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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.

Galilean decompositions of the Lorentz transformation

The background for this post is here.

For space with time (3+1):

The gamma transformation (matrix Γ) expresses the time dilation of clocks and length contraction of rods with a relative speed:

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

Use vector (t  x)T. The gamma transformation is conjugate to the Lorentz boost (matrix Λ) by the Galilean transformations (G, GT), i.e., GTΓG = Λ:

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

or

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

This expands to

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

The matrix second from the right represents the Tangherlini transformation (or inertial synchronized Tangherlini transformation).

For time with space (1+3):

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Lorentz transformation derivations

What follows are four derivations of the Lorentz transformation from the complete Galilei (Galilean) transformations in space with time (3+1) and time with space (1+3). Their intersection is linear space and time (1+1), which is the focus of the derivations. The other dimensions may be reached by rotations in space or time.

I. Space with Time (3+1)

Consider two inertial frames of reference O and O′, assuming O to be at rest while O′ is moving with velocity v with respect to O in the positive x-direction. The origins of O and O′ initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point P on a spherical wavefront at a distance x and x′ from the origins of O and O′ respectively. According to the second postulate of the special theory of relativity the speed of light, c, is the same in both frames, so for the point P:

x = ct, and x′ = ct′.

A. Time velocity

Define velocity v as the time velocity vt = ds/dt. Consider the standard Galilean transformation of ct and x with a factor γ, which is to be determined and may depend on β, where β = v/c:

x′ = γ(x − vt) = γ(x − βct) = γx(1 − β).

The inverse transformation is the same except that the sign of β is reversed:

x = γ(x′ + vt′) = γ(x + βct) = γx′(1 + β).

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Lorentz factor from light clocks

Space and time are inverse perspectives on motion. Space is three dimensions of length. Time is three dimensions of duration. Space is measured by a rigid rod at rest, whereas time is measured by a clock that is always in motion relative to itself.

This is illustrated by deriving the Lorentz factor for time dilation and length contraction from light clocks. The first derivation is in space with a time parameter and the second is in time with a space parameter (stance).

The first figure above shows frame S with a light clock in space as a beam of light reflected back and forth between two mirrored surfaces. Call the height between the surfaces that the light beam travels distance h. Let one time cycle Δt = 2h/c or h = cΔt/2, with speed of light c, which is the maximum speed.

The second figure shows frame with the same light clock as observed by someone moving with velocity v relative to S. Call the length of each half-cycle d, and call the length of the base of one cycle in space b.

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Proper and improper rates

The independent quantity in a proper rate is the denominator. The independent quantity in an improper rate is the numerator. If a rate is multiplied by a quantity with the units of the independent quantity and the result has the units of the dependent quantity, it is proper. Otherwise, it is improper.

A proper rate becomes improper if the proper rate is inverted. An improper rate becomes proper if the improper rate is inverted. If two or more improper rates are added, each must first be inverted. The result of adding proper rates must be inverted again to return to the original improper rate. This is harmonic addition:

\frac{b}{a_{1}}+\frac{b}{a_{2}} \Rightarrow \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right)^{-1}

If the addition of improper rates is divided by the number of addends so that it is the average or arithmetic mean of the inverted rates, then the result inverted is the harmonic mean:

\frac{1}{2}\left (\frac{b}{a_{1}}+\frac{b}{a_{2}} \right ) \Rightarrow \left ( \frac{1}{2} \left (\frac{a_{1}}{b}+\frac{a_{2}}{b} \right ) \right)^{-1}

Time speed is the speed of a body measured by the distance traversed in a known time, which is a proper rate because the independent quantity, time, is the denominator. Space speed is the speed of a body measured by the time it takes to transverse a known distance, which is an improper rate because the independent quantity, distance, is the numerator. Space speeds are averaged by the harmonic mean and called the space mean speed. The time mean speed is the arithmetic mean of time speeds.

