Notes on Relativity and Cosmology - Physics Department, UCSB

82 CHAPTER 3. EINSTEIN AND INERTIAL FRAMES
not sure what the slope is, you may find it helpful to use the results of
part (a) and your value of the parameter b to compute the slope.
Now that you have done part (b), recall that two intersecting lines uniquely
determine a point. So, if we could find another line whose equation we
know (in terms of our coordinates), we could solve the two equations together
to find the coordinates of event B. (Note that Bob’s worldline won’t
work for this, because we don’t yet know what his speed is in our reference
frame.) What other line could we use? Well, we know that event B occurs
a distance v BA (1sec) to Alice’s right in Alice’s frame of reference. This
means that it lies along the line x A = v BA :
Us
Alice
x =0
us x =0
A
Bob
B
Light Ray
t =1
A
t
A
=0
t =1
us
t =0
us
x =v
A BA
(c) Find the values of m and x us,0 for which the equation
t us = m(x us − x us,0 )
describes the line x A = v BA . Note that m is the slope of the line and x us,0
is the place where the line of interest intersects the line t us = 0. It may
also be useful to note that, if Alice holds a rod of proper length v BA and
carries it with her, holding it at one end, then the line x A = v BA is the
worldline of the other end.
(d) Solve the equations from (b) and (c) to find the coordinates of event
B.