iSoul In the beginning is reality.

Interchange of length and duration

August 16, 2019 12:01 pm ⋅

Length and duration are independent measures of the extent of motion. They are measured by comparing them with a uniform reference motion. Although uniform linear motion is simpler in theory, uniform circular motion is simpler in practice – and essential for unstopped motion. With one addition, the classic circular clock with hands serves as a reference motion. The addition is to mark the circumference in length units along with the duration units of the angles between the hands and the vertical.

Galileo uses horizontal uniform linear motion to mark length and duration below (from his Dialogues Concerning Two New Sciences, Fourth Day):

Galileo parabola

The horizontal uniform motion of a particle coming from the right at a-b is continued with b-c-d-e as the horizontal component of the particle descending with uniform acceleration b-o-g-l-n. Because the horizontal motion is uniform, it can represent either length or duration of the motion. The vertical component represents the dependent variable, which has the form of a parabola.

To interchange length and duration in an equation with a parametric function of time requires five steps: (1) replace length components with their radius, which becomes the stance; (2) switch time and stance, that is, switch the independent and dependent variables; (3) linearize stance, that is, break its dependent relation; (4) bring time under a functional relation with the new parameter, stance; and (5) expand time to include angular components. Functions are inverted and the independent and dependent status of variables is switched. An inversion and a kind of re-inversion return to the same function.

In the example above, the horizontal uniform motion which was taken by Galileo to represent time is re-conceived to represent the independent length variable, stance. The constant acceleration of the vertical component is re-conceived to represent the dependent duration variable with constant legerity. The quadratic sequence in units of length becomes a sequence in units of duration at a constant rate.

The result of this interchange process is that the equations of motion for length and duration are interchangeable without functional change. All of the equations of physics in terms of parametric functions of time may be adopted as parametric functions of stance. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and stance.

Posted in: Knowing, length & duration, science ⋅ Tagged: ,

Galileo’s reciprocity

August 13, 2019 1:02 pm ⋅

From Galileo’s Dialogue Concerning the Two Chief World Systems, translated by Stillman Drake (UC Press, 1967):

Salv. Now imagine yourself in a boat with your eyes fixed on point of the sail yard. Do you think that because the boat is moving along briskly, you will have to move your eyes in order to keep your vision always on that point of the sail yard and to follow its motion?

Simp. I am a sure that I should not need to make any change at all; not just as to my vision, but if I had aimed a musket I should never have to move it a hairsbreadth to keep it aimed, no matter how the boat moved.

Salv. And this comes about because the motion the ship confers upon the sail yard, it confers upon you and also upon your eyes, so that you need not move them a bit in order to gaze at the top of the sail yard, which consequently appears motionless to you. [And the rays of vision go from the eye to the sail yard just as if a cord were tied between the two ends of the boat. Now a hundred cords are tied at different fixed points, each of which keeps its place whether the ship moves or remains still.] p.249-250

Galileo is portraying motion as viewed by a human observer. The implication is that the observer in another ship would be observing the same kinds of things. Then two observers in motion with respect to one another who observe one another must face one another. That is, they are positioned opposite one another, effectively each turned 180º from the other.

This is the Kinematic Equivalence Principle, the convention that an observed frame has the opposite orientation of the frame from which it is observed, which ensures that corresponding velocities are equal.

Posted in: Knowing, relativity, science ⋅ Tagged: ,

Relativity of orientation

August 12, 2019 1:51 pm ⋅

The Principle of Relativity states that the laws of physics are the same in all inertial frames of reference (IRF). Since a frame of reference includes an orientation, that is, a convention as to which rectilinear semi-axes are positive (and so which are negative). Since there is no preferred frame of reference, each frame has its own orientation, not the orientation of a particular frame. That means IRF orientations are what is called “body-fixed” orientations.

A frame of reference is called “body-fixed” if it is conceptually attached to a rigid body, such as a vehicle, watercraft, aircraft, or spacecraft. Body-fixed frames are inertial frames if the body to which the frame is affixed is in inertial motion. The body is usually referenced in anthropomorphic terms, such as its left, right, face, or back, although some craft have their own terms, notably, ships with port, starboard, fore, and aft.

