Notes on Relativity and Cosmology - Physics Department, UCSB

5.2. THE UNIFORMLY ACCELERATED FRAME 129
Future Acceleration Horizon
Signals from this region can
never reach the rocket
(‘‘end of the world’’)
The Light rays from
this event never
catch up with the rocket
The rocket can never
send signals to this region
Past Acceleration Horizon
Note the existence of the light ray marked “future acceleration horizon.” It
marks the boundary of the region of spacetime from which the uniformly accelerated
observer can receive signals, since such signals cannot travel faster than
c. This is an interesting phenomenon in and of itself: merely by undergoing
uniform acceleration, the rocket ship has cut itself off from communication with
a large part of the spacetime. In general, the term ‘horizon’ is used whenever an
object is cut off in this way. On the diagram above there is a light ray marked
“past acceleration horizon” which is the boundary of the region of spacetime to
which the uniformly accelerated observer can send signals.
When considering inertial observers, we found it very useful to know how to draw
their lines of simultaneity and their lines of constant position. Presumably, we
will learn equally interesting things from working this out for the uniformly
accelerating rocket.
But, what notion of simultaneity should the rocket use? Let us define the
rocket’s lines of simultaneity to be those of the associated momentarily comoving
inertial frames. It turns out that these are easy to draw. Let us simply
pick any event A on the uniformly accelerated worldline as shown below. I have
also marked with a Z the event from which the worldline maintains a constant
proper distance.
Z
A
Recall that a boost transformation simply slides the events along the hyperbola.
This means that we can find an inertial frame in which the above picture looks
like this: