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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.

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´ (left). Then, as the light clock in K´ moves relative to K with uniform motion at velocity v (right), from K observes the following:

light clocks, at rest & in motion

The left illustration shows one cycle length of the light path (i.e., wavelength), L, and one cycle duration (i.e., period), T, at rest in reference frames K, K´ (left). 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 points 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 instants of one cycle t.

Following Ugarov: Observing clock time rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock time from one frame with readings of several clock times from another frame, because two clock times from different reference frames occur at the same point in space only once. In one of the frames there must be at least two clock times which are supposed to be synchronized. For the sake of definiteness we shall be comparing one clock time, t<, from the frame K´ with two clock times from the frame K, at the point in the beginning and end of a cycle.

Let a clock and a light source be located at the origin O´ of the frame K´. A mirror is set perpendicular to the L axis at the distance L/2 from the light source (and the clock). A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin point O´ with the period TL/c. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and the mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origins, O and O´, the reflection from the mirror will occur at another point x/2 of the frame K and the reception of the reflected signal at the point x of the axis. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the path travelled by light in the frame K is greater than that in the frame K´, one can expect that the period T between the sending and reception of the signal, when measured in the frame K, will be greater than t. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different points of space. The period T between these two events in the frame K will be measured in this case by the two clocks removed from each other by the distance vt along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(x</2)² = (ct</2)² = (vt/2)² + (L/2)².

Given t< = t and collecting t< from this equation, we get

t<²(1 − v²/c²) = (L/c)²,

  t<= (L/c)/√(1 − v²/c²) = γ (L/c)

where γ = 1/√(1 − v²/c²).

Considering that L/c = T, then

t< = γ T.

Since both events occurred at the same point in the frame K´, they were registered by means of the same clock. A time interval between events registered by means of the same clock (which implies that the events occurred at the same point of space) is referred to as a proper-time interval between these events. Of course, a time interval of which the initial and the final moments are registered at different points of the reference frame and, consequently, by means of different clocks will not be a proper-time interval between events.


Following Ugarov but with Euclidean time: Observing clock stance rates in the two frames K and K´ moving relative to each other, one can only compare the reading of one clock stance from one frame with readings of several clock stances from another frame, because two clock stances from different reference frames occur at the same instant in time only once. In one of the frames there must be at least two clock stances which are supposed to be synstancized. For the sake of definiteness we shall be comparing one clock stance x from the frame K´ with two clock stances from the frame K, at the instant of the first and last points of a cycle.

Let a clock and a light source be chronated at the origin instant O´ of the frame K´. A mirroring event occurs parallel to the t⊥ axis at the distime T/2 from the light source (and the clock) perpendicular to the relative motion. A light signal is transmitted from the source to the mirror from which it is reflected back and returns to the origin instant O´ with the wavelength L = cT. Both the light source and the mirror are at rest in the frame K´ and the signal travels there and back along the same straight line.

Now let us consider the propagation of the same signal in the frame K relative to which the source and mirror move to the right together with the frame K´ at the velocity v. Although the signal was sent from the two coincident origin instants, O and O´, the reflection from the mirror will occur at another instant t/2 of the frame K and the reception of the reflected signal at the instant t. In this way the path of the signal in the frame K traces out two sides of an equilateral triangle.

As the time path travelled by light in the frame K is greater than that in the frame K´, one can expect that the wavelength L between the sending and reception of the signal, when measured in the frame K, will be greater than x<. Indeed, the observer from the frame K will certify that the two events, i.e. the emitting of light from the origin O´ and its return to the origin O´, occur at the two different instants of time. The wavelength L between these two events in the frame K will be measured in this case by the two clock stances removed from each other by the distime x/v along the motion direction. The velocity of light is equal to c in all reference frames. Therefore, we obtain:

(t</2)² = (x</2v)² = (x/2c)² + (T/2)².

Given x< = x and collecting x< from this equation, we get

x<²(1 − v²/c²) = (cT)²,

x<= (cT)²/√(1 − v²/c²) = γ (cT),

where γ = 1/√(1 − v²/c²).

Considering that cT = L, then

x< = L/γ.

Since both events occurred at the same instant in the frame K´, they were registered by means of the same clock stance. A length interval between events registered by means of the same clock stance (which implies that the events occurred at the same instant of time) is referred to as a proper-length interval between these events. Of course, a length interval of which the initial and the final moments are registered at different instants of the reference frame and, consequently, by means of different clock stances will not be a proper-length interval between events.

~

The moving light clock has x< = vt. and x< = ct<. Note: if c = ∞, then t|| = 0; x< = x; and t< = t. If v = 0, then t = 0; x< = x||; and t< = t||.

From the Euclidean metric for space we have: (x</2)² = (x/2)² + (x||/2)² . Combine this with the above to get:

(ct</2)² = (vt/2)² + (ct||/2)².

Divide by c² to get:

(t<)² = (βt)² + (t||)².

If x< = x above, then

(x<)² = (x</β)² + x||², or

(x<)² (1 − 1/β²) = x||².

From the Euclidean metric for time we have: (t2)² = (t)² + (t||)². Combine this with the above to get:

(x</c)² = (x/v)² + (x||/c)².

Multiply by c² to get:

(x<)² = (x/β)² + x||²,

which is a weighted metric.

Can we infer x|| = ct||?

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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Old style liberalism

The following excerpt is from “Liberalism, Old Style” by Milton Friedman, published in the 1955 Collier’s Year Book, pp. 360-363. New York: P.F. Collier & Son, 1955. Reprinted in The Indispensable Milton Friedman, Essays on Politics and Economics, edited by Lanny Ebenstein, pp. 11-24. Washington, D. C.: Regnery Publishing, 2012 (see here).

Liberalism, as it developed in the seventeenth and eighteenth centuries and flowered in the nineteenth, puts major emphasis on the freedom of individuals to control their own destinies. Individualism is its creed; collectivism and tyranny its enemy. The state exists to protect individuals from coercion by other individuals or groups and to widen the range within which individuals can exercise their freedom; it is purely instrumental and has no significance in and of itself. Society is a collection of individuals and the whole is no greater than the sum of its parts. The ultimate values are the values of the individuals who form the society; there are no super-individual values or ends. Nations may be convenient administrative units; nationalism is an alien creed.

In politics, liberalism expressed itself as a reaction against authoritarian regimes. Liberals favored limiting the rights of hereditary rulers, establishing democratic parliamentary institutions, extending the franchise, and guaranteeing civil rights. They favored such measures both for their own sake, as a direct expression of essential political freedoms, and as a means of facilitating the adoption of liberal economic measures.

In economic policy, liberalism expressed itself as a reaction against government intervention in economic affairs. Liberals favored free competition at home and free trade among nations. They regarded the organization of economic activity through free private enterprise operating in a competitive market as a direct expression of essential economic freedoms and as important also in facilitating the preservation of political liberty. They regarded free trade among nations as a means of eliminating conflicts that might otherwise produce war. Just as within a country, individuals following their own interests under the pressures of competition indirectly promote the interests of the whole; so, between countries, individuals following their own interests under conditions of free trade, indirectly promote the interests of the world as a whole. By providing free access to goods, services, and resources on the same terms to all, free trade would knit the world into a single economic community.

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