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. Unless specified otherwise, an observer here shall mean an inertial observer.

*Space* is the structure of places 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 structure of times 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 an imaginary lattice of rigid rods at rest relative to an observer that enables every place point of space to be uniquely identified by coordinates. A timeframe of reference is an imaginary lattice of rigid rods in uniform motion relative to an observer that enables every time point of trime to be uniquely identified by coordinates. 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 a rest frame and a motion frame associated with it. An observer’s rest frame is three-dimensional and their motion frame is effectively one-dimensional, that is, only one dimension is used.

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.

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 K_{1} and K_{2} are interchanged. See Figure 2.

If observer P_{1} has a rest frame that is a space frame and a motion frame that is a time frame, then the space and time frames form a *space+time framework* (3+1) for P_{1}. If an observer P_{2} has a rest frame that is a time frame and a motion frame that is a space frame, then the time and space frames form a *time+space framework* (1+3) for P_{2}.

The time rate of motion velocity vector is defined in space+time. The space rate of motion 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}.

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:

Since the rates of motion are constant, we may define *β* = *v*/*c* = *u*/*c* = *k*/ℓ = *k*/*w* (with *c* = 1/*k* = the speed of light in a vacuum), 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.