Defining multi-dimensional time

The revised SI metric system is based on seven defined constants and seven base units. One defined constant is c, the (round-trip mean) speed of light in a vacuum, is defined as exactly 299 792 458 metres per second (before metres and seconds are defined).

The unit of time, the second, is defined as (2019):

The second is defined by taking the fixed numerical value of the caesium frequency ∆ν, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s−1

The unit of length, the metre, is defined as (2019):

The metre is defined by taking the fixed numerical value of the speed of light in vacuum, 𝒸, to be 299 792 458 when expressed in the unit m s−1, where the second is defined in terms of the caesium frequency ∆ν.

In other words, a metre equals one second times c metres per second: (1 m) = c (1 s). Time is defined first, and length in terms of it, which is one reason time is the default independent variable. A secondary definition of a second would be: a second equals one metre divided by c metres per second: (1 s) = (1 m) / c.

Another unit needed for multi-dimensional time, the radian, remains the same:

The radian is the coherent unit for plane angle. One radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius.

Let us postulate the existence of free particles, which are free from interference or assistance in their motion. Based on extrapolation from experiments, these free particles move in indefinitely long straight lines, which leads to Newton’s first law of motion.

Every body perseveres in its state of being at rest or of moving straight forward, except insofar as it is compelled to change its state by forces impressed. [Cohen and Whitman, 416]

Ludwig Lange [The European Physical Journal H, volume 39 (214), pages 251–262] has these definitions and theorems:

Definition I. An “inertial system” is any coordinate system of the kind that in relation to it three points P, P′, P″, projected from the same space point and then left to themselves – which, however, may not lie in one straight line – move on three arbitrary straight lines G, G′, G″ (e.g., on the coordinate axes) that meet at one point.

Theorem I. In relation to an inertial system the path of an arbitrary fourth point, left to itself, is likewise rectilinear.

Definition II. An “inertial timescale” is any timescale in relation to which one point, left to itself (e.g., P), moves uniformly with respect to an inertial system.

Theorem II. In relation to an inertial timescale any other point, left to itself, moves uniformly in its inertial path

In order to measure motions, there needs to be a reference motion, which most simply would be the straight-line motion of a free particle. However, since the rate of motion is relative, two particles are needed: the first particle as a reference particle and the second particle as a uniform motion with a defined rate of motion relative to the first particle. These two particles are independent of the first three particles, and so may be moved to any position on the frame of reference.

Two free particles starting together and moving in straight lines form the basis for a two-dimensional coordinate system. Three free particles starting together and moving in non-coplanar straight lines form a basis for a three-dimensional coordinate system.

Each free particle continues to move indefinitely, which since is physically independent of all other variables cannot be stopped, although reference points along its motion may be designated.

Two kinds of reference frames can be developed from these definitions:

(1) Three axes measured by length, which are called spatial axes, with an independent elapsed time (duration), called space-time; or

(2) Three axes measured by time (duration), which are called temporal axes, with an independent elapsed distance, called time-space.

Since length and time measures are related by the constant c, they may be converted into one another. It is possible for only length or only time measure to be used as long as the independent and dependent variables are carefully distinguished. But (1) is a frame for 3D space with independent time and (2) is a frame for 3D time with independent distance.