Notes on Relativity and Cosmology - Physics Department, UCSB

34 CHAPTER 1. SPACE, TIME, AND NEWTONIAN PHYSICS
A
V
BA
t
V
CB
t
V
CA
B
t
C
time t
WLOG, take t=0 here
At time t, the separation between A and C is V CA t, but we see from the diagram
that it is is also V CB t + V BA t. Canceling the t’s, we have
V CA = V CB + V BA . (1.2)
QED
⋆ ⋆ ⋆ Now, our instructions about how to draw the diagram (from the facts that
our ideas about time and position are well-defined) came from assumptions
T and S, so the Newtonian formula for the addition of velocities is a logical
consequence of T and S. If this formula does not hold, then at least one of T
and S must be false! Of course, I have still not given you any real evidence
to doubt (1.1) – I have only heavily foreshadowed that this will come. It is a
good idea to start thinking now, based on the observations we have just made,
about how completely any such evidence will make us restructure our notions
of reality.
• Q: Where have we used T?
• A: In considering events at the same time (i.e., at time t on the diagram
above).
• Q: Where have we used S?
• A: In implicitly assuming that d BC is same as measured by anyone (A,B,
or C).
1.5 Newton’s Laws: Are all reference frames
equal?
The above analysis was true for all reference frames. It made no difference how
the clock that defines the reference frame was moving.