{1}}}.}$

Since rapidities, contrary to velocities, are additive, they are useful for computing the total ${\displaystyle \Delta v}$ of a multistage rocket.

It is clear from the above calculations that a relativistic rocket would likely need to be antimatter-fired. 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[3]) 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 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 (hence several petawatts 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.[4] 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.[5]

• The star flight handbook, Matloff & Mallove, 1989. Also See on the Bussard ramjet page, under the related inventions section.
• Mirror matter: pioneering antimatter physics, Dr. Robert L Forward, 1986

• 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
• The Relativistic Photon Rocket