# Hyperbolic motion (relativity)

Hyperbolic motion can be visualized on a Minkowski diagram, where the motion of the accelerating particle is along the ${\displaystyle x}$-axis. Each hyperbola is defined by ${\displaystyle X=c^{2}/\alpha .}$

Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.

Regarding the historical development, Hermann Minkowski (1908) showed the relation between a point on a worldline and the magnitude of four-acceleration and a "curvature hyperbola" (German: Krümmungshyperbel).[1] Max Born (1909) subsequently coined the term "hyperbolic motion" (German: Hyperbelbewegung) for the case of constant magnitude of four-acceleration, then provided a detailed description for charged particles in hyperbolic motion, and introduced the corresponding "hyperbolically accelerated reference system" (German: hyperbolisch beschleunigtes Bezugsystem).[2] For early reviews see the textbooks by Max von Laue (1911, 1921)[3] or Wolfgang Pauli (1921).[4] See also Gourgoulhon (2013)[5] and Acceleration (special relativity)#History.

## Worldline and Formulas

The proper acceleration ${\displaystyle \alpha }$ of a particle is defined as the acceleration that a particle "feels" as it accelerates from one inertial reference frame to another. If the proper acceleration is directed parallel to the line of motion, it is related to the ordinary three-acceleration in special relativity ${\displaystyle a=du/dt}$ by

${\displaystyle \alpha =\gamma ^{3}a={\frac {1}{\left(1-u^{2}/c^{2}\right)^{3/2}}}{\frac {du}{dt}},}$

where ${\displaystyle u}$ is the instantaneous speed of the particle, ${\displaystyle \gamma }$ the Lorentz factor, ${\displaystyle c}$ is the speed of light, and ${\displaystyle t}$ is time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time ${\displaystyle t}$ as well as proper time ${\displaystyle \tau }$. For simplification, all initial values for time, location, and velocity can be set to 0, thus:[3][4][6][7][8]

{\displaystyle {\begin{array}{c|c}{\begin{aligned}u(t)&={\frac {\alpha t}{\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}}\\&=c\tanh \left(\operatorname {arsinh} {\frac {\alpha t}{c}}\right)\\x(t)&={\frac {c^{2}}{\alpha }}\left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}-1\right)\\&={\frac {c^{2}}{\alpha }}\left(\cosh \left(\operatorname {arsinh} {\frac {\alpha t}{c}}\right)-1\right)\\c\tau (t)&={\frac {c^{2}}{\alpha }}\ln \left({\sqrt {1+\left({\frac {\alpha t}{c}}\right)^{2}}}+\alpha t\right)\\&={\frac {c^{2}}{\alpha }}\operatorname {arsinh} {\frac {\alpha t}{c}}\end{aligned}}&{\begin{aligned}u(\tau )&=c\tanh {\frac {\alpha \tau }{c}}\\\\x(\tau )&={\frac {c^{2}}{\alpha }}\left(\cosh {\frac {\alpha \tau }{c}}-1\right)\\\\ct(\tau )&={\frac {c^{2}}{\alpha }}\sinh {\frac {\alpha \tau }{c}}\\\\\end{aligned}}\end{array}}}

The function ${\displaystyle x(t)}$ gives ${\displaystyle x^{2}-c^{2}t^{2}=c^{4}/\alpha ^{2},}$ which is a hyperbola in time and the spatial location variable ${\displaystyle x}$. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:[9][10][11]

