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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 instantaneously, which is what the Galilean transformation does.

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