iSoul In the beginning is reality

Circular orbits

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, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, 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:

  1. Each planet orbit the Sun in a spatially circular path.
  2. The Sun is at the center of mass of each planet’s orbit.
  3. The speed of each planet is constant.

Let’s follow the exposition given in Elements of Newtonian Mechanics by J.M. Knudsen and P.G. Hjorth (Springer, 1995), starting on page 6. Because the speed is constant, the acceleration follows the equation for uniform circular motion derived previously:

a = / Rs,

in which v is the speed and a is the acceleration. We have from the definition of speed:

v = S / T = 2πRs / T.

Elimination of v from these equations leads to

aRs / , or

a ∝ S / .

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

.

Combining this with the equation for acceleration yields

a ∝ 1 / Rs², or

a ∝ 1 / , or

a ∝ 1 / T4/3, or

a ∝ 1 / Rt4/3.

Inserting the first acceleration into Newton’s second law leads to:

Fm / Rs²,

with force, F, and mass, m. The force is directed toward the Sun, with a magnitude inversely proportional to the square of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite force from each planet toward the Sun. Which is to say:

FM / Rs².

for mass, M. The combined law of gravitation is thus:

F = GmM/Rs²,

for some constant G.

Now consider gravity at the surface of the Earth. Newton showed that the gravitational force of a body would be the same if the mass were concentrated at its center, so the above equation can be used with the radius of the Earth, ρ²:

F = GmM/ρ² = mg,

with g as the acceleration of gravity on Earth. Then

g = GM/ρ².

If the known values of G, M, and ρ are inserted into this equation, the result is g = 9.8 m/s².


Let us now modify this derivation so that distance is the independent variable instead of time. This is better called a law of levitation since it is naturally directed toward the smaller mass. We have then three propositions:

  1. The Sun orbits each planet in a temporally circular path.
  2. The Sun is at the center of vass of its orbit.
  3. The pace of the Sun is constant.

Because the pace is constant, the prestination follows the equation for uniform circular motion derived previously:

b = / Rt,

in which u is the pace and b is the prestination. Again, distinguish the spatial and temporal versions of the radius, R, by using Rs and Rt. Then S = 2πRs and T = 2πRt. Also, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

We have from the definition of pace:

u = T / S = 2πRt / S.

Elimination of u from these equations leads to

bRt / , or

bT / , or

bT / Rs².

Kepler’s third law states that the orbital period, T, and the semi-major axis, A, are related as . For circular orbits this becomes

Rs³, or

, or

Rt².

Combining this with the equation for prestination yields

b ∝ 1 / Rt1/3, or

b ∝ 1 / T1/3, or

b ∝ 1 / S1/2, or

b ∝ 1 / Rs1/2.

Inserting the latter prestination into Newton’s second law in the form of surge, Γ, gives:

Γ ∝ ℓ / Rs1/2,

with surge, Γ, and vass, ℓ. The surge is directed away from the Sun, with a magnitude inversely proportional to the square root of the distance from the Sun.

Because of Newton’s third law, there is an equal and opposite surge toward each planet from the Sun. Which is to say:

Γ ∝ L / Rs1/2,

for vass, L. The combined law of levitation is thus:

Γ = HℓL/Rs1/2,

for some constant H. Since this is a function of the inverse square root instead of the inverse square, it is more sensitive to changes in Rs, as with a body in radial motion.

Consider levity at the surface of the Earth, using the above equation with the radius of the Earth, ρ:

Γ = HℓL/ρ1/2 = nh,

with h as the prestination of levity on Earth. Then

q = HL/ρ1/2.

The values for H, L, and ρ may be inserted into this equation to determine the value of h.

Foundations of mechanics for time-space

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 time-space. To do so requires terms and correspondences in addition to switching space and time:

space → time space, time → space length, particle → eventicle, body → time body, instant → spot, clock → odologe, reference frame → reference timeframe.

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


To begin with, we recognize two kinds of entities, eventicles and time bodies which are composed of eventicles. Given a time body, R, called a reference timeframe, each eventicle has a geometrical property called its time position with respect to R. We characterize this property indirectly by introducing the concept of Time Space. For each reference timeframe R, a time space T is defined by the following postulates:

  1. T is a 3-dimensional Euclidean space.
  2. The time position (with respect to R) of any eventicle can be represented as a point in T.

The first postulate specifies the mathematical structure of a time space 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 time positions of distinct eventicles. Let us call it the Law of Temporal Order.

Notice that this law asserts that every eventicle has a property called time position and it specifies properties of this property. But it does not tell us how to measure time position. Measurement is a separate matter, since it entails correspondence rules as well as laws. In actual practice the reference timeframe is often fictitious, though it is related indirectly to a physical time body. Our discussion is simplified by feigning that the reference timeframe is always a real time body.

