Galilean Transformation: The Equations Linking Two Reference Frames

When two people measure the position of the same object from two different vantage points, they get different numbers while describing the same event. The Galilean transformation is the system of four equations that converts these readings from one observer to another, provided their relative speed stays constant, with neither one accelerating or veering off a straight line. It is the idea underneath nearly all the physics taught before relativity: an inertial reference frame is, in this scheme, one of those two observers who feel no mysterious force pulling them sideways.

The Galilean transformation formula

Consider two reference frames that coincide at the starting instant, with the second frame (the “primed” one) moving along the x axis at constant speed v relative to the first. The four equations are:

• x’ = x - vt
• y’ = y
• z’ = z
• t’ = t

The sideways coordinates, y and z, do not change: the motion happens only along x. The time t’ equals t, meaning the same clock applies to both observers. From these equations also follows the rule for velocities: if an object moves at speed u in the first frame, its speed measured in the second frame is u’ = u - v.

Worked example: a field hockey player runs toward the goal at 5 meters per second and strikes the ball, which leaves her stick at 30 meters per second relative to her. The goalkeeper, standing still on the field, does not see the 5 or the 30 separately: she sees the ball arrive at 35 meters per second, the sum of the two speeds. Every speed in this calculation only makes sense once you say relative to what it is measured.

Why Galileo got there first

The Galilean principle of relativity states that mechanical experiments give the same result in a frame at rest and in a frame moving at constant velocity in a straight line. Dropping an object, for instance, produces the same outcome whether you stand on a motionless floor or on one gliding forward at constant speed. Galileo used this to tackle an enormous problem for his time: if the Earth orbits the Sun, why don’t we feel it?

The answer, under this principle, is that there is no reason we should feel it. Uniform motion leaves no trace in everyday experiments, so the absence of any sensation of movement does not prove the Earth is standing still. Galileo could not reach a decisive proof of the Earth’s motion through the solar system: what he showed instead is that the Copernican hypothesis does not contradict daily experience, which for his era was already a powerful argument against anyone who assumed the Earth’s motion had to be felt somehow.

How velocities add up

Classical velocity addition, the rule that follows from the Galilean transformation, matches the intuitive way people normally think about motion: velocities add up like ordinary numbers, with no correction needed. That is why the calculation in the player example feels almost too obvious to spell out: Galilean physics lines up with everyday common sense.

This same addition rule is also what keeps the Galilean transformation consistent with Newton’s laws of motion: acceleration, force and mass stay unchanged when switching from one inertial reference frame to another.

Time, length and the simultaneity of two events also stay unchanged under it.

Where the formula stops working

Classical relativity, and with it the Galilean transformation, is an approximation valid when the speeds involved stay below roughly 1% of the speed of light, that is below 3,000 kilometers per second. Within that range, which covers practically every speed encountered in daily life, the equations give correct results with a negligible margin of error. That is why no one driving a car or throwing a ball has ever needed to correct their math with special relativity. The error introduced by the classical approximation grows as speed approaches that threshold, and past a certain point it stops being negligible.

Beyond that threshold the symmetry breaks down: time can no longer be treated as the same for every observer, and the Lorentz transformation takes over, at the core of Einstein’s special relativity. The two frameworks are not in competition in the usual sense of the word, because the Galilean transformation still holds, to an excellent approximation, in every slow-speed case that Lorentz treats as a limiting case.