Notes on Relativity and Cosmology - Physics Department, UCSB

122 CHAPTER 5. ACCELERATING REFERENCE FRAMES . . .
date, this would seem to be a bit odd.
Also, on further reflection, one realizes that this notion of acceleration depends
strongly on the choice of inertial frame. The dv part of a involves subtracting
velocities, and we have seen that plain old subtraction does not in fact give the
relative velocity between two inertial frames. Also, the dt part involves time
measurements, which we know to vary greatly between reference frames. Thus,
there is no guarantee that a constant acceleration a as measured in some inertial
frame will be constant in any other inertial frame, or that it will in any way
“feel” constant to the object that is being accelerated.
5.1.1 Defining uniform acceleration
What we have in mind for uniform acceleration is something that does in fact feel
constant to the object being accelerated. In fact, we will take this as a definition
of “uniform acceleration.” Recall that we can in fact feel accelerations directly:
when an airplane takes off, a car goes around a corner, or an elevator begins to
move upward we feel the forces associated with this acceleration (as in Newton’s
law F=ma). To get the idea of uniform acceleration, picture a large rocket in
deep space that burns fuel at a constant rate. Here we have in mind that this
rate should be constant as measured by a clock in the rocket ship. Presumably
the astronauts on this rocket experience the same force at all times.
Now, I admitted a few minutes ago that Newton’s laws will need to be modified
in relativity. However, we know that Newton’s laws hold for objects small
velocities (much less than the speed of light) relative to us. So, it is OK to use
them for slowly moving objects. We will suppose that these laws are precisely
correct in the limit of zero relative velocity.
So, how can we keep the rocket “moving slowly” relative to us as it continues
to accelerate? We can do so by continuously changing our own reference frame.
Perhaps a better way to say this is that we should arrange for many of our
friends to be inertial observers, but with a wide range of velocities relative to
us. During the short time that the rocket moves slowly relative to us, we use our
reference frame to describe the motion. Then, at event E 1 (after the rocket has
sped up a bit), we’ll use the reference frame of one of our inertial friends whose
velocity relative to us matches that of the rocket at event E 1 . Then the rocket
will be at rest relative to our friend. Our friend’s reference frame is known as
the momentarily co-moving inertial frame at event E 1 . A bit later (at event
E 2 ), we will switch to another friend, and so on.