- #1
JTFreitas
- 18
- 3
- Homework Statement
- Consider an object whose motion is described by ##x(t) = \frac{c}{\kappa}(\sqrt{1+\kappa^2 t^2} -1)##, where ##c## is the speed of light and ##\kappa## is some constant. Express ##x## and ##t## as a function of proper time.
- Relevant Equations
- $$\frac{dt}{d \tau} = \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$
The object moves solely on the $x$-axis, hence I calculated its speed to be $v_x = \frac{dx}{dt} = \frac{c \kappa t}{\sqrt(1+\kappa ^{2} t^2$ Because its speed is not constant, I suppose the Lorentz factor $\gamma = \gamma (t)$, and by plugging in the velocity, I obtain $\gamma = \sqrt(1+ \kappa ^2 t^2)$
I was told to integrate the relationship between $d\tau$ and $dt$ which is just $d\tau = \frac{dt}{\gamma}$. However, integrating this to obtain $\tau$ as a function of $t$ yields a transcendental function that can't exactly be inverted (to express $t$ in terms of $\tau$), and I am stuck in what to try here, in order to obtain the trajectory $x(\tau)$.
Edit: probably obvious but $\tau$ is the proper time.
I was told to integrate the relationship between $d\tau$ and $dt$ which is just $d\tau = \frac{dt}{\gamma}$. However, integrating this to obtain $\tau$ as a function of $t$ yields a transcendental function that can't exactly be inverted (to express $t$ in terms of $\tau$), and I am stuck in what to try here, in order to obtain the trajectory $x(\tau)$.
Edit: probably obvious but $\tau$ is the proper time.