Visualizing Galilean relativity

Figure 1: Galileo's ship

This post is part of a series on visualizing relativity.

In 1632, Galileo gave a famous example of a ship travelling at constant velocity, without rocking, on a smooth sea. His claim was that any observer below the deck would not be able to tell whether the ship was moving or stationary.


In Figure 1, an experimenter releases a ball which drops vertically between time t1 and time t2. Does this imply that the boat must be stationary in the water? No, because if the boat is moving at a constant velocity, the ball will also move with that constant velocity (in the boat's direction of movement) and land in the same place, relative to the boat. Thus the experimenter is unable to tell whether the boat is moving at a constant velocity or is stationary.

This result is captured in the following principle:

Galilean relativity: The laws of motion are the same in all inertial frames

An inertial frame is simply a reference frame that is either at rest or moving at a constant velocity (i.e., not accelerating).

Note, however, that from the reference frame of the water (as indicated by the fish at rest between time t1 and time t2), the ball will follow a curved (parabolic) path. That is a measurable relativistic effect (in the Galilean sense).

Figure 2: Adding velocities

In Figure 2, a train is travelling at 100km/h and Bob (on the train) shoots an arrow at 200km/h toward the target. At the same time, Alice (on the ground) also shoots an arrow at 200km/h toward the target.

According to Alice, the velocity of Bob's arrow is the velocity of the train plus the velocity of the arrow according to Bob. That is, Alice would measure the velocity of Bob's arrow to be 100km/h + 200km/h = 300km/h. Thus Bob's arrow would hit the target before hers, even though both arrows had the same velocity in their respective frames of reference.

Figure 3: Adding velocities (light)

In Figure 3, Bob fires a laser that emits photons travelling at the speed of light (c) toward the target. Alice fires her laser at the same time. Alice, it seems, would measure the velocity of Bob's light to be c + 100km/h, and Bob's laser light hitting the target before hers.

However that is not what happens! Instead, the light from both lasers hits the target at the same instant and Alice measures the velocity of Bob's light to be c, just as Bob does.

This surprising result lead Einstein to extend Galilean relativity to include the following postulate:

The Principle of Invariant Light Speed: The speed of light is the same in all inertial frames

Galilean relativity and the principle of invariant light speed together comprise Einstein's special theory of relativity. As you may know, this gave rise to a number of unexpected consequences, including relativity of simultaneity, time dilation, length contraction and a new equation for adding velocities.

The next post will explore those consequences and present a way to intuitively visualize special relativity.