here - Department of Physics, HKU

CHAPTER 1. PRE-SPECIAL RELATIVITY PHYSICS 4
y
y’
O
O’
x
x’
z z’
Figure 1.2: The relation between two frames.
coordinate to indicate an event. To visualize it, we usually speak of the
explosion of a bomb or the moment when and where we turn on a flashlight.
Note that the same event could have different coordinates with respect to
different frames, while there is no disagreement on the nature of the event.
For example, everyone will agree whether it is an explosion of a bomb or only
the light-up of a flashlight.
We will talk about the Galilean transformations after we discuss a few
more terms. An observer is a being that makes measurements of, among
others, lengths and durations. In relativity, we have to be very careful about
the motion of the observer. The rest frame (or co-moving frame) of a body
or an observer is the frame where the body is at rest. The measurement
made in this frame is usually described by the word “proper.” For example,
the proper length of a rod is the length of the rod as measured by a ruler
at rest (or moving) with the rod. The proper time of a process is the time
taken as measured by a clock at rest (or moving) with the process.
Now, suppose there are two frames S and S ′ , one moving with constant
velocity with respect to another. We can choose the coordinate systems such
that the x and x ′ -axes are aligned with the velocity and their origins coincide,
Fig. 1.2. Hence, for the origins, we have
t ′ = t = 0
x ′ = x = 0
y ′ = y = 0
z ′ = z = 0 . (1.7)
How are the relations between the coordinates of other points?
t ′ = t (1.8)
x ′ = x − vt (1.9)
y ′ = y (1.10)
z ′ = z . (1.11)