Consider the following scenario of cars in five lanes, oriented so that their forward direction is positive, with their unsigned speeds shown relative to the two parked cars in the middle lane:Six cars in five lanesCompare the direction of cars B, C1, C2, and D according to the frames attached to the five cars:

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Posted in: Knowing, relativity, science ⋅ Tagged: ,

Velocity reciprocity clarified

August 5, 2019 4:54 pm ⋅

This is a follow-on to posts here and here.

It is common to derive the Lorentz transformation assuming velocity reciprocity, which seems to say that if a body at rest in frame of reference S´ is observed from a frame of reference S that travels with relative velocity +v, then a body at rest in frame of reference S will be observed from the frame of reference S´ to be traveling with velocity –v. But that’s not the case.

Consider the typical scenario in which a person standing on the earth (embankment, station) with frame of reference S observes a person sitting in a railway car with frame of reference S´. Say they are both waving their right hands and their frame of reference follows a right-hand orientation: the positive direction is toward their right.

Person waves to train

The first illustration shows the scenario from behind the observer standing on the earth in frame S, who observes the passenger sitting in the train moving to their right with velocity +v. The scenario is typically presented from only this perspective, that of an observer at rest in frame A, even if the perspective of an observer at rest in frame S´ is described.

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Posted in: Knowing, relativity, science ⋅ Tagged: ,

Galilean relativity defended

August 3, 2019 4:50 pm ⋅

Galilean relativity is a relational theory of motion as a function of time, which leads to the Galilean transformation. Here is a defense of Galilean relativity from two postulates:

(1) The Galilean principle of relativity, which states that the laws of mechanics are invariant under a Galilean transformation.

(2) A convention that rectilinear coordinates for frames of reference follow the right-handed rule: the unit vectors i, j, and k are related as i × j = k.

The Galilean transformation for constant motion on the x axis is x´ = x – vt,  and t´ = t. Postulate (2) means if the extended right-hand thumb points to the positive X axis and the extended right-hand first finger points to the positive Y axis, then the right-hand middle finger points orthogonally to the positive Z axis.

The standard configuration for derivations of the Lorentz transformation consists of two inertial frames of reference moving relative to each other at constant velocity, with Cartesian coordinates such that the x and x′ axes are collinear facing the same direction:

Axes with same orientation

In this case the velocity of S´ relative to S is +v and the velocity of S relative to S´ is –v. This is called the principle of velocity reciprocity.

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Posted in: Knowing, relativity, science ⋅ Tagged: ,

Conventions and properties

August 3, 2019 10:08 am ⋅

Everything in science is a combination of conventions and properties. For example, frames of reference have certain conventions in common and particular properties that each individual frame has. The definition of a frame of reference is the first convention. Every frame of reference has an origin and at least the possibility of one or more coordinate axes. But the particular origin of a frame need not be in common with other frames; it is a particular property of one frame.

Definitions and postulates are conventions. Stipulations and measurements are properties. Physical laws are conventions with the appropriate supporting definitions and postulates. Interpretations of events become conventions when they are widely accepted.

The SI metric system is the international convention for measurement (i.e., metrology). Individual measurements are properties of things. Kinematics and dynamics have a convention for simultaneity (as well as simulstanceity). The orientation of orthogonal axes follows a convention for the order of the axes and the direction of positivity.

Posted in: Knowing, science ⋅ Tagged: ,

Two principles of velocity reciprocity

July 27, 2019 8:31 pm ⋅

Velocity reciprocity in relativity theory is the relation between two observers, each associated with a frame of reference and moving at different, but constant, velocities. That is, an observer-frame S observes another observer-frame S´ traveling with velocity +v relative to observer-frame S. A velocity reciprocity relation concerns the velocity of S that is observed by S´. Einstein’s principle of velocity reciprocity states that each velocity is the same magnitude (speed) but is in the opposite direction. That is, the velocity of S observed by S´ is –v.

Two frames with same orientation

Einstein’s principle of velocity reciprocity reads

We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v. Ref.

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Length and duration

July 25, 2019 7:51 pm ⋅

Although it has seemed natural to speak of “space and time”, that is a confused designation of length and duration, as well as their metrics, stance and time (see glossary here). Space is the space of motion, so length space is only one side; the other side is duration space. Speaking consistently is challenging but a must in order to bring clarity to a confused subject.

