Express this trajectory in terms of proper time

[JTFreitas]

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.

 

[PeroK]

You need two dollar signs to delimit Latex.

 

[PeroK]

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.

 

[Ibix]

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 fixed your maths - use #[/color]# to delimit inline maths, not $, and use $$ for "paragraph" maths. Also \sqrt needs curly brackets, not parentheses.

Your working seems to be correct. What did you get when you tried to integrate? I get a fairly straightforward expression. If you got an inverse sinh plus a term involving a square root, you solved $d\tau=\gamma dt$ instead of $dt/\gamma$.

 

[JTFreitas]

I fixed your maths - use ## to delimit inline maths, not ,anduse,anduse$ for "paragraph" maths. Also \sqrt needs curly brackets, not parentheses.

Your working seems to be correct. What did you get when you tried to integrate? I get a fairly straightforward expression. If you got an inverse sinh plus a term involving a square root, you solved dτ=γdt instead of dt/γ.

Thanks for the fix. I saw the LaTeX guide now, but my brain typed with the usual inline.

Regarding the work: I found the integral through a formula: $\ln{\kappa t + \kappa^2 t^2}$, but I just found out that the same integral can also be written as $\sinh^{-1}(\kappa t)$. This does make the rest of the problem much easier. I am still not used to the type of functions that are common in relativity. In great part I just need pointers to make sure I am in the right path. Thank you!