Accelerating away from earth, time such that a light beam won't catch up with you

1. Nov 18, 2012

zardiac

1. The problem statement, all variables and given/known data
You start at t=0 at rest on earth and accelerate with uniform acceleration a away form earth.
Find a point in time $t_0$ such that when a beam emitted from earth at $t>t_0$won't catch up.

2. Relevant equations
$x(t)=c^2/a(\sqrt{1+\frac{a^2}{c^2}t^2}-1)$

3. The attempt at a solution
I think that light travel with velocity c. So if the beam is emitted at $t=t_1$ then at time t, the beam have traveled $c(t-t_1)$. So I try to find the solution for $x(t)=c(t-t_1)$, and I end up with the following expression for $t$:
$t=\frac{a}{2c}\frac{t_1(2-a/c t_1)}{(a/c - a^2/c^2 t_1)}$

According to this the time would be negatic in the intervall $t_1=c/a$ and $t_1=2c/a$ So I think in this intevall the beam won't be able to catch up, but after $t_1=2c/a$ the time becomes positive again, which I don't know how to interpret.
Am I approaching this problem the wrong way?

2. Nov 18, 2012

voko

Think about it this way: will you ever accelerate to a speed greater than that of light? If not, how can you possibly outrun light?

3. Nov 18, 2012

zardiac

Well it is problem 3.9 in D'inverno Introducing Einsteins relativity. I agree that it seem impossible but the problem statement is that if you get a large enough headstart the light won't catch up.

4. Nov 18, 2012

voko

What assumptions are you supposed to make?

5. Nov 18, 2012

SteamKing

Staff Emeritus
negatic?

6. Nov 19, 2012

vela

Staff Emeritus
You should be able to show that your world line is a hyperbola. Find its asymptotes.

7. Nov 19, 2012

TSny

Note t approaches ∞ as the denominator on the right approaches 0.

8. Nov 26, 2012

Myslius

Let's say you keep uniform acceleration a, relative to the stationary observer. After c/a time you will be moving at the speed of light. To keep uniform acceleration you need infinite amount of energy. I think the answer is c/a, just the problem is that you can't keep uniform acceleration.

9. Nov 26, 2012

zardiac

10. Nov 28, 2012

vela

Staff Emeritus
You've misinterpreted the problem. The acceleration is uniform relative to the moving observer. As you noted, you can't have a uniform acceleration relative to the stationary observer indefinitely.