KAHIBARO
Table of Contents
Everyday idea of relativity before Einstein
Galilean relativity is the classical idea that the laws of mechanics are the same for all observers who move at constant velocity relative to one another. It belongs to pre Einstein physics and works very well for ordinary speeds, much smaller than the speed of light.
The central question is simple. If you perform a mechanical experiment inside a smoothly moving train, will the result be different from the same experiment done in a station at rest? Galilean relativity says no, as long as the train moves with constant speed in a straight line. Inside the train, the mechanical laws behave exactly as they do in the station.
This idea was a major step in understanding motion. It tells us that there is no special state called absolute rest that can be detected by mechanical experiments alone.
Inertial frames and the principle of relativity
Galilean relativity applies to inertial reference frames. These are frames that move at constant velocity relative to one another, without acceleration.
If frame $S'$ moves with constant velocity relative to frame $S$, then both are inertial frames. In Galilean relativity, Newtonian mechanics has the same form in both frames.
A common example is a ship moving smoothly across calm water. If you drop a ball from the top of a mast, a person on the ship sees it fall straight down to the base of the mast. A person on the shore sees the ball move forward while falling. The two observers describe different paths, but both descriptions are correct in their own frames.
Important principle of Galilean relativity:
All inertial frames are equivalent for the laws of classical mechanics.
There is no preferred inertial frame and no experiment in mechanics can identify absolute rest.
Galilean transformations
To compare measurements in two inertial frames, classical physics uses the Galilean transformation.
Suppose frame $S'$ moves in the positive $x$ direction with constant speed $v$ relative to frame $S$. If the origins coincide at time $t = 0$, then the coordinates are related by
$$
x' = x - vt
$$
$$
y' = y
$$
$$
z' = z
$$
$$
t' = t
$$
The last equation is especially important. In Galilean relativity, time is absolute. All observers agree on the same time interval.
This means that if one observer says two events happen 3 seconds apart, every other inertial observer also says they happen 3 seconds apart.
Galilean transformation:
$$
x' = x - vt, \quad y' = y, \quad z' = z, \quad t' = t
$$
Classical assumption:
$$
t' = t
$$
This is the statement of absolute time in Newtonian physics.
Velocity transformation
Because position changes from one frame to another, velocity also changes.
Starting from
$$
x' = x - vt
$$
and using the fact that $t' = t$, we differentiate with respect to time:
$$
u_x' = u_x - v
$$
For the other components,
$$
u_y' = u_y
$$
$$
u_z' = u_z
$$
So velocities add and subtract in the simple classical way.
If a person walks forward inside a train at speed $u_x'$ relative to the train, and the train moves at speed $v$ relative to the ground, then the person's speed relative to the ground is
$$
u_x = u_x' + v
$$
This is called Galilean velocity addition.
| Quantity | Transformation from $S$ to $S'$ |
|---|---|
| Position in $x$ | $x' = x - vt$ |
| Position in $y$ | $y' = y$ |
| Position in $z$ | $z' = z$ |
| Time | $t' = t$ |
| Velocity in $x$ | $u_x' = u_x - v$ |
| Velocity in $y$ | $u_y' = u_y$ |
| Velocity in $z$ | $u_z' = u_z$ |
Acceleration in Galilean relativity
If we differentiate velocity once more with respect to time, we get acceleration. Since $v$ is constant between inertial frames,
$$
a_x' = a_x
$$
$$
a_y' = a_y
$$
$$
a_z' = a_z
$$
So acceleration is the same in all inertial frames connected by Galilean transformations.
This is why Newton's second law keeps the same form in classical mechanics. If mass is also the same for all observers, then
$$
\vec{F} = m\vec{a}
$$
has the same structure in every inertial frame.
In Galilean relativity, for frames moving at constant relative velocity,
$$
\vec{a}' = \vec{a}
$$
Therefore Newtonian mechanics is invariant under Galilean transformations.
A simple physical example
Imagine a train moving at $20 \, \text{m/s}$ relative to the ground. A passenger throws a ball forward at $5 \, \text{m/s}$ relative to the train.
For the passenger, the ball's speed is
$$
u' = 5 \, \text{m/s}
$$
For a person standing on the ground, the ball's speed is
$$
u = u' + v = 5 + 20 = 25 \, \text{m/s}
$$
Both measurements are correct. They are made in different inertial frames.
Now suppose the passenger throws the ball backward at $5 \, \text{m/s}$. Then for the ground observer,
$$
u = -5 + 20 = 15 \, \text{m/s}
$$
Again, the descriptions differ, but the laws of mechanics remain the same.
What Galilean relativity means physically
Galilean relativity tells us that motion is relative. We do not speak meaningfully of motion alone, only motion relative to something.
A car may be at rest relative to its driver, moving relative to the road, and moving even faster relative to a distant airplane. None of these descriptions is more correct than the others by itself. What matters is the chosen frame.
This classical idea explains why smooth constant motion is difficult to detect from inside a closed room. If there are no outside clues and no acceleration, many ordinary experiments behave exactly the same.
Limits of Galilean relativity
Galilean relativity is extremely successful for everyday mechanics, but it does not correctly describe light and very high speeds.
Its key assumptions include absolute time and simple velocity addition. Later experiments showed that these assumptions fail when dealing with electromagnetic phenomena and speeds close to the speed of light. That is why special relativity was needed.
In this chapter, the important point is that Galilean relativity is the classical starting point. It captures how space, time, and motion were understood before Einstein.
Visualizing two moving frames
Final takeaway
Galilean relativity is the classical principle that all inertial frames are equivalent for mechanics. It uses the Galilean transformation, assumes absolute time, and leads to simple velocity addition. It works well for ordinary motion, and it forms the historical foundation from which special relativity later emerged.
Core ideas to remember:
$$
x' = x - vt, \quad t' = t
$$
$$
u_x' = u_x - v
$$
$$
\vec{a}' = \vec{a}
$$
Galilean relativity is valid for classical mechanics in inertial frames and for speeds much smaller than the speed of light.