We turn now to the problem of formulating the scientific concept of space length. We begin with the idea that space length is a measure of motion, and motion is a change of time position with respect to a given reference timeframe. The concept of space length embraces two distinct relations: spatial order and temporally remote coincidence. To keep this clear we introduce each relation with a separate postulate.

First we formulate the Law of Spatial Order:

The motion of any eventicle with respect to a given reference timeframe can be represented as an orbit in time space.

This postulate has a semantic component as well as a mathematical one. It presumes that each eventicle has a property called motion and attributes a mathematical structure to that property by associating it with an orbit in time space. Recall that an orbit is a continuous, oriented curve. Thus, an eventicle’s orbit in time space represents an ordered sequence of time positions. We call this order a spatial order, so we have attributed a distinct spatial order to the motion of each eventicle.

To define a physical, space length scale as a measure of motion, we select a moving eventicle which we call an eventicle odologe. We refer to each successive time position of this eventicle as a spot. We define the space length interval Δs between two spots by

Δs = cΔt,

where c is a positive numerical constant and Δt is the arc time of the odologe’s orbit between the two spots. Our measure of space length is thus related to the measure of duration in time space.

To use this space length scale as a measure for the motions of other eventicles, we need to relate the motions of eventicles at different time positions. The necessary relation can be introduced by postulating the

Law of Coincidence:

At every spot, each eventicle has a unique time position.

This postulate determines a correspondence between the points on the orbit of any eventicle and points on the orbit of an eventicle odologe. Therefore, every eventicle orbit can be parametrized by a space length parameter defined on the orbit of an eventicle odologe.

Note that this postulate does not tell us how to determine the time position of a given eventicle at any spot. That is a problem for the theory of measurement.

So far our laws permit orbits which are nondifferentiable at isolated points or even at every point. These possibilities will be eliminated by Newton’s laws which require differentiable orbits. We include in the class of allowable orbits, orbits which consist of a single time point through some space length interval. An eventicle with such an orbit is said to be fixed with respect to the given reference timeframe through that space length interval. Of course, we require that the eventicles composing the reference timeframe itself be fixed with respect to each other, so the reference timeframe can be regarded as a rigid time body.

Note that the pace of an eventicle is just a comparison of the eventicle’s distimement to the distimement of an eventicle odologe. The pace of a eventicle odologe has the constant value 1/c = Δt/Δs, so the odologe moves uniformly by definition. In principle, we can use any moving eventicle as an odologe, but the dynamical laws we introduce later suggest a preferred choice. Any moving eventicle defines a periodic process, because it moves successively over time intervals of equal length. It should be evident that any real odologe can be accurately modeled as an eventicle odologe. By regarding the eventicle odologe as the fundamental kind of odologe, we make clear in the foundations of physics that the scientific concept of space length is based on an objective comparison of motions.

We now have definite formulations of time space and space length, so we can define a reference system as a representation x for the possible time position of any eventicle at each space length r in some space length interval. Each reference system presumes the selection of a particular origin for space and time space and particular choices for the units of distance and duration, so each time position and space length is assigned a definite numerical value. The term “reference system” is sometimes construed as a system of procedures for constructing a numerical representation of time space and space length.

After we have formulated our dynamical laws, it will be clear that certain reference systems called alacrital systems have a special status. Then it will be necessary to supplement our Law of Coincidence with a postulate that relates coincident points in different alacrital systems. That is the critical postulate that distinguishes classical mechanics from special relativity, but we defer discussion of it until we are prepared to handle it completely. It is mentioned now, because our formulation of time space and space length will not be complete until such a postulate is made.

It is convenient to summarize and generalize our postulates with a single law statement, the Zeroth (or Temporospatial) Law of Physics:

Every real time body has a continuous history in time space and space length.

Distance, duration, and angles

Let’s follow the orbit of a particle or the route of a vehicle as a curvilinear function with associated directions at every point. Measurement produces travel distance r, travel time t, with directions θ and φ. The directions may be considered as functions of either travel distance or travel time: θr, φr, θt, or φt. There are accordingly four possibilities:

(r, t, θr, φr), (r, t, θt, φt), or (t, r, θr, φt), or (t, r, θt, φr).

The latter two may be made equal by a change of convention for measuring the angle. These may be represented rectilinearly as:

(t, rx, ry, rz), (r, tx, ty, tz), (rw, rx, ty, tz), or (tw, tx, ry, rz).

The latter two may be made equal by a change of convention for the axes.

