Galilean Transformation

According to the correspondence principle, the equations of special relativity must reduce to the corresponding classical equations in the limit of small velocities. Therefore, the Lorentz transformation equations must reduce to the following Galilean transformation equations in the limit of small velocities.

Galilean Transformation Equations	Inverse Galilean Transformation Equations
x‘ = x – vt	x = x' + vt’
y’ = y	y = y’
z’ = z	z = z’
t’ = t	t = t’

Notice that the mixing of space with time in the Galilean transformation equations occurs only in the x-equations. As a result, in classical physics time is universal, the same for all observers. Space does differ from one observer to another, but there is no stretching or compression factor (g) as in relativity. In fact, the only difference in space is a uniform translation factor vt resulting from the relative motion of the observers. Therefore, classical physics also considers space to be universal, the same for all observers except for the relative translational motion.

Further Study:

The Galilean transformation equations and their implications are discussed in greater detail on page 4 in Serway and on pages 3-4 in Tipler.

Questions:

Q1. Which of the following quantities are invariant with respect to a Galilean transformation? (A) Position x (B) Velocity u (C) Acceleration a (D) Two of these (E) Three of these.
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Problems:

P1. Applying the Correspondence Principle
Show that the Lorentz transformation equations reduce to the Galilean transformation equations in the limit of small velocities. Hint: An easy way to reduce relativistic equations to their corresponding classical equations is to assume that the speed of light c is equal to infinity. Then any term containing c or c² in the denominator immediately goes to zero. Of course, a more rigorous procedure is to consider the case where v << c so that v/c approaches a negligible value essentially equal to zero.

P2. The Classical Limit
At what speed do the classical equations begin to break down and relativistic equations become necessary? In other words, how fast must an object travel before the classical equations are no longer considered valid? Hint: Determine the speed at which the gamma factor differs from unity by specified amount. For example, relativistic effects should be obvious by the time g = 1.1, they should be visually discernible by the time g = 1.01, they should easily be measurable if gamma differs from unity by one part in 10⁵, and they should just barely be detectable if gamma differs from unity by one part in 10¹⁰.

P3. Newtonian Relativity
Prove that Newton's three laws of mechanics assume the same form in all inertial frames. (Therefore, the laws of mechanics cannot be used to detect motion through space or to define a preferred reference frame in accordance with the principle of relativity.) Hint: The Galilean transformations pertain to two arbitrary inertial frames, so any law that is invariant under a Galilean transformation takes on the same form in all inertial frames. Therefore, you can assume F = m a is valid in system S and use the Galilean transformation equations to transform that law to system S'. If you get the same form of the law in system S', namely, F = m a', you can conclude that Newton's second law takes on the same form in all inertial frames. Repeating the process for Newton's other two laws finishes the proof. (For additional insight on this topic, see pages 3-4 in Serway and pages 3-4 in Tipler.)

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