Velocity normalized by the speed of light is proper because the invariant speed of light is independent. The speed of light divided by a velocity is improper and must be added harmonically. Lenticity normalized by an hypothesized maximum pace is proper, but if the lenticity is divided by the pace of light, it is improper.

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Space with time and their dual

For the first post in this series see here.

Space with time (3+1)

Space is that which is measured by length; time is that which is measured by duration. There are three dimensions of length and one dimension (or parameter) of duration. Direction in space is measured by an angle, which is part of a circle.

Spatial rates are dependent on another variable, usually interval of time (distime).

Time is that which is measured by duration. Events are ordered by time. Time as an independent variable decreases from the past to the present and increases from the present to the future.

Temporal rates are dependent on another variable, usually the interval of space (stance).

Dual: time with space (1+3)

The dual of space with time is time with space. The dual of space is time and the dual of time is space. Space corresponds to time and time corresponds to space.

Time is that which is measured by duration; space is that which is measured by length. There are three dimensions of duration and one dimension (or parameter) of length. Direction in time is measured by a turn, which is part of a rotation.

Temporal rates are dependent on another variable, usually interval of space (stance).

Space is that which is measured by length. Events are ordered by length (stance). Space as an independent variable decreases from a past there to here and increases from here to a future there.

Spatial rates are dependent on another variable, usually the interval of time (distime). Read more →

Set theory and logic and their dual

(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that each have duals. Let us begin with the standard approaches to these three topics, and then define duals to each of them. To some extent, the original and the dual may be used together.

(1) Set theory and logic

A set is defined by its elements or members. Its properties may also be known or specified, but what is essential to a set is its members, not its properties. The notation for “x is an element of set S” is “x ∈ S”. A subset is a set whose members are all within another set: “s is a subset of S” is “s ⊆ S”. If subset s does not (or cannot) equal S, then it is a proper subset: “s ⊂ S”.

The null set (∅) is a unique set defined as having no members. That is paradoxical but not contradictory. A universal set (Ω) is defined as having all members within a particular universe. An unrestricted universal set is not defined because it would lead to contradictions.

The complement of a set (c) is the set of all elements within a particular universe that are not in the set. A union (∪) of sets is the set containing all members of the referenced sets. An intersection (∩) of sets is defined as the set whose members are contained in every referenced set.

Set theory has a well-known correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to union, and conjunction (AND, ∧) corresponds to intersection. Material implication (→) corresponds to “is a subset of”. Contradiction corresponds to the null set, and tautology corresponds to the universal set.

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Harmonic conversion of space and time

As noted here, there are two kinds of mean rates: the time mean and the space mean. If the denominators have a common time interval, the time mean is the arithmetic mean and the space mean is the harmonic mean. If the denominators have a common space interval (stance), the space mean is the arithmetic mean and the time mean is the harmonic mean.

For example, light reflected back from a mirror at known distance forms two successive trips whose mean rate is the space mean pace. Several measurements with the same apparatus have a time mean pace. The mean speed is the inverse of the mean pace.

The general principle is that quantities with independent time (such as velocity) and a common time interval use ordinary algebra but such quantities with a common space interval use harmonic algebra. Alternately, quantities with independent space (stance) such as lenticity and a common space interval use ordinary algebra but such quantities with a common time interval use harmonic algebra.

In other words, quantities over the same interval with independent variables use ordinary algebra but quantities with different independent variables use harmonic algebra to convert between them.

For example, addition of velocities with a common time interval use ordinary vector addition (e.g., u + v) but addition of velocities with a common space interval use harmonic vector addition (e.g., ((1/u) + (1/v))−1 ≡ ((u+v)/u·v)−1 with u, v, u·v, u+v0.

The relativity parameter γ is based on a (3+1) spatial frame. The parameter γ in a temporal frame with a common time interval (k ≡ 1/c and ≡ 1/v) is:

γ² = (1 − v·v/c²)−1 ⇒ (1 − ℓ·ℓ/k²)−1.