Three possibilities remain: (3D space + 1D time), (1D space + 3D time), or (2D space + 2D time).

An example of the third possibility would be a traveler who measured their horizontal angle relative to magnetic north and their vertical angle relative to the sun. Since magnetic north is (approximately) fixed, it serves to measure the horizontal angle spatially. Since the sun’s position continually changes, it serves to measure the vertical angle temporally. The result is (2+2) with (r, θr) and (t, φt).

Or one could do the opposite and measure the horizontal angle temporally, as with a sundial, and the vertical angle spatially, as with a theodolite. The result is (2+2) with (t, θt) and (r, φr).

If both angles are measured relative to a fixed point, then the result is (3+1) or (t, r, θr, φr). If both angles are measured relative to a moving point, then the result is (r, t, θt, φt). The moving point should be moving at a constant rate, or at least a constant acceleration.

If three coordinates are measured relative to a fixed axis, then the result is (1+3) or (t, rx, ry, rz). If three coordinates are measured relative to a rotating axis, then the result is (r, tx, ty, tz). The moving axis should be moving at a constant rate, or at least a constant acceleration.

The potential reality of (r, t, θr, φr, θt, φt) collapses to one of the possibilities above in the act of measurement. The potential reality of (rx, ry, rz, tx, ty, tz) collapses to one of the rectilinear possibilities above in the act of measurement.

Center of vass

The vass is to time (duration) as the mass is to space (distance). As noted before here, the vass can be measured by a similar procedure as the mass. The mass and vass are inverses with opposite uses.

center of mass

The center of mass is the point that two or more particles (point masses) are balanced (or one large mass is balanced). For two particle masses, m1 and m2 that are located at points x1 and x2, respectively:

Center of Mass (CM) = (m1x1 + m2x2)/(m1 + m2),

which is the weighted arithmetic mean with the masses as the weights. This is similar to the momentum, in which the velocity is weighted by the mass: mv.

The center of vass is the point in time that two or more particle vasses are balanced. For two particle vasses, ℓ1 and ℓ2 that are located at points in time t1 and t2, respectively:

Center of Vass (CV) = (ℓ1t1 + ℓ2t2)/(ℓ1 + ℓ2) = ((t1/m1 + t2/m2)/(1/m1 + 1/m2)),

which is the weighted arithmetic mean with the vasses as the weights. Compare the celentum, in which the celerity is weighted by the vass: ℓu.

In order to generalize this, let’s use the derivation of the center of mass, as in Knudsen and Hjorth’s Elements of Newtonian Mechanics, chapter 9. Start with the time position vector, T, to find the center of vass for a system of particle vasses:

L Tcv = Σ ℓi ti,

where ℓi and ti are the vass and the time position vector of the ith particle vass, and L = Σ ℓi is the total vass of the system. Then differentiate with respect to space length to get

L Tcv´ = Σ ℓi ti´= Σ qi := Q,

where the total linear celentum of the system is denoted Q. In other words, the total linear celentum Q of a system of particle vasses is the same as that of a particle vass with vass L moving the the celerity of the center of vass. This is also stated as

Q = L ucv,

where ucv is the celerity of the center of vass.

Equations of motion in space-time and time-space

First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = t, position = r, initial position = r(t0) = r0, velocity = v, initial velocity = v(t0) = v0, v = |v| = speed, and acceleration = a.

First equation of motion

v = ∫ a dt = v0 + at

Second equation of motion

r = ∫ (v0 + at) dt = r0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (rr0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (rr0)).


Next, here is a derivation of the time-space equations of motion, in which prestination is constant. Let position (displacement) = r, time = t, initial time = t(r0) = t0, celerity = u, initial celerity = u(r0) = u0, u = |u| = pace, and prestination = b.

First equation of motion

u = ∫ b dr = u0 + bt

Second equation of motion

t = ∫ (u0 + br) dr = t0 + u0r + ½br²

Third equation of motion

From u² = uu = (u0 + br) ∙ (u0 + br) = u0² + 2r(bu0) + b²r², and

(2b) ∙ (tt0) = (2b) ∙ (u0r + ½br²) = 2r(bu0) + b²r² = u² ‒ u0², it follows that

u² = u0² + 2(b ∙ (tt0)).

Christianity and science

A good summary of the myth of a long-running conflict between Christianity and science is in Timothy Larsen’s “War is Over, If You Want It” (September 2008). This warfare myth was invented in the 19th century by people such as TH Huxley who either should have known better or were purposely stirring up animosity. It is composed of individual myths that “support” it, such as the myth that Christians thought the earth was flat in the Middle Ages or the myth that Christians opposed the use of anesthesia during childbirth in the 19th century.

