Page 159 - Intro to Tensor Calculus

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Figure 1.5-3. Relative motion of two inertial systems.

The substitution ρ = 1 can be used to represent the equation (1.5.142) in the form
u
2
du 2 2GM E
+ u − u + =0 (1.5.143)
dφ h 2 h 2
which is a form we will return to later in this section. Note that we can separate the variables in equations
(1.5.142) or (1.5.143). The results can then be integrate to produce the equation (1.5.130).
Newton also considered the relative motion of two inertial systems, say S and S. Consider two such
systems as depicted in the figure 1.5-3 where the S system is moving in the x−direction with speed v relative
to the system S.
For a Newtonian system, if at time t = 0 we have clocks in both systems which coincide, than at time t
apoint P(x, y, z)inthe S system can be described by the transformation equations

x =x + vt x =x − vt
y =y y =y
or (1.5.144)
z =z z =z

t =t t =t.

These are the transformation equation of Newton’s relativity sometimes referred to as a Galilean transfor-
mation.
Before Einstein the principle of relativity required that velocities be additive and obey Galileo’s velocity
addition rule
V P/R = V P/Q + V Q/R . (1.5.145)