{\displaystyle {\begin{aligned}u(t)&={\frac {u_{0}\gamma _{0}+\alpha t}{\sqrt {1+\left({\frac {u_{0}\gamma _{0}+\alpha t}{c}}\right)^{2}}}}\quad \\&=c\tanh \left(\operatorname {arsinh} \left({\frac {u_{0}\gamma _{0}+\alpha t}{c}}\right)\right)\\x(t)&=x_{0}+{\frac {c^{2}}{\alpha }}\left({\sqrt {1+\left({\frac {u_{0}\gamma _{0}+\alpha t}{c}}\right)^{2}}}-\gamma _{0}\right)\\&=x_{0}+{\frac {c^{2}}{\alpha }}\left(\cosh \left(\operatorname {arsinh} \left({\frac {u_{0}\gamma _{0}+\alpha t}{c}}\right)\right)-\gamma _{0}\right)\\c\tau (t)&=c\tau _{0}+{\frac {c^{2}}{\alpha }}\ln \left({\frac {{\sqrt {c^{2}+\left(u_{0}\gamma _{0}+\alpha t\right){}^{2}}}+u_{0}\gamma _{0}+\alpha t}{\left(c+u_{0}\right)\gamma _{0}}}\right)\\&=c\tau _{0}+{\frac {c^{2}}{\alpha }}\left(\operatorname {arsinh} \left({\frac {u_{0}\gamma _{0}+\alpha t}{c}}\right)-\operatorname {artanh} \left({\frac {u_{0}}{c}}\right)\right)\\\hline \\u(\tau )&=c\tanh \left(\operatorname {artanh} \left({\frac {u_{0}}{c}}\right)+{\frac {\alpha \tau }{c}}\right)\\x(\tau )&=x_{0}+{\frac {c^{2}}{\alpha }}\left(\cosh \left(\operatorname {artanh} \left({\frac {u_{0}}{c}}\right)+{\frac {\alpha \tau }{c}}\right)-\gamma _{0}\right)\\ct(\tau )&=ct_{0}+{\frac {c^{2}}{\alpha }}\left(\sinh \left(\operatorname {artanh} \left({\frac {u_{0}}{c}}\right)+{\frac {\alpha \tau }{c}}\right)-{\frac {u_{0}\gamma _{0}}{c}}\right)\end{aligned}}}

## References

1. ^ Minkowski, Hermann (1909). "Raum und Zeit. Vortrag, gehalten auf der 80. Naturforscher-Versammlung zu Köln am 21. September 1908." [Space and Time]. Jahresberichte der Deutschen Mathematiker-Vereinigung. Leipzig.
2. ^ Born, Max (1909). "Die Theorie des starren Körpers in der Kinematik des Relativitätsprinzips" [The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity]. Annalen der Physik. 335 (11): 1–56. doi:10.1002/andp.19093351102.
3. ^ a b von Laue, M. (1921). Die Relativitätstheorie, Band 1 (fourth edition of "Das Relativitätsprinzip” ed.). Vieweg. pp. 89–90, 155–166.; First edition 1911, second expanded edition 1913, third expanded edition 1919.
4. ^ a b Pauli, W. (1921). "Die Relativitätstheorie". Encyclopädie der mathematischen Wissenschaften. 5.2. pp. 626–628, 647–648. New edition 2013: Editor: Domenico Giulini, Springer, 2013 ISBN 3642583555.
5. ^ Gourgoulhon, E. (2013). Special Relativity in General Frames: From Particles to Astrophysics. Springer. p. 396. ISBN 3642372767.
6. ^ Møller, C. (1955). The theory of relativity. Oxford Clarendon Press. pp. 74–75.
7. ^ Rindler, W. (1977). Essential Relativity. Springer. pp. 49–50. ISBN 354007970X.
8. ^ PhysicsFAQ (2016), "Relativistic rocket", see external links
9. ^ Gallant, J. (2012). Doing Physics with Scientific Notebook: A Problem Solving Approach. John Wiley & Sons. pp. 437–441. ISBN 0470665971.
10. ^ Müller, T., King, A., & Adis, D. (2006). "A trip to the end of the universe and the twin "paradox"". American Journal of Physics. 76 (4): 360–373. arXiv:. doi:10.1119/1.2830528.
11. ^ Fraundorf, P. (2012). "A traveler-centered intro to kinematics": IV–B. arXiv:.