- #1
Kiwithepike
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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) = \frac{\sqrt{b^{2}+t^{2}}}{\sqrt{1-\frac{t^{2}}{\sqrt{b^{2}+t^{2}}}}}
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.
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