Galilean relativity and its limits

Inertial frames and classical intuition

An inertial frame is a reference frame in which an object with no net force moves at constant velocity. Classical mechanics assumes that the laws of motion have the same form in all inertial frames. This idea is called Galilean relativity.

If you perform a mechanics experiment inside a smoothly moving train, such as tossing a ball straight upward, the ball returns to your hand just as it would in a station at rest. The motion looks different to someone standing outside, but both observers can describe the same physical event using Newton's laws.

Galilean transformations

Suppose frame $S'$ moves at constant speed $v$ in the positive x-direction relative to frame $S$. Classical physics relates coordinates by

$x'=x-vt$

$y'=y$

$z'=z$

$t'=t$

The last equation is the key classical assumption: time is universal. All observers agree on the time between events, even if they disagree about positions.

Velocities transform by subtraction:

$u'*x=u_x-v$

If you walk forward inside a train moving at $20 m/s$ and your walking speed relative to the train is $2 m/s$, someone on the ground says you move at $22 m/s$.

Newton's laws under Galilean transformations

Acceleration is the same in both frames if the relative velocity between frames is constant:

$a'*x=a_x$

Since Newton's second law is

ec{F}=mec{a}

it has the same form in both inertial frames. This is why Galilean relativity works so well for ordinary mechanics. No inertial frame is mechanically special.

The problem with light

In the nineteenth century, Maxwell's equations showed that electromagnetic waves travel at speed

c=rac{1}{sqrt{mu_0epsilon_0}}

This speed matched the measured speed of light. But Galilean velocity addition suggests that if light travels at speed $c$ in one frame, then an observer moving toward the light should measure $c+v$, and an observer moving away should measure $c-v$.

That is not what Maxwell's equations seem to allow. The equations predict one speed $c$, determined by electric and magnetic constants, not by the motion of the source or observer.

The conflict

Physicists faced a choice. Perhaps Maxwell's equations were valid only in one preferred frame, such as the rest frame of a hypothetical aether. Or perhaps the Galilean transformations were not exact for high-speed motion and electromagnetic phenomena.

The first option preserved ordinary ideas about time and space but required a special medium for light. The second option required a radical revision: time and space themselves might depend on the observer.

Low-speed success

Galilean relativity is not useless. It is an excellent approximation when speeds are much smaller than light speed:

$vll c$

For cars, trains, baseballs, and planets moving slowly compared with $c$, relativistic corrections are tiny. Special relativity does not simply discard classical mechanics; it contains it as a low-speed limit.

The big idea

Galilean relativity says the laws of mechanics are the same in all inertial frames and assumes universal time. Its transformations work beautifully at ordinary speeds. But Maxwell's prediction of a fixed light speed conflicts with Galilean velocity addition, forcing a deeper theory of space, time, and motion.