The parameter γ in a temporal frame with a common space interval (stance) is:

γ² = (1 − v·v/c²)−1 ⇒ H(1 − ℓ·ℓ/k²)−1 = ((1 − k²/ℓ·ℓ)−1)−1 = (1 − k²/ℓ·ℓ) ≡ (1 − v·v/c²) = 1/γ².

Interchanging space and time

The space-time exchange invariance, as stated by J. H. Field (see here) has an implicit second part. In addition to (1) the exchange (or interchange) of space and time coordinates, there is (2): the exchange of linear and harmonic algebra for ratios. Harmonic algebra is described here.

This is seen in the different averaging methods for velocities that differ spatially vs. velocities that differ temporally. If two vehicles take the same route, their average velocity is their arithmetic mean (u + v)/2, but if one vehicle has velocity u going and velocity v returning, then the average velocity is their harmonic mean 2/(1/u + 1/v). However, if one vehicle has pace u going and pace v returning, then the average pace is their arithmetic mean, but if two vehicles take the same route, their average pace is their harmonic mean.

Space and time are related to each other as covariant and contravariant components. If space is covariant, then time is contravariant, and if time is covariant, then space is contravariant.

The equations of space-time (3+1) and time-space (1+3) physics are symmetric to one another with the interchange of space and time dimensions. The equations of spacetime (4D) physics is self-symmetric. The interchange of space and time dimensions produces equivalent 4D equations.

To interchange the space and time coordinates, take these steps: For the equations of classical physics, (1) ensure either space or time is a parameter, (2) interchange one dimension with the parameter, and (3) expand the single dimension into three dimensions. For the equations of relativistic physics, (1) ensure there is a symmetry between space and time dimensions, (2) interchange one space and time dimension but leave dimensionless quantities unchanged, and (3) expand the single dimension into three dimensions.

These steps reflect the difference between Galileo’s and Einstein’s relativity. Galileo transforms one frame into another frame but does not combine frames as Einstein’s does. For example, Einstein requires all frames to have the same orientation, but Galileo accepts frame-specific orientations such as the right-hand rule.

The Galilean transformation represents the addition and subtraction of velocities as vectors. The dual Galilean transformation represents the addition and subtraction of lenticities as vectors. The Lorentz transformation represents the combination of Galilean and dual-Galilean transformations, as previously shown.

One and two-way transformations

The transformation of Galileo is a one-way transformation, i.e., it uses only the one-way speed of light, which for simplicity is assumed to be instantaneous. The transformation of Lorentz is a the two-way transformation, which uses the universal two-way speed of light. The following approach defines two different one-way transformations, which combine to equal the two-way Lorentz transformation. Note that β = v/c; 1/γ² = 1 − β²; and γ = 1/γ + β²γ.

Galilean transformation:  {x}' \mapsto x-vt;\; \; {t}' \mapsto t.

Dual Galilean transformation:  {x}' \mapsto x;\; \; {t}' \mapsto t-wx.

These could be combined with a selection factor κ of zero or one:

{x}' \mapsto x - \epsilon vt;\; \; {t}' \mapsto t-(1-\epsilon )wx.

Lorentz transformation (boost): {x}' \mapsto \gamma (x-vt);\; \; {t}' \mapsto \gamma (t-vx/c^{2}).

General Lorentz boost (see here): {x}' \mapsto \gamma (x-vt);\; \; {t}' \mapsto \gamma(t-k^{2}vx)

with \gamma =\left (1-\frac{v^{2}}{c^{2}} \right )^{-1}  and k = 1/c for the Lorentz boost.

General dual Lorentz boost:  {x}' \mapsto \gamma_{2} (x-kwt);\; \; {t}' \mapsto \gamma_{2} (t-wx)

with \gamma_{2} =\left(1-\frac{w^{2}}{k^{2}} \right)^{-1}and k = 1/c.

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