Proper and improper rates

The independent quantity in a proper rate is the denominator. The independent quantity in an improper rate is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is reciprocal addition:

\frac{b}{a_{1}}+\frac{b}{a_{2}} \Rightarrow \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right)^{-1}

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the harmonic mean:

\frac{1}{2}\left (\frac{b}{a_{1}}+\frac{b}{a_{2}} \right ) \Rightarrow \left ( \frac{1}{2} \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right ) \right)^{-1}

Time speed is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. Space speed is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the mean space speed. The mean time speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added in parallel. Lenticity normalized by an hypothesized maximum pace is proper, but if the lenticity is divided by the pace of light, it is improper.

Galilean Transformations

In the standard configuration of relativity, the Galilean transformation is

\begin{pmatrix} {t}'\\ {x}' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -v & 1 \end{pmatrix} \begin{pmatrix} t \\ x \end{pmatrix} = \begin{pmatrix} t \\ -vt+x \end{pmatrix}

This is the proper Galilean transformation (PGT) because the velocity is proper as a coefficient of time. Normalize the transformation with the speed of light, c, and let β = v/c:

\begin{pmatrix} {ct}'\\ {x}' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} = \begin{pmatrix} ct \\ -\beta ct+x \end{pmatrix}

The transpose of the proper Galilean transformation is the improper transposed Galilean transformation (ITGT), which interchanges the space and time coordinates:

\begin{pmatrix} {ct}'\\ {x}' \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} = \begin{pmatrix} ct-\beta x \\ x \end{pmatrix}

It is improper because the normalized velocity is a coefficient of space. To make it proper change the velocity to lenticity, w, normalize by an hypothesized maximum pace, k, and let ρ = w/k:

\begin{pmatrix} {t}'\\ {kx}' \end{pmatrix} = \begin{pmatrix} 1 & -\rho \\ 0 & 1 \end{pmatrix} \begin{pmatrix} t \\ kx \end{pmatrix} = \begin{pmatrix} t-\rho kx \\ kx \end{pmatrix}

This is the proper transposed Galilean transformation (PTGT). Its transpose is the improper Galilean transformation (IGT):

\begin{pmatrix} {t}'\\ {kx}' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\rho & 1 \end{pmatrix} \begin{pmatrix} t \\ kx \end{pmatrix} = \begin{pmatrix} t \\ -\rho t + kx \end{pmatrix}

This is improper because the normalized lenticity is the coefficient of time. Change ρ to β and replace k with c to make this into the improper transposed Galilean transformation above.