Notes on Relativity and Cosmology - Physics Department, UCSB

5.2. THE UNIFORMLY ACCELERATED FRAME 131
And, of course, all of the lines of simultaneity pass through event Z where the
horizons cross. These strange-sounding features of the diagram should remind
you of the weird effects we found associated with Gaston’s acceleration in our
discussion of the twin paradox in section 4.2.
As with Gaston, one is tempted to ask “How can the rocket see things running
backward in time in the left-most region?” In fact, the rocket does not see, or
even know about, anything in this region. As we mentioned above, no signal
of any kind from any event in this region can ever catch up to the rocket. As
a result, this phenomenon of finding things to run backwards in time is a pure
mathematical artifact and is not directly related to anything that observers on
the rocket actually notice.
5.2.2 Friends on a Rope
In the last section we uncovered some odd effects associated with the the acceleration
horizons. In particular, we found that there was a region in which
the lines of simultaneity seemed to run backward. However, we also found that
the rocket could neither signal this region nor receive a signal from it. As a
result, the fact that the lines of simultaneity run backward here is purely a
mathematical artifact.
Despite our discussion above, you might wonder if that funny part of the rocket’s
reference frame might somehow still be meaningful. It turns out to be productive
to get another perspective on this, so let’s think a bit about how we might
actually construct a reference frame for the rocket.
Suppose, for example, that I sit in the nose (the front) of the rocket. I would
probably like to use our usual trick of asking some of my friends (or the students
in class) to sit at a constant distance from me in either direction. I would then
try to have them observe nearby events and tell me which ones happen where.
We would like to know what happens to the ones that lie below the horizon.
Let us begin by asking the question: what worldlines do these fellow observers
follow?
Let’s see.... Consider a friend who remains a constant distance ∆ below us as
measured by us; that is, as measured in the momentarily co-moving frame of reference.
This means that this distance is measured along our line of simultaneity.
But look at what this means on the diagram below: