iSoul In the beginning is reality.

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.

Vectors and Functions in Space and Time

A pdf version of this post is here.

The time velocity of an n-dimensional vector variable or vector-valued function Δx per unit of an independent scalar variable t equals

Similarly, the space lenticity with Δt and Δx, respectively:

The rate of a scalar variable Δx per unit of an independent n-dimensional vector variable or vector-valued function Δt equals

The same except with Δt and Δx, respectively:

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We the Society

The Preamble to the U.S. Constitution famously begins, “We the People …” The state is based on the will of the people, which is properly discerned by representatives of the people meeting together. “We, therefore, the Representatives of the united States of America, in General Congress, Assembled, …” states the Declaration of Independence.

But what does “the People” signify? Is it merely the aggregation of individuals in a region? That would justify “majority rules” and other aspects of democracy. But what brings people together, what gives them a common interest? Is it not because they constitute a society? Yes, a mere set of individuals lacks the solidarity required to be a people. It takes more than people to be the People. It takes a society.

To properly “ordain and establish” (the Preamble again) a constitution for the state requires a society. Which is to say, a republic is constituted to serve society. That “government of the people, by the people, for the people” (Abraham Lincoln) is a government of society, by society, and for society. It is a state that serves society.

Society is the source for the state’s authority. The authority of the state depends on the society. The state has no independent authority of its own. “We, therefore, the Representatives of the united States of America, in General Congress, Assembled, appealing to the Supreme Judge of the world for the rectitude of our intentions, do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, …” (the Declaration again).

The state is not instituted to change society or overrule society in general. Certainly, some elements of society need to change such as the criminal element. But that does not give the state authority to take the role of master over society as a whole. In a republic the state serves society. The state that seeks to dominate society in general is a tyrannical state.

The state in a republic always acknowledges the prior existence and authority of society. All actions of the state, all laws and interpretations of laws reference the society. If there is conflict within society, the state does not step in and pick a side. The state is neutral regarding the ways of society, neither endorsing nor opposing them. As society changes, the state will change because the people have changed.

Politics today is often between those who want the state to change society and those who want the state to serve society. Those who want the state to change society are promoting tyranny. Those who want the state to serve society are promoting democracy. The choice is tyranny or democracy. Let democracy succeed; let the people rule! Let the state serve society!

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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Home is the horizon

As there is an inverse or harmonic algebra, so there is an inverse geometry, an inverse space. Home, the origin, is the horizon, the ends of the earth, and beyond that, the celestial equator, the heavens. We may attempt to journey to the centre of the earth with Jules Verne, but we’ll never make it because it is infinitely far away. We cannot plumb the ultimate depths within, the deep well of the heart. At the centre of it all is a bottomless pit, the hell of eternal darkness.

concentric spheres

The geometric inverse is with respect to a circle or sphere:

circle with line segmentBy Krishnavedala

P’ is the inverse of P with respect to the circle. The inverse of the centre is the point at infinity.

The order of events in this geometry is their distance from the horizon, not the centre. The return to home is the end of events, the final event. The later the event, the better, since it is closer to the end, to home.

The destination is where we’ve come from and where we return. It is a round trip, a circuit, a cycle of life and change.

What we call the beginning is often the end
And to make an end is to make a beginning.
The end is where we start from. T. S. Eliot, Little Gidding

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 →

Number and algebra and their dual

For the first post in this series, see here.

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

(2) Number and algebra

The concept of counting and number is as universal as language, though the full definition of number did not occur until the 19th century. Algebra came to the West from India and Arabia in the Middle Ages but its formal definition did not occur until the 19th century. Abstract algebra also began in the 19th century.

The basic rules of algebra are as follows: addition and multiplication are commutative and associative; multiplication distributes over addition; addition and multiplication have identities and inverses with one exception: there is no multiplicative inverse for zero.

An idea of infinity comes from taking the limit of a number as its value approaches zero: ∞ ∼ 1/x as x → 0. Infinity can be partially incorporated via limits.

Dual: harmonic numbers

An additive dual can be defined by negating every number. A more interesting dual comes from taking the multiplicative dual of every number. This latter case can be called harmonic numbers and harmonic algebra because of its relation to the harmonic mean.

The harmonic isomorphism relates every number x to its harmonic dual by H(x) := 1/y. The dual of zero is ∞.

For harmonic algebra: see here.

Harmonic algebra is the multiplicative inverse of ordinary algebra. There is a sense in which harmonic algebra counts down rather than up. Zero in harmonic numbers is like infinity in ordinary numbers. Larger harmonic numbers correspond to smaller ordinary numbers. Smaller harmonic numbers correspond to larger ordinary numbers.