Notes on Relativity and Cosmology - Physics Department, UCSB

128 CHAPTER 5. ACCELERATING REFERENCE FRAMES . . .
Dividing these two equations we have
v = dx
dt
= c tanh(ατ/c); (5.8)
i.e., θ is indeed ατ/c along this curve. Now, we must show that τ is the proper
time along the curve. But
dpropertime 2 = dt 2 − 1 c 2 dx2 = α2 d 2
c 4 dτ2 . (5.9)
So, we need only choose d such that αd/c 2 = 1 and we are done. Thus, d = c 2 /α.
In summary,
If we start a uniformly accelerated object in the right place (c 2 /α away from
the origin), it follows a worldline that remains a constant proper distance (c 2 /α)
from the origin.
For a general choice of starting location (say, x 0 ), it follows a worldline that
remains a constant proper distance c2 α
from some other event. Since it is sometimes
useful to have this more general equation, let us write it down here:
x − x 0 = c2 α
( ( ατ
) )
cosh − 1 . (5.10)
c
5.2 Exploring the uniformly accelerated reference
frame
We have now found that a uniformly accelerating observer with proper acceleration
α follows a worldline that remains a constant proper distance c 2 /α away
from some event. Just which event this is depends on where and when the
observer began to accelerate. For simplicity, let us consider the case where this
special event is the origin. Let us now look more closely at the geometry of the
situation.
5.2.1 Horizons and Simultaneity
The diagram below shows the uniformly accelerating worldline together with a
few important light rays.