This post continues from the previous post here.

All motion is axial. It is a principle of kinematics that every motion is composed of the simple motions of translation and rotation. These simple motions are either along an axis (translation) or around an axis (rotation).

Here is a table of these two simple motions:

Length measurement is the ratio of the target length to a standard unit of length, such as one metre. Time measurement is the ratio of the target angle to a standard unit of angle, such as one revolution.

Space is a manifold of three dimensions of translation. Time is a manifold of three dimensions of rotation, which are three axes. These comprise six axes of motion, or six degrees of freedom.

The space and time exchange postulate means that the kinematic variables of translation and rotation are symmetric in the equations of motion. That is, a dual equation of motion is formed by exchanging space and time variables.

As described in the previous post here, the three dimensions of motion are axes for traveling along (length) or revolving around (time).

A measure of motion may be either (1) dependent on the the target motion, or (2) independent of the target motion. A measure that is independent is either available prior to or separately from the target motion. For example, an independent measure may be determined by agreement, such as the length of a race, or it may measure another motion, such as the motion of a clock, which is then correlated with the target motion.

A standard clock measures time because it measures rotations around an axis as an angle. A length clock measures rotations about an axis as a length. With constant rates of rotation constant, there is a fixed ratio between the two kinds of clock.

A device that measures its own internal motion may be called an autometer. A clock is an example of an autometer. The internal motion of an autometer can be correlated with a target motion. For clocks this is called synchronization. For a length clock this is called symmacronization.

An odometer is a measurement device that depends on its target motion. The standard odometer measures length of travel. A time odometer, or trip-timer, measures time of travel. A trip-timer is a stopwatch that is on only while the target motion takes place. If there is a stop in the target motion, then the trip-timer also stops. So the trip-timer measures time of motion rather than elapsed time.

A device that measures a quantity of motion need not be attached to the moving body. The theory of relativity deals with the remote measurement of quantities of motion. A device that is attached to the moving body produces proper measures such as proper length or proper time.

As a thought experiment, consider a rifle bullet, conceived of as an inertial projectile, fired at a target. Let the bullet itself be a source of measurement units: there is the length of the bullet and the rotation of the bullet. The extent of the motion of the bullet to the target could then in principle be measured as a number of bullet lengths and a number of rotations.

Length is the number of bullet lengths. Time is the number of bullet rotations. Thus length is essentially a linear measure and time is essentially a rotational measure. Length is generalized as the correspondence of the motion to a linear object, a rigid rod or ruler, which forms the basis of space. Time is generalized as the correspondence of the motion to a rotational object, a clock, which forms the basis of (abstract) time.

A rifle bullet provides a way to conceive of motion coordinates. Consider individually labeled rifle bullets continually fired from a common origin toward three orthogonal directions. The coordinates of a particular motion are then the labels on the coordinates that correspond to the motion. This means there are three pairs of coordinates: two for each bullet. These axes are the six degrees of freedom.

Motion conceived as a length function of time means that each for each rotational coordinate there corresponds its paired length coordinate. Motion conceived as a time function of length means that for each length coordinate there corresponds one paired rotation coordinate.

In conclusion, there are three dimensions of mobility. There are three dimensions for each measure of the extent of motion, which totals six dimensions. For ordinary purposes, the three dimensions of motion are sufficient, with space and time kept separate. But for science, which seeks a unified treatment, space and time should be united into six dimensions.

Victor Yakovenko has a derivation (see here) of the Lorentz Transformation (LT) in which he uses “only the equivalence of all inertial reference frames and the symmetries of space and time.” Because of the use of (spatial) reference frames and velocity, this is not completely symmetric. As we have seen, there is a dual Lorentz Transformation. Let us follow Yakovenko’s derivation but with reference timeframes and legerity (matrix forms omitted).

1) Let us consider two inertial reference timeframes P and P´. The reference timeframe P´ moves relative to P with legerity u along the t₁ ≡ t axis. We know that the coordinates t₂ and t₃ perpendicular to the legerity are the same in both reference timeframes: t₂ = t₂´ and t₃ = t₃´. So, it is sufficient to consider only transformation of the coordinates x and t from the reference timeframe P to x´ = f_x(x; t) and t´ = f_t(x; t) in the reference timeframe P´.

From translational symmetry of space and time, we conclude that the functions f_x(x, t) and f_t(x, t) must be linear functions. Indeed, the relative distances between two events in one reference timeframe must depend only on the relative distances in another timeframe:

 t´₁ – t´₂ = f_t(x₁ – x₂, t₁ – t₂),     x´₁ – x´₂ = f_x(x₁ – x₂, t₁ – t₂).          (1)

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