iSoul In the beginning is reality.

Lorentz in spacetime and timespace

This post builds on the previous posts here and here, and follows the approach of J.-M. Levy here.

First, consider the Lorentz transformation in spacetime along the x axis per Levy’s section III:

Let us now envision two frames in ‘standard configuration’ with having velocity v with respect to K and let x, t (resp. x´, t´) be the coordinates of event M in the two frames. Let O and be the spatial origins of the frames; O and cöıncide at time t = = 0.

Here comes the pretty argument: all we have to do is to express the relation

OM = OO´ + O´M (equation 5)

between vectors (which here reduce to oriented segments) in both frames.

In K, OM = x, OO´ = vt and O´M seen from K is /γ since is O′M as measured in . Hence a first relation is:

x = vt + /γ (equation 6)

In , OM = x/γ since x is OM as measured in K, OO′ = vt´ and O′M = . Hence a second relation:

x/γ = vt´+ x´ (equation 7)

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Light clocks with multidimensional time

A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979).

A light clock is a device with an emission-reflection-reception cycle of light that registers the current time and stance in units of cycle length and duration. Consider two identical light clocks, at first in their reference frames at rest, K, K´. Then, as the light clock in K´ moves transversely relative to K with uniform motion at velocity v (right), from K one observes the following:

transverse light clock

The illustration above shows one cycle length of the light path (i.e., wavelength), X, on the left and one cycle duration (i.e., period), T, on the right at rest in reference frames K, K´. For the reference frame K´, in motion relative to reference frame K, call the arc length of one cycle of the light path x. Call the distance between the beginning and ending spacepoints of one cycle x. For the reference frame K´ relative to reference frame K, call the arc time of one cycle of the light path t. Call the distime between the beginning and ending timepoints of one cycle t.

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Clock-rods

A clock-rod is a linear or planar clock with a parallel rod attached to it. A mechanical or electronic clock-rod might look like this:

clock-rod

Three clock-rods mutually perpendicular would measure length and duration in all directions.

A light clock-rod is conceptually like this:

light clock-rod

The clock and rod are parallel to each other so that parallel or perpendicular motion would change either the measurements of either the clock or the rod but not both. A complete harmonic cycle is not affected by motion:

light clock moving longitudinally

The +vt/2 increased distance of the first half-cycle is offset by the −vt/2 decreased distance of the second half-cycle. Likewise for the times +2d/v and −2d/v.

Kinematic proofs

Displacement with time: displacement s, time t, velocities v1 and v, acceleration a:

To prove: v = v1 + at

a = (vv1) / t    by definition

at = (vv1)     multiply by t

v = v1 + at

To prove: s = v1t + ½ at2

vavg = s / t          by definition

vavg = (v1 + v) / 2        by definition

s / t = (v1 + v) / 2       combining these two

s = (v1 + v) t / 2         multiply by t

s = (v1 + (v1 + at)) t / 2       from above

s = (2v1 + at) t / 2

s = v1t + ½ at2

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Space as time and time as space

Galileo parabola

Galileo used the length of uniform motion as a measure of duration, i.e., time (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914):

Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at the end of the time-interval bc finds itself at the point i. p.199

Without getting into the details of the Figure 108, notice the shift of language: “the body moves from b to c” [i.e., a length-interval], then “the time-interval bc“. Galileo uses a length interval to measure a time-interval, which is justified since the motion is “with uniform speed”.

Let there be a ball dropped out the window by a passenger on a train in uniform motion. Consider the following four scenarios, in which the length or duration of a uniform motion is measured: (1) looking down above the moving ball, measuring the length of fall; (2) looking down above the moving ball, measuring the (uniform) duration of fall; (3) looking from the side, measuring the length of motion in two dimensions; and (4) looking from the side, measuring the (uniform) duration of motion in two dimensions.

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Metric postulates for time geometry

Geometry was developed by the ancient Greeks in the language of length, but it is an abstraction that may be applied to anything that conforms to its definitions and axioms. Here we apply it to duration. We will use Brossard’s “Metric Postulates for Space Geometry” [American Mathematical Monthly, Vol. 74, No. 7, Aug.-Sep., 1967, pp. 777-788], which generalizes the plane metric geometry of Birkhoff to three dimensions. First, Brossard:

Primitive notions. Points are abstract undefined objects. Primitive terms are: point, distance, line, ray or half-line, half-bundle of rays, and angular measure. The set of all points will be denoted by the letter S and some subsets of S are called lines. The plane as well as the three-dimensional space shall not be taken as primitive terms but will be constructed.

Axioms on points, lines, and distance. The axioms on the points and on the lines are:

E1. There exist at least two points in S.

E2. A line contains at least two points.

E3. Through two distinct points there is one and only one line.

E4. There exist points not all on the same line.

A set of points is said to be collinear if this set is a subset of a line. Two sets are collinear if the union of these sets is collinear. The axioms on distance are:

D1. If A and B are points, then d(AB) is a nonnegative real number.

D2. For points A and B, d(AB) = 0 if and only if A = B.

D3. If A and B are points, then d(AB) = d(BA).

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Length and duration in time and space

Length and duration are defined by their measurement. Length is that which is measured by a rigid rod or its equivalent.

Length is “extension in space” (Dict. of Physics).

Duration is that which is measured by a clock or its equivalent.

Duration is “time measured by a clock or comparable mechanism” (Dict. of Physics).

