Finding t and x in terms of proper time?

[Physicsphysics]

Homework Statement

A particle is moving along x, uniformly accelerated at a=g=constant.
(a) find x and t as a function of proper time (provided at t=0, x=0 and v=0)
Hint: (now a and u are 4-vectors) consider u and a. What are a.a, u.u and a.u? Use these to find the particle's 4-velocity and integrate to find position.

Relevant Equations

Still 4-velocities
u.u=1
a.u=0
u=(γ,γv)
a=(γ[SUP]4[/SUP]v.a, γ[SUP]2[/SUP]a + γ[SUP]4[/SUP](v.a)v)
On the right hand side, v and a are three vectors

I tried finding a.a (four vector inner product) and I got to γ⁴{(v.a)²(1-γ⁴v.v - 2γ²) - a.a}, where again a and v are three vectors on the rhs (sorry to be confusing). a.a = g² since it's a constant.

I have no idea where to go from here to find the time and position. Please help!

 

[TSny]

Homework Statement:: A particle is moving along x, uniformly accelerated at a=g=constant.

By"uniformly accelerated", I suspect that they mean that ##a^1 = g## in the instantaneous rest frame of the particle. So, if you were to move with the particle you would always experience ##1 g## of acceleration.

Relevant Equations:: Still 4-velocities
u.u=1
a.u=0
u=(γ,γv)
a=(γ⁴v.a, γ²a + γ⁴(v.a)v)

Instead of the last equation, you can use the fact that a.a is an invariant. Thus, a.a in the fixed inertial frame must equal a.a in the instantaneous rest frame of the particle. This allows you to relate ##a^0## and ##a^1## in the fixed frame.