Hyperbolic Motion: SR Homework Solutions

[Kiwithepike]

Homework Statement

Consider a particle in one-dimensional so called hyperbolic motion
x(t)=$\sqrt{b^{2}+t^{2}}$
where b is a constant.

a) Find$\gamma$(t).
b) Find the proper time $\tau$(t). (assume that $\tau$=0 when t = 0
c) Find x and v$_x$ as functions of the propertime $\tau$.
d) FInd the 4-velocity u$^{\mu}$.

The Attempt at a Solution

A) ok to begin I took the derivative of x(t) to get velocity. tuned out to be t(b$^{2}$+t$^{2}$)$^{-1/2}$.
soo therefor $\gamma$(t) = [itex]\frac{$\sqrt{b^{2}+t^{2}}$}{$\sqrt{1-\frac{t^{2}}{\sqrt{b^{2}+t^{2}}}}$}[/itex]

b) so now $\tau$(0) = $\sqrt{t^{2}-(b^{2}+t^{2}}$
$\tau$(0) = $\sqrt{0^{2}-(b^{2}+t^{0}}$ = 0
$\tau$(0) = $\sqrt{-b^{2}}$ = 0
so would b = 0?
this is where I'm getting lost.
c) x as a function of \tau would be $\sqrt{t^{2}-\tau^{2}}$=x?
where does v$_x$ come in? would i solve v(t) for t^2?

d) I know the 4 vector for u$^{\mu}$ is (u^0,u^1,u^2,u^3) and the roattional lorrentz for hyperbolic is
|t'| = |cosh$\varphi$ -sinh$\varphi$ |
|x'| |-sinh$\varphi$ cosh$\varphi$ |

where tanh$\varphi$=v
where cosh$\varphi$= $\gamma$

where do i go from here? Thanks for all the help.

 

[George Jones]

b) Find the proper time $\tau$(t). (assume that $\tau$=0 when t = 0

Try using $d\tau^2 = dt^2 - dx^2$.