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Space and time as frames

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 Ks with axes a1, a2, and a3, that is at rest relative to observer P1 is a rest frame of P1. A frame of reference Kt with axes 1, 2, and 3, that is in motion on the coincident a1-a´1 axis at a uniform rate relative to observer P1 is an a1-axis motion frame of P1. In the inverse case, the frame Kt is at rest relative to observer P2 and so is a rest frame of P2, and the frame Ks 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. See Figure 1.

Two frames

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.

Two frames plus

Figure 2

As an example, consider an observer P1 with space frame Ks at rest relative to the Earth and time frame Kt moving at a constant rate relative to the Earth. Thus observer P1 has a space+time framework. Consider an observer P2 at rest relative to time frame Kt and in uniform motion relative to Ks. Thus observer P2 has a time+space framework. Note that the space and time frames of Ks and Kt 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 x1, x2, and x3, and the second space frame has coordinates x1´, x2´, and x3´, and the second frame is moving with time velocity v relative to the first frame along the x1-x1´ axis, then the coordinate transformation is as follows:

x1´x1 − vt1, x2´ = x2, x3´ = x3t1´ = t1, t2´ = t2, and t3´ = t3.

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

t1´ = t1ux1/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 the first time frame has coordinates t1, t2, and t3, and the second time frame has coordinates t1´, t2´, and t3´, and the second frame is moving with space lenticity ℓ relative to the first frame along the t1-t´1 axis, then the coordinate transformation is as follows:

t1´ = t1x1/ℓ, t2´ = t2, t3´ = t3, x1´ = x1, x2´ = x2, and x3´ = x3.

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

x1´x1 − k²t1/w, x2´ = x2, x3´ = x3t1´ = t1, t2´ = t2, and t3´ = t3.

In matrix terms, these are variations on the Galilean transformation:

\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 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:

\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.

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