Here is a diagram of the combinations of length and duration, stance and time (click to enlarge):

Length-duration diamond

The extent of motion is measured by length and duration. The space of motion has three dimensions. Length space and duration space therefore have three dimensions as well. The space of motion is  represented by an ordered pair of length space and duration space coordinates. If there is absolute inter-convertibility between length and duration, then the space of motion is a six-dimensional space of either length units or duration units.

The metric of length is distance and the metric of duration is distime (though often called time). Stance is the distance of a point from the origin in length space. Time is the distime of an instant from the origin in duration space. Events are located in length space and chronated in duration-space.

As one can speak of distance in three-dimensional length space, one can speak of distime in three-dimensional duration space. In this sense, time is a three-dimensional concept. There are three dimensions of duration rather than three dimensions of time per se, but time has a three-dimensional aspect.

Posted in: Knowing, length & duration ⋅ Tagged: ,

Sex and gender

July 25, 2019 2:11 pm ⋅

Let us untangle the words sex and gender, which have become so confused and adulterated. In the past they were almost synonymous but today are quite different. The Online Etymology Dictionary notes:

As sex (n.) took on erotic qualities in 20c., gender came to be the usual English word for “sex of a human being,” in which use it was at first regarded as colloquial or humorous.

Academic theorists have successfully tied gender to culture, which can change with one’s place, time, or personal inclinations. Today there is no reason to use gender as a substitute for sex, for example on forms. Also one should not say sex when one means gender.

Let us then accept the simple distinction between sex, a biological term, and gender, a cultural term. In accord with this, let us distinguish male and female sexes from masculine and feminine genders. The former is fixed at conception, recorded at birth, and an aspect of physical embodiment throughout life.

Let us also acknowledge that there is no simple relation between male and female on the one hand, and masculine and feminine on the other hand. We could posit as a definition that masculine means the cultural expectations for males, and feminine means the cultural expectations for females.

But it would be simplistic to speak as though males are always masculine and females are always feminine. We are all a mix of masculine and feminine aspects, although their definition suggests that most males are mostly masculine and most females are mostly feminine.

What if a male is mostly feminine or a female is mostly masculine? Some societies would not permit it, but most find a way of accommodation. This has been the source of humor as, for example, the not-so masculine husband with the not-so feminine wife who appear to act in socially expected ways but really do not.

What if a male is so feminine they want to be addressed with a feminine pronoun, or a female is so masculine they want to be addressed with a masculine pronoun? On the face of it, masculine behavior and masculine pronouns might seem to go together, as might feminine behavior and feminine pronouns. But that would make society captive to personal behaviors that have changed and may change again. One purpose of culture, that is, social conventions, is to make things simpler for society. Conforming society to the changed and changing behaviors of individuals would make society the pawn of individuals. That would be counter to social solidarity.

So let us use pronouns that correspond to sex, not gender. A similar argument holds for what should be obvious, public accommodations such as public bathrooms that are related to biological functioning, in which public safety and comfort are also at issue.

Posted in: Relating ⋅ Tagged:

What Galileo really demonstrated

July 24, 2019 6:00 pm ⋅

Galileo Galilei’s inclined plane experiment is described in his work Dialogues Concerning Two New Sciences, which I quote from the Dover edition. He speaks (through his character Salviati) of “those sciences where mathematical demonstrations are applied to natural phenomena, as is seen in the case of perspective, astronomy, mechanics, music, and others where the principles, once established by well-chosen experiments, become the foundations of the entire superstructure.” (p.178) This is the ancient method of science that Galileo applied to experiments, establishing the foundation of modern science.

Galileo states his Theorem II, Proposition II as:

The spaces described [i.e., traced] by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. (p.174 or p.142 on the OLL edition)

But it has just been proved that so far as distances traversed are concerned it is precisely the same whether a body falls from rest with a uniform acceleration or whether it falls during an equal time-interval with a constant speed which is one-half the maximum speed attained during the accelerated motion.

Then he describes his experiment:

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Posted in: Knowing, relativity, science ⋅ Tagged: ,