Notes on Relativity and Cosmology - Physics Department, UCSB

Chapter 5
Accelerating Reference
Frames in Special Relativity
We have now reached an important point in our study of relativity. Although I
know that many of you are still absorbing it, we have learned the basic structure
of the new ideas about spacetime, how they developed, and how they fit with the
various pieces of experimental data. We have also finished all of the material in
Einstein’s Relativity (and in fact in most introductions) associated with so-called
‘special relativity.’ You may well be wondering, “What’s next?”
One important subject with which we have not yet dealt is that of “dynamics,”
or, “what replaces Newton’s Laws in post-Einstein physics?” I would like to
discuss this in some depth, both for its own sake and because it will provide
a natural transition to our study of General Relativity and gravity. However,
there is something else that we must discuss first. Recall, for example, that
Newton’s second Law (F=ma), the centerpiece of pre-relativistic physics, involves
acceleration. Although we have to some extent been able to deal with
accelerations in special relativity (as in the twin paradox), we have seen that
accelerations produce further unexpected effects. We need to study these more
carefully before continuing onward. So, for most of this chapter we are going
to carefully investigate the simple but illustrative special case known as ‘uniform’
acceleration. We’ll save true discussion of dynamics (forces and such) for
chapter 6.
5.1 The Uniformly Accelerating Worldline
Now, what do I mean by ‘uniform’ acceleration? One might at first think
that this means that the acceleration a = dv/dt of some object is constant, as
measured in some inertial frame. However, this would imply that the velocity
(relative to that frame) as a function of time is of the form v = v 0 + at. One
notes that this eventually exceed the speed of light. Given our experience to
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