Larsen references Frank M. Turner’s “Contesting Cultural Authority” (Cambridge, 1993), as someone who “persuasively argued that the notion of a conflict between theology and science was generated as part of a campaign of professionalization by would-be scientists.” (p.150) It’s almost forgotten today, but the profession of a scientist didn’t exist until the late 19th century. Before that, science was developed by amateurs (including clerics) who had the leisure and interest. TH Huxley and others fought against such people because they stood in the way of a new class of professional scientists.

Although the warfare meme is vastly exaggerated, there are enough misunderstandings that the opposite idea of integration isn’t realistic. For example, it is said that many Christians quickly accepted Darwin’s theory of evolution in the 19th century and later. But what is overlooked is the fact that Christians misunderstood Darwin and substituted their own ideas of evolution by law or miracle.  Theistic evolution is common among Christians who either insert a law-bound version for Darwin’s undirected version or else invent undetectable miracles that make it God-directed.

Many have noted that modern science developed in a Christian matrix. If science jettisons its Christian roots, it loses a reason to expect an ordered universe that can be understood by human beings. It may either adopt a multiverse that just happens to have order in one universe or drift toward non-causal explanations in a chaotic universe.

Some scientists want to deepen the Christian roots of science rather than cut them off. They are mostly creationists or intelligent design proponents. Those who follow TH Huxley will have nothing of it. But some are willing to entertain new proposals. As the modern era comes to a close, we can expect that modern science will change into something else.

Derivation of Newton’s second law

It is often said that Newton’s laws are laws of nature, which can only be determined by observation. That’s true in the sense that the definitions required are based on inductive reasoning. However, once these definitions are in hand, it should be a deductive science. This is classical science, in the sense of Euclid, Archimedes, Plato, and Aristotle.

Here is a derivation of Newton’s second law in space-time, with distance, s, time, t, and mass, m:

Velocity, v := ds/dt.

Acceleration, a := dv/dt.

Mass flow rate, := dm/dt.

Weighted distance, ș := ms.

Momentum, p := dș/dt = d(ms)/dt = (mds + sdm)/dt = m(ds/dt) + s(dm/dt) = mv + sṁ.

If mass is constant, then p = mv.

Force, F  := dp/dt = d(mv + sṁ)/dt = d(mv)/dt + d(sṁ)/dt = (mdv/dt + vdm/dt) + (sd/dt + ds/dt) = ma + vṁ + sṃ + ṁvma + 2ṁv + sṃ, where = d/dt.

If mass is constant, then F = ma.

Here is a derivation of Newton’s second law in time-space, with vass, n = 1/m:

Celerity, u := dt/ds.

Prestination, b := du/ds.

Vass flow rate,  := dn/ds.

Weighted durationț := nt.

Celentum, q := dț/ds = d(nt)/ds = (ndt + tdn)/ds = n(dt/ds) + t(dn/ds) = nu + tṅ.

If vass is constant, then q = nu.

SurgeΓ := dq/ds = d(nu + tṅ)/ds = d(nu)/ds + d(tṅ)/ds = (ndu/ds + udn/ds) + (td/ds + dt/ds) = nb + uṅ + tṇ + ṅu = nb + 2ṅu + tṇ, where  = d/ds.

If vass is constant, then Γ = nb.

Transformations with two speeds of light

I’ve discussed before how there may be two different one-way speeds of light since they are matters of convention. See here and here. This post is concerned with the transformation between two observers in such a case. I continue to work with the standard configuration, see here.

The standard transformation of reference frames begins with two frames in uniform relative motion along one axis (usually called x). Here we take the spatial axis to be the r-axis, which parallels the spatial axis of motion. Similarly, the temporal axis is taken to be the t-axis, which parallels the temporal axis of motion.

The notation here is indifferent as to the existence of other dimensions. If they exist, they are orthogonal to the direction of motion, whether spatial or temporal, and their corresponding values are the same for both frames. One can generalize the results here to other directions by rotation.

The two frames are differentiated by primed and unprimed letters. They coincide at time t = 0 and their relative speed is v. The one-way speed of light is c1 in one direction and c2 in the opposite direction such that the round-trip speed equals c, the standard speed of light in a vacuum. That is, the harmonic mean of c1 and c2 equals c.

The trajectory of a reference particle or wave that travels at the speed of light follows these equations in both frames:

r =  c1t or r/c1 = t and r′ = c2t′ or r′/c2 = t′.

Consider a point event such as a flash of light that is observed from each reference frame. How are its coordinates in each frame related?