Time and space and defined as concepts. Time is a local uniform motion that indicates the length or duration of local events by convention.

Time is “The dimension of the physical universe which, at a given place, orders the sequence of events.” (Dict. of Physics).

Space is a three-dimensional expanse whose extent is measured by length and duration.

“Space, a boundless, three-dimensional extent in which objects and events occur and have relative position and direction.” (Encyclopedia Britannica)

The difference between length and duration is the relation between the observed and the observed, the measurand. The question is, which one is the reference quantity and which one is the measured quantity. If the observer is (or has) the reference, and the observed is the measurand, then the value measured is length. If the observer is the measurand, and the observed is (or has) the reference, then the value measured is duration.

Time is derived from (1) three orthogonal uniform motions, or (2) one orthogonal uniform motion in which the three orthogonal uniform motions are components, or (3) the distance from the origin of the one orthogonal uniform motion in (2). In both (1) and (2) time is derived from three-dimensional, whereas in (3) time is one-dimensional or a scalar.

Space is derived from (1) three orthogonal uniform motions, or (2) one orthogonal uniform motion in which the three orthogonal uniform motions are components, or (3) the distance from the origin of the one orthogonal uniform motion in (2). In both (1) and (2) space is derived from three-dimensional, whereas in (3) time is one-dimensional or a scalar.

The difference between time and space is the relation between the observer and observed, or reference and measurand. If the reference motion is at rest in the observer’s frame, then what is measured is the length of motion of an observed body. If the reference motion is at rest in the observed frame, then what is measured is the duration of motion of an observed body.

Two forms of time and space

The SI metric base unit of length is the metre. The SI base unit of duration is the second. Other units of length are the kilometre, millimetre, inch, foot, mile, etc. Other units of duration are the minute, hour, day (sidereal, solar, etc.), year, etc.

Duration time is also called time. Length time may be called stance. Length space is also called space. Duration space may be called chronotopy.

Time is an independent variable measured in units of length or duration. Space is a dependent geometry of positions measured in units of length or duration.

Time is indicated by a reference uniform motion, which in units of duration is called a clock and in units of length is called a metreloge. The reference rate of uniform motion is a convention that enables one to convert length to duration and vice versa. By appropriate choice of units, this rate may equal one, in which case length and duration may be interchanged in calculations.

In uniform motion the spaces covered are proportional to the time elapsed. Independent space and time are two sides of the same coin.

The independence of time allows it to be selected independent of other variables, as in setting an appointment, a tempo for music, or the duration of a game. Its independence and uniform motion allows time to change by some linear rule.

Kinds of rights

Human rights are the political rights people have because they are human beings. They apply equally to all because of their common humanity. There are several statements of human, or natural, rights. The United Nations Universal Declaration of Human Rights issued in 1948 is a statement of human rights.

Developmental rights are the political rights people have because of their stage of development, notably, childhood or adulthood. The rights of children differ from the rights of adults because of the differences between children and adults. Children require adult parenting, whereas adults do not. Adults can marry, whereas children cannot.

Parenting is a developmental right because it concerns the stage of development. Adults parent children, not the other way around. In the confused times of today, developmental rights are confused with human rights and children are treated as adults.

Sexual rights are the political rights people have because of their sex, that is, male and female. Sex is also called gender, although gender is a grammatical term, whereas sex is a biological term. The rights of males differ from the rights of females because of the differences between males and females. An unmarried male adult has the right to marry any unmarried female adult. An unmarried female adult has the right to marry any unmarried male adult.

Marriage is a sexual right because it concerns a sexual relationship that naturally leads to the birth of offspring. In the confused times of today, sexual rights are confused with human rights and the sexes are treated as if they did not exist.

Woe to the society that confuses childhood with adulthood and male with female. That society will learn the hard way the importance of developmental and sexual differences.

Communitarianism

This post is a parallel contrast to the previous post on Old style liberalism.

Communitarianism puts major emphasis on the freedom of communities to control their own destinies. Communitarianism is its creed; individualism and alienation its enemy. The state exists to protect communities from coercion by other communities or individuals and to widen the range within which communities can exercise their freedom; it is purely instrumental and has no significance in and of itself. Society is a collection of communities and the whole is no greater than the sum of its parts. The ultimate values are the values of the communities who form the society; there are no sub- or super-community values or ends. Nations are convenient social units; patriotism is a part of its creed.

In politics, communitarianism expresses itself as a reaction against individualistic regimes. Communitarians favored limiting the rights of individuals, establishing geocratic governing institutions, limiting the franchise, and moderating civil rights. They favor such measures both for their own sake, as a direct expression of essential political freedoms, and as a means of facilitating the adoption of communitarian economic measures.

In economic policy, communitarianism expresses itself as a reaction against individuals controlling economic affairs. Communitarians favor free cooperation at home and among nations. They regard the organization of economic activity through free private enterprise operating in a competitive market as a limited expression of essential economic freedoms and as unimportant in facilitating the preservation of political liberty. They regard cooperation among nations as a means of eliminating conflicts that might otherwise produce war. Just as within a country, communities following their own interests under the influence of cooperation indirectly promote the interests of the whole; so, between countries, communities following their own interests under conditions of cooperation, indirectly promote the interests of the world as a whole. By providing common access to goods, services, and resources on the same terms to all, cooperation would knit the world into a single economic community.

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