## The Galilean transformation

The mathematical method of comparing how things appear to observers in different frames of reference is called a transformation of coordinates from a set of axes centred in one frame of reference to another.

The Galilean transformation is the 'normal', classical transformation between two inertial frames, such as a ship which is stationary at the dockside, or a ship which is moving with a speed v in the x direction. The Galilean transformation merely involves the addition of a term to the x coordinate which is changing with time at a constant rate.

The Galilean transformation gives the relationship between space coordinates in frames S and S'; it also states the 'obvious' fact that the time is the same in both.

When we compare phenomena within two different frames of reference, this transformation gives results which are in line with Newton's laws of motion in both frames. Forces are proportional to changes in velocities (accelerations) and the addition of a constant component to all velocities will leave the behaviour of physical systems unaffected. Insects and fishes dart about in random fashion, and water drops fall from one bottle into the other, just the same as before. However, looking from the outside, the magnitudes of velocities in another frame will appear to be different. For example, fish in a tank being transported in a cargo jet will all have the systematic velocity vector of the plane added to their random dart velocities.

If two observers view the same object, each using a different frame of reference, and these frames of reference are in relative motion, they will not agree on the speed of the object. It will move at different speeds relative to the two observers.

xX |
= x - vt |

y |
= y |

z' |
= z |

t' |
= t |

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