Space and time as frames

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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 R3. A place point (or placepoint) is a (spatial) point. The space origin is a reference place point. The location of a place point is the space vector to it from the space origin. Chron (3D time) is the geometry of times and durations in R3. A time point (or timepoint) is an instant. The time origin is a reference time point. The chronation of a time point is the chron 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 relentation 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 K1 with axes a1, a2, and a3, that is a rest frame of observer P1, and let there be a motion frame K2 with axes a´1, a´2, and a´3, that is a motion frame of P1 along the coincident a1-a´1 axis. See Figure 1.

Two frames
Figure 1

In the inverse case, the frame K2 is at rest relative to observer P2 and so is a rest frame of P2, and the frame K1 is a motion frame of P2. In other words, the rest frame of P1 is the motion frame of P2, and the rest frame of P2 is the motion frame of P1. The rest and motion frames of P1 and P2 are interchanged. See Figure 2.

Two frames
Figure 2

If observer P1 has a space frame and a time frame, then they form a space+time framework (3+1) for P1. If an observer P2 has a time frame and a space frame, then they form a time+space framework (1+3) for P2. The time velocity vector is defined for space+time. The space lenticity vector is defined for 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 P1 has space frame K1 with coordinates x1, x2, and x3, and observer P2 has space frame K2 with coordinates x1´, x2´, and x3´, and frame K2 is moving with time velocity v relative to frame K1 along the x1-x1´ axis, then the coordinate transformation is as follows (see Figure 1):

x1´ = x1 − vt1, x2´ = x2, x3´ = x3, t1´ = t1, t2´ = t2, and t3´ = t3.

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

If frame K2 is moving with space velocity u relative to frame K1 along the t1-t1´ axis, then the coordinate transformation is as follows:

t1´ = t1 − ux1/c², t2´ = t2, t3´ = t3, x1´ = x1, x2´ = x2, and x3´ = x3.

The coordinates of a time+space framework can be transformed to another time+space framework with a known relative lenticity. If observer P1 has time frame K1 with coordinates t1, t2, and t3, and observer P2 has time frame K2 with coordinates t1´, t2´, and t3´, and frame K2 is moving with space lenticity ℓ relative to frame K1 along the t1-t´1 axis, then the coordinate transformation is as follows (see Figure 2):

t1´ = t1 − x1/ℓ, t2´ = t2, t3´ = t3, x1´ = x1, x2´ = x2, and x3´ = x3.

If frame K2 is moving with time lenticity w relative to frame K1 along the x1-x1´ axis, then the coordinate transformation is as follows:

x1´ = x1 − k²t1/w, x2´ = x2, x3´ = x3, t1´ = t1, t2´ = t2, and t3´ = t3.

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

\begin{pmatrix} t'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -v & 1 \end{pmatrix} \begin{pmatrix} t\\ x \end{pmatrix}           \begin{pmatrix} t'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & -u/c^{2} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} t\\ x \end{pmatrix}

\begin{pmatrix} x'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1/\l & 1 \end{pmatrix} \begin{pmatrix} x\\ t \end{pmatrix}        \begin{pmatrix} x'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & -k^{2}/w \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ t \end{pmatrix}

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

\begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}          \begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}

\begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} kx\\ t \end{pmatrix}       \begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} kx\\ t \end{pmatrix}

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:

 \begin{pmatrix} ct'\\ x' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & 1/\gamma \end{pmatrix} \begin{pmatrix} ct\\ x \end{pmatrix}

\begin{pmatrix} kx'\\ t' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} kx\\ x \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & 1/\gamma \end{pmatrix} \begin{pmatrix} kx\\ x \end{pmatrix}

The result in both cases is the Lorentz transformation.