An *observer* is a body capable of use as a measurement apparatus. An *inertial observer* is an observer in inertial motion, i.e., one that is not accelerated with respect to an inertial system. An observer here shall mean an inertial observer.

An observer makes measurements relative to a frame of reference. A *frame of reference* is a physical system relative to which motion and rest may be measured. An *inertial frame* is a frame in which Newton’s first law holds (a body either remains at rest or moves in uniform motion, unless acted upon by a force). A frame of reference here shall mean an inertial frame.

A *rest frame* of observer P is a frame at rest relative to P. A *motion frame* of observer P is a frame in uniform motion relative to P. Each observer has at least one rest frame and at least one motion frame associated with it. An observer’s rest frame is three-dimensional, but their motion frame is effectively one-dimensional, that is, only one dimension is needed.

*Space* is the geometry of places and lengths in R^{3}. A *place point* (or *placepoint*) is a point in space. The space origin is a reference place point in space. The *location* of a place point is the space vector to it from the space origin. *Trime* (3D time) is the geometry of times and durations in R^{3}. A *time point* (or *timepoint*) is a point in trime. The time origin is a reference time point in trime. The *chronation* of a time point is the trime vector to it from the time origin.

A frame of reference is *unmarked* if there are no units specified for its coordinates. A frame of reference is *marked* by specifying (1) units of either length or duration for its coordinates and (2) an origin point. A *space frame* of observer P is a rest frame of P that is marked with units of length. A *time frame* of observer P is a motion frame of P that is marked with units of duration.

Speed, velocity, and acceleration require an independent motion frame. Pace, lenticity, and retardation require an independent rest frame. These independent frames are standardized as clocks or odologes so they are the same for all observers.

Let there be a frame K_{1} with axes *a*_{1}, *a*_{2}, and *a*_{3}, that is a rest frame of observer P_{1}, and let there be a motion frame K_{2} with axes *a´*_{1}, *a´*_{2}, and *a´*_{3}, that is a motion frame of P_{1} along the coincident *a*_{1}*-a´*_{1 } axis. See Figure 1.

In the inverse case, the frame K_{2} is at rest relative to observer P_{2} and so is a rest frame of P_{2}, and the frame K_{1} is a motion frame of P_{2}. In other words, the rest frame of P_{1} is the motion frame of P_{2}, and the rest frame of P_{2} is the motion frame of P_{1}. The rest and motion frames of P_{1} and P_{2} are interchanged. See Figure 2.

If observer P_{1} has a space frame and a time frame, then they form a *space+time framework* (3+1) for P_{1}. If an observer P_{2} has a time frame and a space frame, then they form a *time+space framework* (1+3) for P_{2}. The time velocity vector is defined in space+time. The space lenticity vector is defined in time+space. If only the linear path of motion is considered, as on a railway, then both rates are defined as scalars in a (1+1) framework.

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The coordinates of a space+time framework can be transformed to another space+time framework with a known relative velocity. If observer P_{1} has space frame K_{1} with coordinates *x*_{1}, *x*_{2}, and *x*_{3}, and observer P_{2} has space frame K_{2} with coordinates *x*_{1}*´*, *x*_{2}*´*, and *x*_{3}*´*, and frame K_{2} is moving with time velocity *v* relative to frame K_{1} along the *x*_{1}*-x*_{1}*´* axis, then the coordinate transformation is as follows (see Figure 1):

*x*_{1}*´* = *x*_{1}* − v**t*_{1}, *x*_{2}*´* = *x*_{2}, *x*_{3}*´ = x*_{3}, *t*_{1}´ = *t*_{1}, *t*_{2}´ = *t*_{2}, and *t*_{3}´ = *t*_{3}.

The unused space coordinates do not change. The time coordinates do not change because they all use a standard motion frame.

If frame K_{2} is moving with space velocity *u* relative to frame K_{1} along the *t*_{1}*-t*_{1}´ axis, then the coordinate transformation is as follows:

*t*_{1}´ = *t*_{1} − *ux*_{1}/c², *t*_{2}´ = *t*_{2}, *t*_{3}´ = *t*_{3}, *x*_{1}*´* = *x*_{1}, *x*_{2}*´* = *x*_{2}, and *x*_{3}*´* = *x*_{3}.

The coordinates of a time+space framework can be transformed to another time+space framework with a known relative lenticity. If observer P_{1} has time frame K_{1} with coordinates *t*_{1}, *t*_{2}, and *t*_{3}, and observer P_{2} has time frame K_{2} with coordinates *t*_{1}*´*, *t*_{2}*´*, and *t*_{3}*´*, and frame K_{2} is moving with space lenticity ℓ relative to frame K_{1} along the *t*_{1}*-t´*_{1} axis, then the coordinate transformation is as follows (see Figure 2):

*t*_{1}´ = *t*_{1} − *x*_{1}/ℓ, *t*_{2}´ = *t*_{2}, *t*_{3}´ = *t*_{3}, *x*_{1}*´* = *x*_{1}, *x*_{2}*´* = *x*_{2}, and *x*_{3}*´* = *x*_{3}.

If frame K_{2} is moving with time lenticity *w* relative to frame K_{1} along the *x*_{1}*-x*_{1}*´* axis, then the coordinate transformation is as follows:

*x*_{1}*´* = *x*_{1}* − k²**t*_{1}/*w*, *x*_{2}*´* = *x*_{2}, *x*_{3}*´ = x*_{3}, *t*_{1}´ = *t*_{1}, *t*_{2}´ = *t*_{2}, and *t*_{3}´ = *t*_{3}.

In matrix terms, these are variations on the Galilean transformation (with *c* = 1/*k* = the speed of light in a vacuum):

Since the motion is uniform, we may define *β* = *v*/*c* = *u*/*c* = *k*/ℓ = *k*/*w*, and simplify these as follows:

Given that the speed *c* is a universal maximum and pace *k* is a universal minimum, the time dilation and length contraction may be derived with the gamma factor (see *here* and *here*), then the these transformations can be combined with a gamma matrix as follows:

The result in both cases is the Lorentz transformation.