(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle of mass m has a position along the x-axis as a function of time given by the equation

u = cgt / (1 + g^{2}t^{2})^{1/2}

where g is a constant and c is the speed of light.

(a) Find the 4-velocity of the particle.

(b) Express x and t as a function of the proper time of the particle.

(c) Find the 4-force acting on the particle. Does it ever exceed the speed of light?

2. Relevant equations

u^{μ}= γ(c, u)

λ = dt / dτ

3. The attempt at a solution

a) Given the first equation, the four-velocity is simply

u^{μ}= γ(c, cgt / (1 + g^{2}t^{2})^{1/2}

I think.

b) To find the position, we take the integral of dx/dt and find

x = (c/g)(1 + g^{2}t^{2})^{1/2}

If we let dt = γdτ, then we can easily see t = γτ.

However, this is a problem, as the particle is accelerating (its second time derivative is not zero) and that means γ must change. And, another problem, is that we cannot sub in

γ = (1 - u^{2}/c^{2})^{-1/2}

because then we have a recursive definition.

c) I know that the four-force is simply mass x second derivative of the four position (or mass x derivative of four-velocity), but I am not too sure how to differentiate it. I also know that the four-force is (F^{0}, F) but I don't know how to find F^{0}:(

PLEASE BE GENTLE. I am a first-year physics student, and my university decided to put general relativity into a first-year course.

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# Homework Help: Four-vectors of a moving particle

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