Notes on Relativity and Cosmology - Physics Department, UCSB

124 CHAPTER 5. ACCELERATING REFERENCE FRAMES . . .
where the E-subscripts remind us that this is to be computed in the momentarily
co-moving inertial frame at event E. Notice the analogy with the definition of
proper time along a worldline, which says that the proper time is the time as
measured in a co-moving inertial frame (i.e., a frame in which the worldline is
at rest).
An important point is that, although our computation of α(E) involves a discussion
of certain reference frames, α(E) is a quantity that is intrinsic to the
motion of the rocket and does not depend on choosing of some particular inertial
frame from which to measure it. Thus, it is not necessary to specify an inertial
frame in which α(E) is measured, or to talk about α(E) “relative” to some
frame. As with proper time, we use a Greek letter (α) to distinguish proper
acceleration from the more familiar frame-dependent acceleration a.
We should also point out that the notion of proper acceleration is also just
how the rocket would naturally measure its own acceleration (relative to inertial
frames). For example, a person in the rocket might decide to drop a rock
out the window at event E. If the rock is gently released at event E, it will
initially have no velocity relative to the rocket – its frame of reference will be
the momentarily co-moving inertial frame at event E. Of course, the rock will
then begin to be left behind. If the observer in the rocket measures the relative
acceleration between the rock and the rocket, this will be the same size (though
in the opposite direction) as the acceleration of the rocket as measured by the
(inertial and momentarily co-moving) rock. In other words, it will be the proper
acceleration α of the rocket.
5.1.2 Uniform Acceleration and Boost Parameters
So, now we know what we mean by uniform acceleration. But, it would be
useful to know how to draw this kind of motion on a spacetime diagram (in
some inertial frame). In other words, we’d like to know what sort of worldline
this rocket actually follows through spacetime.
There are several ways to approach this question, but I want to use some of
the tools that we’ve been developing. As we have seen, uniform acceleration
is a very natural notion that is not tied to any particular reference frame. We
also know that, in some sense, it involves a change in velocity and a change
in time. One might expect the discussion to be simplest if we measure each
of these in the most natural way possible, without referring to any particular
reference frame. What do you think is the natural measure of velocity (and the
change in velocity)? By natural, I mean something associated with the basic
structure (geometry!) of spacetime. What do you think is a natural measure of
the passage time? Stop reading for a moment and think about this.
As we discussed last week, the natural way to describe velocity (in terms of, say,
Minkowskian geometry) is in terms of the associated boost parameter θ. Recall
that boost parameters really do add together (eq. 4.17) in the simple, natural
way. This means that when we consider a difference of two boost parameters
(like, say, in ∆θ or dθ), this difference is in fact independent of the reference