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Tag Archives: Physics

Circular orbits

*** Revised from June 2017 ***

With circular motion there is a radius and circumference that may be measured as distance or duration. Call the spatial circumference S, and the temporal circumference T, which is known as the period. Distinguish the spatial and temporal versions of the radius: R, for space, and Q, for time. Then S = 2πR and T = 2πQ. Also, R = Qv, and Q = Ru, with speed, v, and pace, u.

Circular orbits are a convenient entry into Kepler’s and Newton’s laws of planetary motion. Copernicus thought the orbits were circular, and most planetary orbits are in fact nearly circular. We have then three propositions from the perspective of the Sun toward each orbiting planet:

  1. Each planet orbits the Sun in a circular path with radius R in 3D space.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is a constant, v.

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Galilean transformation expanded

The Galilean transformation is typically presented for motion in direction of the x-axis, with the other axes unchanged:

x´ = xvt, y´ = y, z´ = z, and t´ = t,

where v is the relative velocity of the observers. This is incompatible with the Lorentz transformation, but more than that, it is inconsistent with the two-way (round-trip) speed of light in a vacuum.

The Lorentz transformation can be made compatible with the round-trip speed of light if light is considered to travel instantaneously to its observer, which is usually the final leg. The speed of light for the other part of the round trip can be inferred so that their harmonic mean equals c, which is the most that is known (see One-way speed of light).

That is, if the speed of light in the Lorentz transformation is allowed to approach infinity, then the transformation will approach the Galilean transformation. Here the Galilean transformation arises as a limit of the Lorentz transformation by the speed of light approaching infinity, rather than usual the relative velocity approaching zero.

The Lorentz transformation for motion in the direction of the x-axis is:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (tvx/c²), with γ = (1 – v²/c²)–1/2,

where γ (gamma) is the Lorentz factor. As c → ∞, γ → 1 and t´ → t. This can only be the case for one part of the light trip, which we’re taking as the last part of the trip.

Why the last part? Because that’s what is observed, directly or reflected in a mirror. And in everyday conversation the place where something is observed to be is spoken of as where it is now. Even with the convention of a constant speed of light, one has to be very pedantic to keep correcting others and oneself by saying that where something is seen to be is in fact where it was in the past.

For a round trip, the speed of light for the part not directly observed can be inferred from the empirical result that the round trip speed equals the constant, c:

x´ = γ (xvt), y´ = y, z´ = z, and t´ = γ (t – 4vx/c²), with γ = (1 – 4v²/c²)–1/2.

The speed of light for the unobserved part is inferred from the necessity that the harmonic mean equals c:

(1/c1 + 1/c2) = 2/c,

where c1c/2 as c2 → ∞. This harmonic mean of speeds is the arithmetic mean of paces. What is actually measured is the pace of light from the independent length traversed in the dependent time.

Length and time parallels

This post continues the parallelism between length and time, and includes some new terms.

Length and time both have base units in SI metric: the meter (or better: metre to distinguish it from a device) and the second. They can both be associated with direction. Length in a direction is from or toward an event place. Time in a direction is from or toward an event time.

Multiple dimensions of length are called space. However, space can mean merely the space between two points. To designate 3D space, let’s use the Latin spatium (space). Analogously, let’s use the term tempium to designate 3D time (cf. Latin tempus, time).

Events ordered by time are in time order. Events ordered by length are in length order. Events ordered by importance could be said to be in magna order.

Things are persistent events. Things have length. Things have three dimensions of length. Events have duration. Events have three dimensions of duration. The extent of space between things is called distance. The extent of tempium between events could be called temstance.

Relative space is divided into here and there; “here I am, there I was, there I will be.” The present tense of space is here. The past or future tense of space is there. Here I am. Some places were traversed in the past. Some places will be traversed in the future.

Relative tempium is divided into now and then; “now I am, then I was, then I will be.” The present tense of tempium is now. The past or future tense of tempium is then. Now I am. Some times were traversed in the past. Some times will be traversed in the future.

Matter is a spatial substance. Figure is a temporal substance. Matter has mass, solidity. Figure has vass, lightness. Many sports move matter, such as a baseball pitcher throwing the ball. A figure skater traces out figures in space and time, that is, spatium and tempium.

Observers and participants

Observers detect objects and events with objects. These objects are essentially passive; they must be made to do things by force and work.

Participants are subjects among subjects, actively engaging in events and making them happen. Subjects participate and participants are subjects.

The perspectives of an observer and a participant are inverses of each other. They are different attitudes. An observer has an attitude of standing apart from the world. A participant has an attitude of being part of the world. The world is placed in different contexts because of the attitude of the contact.

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Foundations of mechanics for 3D space or 3D time

The first edition of New Foundations for Classical Mechanics (1986) by David Hestenes included “Foundations of Mechanics” as Chapter 9. This was removed for lack of space in the second edition, but is available online as a pdf here. This space-time foundation may serve as a guide for the foundation of mechanics for either space-time or time-space. To do so requires introducing abstract terminology, notably:

position space → position geometry; time → event order; particle → point body; instant → point event; clock → event order indicator; simultaneity → correspondence; reference frame → frame.

