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.

A frame of reference *associated with* (or “of”) an observer P moves with P. A frame of reference not associated with observer P does not move with 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 one-dimensional, that is, only one dimension is used.

A frame of reference is *unmarked* if there are no units specified for it. A frame of reference is *marked* by specifying its coordinates in units of either length or duration. A *space frame* of observer P is a frame of reference at rest relative to P that is marked with units of length. A *time frame* of observer P is a frame of reference in uniform motion relative to P that is marked with units of duration.

A frame of reference K_{s} with axes *a*_{1}, *a*_{2}, and *a*_{3}, that is at rest relative to observer P_{1} is a *rest frame* of P_{1}. A frame of reference K_{t} with axes *a´*_{1}, *a´*_{2}, and *a´*_{3}, that is in motion on the coincident *a*_{1}*-a´*_{1 } axis at a uniform rate relative to observer P_{1} is an *a*_{1}-axis *motion frame* of P_{1}. In the inverse case, the frame K_{t} is at rest relative to observer P_{2} and so is a rest frame of P_{2}, and the frame K_{s} 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}. See Figure 1.

If an observer has a rest frame that is a space frame and a motion frame that is a time frame, then the two frames form a *space+time framework* for the observer. If an observer has a rest frame that is a time frame and a motion frame that is a space frame, then the two frames form a *time+space framework* for the observer. The time rate of motion is defined in space+time. The space rate of motion is defined in time+space. If only the linear path of motion is considered, as on a railway, then both rates are defined.

As an example, consider an observer P_{1} with space frame K_{s} at rest relative to the Earth and time frame K_{t} moving at a constant rate relative to the Earth. Thus observer P_{1} has a space+time framework. Consider an observer P_{2} at rest relative to time frame K_{t} and in uniform motion relative to K_{s}. Thus observer P_{2} has a time+space framework. Note that the space and time frames of K_{s} and K_{t} are interchanged. See Figure 2.

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

*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 the second frame is moving with space velocity *u* relative to the first frame 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 the first time frame has coordinates *t*_{1}, *t*_{2}, and *t*_{3}, and the second time frame has coordinates *t*_{1}*´*, *t*_{2}*´*, and *t*_{3}*´*, and the second frame is moving with space lenticity ℓ relative to the first frame along the *t*_{1}*-t´*_{1} axis, then the coordinate transformation is as follows:

*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 the second frame is moving with time lenticity *w* relative to the first frame 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.