The basic relations are: r′ = rtv = r (1 – v/c1) and t′ = t (1 – v/c1). Next a factor, γ, is included in the transformation equations:

r′ = γ (rvt) = γr (1 – v/c1), and

t′ = γt (1 – v/c1),

with equal values for the other corresponding primed and unprimed coordinates. The inverse transformations use the other speed of light:

r = γ (r′ + vt′) = γr′ (1 + v/c2), and

t = γt′ (1 + v/c2).

Multiply each corresponding pair together to get:

rr′ = γ²rr′ (1 – v/c1)(1 + v/c2).

Dividing out rr′ yields:

1 = γ2 (1 – v/c1)(1 + v/c2).

Solving for γ leads to:

γ = [(1 – v/c1)(1 + v/c2)]–1/2, which applies if |v| < |c|.

This is the Lorentz transformation with two one-way speeds of light.

In the extreme case, the speed in one direction is instantaneous and in the other direction is c/2, which leads to

γ = (1 – 2v/c)–1/2.

And so

r′ = γ (rvt) = r (1 – 2v/c)1/2, and

t′ = γt (1 – 2v/c) = t (1 – 2v/c)1/2.

Clock race

This post continues previous ones contrasting ancient and modern space and time, such as here.
Nordic Sun Chariot in Bronze
The above bronze-age depiction of the Sun on a chariot shows a common image from antiquity: the Sun crossing the heavens daily. The path of the Sun was also described as traversing a celestial circle (or sphere) and going around a racecourse. These images show that the clock of the Sun was considered as covering more than one dimension, in contrast to the modern concept of scalar duration.

Psalm 19
1 The heavens declare the glory of God,
and the sky above proclaims his handiwork.
2 Day to day pours out speech,
and night to night reveals knowledge.
3 There is no speech, nor are there words,
whose voice is not heard.
4 Their voice goes out through all the earth,
and their words to the end of the world.
In them he has set a tent for the sun,
5 which comes out like a bridegroom leaving his chamber,
and, like a strong man, runs its course with joy.
6 Its rising is from the end of the heavens,
and its circuit to the end of them,
and there is nothing hidden from its heat.

These are more examples of the interchange of travel distance with travel time that occurred in the transition from ancient to modern thinking. We can undo this interchange and find an alternative way of conceiving motion.

Centripetal prestination

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, and the angle of motion, θ, by using Rs and Rt, and θs and θt, respectively. Then S = 2πRs and T = 2πRt. Also, Δs = Rs Δθs, Δt = Rt Δθt, Rs = Rtv, and Rt = Rsu, with velocity, v, and celerity, u.

What is the acceleration that occurs in uniform circular motion? Christiaan Huygens was the first to answer this in 1658. Here is a simple derivation:

An object in uniform circular motion traverses a circle at constant speed, v. Its spatial position can be represented by a vector, Rs, which changes its angle but not its magnitude. The distance traversed in one cycle is S = 2πRs. The period or duration of one cycle is T = 2πRt = 2πRs/v.

The velocity vector of this object can also be represented by a vector that changes its angle but not its magnitude. The accumulated change in velocity is 2πv. The magnitude of the acceleration is the change in velocity divided by the duration:

a = 2πv / T = 2πv / (2πRs/v) = / Rs.

Another derivation uses a diagram such as this:

centripetal acceleration diagram

Substituting Rs for r and θs for θ, the derivation is as follows:

Δs = Rs Δθs

v = |Δs| / |Δt|, and

a = |a| = |Δv| / |Δt| = v θs| / |Δt| = v s| / (Rst|) = t| / (Rst|) = / Rs.

Additionally,

a = / Rs = / (Rtv) = v / Rt = Rs / Rt².


What is the prestination that occurs in uniform circular motion? A simple derivation follows the first method above:

An object in uniform circular motion traverses a circle at constant pace, u. Its time position can be represented by a vector, Rt, which changes its angle but not its magnitude. The period or duration of one cycle is T = 2πRt. The distance traversed by one cycle is S = 2πRs = 2πRt/u.

The celerity vector of this object can also be represented by a vector that changes its angle but not its magnitude. The accumulated change in celerity is 2πu. The magnitude of the prestination is the change in celerity divided by the distance:

b = 2πu / S = 2πu / (2πRt/u) = / Rt.

A second derivation uses a diagram similar to the one above with these substitutions: rRt, θ → θt, st, and vu. Then,

Δt = Rt Δθt,

u = |Δt| / |Δs|, and

b = |b| = |Δu| / |Δs| = uθt| / |Δs| = ut| / (Rts|) = u² |Δs| / (Rts|) = / Rt.

Additionally,

b = / Rt = / (Rsu) = u / Rs = Rt / Rs².