The application of this abstract theory is to interpret the 3D position geometry with event order as either 3D position space with temporal event order (space-time) or 3D position time with spatial event order (time-space). It could also be applied to derivatives or integrals of these, e.g., a velocity space.

Let’s focus on section 2 “The Zeroth Law of Physics” and start with the second paragraph on page 8, revising it for 3D space or 3D time:


To begin with, we recognize two kinds of bodies, point bodies and bodies which are composed of point bodies. Given a body R called a frame, each point body has a geometrical property called its position with respect to R. We characterize this property indirectly by introducing the concept of 3D Position Geometry, or Relative Geometry, if you prefer. For each frame R, a position geometry P is defined by the following postulates:

  1. P is a 3-dimensional Euclidean geometry.
  2. The position (with respect to R) of any point body can be represented as a point in P.

The first postulate specifies the mathematical structure of a 3D position geometry while the second postulate supplies it with a physical interpretation. Thus, the postulates define a physical law, for the mathematical structure implies geometrical relations among the positions of distinct point bodies. Let us call it the Law of Geometric Order.

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Transforming 3D space into 3D time

There is a symmetry between space and time. As one can transform an observation by rectilinear motion (translation), or by rotation, or by a timeline change, so one can transform 3D space into an equivalent 3D time. This is not a continuous change so don’t expect a simple equation. There are four things that must be done to transform 3D space into 3D time, that is, 3+1 spacetime into 1+3 timespace:

(1) The ordering of events should be switched between a timeline (1D time order) and a placeline (1D space order). So a measurement of time, such as the duration from a reference event, should be switched with a measurement of place, such as the distance from a reference event.

(2) Scalars should be inverted: speed ⇒ pace, mass ⇒ 1/mass = vass, energy ⇒ 1/energy = invergy, work ⇒ 1/work = invork, etc.

(3) Vectors that are ratios of base units or products of base units should switch their numerators and denominators such that (a) the denominator becomes a magnitude of the former numerator and (b) the numerator becomes the vector with units of the former denominator: velocity ⇒ legerity, momentum ⇒ fulmentum, etc. This is similar to an inversion since s/t ⇒ t/s = (1/s)/(1/t).

(4) Other units should be derived from these, with new rates relative to the timeline for 3D space and the placeline for 3D time: acceleration ⇒ expedience, force ⇒ rush, power ⇒ exertion, etc.

There should be no time vectors in 3D space and no space vectors in 3D time. The distance from a reference place and duration from a reference event should be the same for both, apart from a change of reference points. The laws of physics should be the same for observation or transportation in each frame.

Timelines and placelines

Events may be ordered in various ways (see here). Events ordered by time form a timeline, which is:

1. a linear representation of important events in the order in which they occurred.
2. a schedule; timetable.

This may be generalized to the following definition:

A timeline is an ordering of events by time or duration.

For example, below is a timeline of a Project Mercury flight:

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From 3D space to 3D time

We observe the sun and the moon traversing the sky. We know that the moon is objectively orbiting the earth but the sun is not. Where then is the sun that is observed traversing the sky in daily and annual cycles? It is not in 3D space. It is in 3D time.

Binary stars orbit their common barycenter. If the sun and earth were the only celestial bodies, it might not be clear as to which was orbiting which. But since there are other planets orbiting the sun, the only objective view is that all the planets are orbiting the sun (more precisely, the barycenter of the solar system, which is in or near the sun).

Compare sun-centered (heliocentric) and earth-centered (geocentric) frames of reference (click to enlarge):

Source

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Direction for expedience and rush

Expedience is the change in legerity per unit of length. Average expedience is (u2u1)/Δs. If the second legerity is faster than the first, the time interval is shorter and the legerity is lower in value, which makes the expedience negative. Since increased motion is the default for expedience (as it is for acceleration), the default direction for expedience is in the direction opposite the motion. Negative expedience, or inexpedience, is in the direction of motion.

Rush, which is vass times expedience, is also in the direction opposite the motion. So the rush of levity is in the opposite direction as well. That is why cyclic motion in 3D time has a centrifugal rush. And that is why there is levity, not gravity, in 3D time although the motion is the same.

Newton’s 3rd law shows that forces (and rushes) come in direction pairs. So it is a matter of convention which direction is the primary one. The force toward the spatial center is primary in 3D space. The rush from the temporal center is primary in 3D time. Both directions are valid but one is primary.

Displacement vs. arc length

As pointed out here, average speed does not equal the magnitude of average velocity. But the instantaneous speed does equal the magnitude of instantaneous velocity. For example, the average velocity of one orbit is zero but the average speed is positive.

Consider a section of a curve as below:

The arc length of this section of the curve is Δs. The displacement is Δr. This with the horizontal and vertical differences Δx and Δy makes a triangle. The Pythagorean theorem gives the hypotenuse of the triangle:

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