Galilean relativity defended

Galilean relativity is a relational theory of motion as a function of time, which leads to the Galilean transformation. Here is a defense of Galilean relativity from two postulates:

(1) The Galilean principle of relativity, which states that the laws of mechanics are invariant under a Galilean transformation.

(2) A convention that rectilinear coordinates for frames of reference follow the right-handed rule: the unit vectors i, j, and k are related as i × j = k.

The Galilean transformation for constant motion on the x axis is x´ = xvt,  and t´ = t. Postulate (2) means if the extended right-hand thumb points to the positive X axis and the extended right-hand first finger points to the positive Y axis, then the right-hand middle finger points orthogonally to the positive Z axis.

The standard configuration for derivations of the Lorentz transformation consists of two inertial frames of reference moving relative to each other at constant velocity, with Cartesian coordinates such that the x and x′ axes are collinear facing the same direction:

Axes with same orientation

In this case the velocity of S´ relative to S is +v and the velocity of S relative to S´ is –v. This is called the principle of velocity reciprocity.

It assumes there is a third frame of reference (which could be S or S´) that includes both frames so that there is a common orientation scheme. For example, the positive x axis could be considered to point East, with both frames of reference on the Earth.

In the Galilean context this configuration is different: the Cartesian coordinates have x and x′ axes that are collinear but facing the opposite direction:

Axes turned 180 degrees

That is, the velocity of S´ relative to S is +v and the velocity of S relative to S´ is also +v. This could be called the principle of velocity equality.

In the derivation of the Lorentz transformation the equations for x´(x, t) and t´(x, t) are combined with the equations for x(x´, t´) and t(x´, t´) in a third frame of reference. However, this cannot be done in the Galilean context because a third frame cannot be assumed.

This means every frame of observation is a separate frame of reference. So for example the speed of light is a one-way speed. The two-way (round-trip) speed cannot be considered without a common frame of reference, which is not required for observation.

The significance of this is that it is fully justified to consider light traveling at an infinite one-way speed, which is what the Galilean transformation does.