Hyperbolic motion (relativity): Difference between revisions

 

:<math>\begin{array}{c|c}

& =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)\\

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}

\\

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{array}</math>

 

The function <math>x(t)</math> gives <math>x^2-c^2t^2=c^4/\alpha^2,</math> which is a hyperbola in time and the spatial location variable <math>x</math>. If instead there are initial values different from zero, the formulas for hyperbolic motion assume the form:<ref>{{Cite book|author=Gallant, J.|title=Doing Physics with Scientific Notebook: A Problem Solving Approach|year=2012|pages=437–441|publisher=John Wiley & Sons|isbn=0470665971}}</ref><ref>{{Cite journal|author=Müller, T., King, A., & Adis, D.|year=2006|title=A trip to the end of the universe and the twin "paradox"|journal=American Journal of Physics|volume=76|issue=4|pages=360–373|arxiv=physics/0612126|doi=10.1119/1.2830528}}</ref><ref name=fraundorf>{{Cite journal|author=Fraundorf, P. |year=2012|title=A traveler-centered intro to kinematics|journal=|volume=|issue=|pages=IV-B|arxiv=1206.2877|doi=}}</ref>

 

& =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(\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)

 

==See also==